TPTP Problem File: SLH0533^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FO_Theory_Rewriting/0012_Saturation/prob_00421_014843__18252364_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1515 ( 536 unt; 232 typ; 0 def)
% Number of atoms : 3810 (1429 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 13121 ( 366 ~; 38 |; 259 &;10714 @)
% ( 0 <=>;1744 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 1903 (1903 >; 0 *; 0 +; 0 <<)
% Number of symbols : 212 ( 209 usr; 22 con; 0-5 aty)
% Number of variables : 3886 ( 108 ^;3630 !; 148 ?;3886 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:52:49.625
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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% Explicit typings (209)
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thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Option__Ooption_Itf__a_J,type,
hd_option_a: list_option_a > option_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__Option__Ooption_Itf__a_J,type,
set_option_a2: list_option_a > set_option_a ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_set_option_a2: list_set_option_a > set_set_option_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__Option__Ooption_Itf__a_J,type,
tl_option_a: list_option_a > list_option_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Oremdups_001t__Option__Ooption_Itf__a_J,type,
remdups_option_a: list_option_a > list_option_a ).
thf(sy_c_List_OremoveAll_001t__Option__Ooption_Itf__a_J,type,
removeAll_option_a: option_a > list_option_a > list_option_a ).
thf(sy_c_Map_Odom_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
dom_op4724496951392727122tion_a: ( option_a > option_option_a ) > set_option_a ).
thf(sy_c_Map_Odom_001t__Option__Ooption_Itf__a_J_001tf__a,type,
dom_option_a_a: ( option_a > option_a ) > set_option_a ).
thf(sy_c_Map_Odom_001tf__a_001tf__a,type,
dom_a_a: ( a > option_a ) > set_a ).
thf(sy_c_Map_Omap__add_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
map_ad299661227452494261tion_a: ( option_a > option_option_a ) > ( option_a > option_option_a ) > option_a > option_option_a ).
thf(sy_c_Map_Omap__add_001t__Option__Ooption_Itf__a_J_001tf__a,type,
map_add_option_a_a: ( option_a > option_a ) > ( option_a > option_a ) > option_a > option_a ).
thf(sy_c_Map_Omap__add_001tf__a_001tf__a,type,
map_add_a_a: ( a > option_a ) > ( a > option_a ) > a > option_a ).
thf(sy_c_Map_Oran_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
ran_op6317565877353657455tion_a: ( option_a > option_option_a ) > set_option_a ).
thf(sy_c_Map_Orestrict__map_001t__Option__Ooption_Itf__a_J_001tf__a,type,
restri3984065703976872170on_a_a: ( option_a > option_a ) > set_option_a > option_a > option_a ).
thf(sy_c_Map_Orestrict__map_001tf__a_001tf__a,type,
restrict_map_a_a: ( a > option_a ) > set_a > a > option_a ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_Itf__a_J,type,
none_option_a: option_option_a ).
thf(sy_c_Option_Ooption_ONone_001tf__a,type,
none_a: option_a ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
some_option_option_a: option_option_a > option6015282985685139280tion_a ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_Itf__a_J,type,
some_option_a: option_a > option_option_a ).
thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
some_set_option_a: set_option_a > option_set_option_a ).
thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_Itf__a_J,type,
some_set_a: set_a > option_set_a ).
thf(sy_c_Option_Ooption_OSome_001tf__a,type,
some_a: a > option_a ).
thf(sy_c_Option_Ooption_Othe_001tf__a,type,
the_a: option_a > a ).
thf(sy_c_Option_Othese_001t__Option__Ooption_Itf__a_J,type,
these_option_a: set_option_option_a > set_option_a ).
thf(sy_c_Option_Othese_001tf__a,type,
these_a: set_option_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
bot_bot_option_a_o: option_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Option__Ooption_Itf__a_J_J,type,
bot_bo5190894291365496758tion_a: filter_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
bot_bo4163488203964334806tion_a: set_option_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
bot_bot_set_option_a: set_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
bot_bo651546871064636272tion_a: set_set_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
ord_le6764313125106342922tion_a: set_option_option_a > set_option_option_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
ord_le1955136853071979460tion_a: set_option_a > set_option_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
ord_le7761976607055303332tion_a: set_set_option_a > set_set_option_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
top_top_option_a_o: option_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
top_to6364489741703166539tion_a: set_op7810783462584961947tion_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
top_to1659475022456381882tion_a: set_option_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
top_to7232205438743699540tion_a: set_set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Saturation_Olift__f__total_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
lift_f7361094991155314338tion_a: ( option_option_a > option_option_a > $o ) > ( option_option_a > option_option_a > option_option_a ) > option6015282985685139280tion_a > option6015282985685139280tion_a > option6015282985685139280tion_a ).
thf(sy_c_Saturation_Olift__f__total_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
lift_f6754278611557336796tion_a: ( option_a > option_a > $o ) > ( option_a > option_a > option_a ) > option_option_a > option_option_a > option_option_a ).
thf(sy_c_Saturation_Olift__f__total_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
lift_f1864078254722121148tion_a: ( set_option_a > set_option_a > $o ) > ( set_option_a > set_option_a > set_option_a ) > option_set_option_a > option_set_option_a > option_set_option_a ).
thf(sy_c_Saturation_Olift__f__total_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
lift_f1381871799933197046_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > set_a ) > option_set_a > option_set_a > option_set_a ).
thf(sy_c_Saturation_Olift__f__total_001tf__a_001tf__a_001tf__a,type,
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thf(sy_c_Saturation_Osemilattice__closure__operator_001t__Option__Ooption_Itf__a_J,type,
semila6266762623024302478tion_a: ( option_a > option_a > option_a ) > ( option_a > option_a > $o ) > option_a > $o ).
thf(sy_c_Saturation_Osemilattice__closure__operator_001tf__a,type,
semila7314208447189752392ator_a: ( a > a > a ) > ( a > a > $o ) > a > $o ).
thf(sy_c_Saturation_Osemilattice__closure__operator_Osmaller__subset_001t__Option__Ooption_Itf__a_J,type,
semila1830171625574221010tion_a: ( option_a > option_a > $o ) > option_a > set_option_a > set_option_a ).
thf(sy_c_Saturation_Osemilattice__closure__operator_Osmaller__subset_001tf__a,type,
semila2432794410337176460bset_a: ( a > a > $o ) > a > set_a > set_a ).
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semila5752222179541100331tion_a: ( option_a > option_a > option_a ) > ( option_a > option_a > $o ) > option_a > $o ).
thf(sy_c_Saturation_Oset__closure__oprator_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
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thf(sy_c_Saturation_Oset__closure__oprator_001t__Option__Ooption_Itf__a_J,type,
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thf(sy_c_Saturation_Oset__closure__oprator_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Saturation_Oset__closure__oprator_001tf__a,type,
set_cl1870446787657330701ator_a: ( a > a > a ) > $o ).
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thf(sy_c_Saturation_Oset__closure__oprator_Oclosurep_001tf__a,type,
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thf(sy_c_Saturation_Oset__closure__partial__oprator_001tf__a,type,
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image_2983976770393306955_set_a: ( set_a > option_set_a ) > set_set_a > set_option_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
image_2200789291716222155tion_a: ( set_a > set_option_a ) > set_set_a > set_set_option_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a: ( a > option_a ) > set_a > set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
image_a_set_option_a: ( a > set_option_a ) > set_a > set_set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
insert605063979879581146tion_a: option_option_a > set_option_option_a > set_option_option_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_Itf__a_J,type,
insert_option_a: option_a > set_option_a > set_option_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
insert_set_option_a: set_option_a > set_set_option_a > set_set_option_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Option__Ooption_Itf__a_J,type,
is_sin3348965821858909752tion_a: set_option_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__Option__Ooption_Itf__a_J,type,
remove_option_a: option_a > set_option_a > set_option_a ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Option__Ooption_Itf__a_J,type,
the_elem_option_a: set_option_a > option_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
member7397271250019230201tion_a: option6015282985685139280tion_a > set_op1225621012726482224tion_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
member5113800082084363315tion_a: option_option_a > set_option_option_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
member6923355609089322771tion_a: option_set_option_a > set_op2222149991272740682tion_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_Itf__a_J_J,type,
member_option_set_a: option_set_a > set_option_set_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
member_set_option_a: set_option_a > set_set_option_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_P,type,
p: a > a > $o ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_f,type,
f: a > a > a ).
thf(sy_v_s____,type,
s2: a ).
% Relevant facts (1277)
thf(fact_0_local_Orefl,axiom,
! [X: a] : ( p @ X @ X ) ).
% local.refl
thf(fact_1_local_Osym,axiom,
! [X: a,Y: a] :
( ( p @ X @ Y )
=> ( p @ Y @ X ) ) ).
% local.sym
thf(fact_2_idem,axiom,
! [X: a] :
( ( f @ X @ X )
= X ) ).
% idem
thf(fact_3_assP,axiom,
! [X: a,Y: a,Z: a] :
( ( p @ X @ ( f @ Y @ Z ) )
=> ( ( p @ Y @ Z )
=> ( p @ ( f @ X @ Y ) @ Z ) ) ) ).
% assP
thf(fact_4_com,axiom,
! [X: a,Y: a] :
( ( p @ X @ Y )
=> ( ( f @ X @ Y )
= ( f @ Y @ X ) ) ) ).
% com
thf(fact_5_dist,axiom,
! [Y: a,Z: a,X: a] :
( ( p @ Y @ Z )
=> ( ( p @ X @ ( f @ Y @ Z ) )
=> ( p @ X @ Y ) ) ) ).
% dist
thf(fact_6_lift__fun_Ofun__left__comm,axiom,
! [Y: option_a,X: option_a,Z: option_a] :
( ( lift_f_total_a_a_a @ p @ f @ Y @ ( lift_f_total_a_a_a @ p @ f @ X @ Z ) )
= ( lift_f_total_a_a_a @ p @ f @ X @ ( lift_f_total_a_a_a @ p @ f @ Y @ Z ) ) ) ).
% lift_fun.fun_left_comm
thf(fact_7_lift__fun_Ofun__left__idem,axiom,
! [X: option_a,Z: option_a] :
( ( lift_f_total_a_a_a @ p @ f @ X @ ( lift_f_total_a_a_a @ p @ f @ X @ Z ) )
= ( lift_f_total_a_a_a @ p @ f @ X @ Z ) ) ).
% lift_fun.fun_left_idem
thf(fact_8_lift__f__total__ass,axiom,
! [X: option_a,Y: option_a,Z: option_a] :
( ( lift_f_total_a_a_a @ p @ f @ X @ ( lift_f_total_a_a_a @ p @ f @ Y @ Z ) )
= ( lift_f_total_a_a_a @ p @ f @ ( lift_f_total_a_a_a @ p @ f @ X @ Y ) @ Z ) ) ).
% lift_f_total_ass
thf(fact_9_lift__f__total__com,axiom,
! [X: option_a,Y: option_a] :
( ( lift_f_total_a_a_a @ p @ f @ X @ Y )
= ( lift_f_total_a_a_a @ p @ f @ Y @ X ) ) ).
% lift_f_total_com
thf(fact_10_lift__f__total__idem,axiom,
! [X: option_a] :
( ( lift_f_total_a_a_a @ p @ f @ X @ X )
= X ) ).
% lift_f_total_idem
thf(fact_11_lift__fun_Oclosure_Obase,axiom,
! [S: option_a,S2: set_option_a] :
( ( member_option_a @ S @ S2 )
=> ( member_option_a @ S @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) ) ) ).
% lift_fun.closure.base
thf(fact_12_lift__fun_Oclosure_Ocases,axiom,
! [A: option_a,S2: set_option_a] :
( ( member_option_a @ A @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
=> ( ~ ( member_option_a @ A @ S2 )
=> ~ ! [S3: option_a,T: option_a] :
( ( A
= ( lift_f_total_a_a_a @ p @ f @ S3 @ T ) )
=> ( ( member_option_a @ S3 @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
=> ~ ( member_option_a @ T @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) ) ) ) ) ) ).
% lift_fun.closure.cases
thf(fact_13_lift__fun_Oclosure_Osimps,axiom,
! [A: option_a,S2: set_option_a] :
( ( member_option_a @ A @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
= ( ? [S4: option_a] :
( ( A = S4 )
& ( member_option_a @ S4 @ S2 ) )
| ? [S4: option_a,T2: option_a] :
( ( A
= ( lift_f_total_a_a_a @ p @ f @ S4 @ T2 ) )
& ( member_option_a @ S4 @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
& ( member_option_a @ T2 @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) ) ) ) ) ).
% lift_fun.closure.simps
thf(fact_14_lift__fun_Oclosure_Ostep,axiom,
! [S: option_a,S2: set_option_a,T3: option_a] :
( ( member_option_a @ S @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
=> ( ( member_option_a @ T3 @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
=> ( member_option_a @ ( lift_f_total_a_a_a @ p @ f @ S @ T3 ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) ) ) ) ).
% lift_fun.closure.step
thf(fact_15_lift__f__totalE,axiom,
! [S: option_a,U: option_a,T3: a] :
( ( ( lift_f_total_a_a_a @ p @ f @ S @ U )
= ( some_a @ T3 ) )
=> ~ ( ? [V: a] :
( S
= ( some_a @ V ) )
=> ! [W: a] :
( U
!= ( some_a @ W ) ) ) ) ).
% lift_f_totalE
thf(fact_16_set__closure__partial__oprator__axioms,axiom,
set_cl4247418117474325792ator_a @ p @ f ).
% set_closure_partial_oprator_axioms
thf(fact_17_lift__fun_Oclosure__idem,axiom,
! [S2: set_option_a] :
( ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
= ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) ) ).
% lift_fun.closure_idem
thf(fact_18_ass,axiom,
member_a @ s2 @ ( image_option_a_a @ the_a @ ( minus_1574173051537231627tion_a @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( image_a_option_a @ some_a @ s ) ) @ ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ).
% ass
thf(fact_19_set__closure__oprator_Oclosure_Ocong,axiom,
set_cl2401587249382337922sure_a = set_cl2401587249382337922sure_a ).
% set_closure_oprator.closure.cong
thf(fact_20_set__closure__oprator_Oclosure_Ocong,axiom,
set_cl4053507043448260238tion_a = set_cl4053507043448260238tion_a ).
% set_closure_oprator.closure.cong
thf(fact_21_set__closure__oprator_Oclosure_Ocong,axiom,
set_cl4093418192473505224tion_a = set_cl4093418192473505224tion_a ).
% set_closure_oprator.closure.cong
thf(fact_22_lift__fun_Oclosure__empty,axiom,
( ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% lift_fun.closure_empty
thf(fact_23_lift__fun_Ocomp__fun__idem__axioms,axiom,
finite6444408709487604428tion_a @ ( lift_f_total_a_a_a @ p @ f ) ).
% lift_fun.comp_fun_idem_axioms
thf(fact_24_pred__closure__to__some__lift__closure,axiom,
! [S: a,S2: set_a] :
( ( member_a @ S @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
=> ( member_option_a @ ( some_a @ S ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( image_a_option_a @ some_a @ S2 ) ) ) ) ).
% pred_closure_to_some_lift_closure
thf(fact_25_some__lift__closure__pred__closure,axiom,
! [T3: a,S2: set_a] :
( ( member_option_a @ ( some_a @ T3 ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( image_a_option_a @ some_a @ S2 ) ) )
=> ( member_a @ T3 @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) ) ) ).
% some_lift_closure_pred_closure
thf(fact_26_lift__fun_Ocomp__fun__commute__axioms,axiom,
finite8922082839177200849tion_a @ ( lift_f_total_a_a_a @ p @ f ) ).
% lift_fun.comp_fun_commute_axioms
thf(fact_27_lift__set__closure__oprator,axiom,
set_cl4772385950493004755tion_a @ ( lift_f_total_a_a_a @ p @ f ) ).
% lift_set_closure_oprator
thf(fact_28_lift__fun_Oclosurep_Obase,axiom,
! [S2: option_a > $o,S: option_a] :
( ( S2 @ S )
=> ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ S ) ) ).
% lift_fun.closurep.base
thf(fact_29_lift__fun_Oclosurep_Ocases,axiom,
! [S2: option_a > $o,A: option_a] :
( ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ A )
=> ( ~ ( S2 @ A )
=> ~ ! [S3: option_a,T: option_a] :
( ( A
= ( lift_f_total_a_a_a @ p @ f @ S3 @ T ) )
=> ( ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ S3 )
=> ~ ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ T ) ) ) ) ) ).
% lift_fun.closurep.cases
thf(fact_30_lift__fun_Oclosurep_Osimps,axiom,
! [S2: option_a > $o,A: option_a] :
( ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ A )
= ( ? [S4: option_a] :
( ( A = S4 )
& ( S2 @ S4 ) )
| ? [S4: option_a,T2: option_a] :
( ( A
= ( lift_f_total_a_a_a @ p @ f @ S4 @ T2 ) )
& ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ S4 )
& ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ T2 ) ) ) ) ).
% lift_fun.closurep.simps
thf(fact_31_lift__fun_Oclosurep_Ostep,axiom,
! [S2: option_a > $o,S: option_a,T3: option_a] :
( ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ S )
=> ( ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ T3 )
=> ( set_cl7484131188691603406tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 @ ( lift_f_total_a_a_a @ p @ f @ S @ T3 ) ) ) ) ).
% lift_fun.closurep.step
thf(fact_32_lift__fun_Oclosure__mono,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( ord_le1955136853071979460tion_a @ S2 @ T4 )
=> ( ord_le1955136853071979460tion_a @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ T4 ) ) ) ).
% lift_fun.closure_mono
thf(fact_33_pred__closure_Ostep,axiom,
! [S: a,S2: set_a,T3: a] :
( ( member_a @ S @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
=> ( ( member_a @ T3 @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
=> ( ( p @ S @ T3 )
=> ( member_a @ ( f @ S @ T3 ) @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) ) ) ) ) ).
% pred_closure.step
thf(fact_34_pred__closure_Osimps,axiom,
! [A: a,S2: set_a] :
( ( member_a @ A @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
= ( ? [S4: a] :
( ( A = S4 )
& ( member_a @ S4 @ S2 ) )
| ? [S4: a,T2: a] :
( ( A
= ( f @ S4 @ T2 ) )
& ( member_a @ S4 @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
& ( member_a @ T2 @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
& ( p @ S4 @ T2 ) ) ) ) ).
% pred_closure.simps
thf(fact_35_pred__closure_Ocases,axiom,
! [A: a,S2: set_a] :
( ( member_a @ A @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
=> ( ~ ( member_a @ A @ S2 )
=> ~ ! [S3: a,T: a] :
( ( A
= ( f @ S3 @ T ) )
=> ( ( member_a @ S3 @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
=> ( ( member_a @ T @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) )
=> ~ ( p @ S3 @ T ) ) ) ) ) ) ).
% pred_closure.cases
thf(fact_36_pred__closure_Obase,axiom,
! [S: a,S2: set_a] :
( ( member_a @ S @ S2 )
=> ( member_a @ S @ ( set_cl7833034999824766768sure_a @ p @ f @ S2 ) ) ) ).
% pred_closure.base
thf(fact_37_set__closure__oprator_Oass,axiom,
! [F: a > a > a,X: a,Y: a,Z: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( F @ X @ ( F @ Y @ Z ) )
= ( F @ ( F @ X @ Y ) @ Z ) ) ) ).
% set_closure_oprator.ass
thf(fact_38_set__closure__oprator_Oass,axiom,
! [F: option_a > option_a > option_a,X: option_a,Y: option_a,Z: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( F @ X @ ( F @ Y @ Z ) )
= ( F @ ( F @ X @ Y ) @ Z ) ) ) ).
% set_closure_oprator.ass
thf(fact_39_set__closure__oprator_Ocom,axiom,
! [F: a > a > a,X: a,Y: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( F @ X @ Y )
= ( F @ Y @ X ) ) ) ).
% set_closure_oprator.com
thf(fact_40_set__closure__oprator_Ocom,axiom,
! [F: option_a > option_a > option_a,X: option_a,Y: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( F @ X @ Y )
= ( F @ Y @ X ) ) ) ).
% set_closure_oprator.com
thf(fact_41_set__closure__oprator_Oidem,axiom,
! [F: a > a > a,X: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( F @ X @ X )
= X ) ) ).
% set_closure_oprator.idem
thf(fact_42_set__closure__oprator_Oidem,axiom,
! [F: option_a > option_a > option_a,X: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( F @ X @ X )
= X ) ) ).
% set_closure_oprator.idem
thf(fact_43_set__closure__oprator_Ointro,axiom,
! [F: a > a > a] :
( ! [X2: a,Y2: a] :
( ( F @ X2 @ Y2 )
= ( F @ Y2 @ X2 ) )
=> ( ! [X2: a,Y2: a,Z2: a] :
( ( F @ X2 @ ( F @ Y2 @ Z2 ) )
= ( F @ ( F @ X2 @ Y2 ) @ Z2 ) )
=> ( ! [X2: a] :
( ( F @ X2 @ X2 )
= X2 )
=> ( set_cl1870446787657330701ator_a @ F ) ) ) ) ).
% set_closure_oprator.intro
thf(fact_44_set__closure__oprator_Ointro,axiom,
! [F: option_a > option_a > option_a] :
( ! [X2: option_a,Y2: option_a] :
( ( F @ X2 @ Y2 )
= ( F @ Y2 @ X2 ) )
=> ( ! [X2: option_a,Y2: option_a,Z2: option_a] :
( ( F @ X2 @ ( F @ Y2 @ Z2 ) )
= ( F @ ( F @ X2 @ Y2 ) @ Z2 ) )
=> ( ! [X2: option_a] :
( ( F @ X2 @ X2 )
= X2 )
=> ( set_cl4772385950493004755tion_a @ F ) ) ) ) ).
% set_closure_oprator.intro
thf(fact_45_set__closure__partial__oprator_Oass,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_a,Y: option_a,Z: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( P @ X @ Y )
=> ( ( P @ Y @ Z )
=> ( ( F @ X @ ( F @ Y @ Z ) )
= ( F @ ( F @ X @ Y ) @ Z ) ) ) ) ) ).
% set_closure_partial_oprator.ass
thf(fact_46_set__closure__partial__oprator_Oass,axiom,
! [P: a > a > $o,F: a > a > a,X: a,Y: a,Z: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( P @ X @ Y )
=> ( ( P @ Y @ Z )
=> ( ( F @ X @ ( F @ Y @ Z ) )
= ( F @ ( F @ X @ Y ) @ Z ) ) ) ) ) ).
% set_closure_partial_oprator.ass
thf(fact_47_set__closure__partial__oprator_Ocom,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_a,Y: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( P @ X @ Y )
=> ( ( F @ X @ Y )
= ( F @ Y @ X ) ) ) ) ).
% set_closure_partial_oprator.com
thf(fact_48_set__closure__partial__oprator_Ocom,axiom,
! [P: a > a > $o,F: a > a > a,X: a,Y: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( P @ X @ Y )
=> ( ( F @ X @ Y )
= ( F @ Y @ X ) ) ) ) ).
% set_closure_partial_oprator.com
thf(fact_49_set__closure__partial__oprator_Osym,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_a,Y: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( P @ X @ Y )
=> ( P @ Y @ X ) ) ) ).
% set_closure_partial_oprator.sym
thf(fact_50_set__closure__partial__oprator_Osym,axiom,
! [P: a > a > $o,F: a > a > a,X: a,Y: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( P @ X @ Y )
=> ( P @ Y @ X ) ) ) ).
% set_closure_partial_oprator.sym
thf(fact_51_mem__Collect__eq,axiom,
! [A: option_option_a,P: option_option_a > $o] :
( ( member5113800082084363315tion_a @ A @ ( collec6688160826630954613tion_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: set_option_a,P: set_option_a > $o] :
( ( member_set_option_a @ A @ ( collect_set_option_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: option_a,P: option_a > $o] :
( ( member_option_a @ A @ ( collect_option_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_option_option_a] :
( ( collec6688160826630954613tion_a
@ ^ [X3: option_option_a] : ( member5113800082084363315tion_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_set_option_a] :
( ( collect_set_option_a
@ ^ [X3: set_option_a] : ( member_set_option_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_option_a] :
( ( collect_option_a
@ ^ [X3: option_a] : ( member_option_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_62_Collect__cong,axiom,
! [P: option_a > $o,Q: option_a > $o] :
( ! [X2: option_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_option_a @ P )
= ( collect_option_a @ Q ) ) ) ).
% Collect_cong
thf(fact_63_set__closure__partial__oprator_OassP,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_a,Y: option_a,Z: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( P @ X @ ( F @ Y @ Z ) )
=> ( ( P @ Y @ Z )
=> ( P @ ( F @ X @ Y ) @ Z ) ) ) ) ).
% set_closure_partial_oprator.assP
thf(fact_64_set__closure__partial__oprator_OassP,axiom,
! [P: a > a > $o,F: a > a > a,X: a,Y: a,Z: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( P @ X @ ( F @ Y @ Z ) )
=> ( ( P @ Y @ Z )
=> ( P @ ( F @ X @ Y ) @ Z ) ) ) ) ).
% set_closure_partial_oprator.assP
thf(fact_65_set__closure__partial__oprator_Odist,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,Y: option_a,Z: option_a,X: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( P @ Y @ Z )
=> ( ( P @ X @ ( F @ Y @ Z ) )
=> ( P @ X @ Y ) ) ) ) ).
% set_closure_partial_oprator.dist
thf(fact_66_set__closure__partial__oprator_Odist,axiom,
! [P: a > a > $o,F: a > a > a,Y: a,Z: a,X: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( P @ Y @ Z )
=> ( ( P @ X @ ( F @ Y @ Z ) )
=> ( P @ X @ Y ) ) ) ) ).
% set_closure_partial_oprator.dist
thf(fact_67_set__closure__partial__oprator_Oidem,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( F @ X @ X )
= X ) ) ).
% set_closure_partial_oprator.idem
thf(fact_68_set__closure__partial__oprator_Oidem,axiom,
! [P: a > a > $o,F: a > a > a,X: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( F @ X @ X )
= X ) ) ).
% set_closure_partial_oprator.idem
thf(fact_69_set__closure__partial__oprator_Orefl,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( P @ X @ X ) ) ).
% set_closure_partial_oprator.refl
thf(fact_70_set__closure__partial__oprator_Orefl,axiom,
! [P: a > a > $o,F: a > a > a,X: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( P @ X @ X ) ) ).
% set_closure_partial_oprator.refl
thf(fact_71_set__closure__oprator_Oclosurep_Obase,axiom,
! [F: a > a > a,S2: a > $o,S: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( S2 @ S )
=> ( set_cl8402473243758428296urep_a @ F @ S2 @ S ) ) ) ).
% set_closure_oprator.closurep.base
thf(fact_72_set__closure__oprator_Oclosurep_Obase,axiom,
! [F: option_a > option_a > option_a,S2: option_a > $o,S: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( S2 @ S )
=> ( set_cl7484131188691603406tion_a @ F @ S2 @ S ) ) ) ).
% set_closure_oprator.closurep.base
thf(fact_73_set__closure__oprator_Oclosurep_Ocong,axiom,
set_cl7484131188691603406tion_a = set_cl7484131188691603406tion_a ).
% set_closure_oprator.closurep.cong
thf(fact_74_set__closure__oprator_Oclosurep_Ostep,axiom,
! [F: a > a > a,S2: a > $o,S: a,T3: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( set_cl8402473243758428296urep_a @ F @ S2 @ S )
=> ( ( set_cl8402473243758428296urep_a @ F @ S2 @ T3 )
=> ( set_cl8402473243758428296urep_a @ F @ S2 @ ( F @ S @ T3 ) ) ) ) ) ).
% set_closure_oprator.closurep.step
thf(fact_75_set__closure__oprator_Oclosurep_Ostep,axiom,
! [F: option_a > option_a > option_a,S2: option_a > $o,S: option_a,T3: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( set_cl7484131188691603406tion_a @ F @ S2 @ S )
=> ( ( set_cl7484131188691603406tion_a @ F @ S2 @ T3 )
=> ( set_cl7484131188691603406tion_a @ F @ S2 @ ( F @ S @ T3 ) ) ) ) ) ).
% set_closure_oprator.closurep.step
thf(fact_76_set__closure__partial__oprator_Ointro,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a] :
( ! [X2: option_a] : ( P @ X2 @ X2 )
=> ( ! [X2: option_a,Y2: option_a] :
( ( P @ X2 @ Y2 )
=> ( P @ Y2 @ X2 ) )
=> ( ! [X2: option_a,Y2: option_a,Z2: option_a] :
( ( P @ Y2 @ Z2 )
=> ( ( P @ X2 @ ( F @ Y2 @ Z2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( ! [X2: option_a,Y2: option_a,Z2: option_a] :
( ( P @ X2 @ ( F @ Y2 @ Z2 ) )
=> ( ( P @ Y2 @ Z2 )
=> ( P @ ( F @ X2 @ Y2 ) @ Z2 ) ) )
=> ( ! [X2: option_a,Y2: option_a] :
( ( P @ X2 @ Y2 )
=> ( ( F @ X2 @ Y2 )
= ( F @ Y2 @ X2 ) ) )
=> ( ! [X2: option_a,Y2: option_a,Z2: option_a] :
( ( P @ X2 @ Y2 )
=> ( ( P @ Y2 @ Z2 )
=> ( ( F @ X2 @ ( F @ Y2 @ Z2 ) )
= ( F @ ( F @ X2 @ Y2 ) @ Z2 ) ) ) )
=> ( ! [X2: option_a] :
( ( F @ X2 @ X2 )
= X2 )
=> ( set_cl936642855594582118tion_a @ P @ F ) ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.intro
thf(fact_77_set__closure__partial__oprator_Ointro,axiom,
! [P: a > a > $o,F: a > a > a] :
( ! [X2: a] : ( P @ X2 @ X2 )
=> ( ! [X2: a,Y2: a] :
( ( P @ X2 @ Y2 )
=> ( P @ Y2 @ X2 ) )
=> ( ! [X2: a,Y2: a,Z2: a] :
( ( P @ Y2 @ Z2 )
=> ( ( P @ X2 @ ( F @ Y2 @ Z2 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( ! [X2: a,Y2: a,Z2: a] :
( ( P @ X2 @ ( F @ Y2 @ Z2 ) )
=> ( ( P @ Y2 @ Z2 )
=> ( P @ ( F @ X2 @ Y2 ) @ Z2 ) ) )
=> ( ! [X2: a,Y2: a] :
( ( P @ X2 @ Y2 )
=> ( ( F @ X2 @ Y2 )
= ( F @ Y2 @ X2 ) ) )
=> ( ! [X2: a,Y2: a,Z2: a] :
( ( P @ X2 @ Y2 )
=> ( ( P @ Y2 @ Z2 )
=> ( ( F @ X2 @ ( F @ Y2 @ Z2 ) )
= ( F @ ( F @ X2 @ Y2 ) @ Z2 ) ) ) )
=> ( ! [X2: a] :
( ( F @ X2 @ X2 )
= X2 )
=> ( set_cl4247418117474325792ator_a @ P @ F ) ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.intro
thf(fact_78_set__closure__oprator_Oclosurep_Ocases,axiom,
! [F: a > a > a,S2: a > $o,A: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( set_cl8402473243758428296urep_a @ F @ S2 @ A )
=> ( ~ ( S2 @ A )
=> ~ ! [S3: a,T: a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( set_cl8402473243758428296urep_a @ F @ S2 @ S3 )
=> ~ ( set_cl8402473243758428296urep_a @ F @ S2 @ T ) ) ) ) ) ) ).
% set_closure_oprator.closurep.cases
thf(fact_79_set__closure__oprator_Oclosurep_Ocases,axiom,
! [F: option_a > option_a > option_a,S2: option_a > $o,A: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( set_cl7484131188691603406tion_a @ F @ S2 @ A )
=> ( ~ ( S2 @ A )
=> ~ ! [S3: option_a,T: option_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( set_cl7484131188691603406tion_a @ F @ S2 @ S3 )
=> ~ ( set_cl7484131188691603406tion_a @ F @ S2 @ T ) ) ) ) ) ) ).
% set_closure_oprator.closurep.cases
thf(fact_80_set__closure__oprator_Oclosurep_Osimps,axiom,
! [F: a > a > a,S2: a > $o,A: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( set_cl8402473243758428296urep_a @ F @ S2 @ A )
= ( ? [S4: a] :
( ( A = S4 )
& ( S2 @ S4 ) )
| ? [S4: a,T2: a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( set_cl8402473243758428296urep_a @ F @ S2 @ S4 )
& ( set_cl8402473243758428296urep_a @ F @ S2 @ T2 ) ) ) ) ) ).
% set_closure_oprator.closurep.simps
thf(fact_81_set__closure__oprator_Oclosurep_Osimps,axiom,
! [F: option_a > option_a > option_a,S2: option_a > $o,A: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( set_cl7484131188691603406tion_a @ F @ S2 @ A )
= ( ? [S4: option_a] :
( ( A = S4 )
& ( S2 @ S4 ) )
| ? [S4: option_a,T2: option_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( set_cl7484131188691603406tion_a @ F @ S2 @ S4 )
& ( set_cl7484131188691603406tion_a @ F @ S2 @ T2 ) ) ) ) ) ).
% set_closure_oprator.closurep.simps
thf(fact_82_set__closure__partial__oprator_Opred__closure_Obase,axiom,
! [P: option_option_a > option_option_a > $o,F: option_option_a > option_option_a > option_option_a,S: option_option_a,S2: set_option_option_a] :
( ( set_cl1309597857828561964tion_a @ P @ F )
=> ( ( member5113800082084363315tion_a @ S @ S2 )
=> ( member5113800082084363315tion_a @ S @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.base
thf(fact_83_set__closure__partial__oprator_Opred__closure_Obase,axiom,
! [P: set_option_a > set_option_a > $o,F: set_option_a > set_option_a > set_option_a,S: set_option_a,S2: set_set_option_a] :
( ( set_cl2851621040977480518tion_a @ P @ F )
=> ( ( member_set_option_a @ S @ S2 )
=> ( member_set_option_a @ S @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.base
thf(fact_84_set__closure__partial__oprator_Opred__closure_Obase,axiom,
! [P: set_a > set_a > $o,F: set_a > set_a > set_a,S: set_a,S2: set_set_a] :
( ( set_cl2751670595400742528_set_a @ P @ F )
=> ( ( member_set_a @ S @ S2 )
=> ( member_set_a @ S @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.base
thf(fact_85_set__closure__partial__oprator_Opred__closure_Obase,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,S: option_a,S2: set_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( member_option_a @ S @ S2 )
=> ( member_option_a @ S @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.base
thf(fact_86_set__closure__partial__oprator_Opred__closure_Obase,axiom,
! [P: a > a > $o,F: a > a > a,S: a,S2: set_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( member_a @ S @ S2 )
=> ( member_a @ S @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.base
thf(fact_87_set__closure__partial__oprator_Opred__closure_Ocong,axiom,
set_cl7151870100604144246tion_a = set_cl7151870100604144246tion_a ).
% set_closure_partial_oprator.pred_closure.cong
thf(fact_88_set__closure__partial__oprator_Opred__closure_Ocong,axiom,
set_cl7833034999824766768sure_a = set_cl7833034999824766768sure_a ).
% set_closure_partial_oprator.pred_closure.cong
thf(fact_89_set__closure__partial__oprator_Opred__closure_Ostep,axiom,
! [P: option_option_a > option_option_a > $o,F: option_option_a > option_option_a > option_option_a,S: option_option_a,S2: set_option_option_a,T3: option_option_a] :
( ( set_cl1309597857828561964tion_a @ P @ F )
=> ( ( member5113800082084363315tion_a @ S @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
=> ( ( member5113800082084363315tion_a @ T3 @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
=> ( ( P @ S @ T3 )
=> ( member5113800082084363315tion_a @ ( F @ S @ T3 ) @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.step
thf(fact_90_set__closure__partial__oprator_Opred__closure_Ostep,axiom,
! [P: set_option_a > set_option_a > $o,F: set_option_a > set_option_a > set_option_a,S: set_option_a,S2: set_set_option_a,T3: set_option_a] :
( ( set_cl2851621040977480518tion_a @ P @ F )
=> ( ( member_set_option_a @ S @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
=> ( ( member_set_option_a @ T3 @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
=> ( ( P @ S @ T3 )
=> ( member_set_option_a @ ( F @ S @ T3 ) @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.step
thf(fact_91_set__closure__partial__oprator_Opred__closure_Ostep,axiom,
! [P: set_a > set_a > $o,F: set_a > set_a > set_a,S: set_a,S2: set_set_a,T3: set_a] :
( ( set_cl2751670595400742528_set_a @ P @ F )
=> ( ( member_set_a @ S @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
=> ( ( member_set_a @ T3 @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
=> ( ( P @ S @ T3 )
=> ( member_set_a @ ( F @ S @ T3 ) @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.step
thf(fact_92_set__closure__partial__oprator_Opred__closure_Ostep,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,S: option_a,S2: set_option_a,T3: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( member_option_a @ S @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
=> ( ( member_option_a @ T3 @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
=> ( ( P @ S @ T3 )
=> ( member_option_a @ ( F @ S @ T3 ) @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.step
thf(fact_93_set__closure__partial__oprator_Opred__closure_Ostep,axiom,
! [P: a > a > $o,F: a > a > a,S: a,S2: set_a,T3: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( member_a @ S @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
=> ( ( member_a @ T3 @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
=> ( ( P @ S @ T3 )
=> ( member_a @ ( F @ S @ T3 ) @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.step
thf(fact_94_set__closure__partial__oprator_Opred__closure_Ocases,axiom,
! [P: option_option_a > option_option_a > $o,F: option_option_a > option_option_a > option_option_a,A: option_option_a,S2: set_option_option_a] :
( ( set_cl1309597857828561964tion_a @ P @ F )
=> ( ( member5113800082084363315tion_a @ A @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
=> ( ~ ( member5113800082084363315tion_a @ A @ S2 )
=> ~ ! [S3: option_option_a,T: option_option_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member5113800082084363315tion_a @ S3 @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
=> ( ( member5113800082084363315tion_a @ T @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
=> ~ ( P @ S3 @ T ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.cases
thf(fact_95_set__closure__partial__oprator_Opred__closure_Ocases,axiom,
! [P: set_option_a > set_option_a > $o,F: set_option_a > set_option_a > set_option_a,A: set_option_a,S2: set_set_option_a] :
( ( set_cl2851621040977480518tion_a @ P @ F )
=> ( ( member_set_option_a @ A @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
=> ( ~ ( member_set_option_a @ A @ S2 )
=> ~ ! [S3: set_option_a,T: set_option_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_set_option_a @ S3 @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
=> ( ( member_set_option_a @ T @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
=> ~ ( P @ S3 @ T ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.cases
thf(fact_96_set__closure__partial__oprator_Opred__closure_Ocases,axiom,
! [P: set_a > set_a > $o,F: set_a > set_a > set_a,A: set_a,S2: set_set_a] :
( ( set_cl2751670595400742528_set_a @ P @ F )
=> ( ( member_set_a @ A @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
=> ( ~ ( member_set_a @ A @ S2 )
=> ~ ! [S3: set_a,T: set_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_set_a @ S3 @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
=> ( ( member_set_a @ T @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
=> ~ ( P @ S3 @ T ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.cases
thf(fact_97_set__closure__partial__oprator_Opred__closure_Ocases,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,A: option_a,S2: set_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( member_option_a @ A @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
=> ( ~ ( member_option_a @ A @ S2 )
=> ~ ! [S3: option_a,T: option_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_option_a @ S3 @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
=> ( ( member_option_a @ T @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
=> ~ ( P @ S3 @ T ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.cases
thf(fact_98_set__closure__partial__oprator_Opred__closure_Ocases,axiom,
! [P: a > a > $o,F: a > a > a,A: a,S2: set_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( member_a @ A @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
=> ( ~ ( member_a @ A @ S2 )
=> ~ ! [S3: a,T: a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_a @ S3 @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
=> ( ( member_a @ T @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
=> ~ ( P @ S3 @ T ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.cases
thf(fact_99_set__closure__partial__oprator_Opred__closure_Osimps,axiom,
! [P: option_option_a > option_option_a > $o,F: option_option_a > option_option_a > option_option_a,A: option_option_a,S2: set_option_option_a] :
( ( set_cl1309597857828561964tion_a @ P @ F )
=> ( ( member5113800082084363315tion_a @ A @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
= ( ? [S4: option_option_a] :
( ( A = S4 )
& ( member5113800082084363315tion_a @ S4 @ S2 ) )
| ? [S4: option_option_a,T2: option_option_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member5113800082084363315tion_a @ S4 @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
& ( member5113800082084363315tion_a @ T2 @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
& ( P @ S4 @ T2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.simps
thf(fact_100_set__closure__partial__oprator_Opred__closure_Osimps,axiom,
! [P: set_option_a > set_option_a > $o,F: set_option_a > set_option_a > set_option_a,A: set_option_a,S2: set_set_option_a] :
( ( set_cl2851621040977480518tion_a @ P @ F )
=> ( ( member_set_option_a @ A @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
= ( ? [S4: set_option_a] :
( ( A = S4 )
& ( member_set_option_a @ S4 @ S2 ) )
| ? [S4: set_option_a,T2: set_option_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_set_option_a @ S4 @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
& ( member_set_option_a @ T2 @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
& ( P @ S4 @ T2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.simps
thf(fact_101_set__closure__partial__oprator_Opred__closure_Osimps,axiom,
! [P: set_a > set_a > $o,F: set_a > set_a > set_a,A: set_a,S2: set_set_a] :
( ( set_cl2751670595400742528_set_a @ P @ F )
=> ( ( member_set_a @ A @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
= ( ? [S4: set_a] :
( ( A = S4 )
& ( member_set_a @ S4 @ S2 ) )
| ? [S4: set_a,T2: set_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_set_a @ S4 @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
& ( member_set_a @ T2 @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
& ( P @ S4 @ T2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.simps
thf(fact_102_set__closure__partial__oprator_Opred__closure_Osimps,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,A: option_a,S2: set_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( member_option_a @ A @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
= ( ? [S4: option_a] :
( ( A = S4 )
& ( member_option_a @ S4 @ S2 ) )
| ? [S4: option_a,T2: option_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_option_a @ S4 @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
& ( member_option_a @ T2 @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
& ( P @ S4 @ T2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.simps
thf(fact_103_set__closure__partial__oprator_Opred__closure_Osimps,axiom,
! [P: a > a > $o,F: a > a > a,A: a,S2: set_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( member_a @ A @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
= ( ? [S4: a] :
( ( A = S4 )
& ( member_a @ S4 @ S2 ) )
| ? [S4: a,T2: a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_a @ S4 @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
& ( member_a @ T2 @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
& ( P @ S4 @ T2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure.simps
thf(fact_104_ac__comp__fun__commute,axiom,
! [F: option_a > option_a > option_a] :
( ! [X2: option_a,Y2: option_a] :
( ( F @ X2 @ Y2 )
= ( F @ Y2 @ X2 ) )
=> ( ! [X2: option_a,Y2: option_a,Z2: option_a] :
( ( F @ X2 @ ( F @ Y2 @ Z2 ) )
= ( F @ ( F @ X2 @ Y2 ) @ Z2 ) )
=> ( finite8922082839177200849tion_a @ F ) ) ) ).
% ac_comp_fun_commute
thf(fact_105_set__closure__oprator__def,axiom,
( set_cl1870446787657330701ator_a
= ( ^ [F2: a > a > a] :
( ! [X3: a,Y3: a] :
( ( F2 @ X3 @ Y3 )
= ( F2 @ Y3 @ X3 ) )
& ! [X3: a,Y3: a,Z3: a] :
( ( F2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
= ( F2 @ ( F2 @ X3 @ Y3 ) @ Z3 ) )
& ! [X3: a] :
( ( F2 @ X3 @ X3 )
= X3 ) ) ) ) ).
% set_closure_oprator_def
thf(fact_106_set__closure__oprator__def,axiom,
( set_cl4772385950493004755tion_a
= ( ^ [F2: option_a > option_a > option_a] :
( ! [X3: option_a,Y3: option_a] :
( ( F2 @ X3 @ Y3 )
= ( F2 @ Y3 @ X3 ) )
& ! [X3: option_a,Y3: option_a,Z3: option_a] :
( ( F2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
= ( F2 @ ( F2 @ X3 @ Y3 ) @ Z3 ) )
& ! [X3: option_a] :
( ( F2 @ X3 @ X3 )
= X3 ) ) ) ) ).
% set_closure_oprator_def
thf(fact_107_set__closure__partial__oprator__def,axiom,
( set_cl936642855594582118tion_a
= ( ^ [P2: option_a > option_a > $o,F2: option_a > option_a > option_a] :
( ! [X3: option_a] : ( P2 @ X3 @ X3 )
& ! [X3: option_a,Y3: option_a] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ Y3 @ X3 ) )
& ! [X3: option_a,Y3: option_a,Z3: option_a] :
( ( P2 @ Y3 @ Z3 )
=> ( ( P2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
=> ( P2 @ X3 @ Y3 ) ) )
& ! [X3: option_a,Y3: option_a,Z3: option_a] :
( ( P2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
=> ( ( P2 @ Y3 @ Z3 )
=> ( P2 @ ( F2 @ X3 @ Y3 ) @ Z3 ) ) )
& ! [X3: option_a,Y3: option_a] :
( ( P2 @ X3 @ Y3 )
=> ( ( F2 @ X3 @ Y3 )
= ( F2 @ Y3 @ X3 ) ) )
& ! [X3: option_a,Y3: option_a,Z3: option_a] :
( ( P2 @ X3 @ Y3 )
=> ( ( P2 @ Y3 @ Z3 )
=> ( ( F2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
= ( F2 @ ( F2 @ X3 @ Y3 ) @ Z3 ) ) ) )
& ! [X3: option_a] :
( ( F2 @ X3 @ X3 )
= X3 ) ) ) ) ).
% set_closure_partial_oprator_def
thf(fact_108_set__closure__partial__oprator__def,axiom,
( set_cl4247418117474325792ator_a
= ( ^ [P2: a > a > $o,F2: a > a > a] :
( ! [X3: a] : ( P2 @ X3 @ X3 )
& ! [X3: a,Y3: a] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ Y3 @ X3 ) )
& ! [X3: a,Y3: a,Z3: a] :
( ( P2 @ Y3 @ Z3 )
=> ( ( P2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
=> ( P2 @ X3 @ Y3 ) ) )
& ! [X3: a,Y3: a,Z3: a] :
( ( P2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
=> ( ( P2 @ Y3 @ Z3 )
=> ( P2 @ ( F2 @ X3 @ Y3 ) @ Z3 ) ) )
& ! [X3: a,Y3: a] :
( ( P2 @ X3 @ Y3 )
=> ( ( F2 @ X3 @ Y3 )
= ( F2 @ Y3 @ X3 ) ) )
& ! [X3: a,Y3: a,Z3: a] :
( ( P2 @ X3 @ Y3 )
=> ( ( P2 @ Y3 @ Z3 )
=> ( ( F2 @ X3 @ ( F2 @ Y3 @ Z3 ) )
= ( F2 @ ( F2 @ X3 @ Y3 ) @ Z3 ) ) ) )
& ! [X3: a] :
( ( F2 @ X3 @ X3 )
= X3 ) ) ) ) ).
% set_closure_partial_oprator_def
thf(fact_109_set__closure__oprator_Oclosure__mono,axiom,
! [F: set_option_a > set_option_a > set_option_a,S2: set_set_option_a,T4: set_set_option_a] :
( ( set_cl6352350534672173747tion_a @ F )
=> ( ( ord_le7761976607055303332tion_a @ S2 @ T4 )
=> ( ord_le7761976607055303332tion_a @ ( set_cl6145747590335520936tion_a @ F @ S2 ) @ ( set_cl6145747590335520936tion_a @ F @ T4 ) ) ) ) ).
% set_closure_oprator.closure_mono
thf(fact_110_set__closure__oprator_Oclosure__mono,axiom,
! [F: set_a > set_a > set_a,S2: set_set_a,T4: set_set_a] :
( ( set_cl948588522047921517_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ S2 @ T4 )
=> ( ord_le3724670747650509150_set_a @ ( set_cl7982219259948975842_set_a @ F @ S2 ) @ ( set_cl7982219259948975842_set_a @ F @ T4 ) ) ) ) ).
% set_closure_oprator.closure_mono
thf(fact_111_set__closure__oprator_Oclosure__mono,axiom,
! [F: a > a > a,S2: set_a,T4: set_a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( ord_less_eq_set_a @ S2 @ T4 )
=> ( ord_less_eq_set_a @ ( set_cl2401587249382337922sure_a @ F @ S2 ) @ ( set_cl2401587249382337922sure_a @ F @ T4 ) ) ) ) ).
% set_closure_oprator.closure_mono
thf(fact_112_set__closure__oprator_Oclosure__mono,axiom,
! [F: option_option_a > option_option_a > option_option_a,S2: set_option_option_a,T4: set_option_option_a] :
( ( set_cl4060464781538930713tion_a @ F )
=> ( ( ord_le6764313125106342922tion_a @ S2 @ T4 )
=> ( ord_le6764313125106342922tion_a @ ( set_cl4053507043448260238tion_a @ F @ S2 ) @ ( set_cl4053507043448260238tion_a @ F @ T4 ) ) ) ) ).
% set_closure_oprator.closure_mono
thf(fact_113_set__closure__oprator_Oclosure__mono,axiom,
! [F: option_a > option_a > option_a,S2: set_option_a,T4: set_option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( ord_le1955136853071979460tion_a @ S2 @ T4 )
=> ( ord_le1955136853071979460tion_a @ ( set_cl4093418192473505224tion_a @ F @ S2 ) @ ( set_cl4093418192473505224tion_a @ F @ T4 ) ) ) ) ).
% set_closure_oprator.closure_mono
thf(fact_114_set__closure__partial__oprator_Olift__set__closure__oprator,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( set_cl4060464781538930713tion_a @ ( lift_f6754278611557336796tion_a @ P @ F ) ) ) ).
% set_closure_partial_oprator.lift_set_closure_oprator
thf(fact_115_set__closure__partial__oprator_Olift__set__closure__oprator,axiom,
! [P: a > a > $o,F: a > a > a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( set_cl4772385950493004755tion_a @ ( lift_f_total_a_a_a @ P @ F ) ) ) ).
% set_closure_partial_oprator.lift_set_closure_oprator
thf(fact_116_set__closure__oprator_Oclosure__empty,axiom,
! [F: set_option_a > set_option_a > set_option_a] :
( ( set_cl6352350534672173747tion_a @ F )
=> ( ( set_cl6145747590335520936tion_a @ F @ bot_bo651546871064636272tion_a )
= bot_bo651546871064636272tion_a ) ) ).
% set_closure_oprator.closure_empty
thf(fact_117_set__closure__oprator_Oclosure__empty,axiom,
! [F: a > a > a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( set_cl2401587249382337922sure_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ) ).
% set_closure_oprator.closure_empty
thf(fact_118_set__closure__oprator_Oclosure__empty,axiom,
! [F: option_option_a > option_option_a > option_option_a] :
( ( set_cl4060464781538930713tion_a @ F )
=> ( ( set_cl4053507043448260238tion_a @ F @ bot_bo4163488203964334806tion_a )
= bot_bo4163488203964334806tion_a ) ) ).
% set_closure_oprator.closure_empty
thf(fact_119_set__closure__oprator_Oclosure__empty,axiom,
! [F: option_a > option_a > option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( set_cl4093418192473505224tion_a @ F @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ) ).
% set_closure_oprator.closure_empty
thf(fact_120_set__closure__partial__oprator_Olift__f__total__idem,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( lift_f6754278611557336796tion_a @ P @ F @ X @ X )
= X ) ) ).
% set_closure_partial_oprator.lift_f_total_idem
thf(fact_121_set__closure__partial__oprator_Olift__f__total__idem,axiom,
! [P: a > a > $o,F: a > a > a,X: option_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( lift_f_total_a_a_a @ P @ F @ X @ X )
= X ) ) ).
% set_closure_partial_oprator.lift_f_total_idem
thf(fact_122_set__closure__partial__oprator_Olift__f__total__com,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_option_a,Y: option_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( lift_f6754278611557336796tion_a @ P @ F @ X @ Y )
= ( lift_f6754278611557336796tion_a @ P @ F @ Y @ X ) ) ) ).
% set_closure_partial_oprator.lift_f_total_com
thf(fact_123_set__closure__partial__oprator_Olift__f__total__com,axiom,
! [P: a > a > $o,F: a > a > a,X: option_a,Y: option_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( lift_f_total_a_a_a @ P @ F @ X @ Y )
= ( lift_f_total_a_a_a @ P @ F @ Y @ X ) ) ) ).
% set_closure_partial_oprator.lift_f_total_com
thf(fact_124_set__closure__partial__oprator_Olift__f__total__ass,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,X: option_option_a,Y: option_option_a,Z: option_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( lift_f6754278611557336796tion_a @ P @ F @ X @ ( lift_f6754278611557336796tion_a @ P @ F @ Y @ Z ) )
= ( lift_f6754278611557336796tion_a @ P @ F @ ( lift_f6754278611557336796tion_a @ P @ F @ X @ Y ) @ Z ) ) ) ).
% set_closure_partial_oprator.lift_f_total_ass
thf(fact_125_set__closure__partial__oprator_Olift__f__total__ass,axiom,
! [P: a > a > $o,F: a > a > a,X: option_a,Y: option_a,Z: option_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( lift_f_total_a_a_a @ P @ F @ X @ ( lift_f_total_a_a_a @ P @ F @ Y @ Z ) )
= ( lift_f_total_a_a_a @ P @ F @ ( lift_f_total_a_a_a @ P @ F @ X @ Y ) @ Z ) ) ) ).
% set_closure_partial_oprator.lift_f_total_ass
thf(fact_126_set__closure__oprator_Oclosure_Osimps,axiom,
! [F: set_option_a > set_option_a > set_option_a,A: set_option_a,S2: set_set_option_a] :
( ( set_cl6352350534672173747tion_a @ F )
=> ( ( member_set_option_a @ A @ ( set_cl6145747590335520936tion_a @ F @ S2 ) )
= ( ? [S4: set_option_a] :
( ( A = S4 )
& ( member_set_option_a @ S4 @ S2 ) )
| ? [S4: set_option_a,T2: set_option_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_set_option_a @ S4 @ ( set_cl6145747590335520936tion_a @ F @ S2 ) )
& ( member_set_option_a @ T2 @ ( set_cl6145747590335520936tion_a @ F @ S2 ) ) ) ) ) ) ).
% set_closure_oprator.closure.simps
thf(fact_127_set__closure__oprator_Oclosure_Osimps,axiom,
! [F: set_a > set_a > set_a,A: set_a,S2: set_set_a] :
( ( set_cl948588522047921517_set_a @ F )
=> ( ( member_set_a @ A @ ( set_cl7982219259948975842_set_a @ F @ S2 ) )
= ( ? [S4: set_a] :
( ( A = S4 )
& ( member_set_a @ S4 @ S2 ) )
| ? [S4: set_a,T2: set_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_set_a @ S4 @ ( set_cl7982219259948975842_set_a @ F @ S2 ) )
& ( member_set_a @ T2 @ ( set_cl7982219259948975842_set_a @ F @ S2 ) ) ) ) ) ) ).
% set_closure_oprator.closure.simps
thf(fact_128_set__closure__oprator_Oclosure_Osimps,axiom,
! [F: option_option_a > option_option_a > option_option_a,A: option_option_a,S2: set_option_option_a] :
( ( set_cl4060464781538930713tion_a @ F )
=> ( ( member5113800082084363315tion_a @ A @ ( set_cl4053507043448260238tion_a @ F @ S2 ) )
= ( ? [S4: option_option_a] :
( ( A = S4 )
& ( member5113800082084363315tion_a @ S4 @ S2 ) )
| ? [S4: option_option_a,T2: option_option_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member5113800082084363315tion_a @ S4 @ ( set_cl4053507043448260238tion_a @ F @ S2 ) )
& ( member5113800082084363315tion_a @ T2 @ ( set_cl4053507043448260238tion_a @ F @ S2 ) ) ) ) ) ) ).
% set_closure_oprator.closure.simps
thf(fact_129_set__closure__oprator_Oclosure_Osimps,axiom,
! [F: a > a > a,A: a,S2: set_a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( member_a @ A @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
= ( ? [S4: a] :
( ( A = S4 )
& ( member_a @ S4 @ S2 ) )
| ? [S4: a,T2: a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_a @ S4 @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
& ( member_a @ T2 @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ) ).
% set_closure_oprator.closure.simps
thf(fact_130_set__closure__oprator_Oclosure_Osimps,axiom,
! [F: option_a > option_a > option_a,A: option_a,S2: set_option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( member_option_a @ A @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
= ( ? [S4: option_a] :
( ( A = S4 )
& ( member_option_a @ S4 @ S2 ) )
| ? [S4: option_a,T2: option_a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( member_option_a @ S4 @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
& ( member_option_a @ T2 @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ) ).
% set_closure_oprator.closure.simps
thf(fact_131_set__closure__oprator_Oclosure_Ocases,axiom,
! [F: set_option_a > set_option_a > set_option_a,A: set_option_a,S2: set_set_option_a] :
( ( set_cl6352350534672173747tion_a @ F )
=> ( ( member_set_option_a @ A @ ( set_cl6145747590335520936tion_a @ F @ S2 ) )
=> ( ~ ( member_set_option_a @ A @ S2 )
=> ~ ! [S3: set_option_a,T: set_option_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_set_option_a @ S3 @ ( set_cl6145747590335520936tion_a @ F @ S2 ) )
=> ~ ( member_set_option_a @ T @ ( set_cl6145747590335520936tion_a @ F @ S2 ) ) ) ) ) ) ) ).
% set_closure_oprator.closure.cases
thf(fact_132_set__closure__oprator_Oclosure_Ocases,axiom,
! [F: set_a > set_a > set_a,A: set_a,S2: set_set_a] :
( ( set_cl948588522047921517_set_a @ F )
=> ( ( member_set_a @ A @ ( set_cl7982219259948975842_set_a @ F @ S2 ) )
=> ( ~ ( member_set_a @ A @ S2 )
=> ~ ! [S3: set_a,T: set_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_set_a @ S3 @ ( set_cl7982219259948975842_set_a @ F @ S2 ) )
=> ~ ( member_set_a @ T @ ( set_cl7982219259948975842_set_a @ F @ S2 ) ) ) ) ) ) ) ).
% set_closure_oprator.closure.cases
thf(fact_133_set__closure__oprator_Oclosure_Ocases,axiom,
! [F: option_option_a > option_option_a > option_option_a,A: option_option_a,S2: set_option_option_a] :
( ( set_cl4060464781538930713tion_a @ F )
=> ( ( member5113800082084363315tion_a @ A @ ( set_cl4053507043448260238tion_a @ F @ S2 ) )
=> ( ~ ( member5113800082084363315tion_a @ A @ S2 )
=> ~ ! [S3: option_option_a,T: option_option_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member5113800082084363315tion_a @ S3 @ ( set_cl4053507043448260238tion_a @ F @ S2 ) )
=> ~ ( member5113800082084363315tion_a @ T @ ( set_cl4053507043448260238tion_a @ F @ S2 ) ) ) ) ) ) ) ).
% set_closure_oprator.closure.cases
thf(fact_134_set__closure__oprator_Oclosure_Ocases,axiom,
! [F: a > a > a,A: a,S2: set_a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( member_a @ A @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ( ~ ( member_a @ A @ S2 )
=> ~ ! [S3: a,T: a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_a @ S3 @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ~ ( member_a @ T @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ) ) ).
% set_closure_oprator.closure.cases
thf(fact_135_set__closure__oprator_Oclosure_Ocases,axiom,
! [F: option_a > option_a > option_a,A: option_a,S2: set_option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( member_option_a @ A @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ( ~ ( member_option_a @ A @ S2 )
=> ~ ! [S3: option_a,T: option_a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( member_option_a @ S3 @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ~ ( member_option_a @ T @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ) ) ).
% set_closure_oprator.closure.cases
thf(fact_136_set__closure__oprator_Oclosure__idem,axiom,
! [F: a > a > a,S2: set_a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( set_cl2401587249382337922sure_a @ F @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
= ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ).
% set_closure_oprator.closure_idem
thf(fact_137_set__closure__oprator_Oclosure__idem,axiom,
! [F: option_option_a > option_option_a > option_option_a,S2: set_option_option_a] :
( ( set_cl4060464781538930713tion_a @ F )
=> ( ( set_cl4053507043448260238tion_a @ F @ ( set_cl4053507043448260238tion_a @ F @ S2 ) )
= ( set_cl4053507043448260238tion_a @ F @ S2 ) ) ) ).
% set_closure_oprator.closure_idem
thf(fact_138_set__closure__oprator_Oclosure__idem,axiom,
! [F: option_a > option_a > option_a,S2: set_option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( set_cl4093418192473505224tion_a @ F @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
= ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ).
% set_closure_oprator.closure_idem
thf(fact_139_set__closure__oprator_Oclosure_Ostep,axiom,
! [F: set_option_a > set_option_a > set_option_a,S: set_option_a,S2: set_set_option_a,T3: set_option_a] :
( ( set_cl6352350534672173747tion_a @ F )
=> ( ( member_set_option_a @ S @ ( set_cl6145747590335520936tion_a @ F @ S2 ) )
=> ( ( member_set_option_a @ T3 @ ( set_cl6145747590335520936tion_a @ F @ S2 ) )
=> ( member_set_option_a @ ( F @ S @ T3 ) @ ( set_cl6145747590335520936tion_a @ F @ S2 ) ) ) ) ) ).
% set_closure_oprator.closure.step
thf(fact_140_set__closure__oprator_Oclosure_Ostep,axiom,
! [F: set_a > set_a > set_a,S: set_a,S2: set_set_a,T3: set_a] :
( ( set_cl948588522047921517_set_a @ F )
=> ( ( member_set_a @ S @ ( set_cl7982219259948975842_set_a @ F @ S2 ) )
=> ( ( member_set_a @ T3 @ ( set_cl7982219259948975842_set_a @ F @ S2 ) )
=> ( member_set_a @ ( F @ S @ T3 ) @ ( set_cl7982219259948975842_set_a @ F @ S2 ) ) ) ) ) ).
% set_closure_oprator.closure.step
thf(fact_141_set__closure__oprator_Oclosure_Ostep,axiom,
! [F: option_option_a > option_option_a > option_option_a,S: option_option_a,S2: set_option_option_a,T3: option_option_a] :
( ( set_cl4060464781538930713tion_a @ F )
=> ( ( member5113800082084363315tion_a @ S @ ( set_cl4053507043448260238tion_a @ F @ S2 ) )
=> ( ( member5113800082084363315tion_a @ T3 @ ( set_cl4053507043448260238tion_a @ F @ S2 ) )
=> ( member5113800082084363315tion_a @ ( F @ S @ T3 ) @ ( set_cl4053507043448260238tion_a @ F @ S2 ) ) ) ) ) ).
% set_closure_oprator.closure.step
thf(fact_142_set__closure__oprator_Oclosure_Ostep,axiom,
! [F: a > a > a,S: a,S2: set_a,T3: a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( member_a @ S @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ( ( member_a @ T3 @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ( member_a @ ( F @ S @ T3 ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ).
% set_closure_oprator.closure.step
thf(fact_143_set__closure__oprator_Oclosure_Ostep,axiom,
! [F: option_a > option_a > option_a,S: option_a,S2: set_option_a,T3: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( member_option_a @ S @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ( ( member_option_a @ T3 @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ( member_option_a @ ( F @ S @ T3 ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ).
% set_closure_oprator.closure.step
thf(fact_144_set__closure__oprator_Oclosure_Obase,axiom,
! [F: set_option_a > set_option_a > set_option_a,S: set_option_a,S2: set_set_option_a] :
( ( set_cl6352350534672173747tion_a @ F )
=> ( ( member_set_option_a @ S @ S2 )
=> ( member_set_option_a @ S @ ( set_cl6145747590335520936tion_a @ F @ S2 ) ) ) ) ).
% set_closure_oprator.closure.base
thf(fact_145_set__closure__oprator_Oclosure_Obase,axiom,
! [F: set_a > set_a > set_a,S: set_a,S2: set_set_a] :
( ( set_cl948588522047921517_set_a @ F )
=> ( ( member_set_a @ S @ S2 )
=> ( member_set_a @ S @ ( set_cl7982219259948975842_set_a @ F @ S2 ) ) ) ) ).
% set_closure_oprator.closure.base
thf(fact_146_set__closure__oprator_Oclosure_Obase,axiom,
! [F: option_option_a > option_option_a > option_option_a,S: option_option_a,S2: set_option_option_a] :
( ( set_cl4060464781538930713tion_a @ F )
=> ( ( member5113800082084363315tion_a @ S @ S2 )
=> ( member5113800082084363315tion_a @ S @ ( set_cl4053507043448260238tion_a @ F @ S2 ) ) ) ) ).
% set_closure_oprator.closure.base
thf(fact_147_set__closure__oprator_Oclosure_Obase,axiom,
! [F: a > a > a,S: a,S2: set_a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( member_a @ S @ S2 )
=> ( member_a @ S @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ).
% set_closure_oprator.closure.base
thf(fact_148_set__closure__oprator_Oclosure_Obase,axiom,
! [F: option_a > option_a > option_a,S: option_a,S2: set_option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( member_option_a @ S @ S2 )
=> ( member_option_a @ S @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ).
% set_closure_oprator.closure.base
thf(fact_149_set__closure__partial__oprator_Olift__f__totalE,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,S: option_option_a,U: option_option_a,T3: option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( ( lift_f6754278611557336796tion_a @ P @ F @ S @ U )
= ( some_option_a @ T3 ) )
=> ~ ( ? [V: option_a] :
( S
= ( some_option_a @ V ) )
=> ! [W: option_a] :
( U
!= ( some_option_a @ W ) ) ) ) ) ).
% set_closure_partial_oprator.lift_f_totalE
thf(fact_150_set__closure__partial__oprator_Olift__f__totalE,axiom,
! [P: a > a > $o,F: a > a > a,S: option_a,U: option_a,T3: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( ( lift_f_total_a_a_a @ P @ F @ S @ U )
= ( some_a @ T3 ) )
=> ~ ( ? [V: a] :
( S
= ( some_a @ V ) )
=> ! [W: a] :
( U
!= ( some_a @ W ) ) ) ) ) ).
% set_closure_partial_oprator.lift_f_totalE
thf(fact_151_set__closure__partial__oprator_Opred__closure__to__some__lift__closure,axiom,
! [P: option_option_a > option_option_a > $o,F: option_option_a > option_option_a > option_option_a,S: option_option_a,S2: set_option_option_a] :
( ( set_cl1309597857828561964tion_a @ P @ F )
=> ( ( member5113800082084363315tion_a @ S @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) )
=> ( member7397271250019230201tion_a @ ( some_option_option_a @ S ) @ ( set_cl7522643762412554196tion_a @ ( lift_f7361094991155314338tion_a @ P @ F ) @ ( image_8975281652752015947tion_a @ some_option_option_a @ S2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure_to_some_lift_closure
thf(fact_152_set__closure__partial__oprator_Opred__closure__to__some__lift__closure,axiom,
! [P: set_option_a > set_option_a > $o,F: set_option_a > set_option_a > set_option_a,S: set_option_a,S2: set_set_option_a] :
( ( set_cl2851621040977480518tion_a @ P @ F )
=> ( ( member_set_option_a @ S @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) )
=> ( member6923355609089322771tion_a @ ( some_set_option_a @ S ) @ ( set_cl2512250281393245550tion_a @ ( lift_f1864078254722121148tion_a @ P @ F ) @ ( image_2571330352900812747tion_a @ some_set_option_a @ S2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure_to_some_lift_closure
thf(fact_153_set__closure__partial__oprator_Opred__closure__to__some__lift__closure,axiom,
! [P: set_a > set_a > $o,F: set_a > set_a > set_a,S: set_a,S2: set_set_a] :
( ( set_cl2751670595400742528_set_a @ P @ F )
=> ( ( member_set_a @ S @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) )
=> ( member_option_set_a @ ( some_set_a @ S ) @ ( set_cl6928935069012605736_set_a @ ( lift_f1381871799933197046_set_a @ P @ F ) @ ( image_2983976770393306955_set_a @ some_set_a @ S2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure_to_some_lift_closure
thf(fact_154_set__closure__partial__oprator_Opred__closure__to__some__lift__closure,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,S: option_a,S2: set_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( member_option_a @ S @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) )
=> ( member5113800082084363315tion_a @ ( some_option_a @ S ) @ ( set_cl4053507043448260238tion_a @ ( lift_f6754278611557336796tion_a @ P @ F ) @ ( image_2132136900116418507tion_a @ some_option_a @ S2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure_to_some_lift_closure
thf(fact_155_set__closure__partial__oprator_Opred__closure__to__some__lift__closure,axiom,
! [P: a > a > $o,F: a > a > a,S: a,S2: set_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( member_a @ S @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) )
=> ( member_option_a @ ( some_a @ S ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ P @ F ) @ ( image_a_option_a @ some_a @ S2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closure_to_some_lift_closure
thf(fact_156_set__closure__partial__oprator_Osome__lift__closure__pred__closure,axiom,
! [P: option_option_a > option_option_a > $o,F: option_option_a > option_option_a > option_option_a,T3: option_option_a,S2: set_option_option_a] :
( ( set_cl1309597857828561964tion_a @ P @ F )
=> ( ( member7397271250019230201tion_a @ ( some_option_option_a @ T3 ) @ ( set_cl7522643762412554196tion_a @ ( lift_f7361094991155314338tion_a @ P @ F ) @ ( image_8975281652752015947tion_a @ some_option_option_a @ S2 ) ) )
=> ( member5113800082084363315tion_a @ T3 @ ( set_cl1739147886471116348tion_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.some_lift_closure_pred_closure
thf(fact_157_set__closure__partial__oprator_Osome__lift__closure__pred__closure,axiom,
! [P: set_option_a > set_option_a > $o,F: set_option_a > set_option_a > set_option_a,T3: set_option_a,S2: set_set_option_a] :
( ( set_cl2851621040977480518tion_a @ P @ F )
=> ( ( member6923355609089322771tion_a @ ( some_set_option_a @ T3 ) @ ( set_cl2512250281393245550tion_a @ ( lift_f1864078254722121148tion_a @ P @ F ) @ ( image_2571330352900812747tion_a @ some_set_option_a @ S2 ) ) )
=> ( member_set_option_a @ T3 @ ( set_cl8037838587962438998tion_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.some_lift_closure_pred_closure
thf(fact_158_set__closure__partial__oprator_Osome__lift__closure__pred__closure,axiom,
! [P: set_a > set_a > $o,F: set_a > set_a > set_a,T3: set_a,S2: set_set_a] :
( ( set_cl2751670595400742528_set_a @ P @ F )
=> ( ( member_option_set_a @ ( some_set_a @ T3 ) @ ( set_cl6928935069012605736_set_a @ ( lift_f1381871799933197046_set_a @ P @ F ) @ ( image_2983976770393306955_set_a @ some_set_a @ S2 ) ) )
=> ( member_set_a @ T3 @ ( set_cl3308208168262416528_set_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.some_lift_closure_pred_closure
thf(fact_159_set__closure__partial__oprator_Osome__lift__closure__pred__closure,axiom,
! [P: option_a > option_a > $o,F: option_a > option_a > option_a,T3: option_a,S2: set_option_a] :
( ( set_cl936642855594582118tion_a @ P @ F )
=> ( ( member5113800082084363315tion_a @ ( some_option_a @ T3 ) @ ( set_cl4053507043448260238tion_a @ ( lift_f6754278611557336796tion_a @ P @ F ) @ ( image_2132136900116418507tion_a @ some_option_a @ S2 ) ) )
=> ( member_option_a @ T3 @ ( set_cl7151870100604144246tion_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.some_lift_closure_pred_closure
thf(fact_160_set__closure__partial__oprator_Osome__lift__closure__pred__closure,axiom,
! [P: a > a > $o,F: a > a > a,T3: a,S2: set_a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( member_option_a @ ( some_a @ T3 ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ P @ F ) @ ( image_a_option_a @ some_a @ S2 ) ) )
=> ( member_a @ T3 @ ( set_cl7833034999824766768sure_a @ P @ F @ S2 ) ) ) ) ).
% set_closure_partial_oprator.some_lift_closure_pred_closure
thf(fact_161_lift__f__total_Osimps_I1_J,axiom,
! [P: a > a > $o,F: a > a > a,Uu: option_a] :
( ( lift_f_total_a_a_a @ P @ F @ none_a @ Uu )
= none_a ) ).
% lift_f_total.simps(1)
thf(fact_162_lift__f__total_Oelims,axiom,
! [X: a > a > $o,Xa: a > a > a,Xb: option_a,Xc: option_a,Y: option_a] :
( ( ( lift_f_total_a_a_a @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xb = none_a )
=> ( Y != none_a ) )
=> ( ( ? [V: a] :
( Xb
= ( some_a @ V ) )
=> ( ( Xc = none_a )
=> ( Y != none_a ) ) )
=> ~ ! [S3: a] :
( ( Xb
= ( some_a @ S3 ) )
=> ! [T: a] :
( ( Xc
= ( some_a @ T ) )
=> ~ ( ( ( X @ S3 @ T )
=> ( Y
= ( some_a @ ( Xa @ S3 @ T ) ) ) )
& ( ~ ( X @ S3 @ T )
=> ( Y = none_a ) ) ) ) ) ) ) ) ).
% lift_f_total.elims
thf(fact_163_lift__f__total_Osimps_I2_J,axiom,
! [P: a > a > $o,F: a > a > a,V2: a] :
( ( lift_f_total_a_a_a @ P @ F @ ( some_a @ V2 ) @ none_a )
= none_a ) ).
% lift_f_total.simps(2)
thf(fact_164_lift__f__total_Osimps_I3_J,axiom,
! [P: a > a > $o,S: a,T3: a,F: a > a > a] :
( ( ( P @ S @ T3 )
=> ( ( lift_f_total_a_a_a @ P @ F @ ( some_a @ S ) @ ( some_a @ T3 ) )
= ( some_a @ ( F @ S @ T3 ) ) ) )
& ( ~ ( P @ S @ T3 )
=> ( ( lift_f_total_a_a_a @ P @ F @ ( some_a @ S ) @ ( some_a @ T3 ) )
= none_a ) ) ) ).
% lift_f_total.simps(3)
thf(fact_165_option_Ocollapse,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( ( some_a @ ( the_a @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_166_lift__fun_Ofold__graph__insertE__aux,axiom,
! [Z: option_a,A2: set_option_a,Y: option_a,A: option_a] :
( ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 @ Y )
=> ( ( member_option_a @ A @ A2 )
=> ? [Y4: option_a] :
( ( Y
= ( lift_f_total_a_a_a @ p @ f @ A @ Y4 ) )
& ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ Y4 ) ) ) ) ).
% lift_fun.fold_graph_insertE_aux
thf(fact_167_insert__Diff__single,axiom,
! [A: option_a,A2: set_option_a] :
( ( insert_option_a @ A @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( insert_option_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_168_Diff__eq__empty__iff,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ( minus_1574173051537231627tion_a @ A2 @ B )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_169_singleton__insert__inj__eq,axiom,
! [B2: option_a,A: option_a,A2: set_option_a] :
( ( ( insert_option_a @ B2 @ bot_bot_set_option_a )
= ( insert_option_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ B2 @ bot_bot_set_option_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_170_singleton__insert__inj__eq_H,axiom,
! [A: option_a,A2: set_option_a,B2: option_a] :
( ( ( insert_option_a @ A @ A2 )
= ( insert_option_a @ B2 @ bot_bot_set_option_a ) )
= ( ( A = B2 )
& ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ B2 @ bot_bot_set_option_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_171_lift__fun_Ofold__graph__insertE,axiom,
! [Z: option_a,X: option_a,A2: set_option_a,V2: option_a] :
( ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( insert_option_a @ X @ A2 ) @ V2 )
=> ( ~ ( member_option_a @ X @ A2 )
=> ~ ! [Y2: option_a] :
( ( V2
= ( lift_f_total_a_a_a @ p @ f @ X @ Y2 ) )
=> ~ ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 @ Y2 ) ) ) ) ).
% lift_fun.fold_graph_insertE
thf(fact_172_not__Some__eq,axiom,
! [X: option_a] :
( ( ! [Y3: a] :
( X
!= ( some_a @ Y3 ) ) )
= ( X = none_a ) ) ).
% not_Some_eq
thf(fact_173_not__None__eq,axiom,
! [X: option_a] :
( ( X != none_a )
= ( ? [Y3: a] :
( X
= ( some_a @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_174_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_175_Diff__insert0,axiom,
! [X: option_a,A2: set_option_a,B: set_option_a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ B ) )
= ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_176_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_177_insert__Diff1,axiom,
! [X: option_a,B: set_option_a,A2: set_option_a] :
( ( member_option_a @ X @ B )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A2 ) @ B )
= ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_178_image__eqI,axiom,
! [B2: option_a,F: option_a > option_a,X: option_a,A2: set_option_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_option_a @ X @ A2 )
=> ( member_option_a @ B2 @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_179_image__eqI,axiom,
! [B2: a,F: option_a > a,X: option_a,A2: set_option_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_option_a @ X @ A2 )
=> ( member_a @ B2 @ ( image_option_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_180_image__eqI,axiom,
! [B2: option_a,F: a > option_a,X: a,A2: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_option_a @ B2 @ ( image_a_option_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_181_image__eqI,axiom,
! [B2: a,F: a > a,X: a,A2: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_182_empty__Collect__eq,axiom,
! [P: option_a > $o] :
( ( bot_bot_set_option_a
= ( collect_option_a @ P ) )
= ( ! [X3: option_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_183_Collect__empty__eq,axiom,
! [P: option_a > $o] :
( ( ( collect_option_a @ P )
= bot_bot_set_option_a )
= ( ! [X3: option_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_184_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_185_all__not__in__conv,axiom,
! [A2: set_option_a] :
( ( ! [X3: option_a] :
~ ( member_option_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_option_a ) ) ).
% all_not_in_conv
thf(fact_186_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_187_empty__iff,axiom,
! [C: option_a] :
~ ( member_option_a @ C @ bot_bot_set_option_a ) ).
% empty_iff
thf(fact_188_subset__antisym,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( ord_le1955136853071979460tion_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_189_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_190_subsetI,axiom,
! [A2: set_option_a,B: set_option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( member_option_a @ X2 @ B ) )
=> ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ).
% subsetI
thf(fact_191_insert__absorb2,axiom,
! [X: option_a,A2: set_option_a] :
( ( insert_option_a @ X @ ( insert_option_a @ X @ A2 ) )
= ( insert_option_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_192_insert__iff,axiom,
! [A: option_a,B2: option_a,A2: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_option_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_193_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_194_insertCI,axiom,
! [A: option_a,B: set_option_a,B2: option_a] :
( ( ~ ( member_option_a @ A @ B )
=> ( A = B2 ) )
=> ( member_option_a @ A @ ( insert_option_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_195_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_196_Diff__idemp,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( minus_1574173051537231627tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ B )
= ( minus_1574173051537231627tion_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_197_Diff__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_198_Diff__iff,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( ( member_option_a @ C @ A2 )
& ~ ( member_option_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_199_DiffI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_200_DiffI,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ A2 )
=> ( ~ ( member_option_a @ C @ B )
=> ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_201_option_Oinject,axiom,
! [X22: a,Y22: a] :
( ( ( some_a @ X22 )
= ( some_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_202_lift__fun_Ofold__graph__determ,axiom,
! [Z: option_a,A2: set_option_a,X: option_a,Y: option_a] :
( ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 @ X )
=> ( ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 @ Y )
=> ( Y = X ) ) ) ).
% lift_fun.fold_graph_determ
thf(fact_203_image__empty,axiom,
! [F: a > option_a] :
( ( image_a_option_a @ F @ bot_bot_set_a )
= bot_bot_set_option_a ) ).
% image_empty
thf(fact_204_image__empty,axiom,
! [F: option_a > a] :
( ( image_option_a_a @ F @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_205_image__empty,axiom,
! [F: option_a > option_a] :
( ( image_7439109396645324421tion_a @ F @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% image_empty
thf(fact_206_empty__is__image,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( bot_bot_set_a
= ( image_option_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_option_a ) ) ).
% empty_is_image
thf(fact_207_empty__is__image,axiom,
! [F: a > option_a,A2: set_a] :
( ( bot_bot_set_option_a
= ( image_a_option_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_208_empty__is__image,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( bot_bot_set_option_a
= ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( A2 = bot_bot_set_option_a ) ) ).
% empty_is_image
thf(fact_209_image__is__empty,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( ( image_option_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_option_a ) ) ).
% image_is_empty
thf(fact_210_image__is__empty,axiom,
! [F: a > option_a,A2: set_a] :
( ( ( image_a_option_a @ F @ A2 )
= bot_bot_set_option_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_211_image__is__empty,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( ( image_7439109396645324421tion_a @ F @ A2 )
= bot_bot_set_option_a )
= ( A2 = bot_bot_set_option_a ) ) ).
% image_is_empty
thf(fact_212_empty__subsetI,axiom,
! [A2: set_option_a] : ( ord_le1955136853071979460tion_a @ bot_bot_set_option_a @ A2 ) ).
% empty_subsetI
thf(fact_213_subset__empty,axiom,
! [A2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ bot_bot_set_option_a )
= ( A2 = bot_bot_set_option_a ) ) ).
% subset_empty
thf(fact_214_image__insert,axiom,
! [F: a > option_a,A: a,B: set_a] :
( ( image_a_option_a @ F @ ( insert_a @ A @ B ) )
= ( insert_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ B ) ) ) ).
% image_insert
thf(fact_215_image__insert,axiom,
! [F: option_a > a,A: option_a,B: set_option_a] :
( ( image_option_a_a @ F @ ( insert_option_a @ A @ B ) )
= ( insert_a @ ( F @ A ) @ ( image_option_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_216_image__insert,axiom,
! [F: option_a > option_a,A: option_a,B: set_option_a] :
( ( image_7439109396645324421tion_a @ F @ ( insert_option_a @ A @ B ) )
= ( insert_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ).
% image_insert
thf(fact_217_insert__image,axiom,
! [X: option_a,A2: set_option_a,F: option_a > a] :
( ( member_option_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_option_a_a @ F @ A2 ) )
= ( image_option_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_218_insert__image,axiom,
! [X: option_a,A2: set_option_a,F: option_a > option_a] :
( ( member_option_a @ X @ A2 )
=> ( ( insert_option_a @ ( F @ X ) @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_219_insert__image,axiom,
! [X: a,A2: set_a,F: a > option_a] :
( ( member_a @ X @ A2 )
=> ( ( insert_option_a @ ( F @ X ) @ ( image_a_option_a @ F @ A2 ) )
= ( image_a_option_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_220_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_221_singletonI,axiom,
! [A: option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singletonI
thf(fact_222_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_223_insert__subset,axiom,
! [X: option_a,A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ B )
= ( ( member_option_a @ X @ B )
& ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_224_Diff__cancel,axiom,
! [A2: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ A2 )
= bot_bot_set_option_a ) ).
% Diff_cancel
thf(fact_225_empty__Diff,axiom,
! [A2: set_option_a] :
( ( minus_1574173051537231627tion_a @ bot_bot_set_option_a @ A2 )
= bot_bot_set_option_a ) ).
% empty_Diff
thf(fact_226_Diff__empty,axiom,
! [A2: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ bot_bot_set_option_a )
= A2 ) ).
% Diff_empty
thf(fact_227_imageI,axiom,
! [X: option_a,A2: set_option_a,F: option_a > option_a] :
( ( member_option_a @ X @ A2 )
=> ( member_option_a @ ( F @ X ) @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_228_imageI,axiom,
! [X: option_a,A2: set_option_a,F: option_a > a] :
( ( member_option_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_option_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_229_imageI,axiom,
! [X: a,A2: set_a,F: a > option_a] :
( ( member_a @ X @ A2 )
=> ( member_option_a @ ( F @ X ) @ ( image_a_option_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_230_imageI,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_231_image__iff,axiom,
! [Z: option_a,F: a > option_a,A2: set_a] :
( ( member_option_a @ Z @ ( image_a_option_a @ F @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_232_image__iff,axiom,
! [Z: a,F: option_a > a,A2: set_option_a] :
( ( member_a @ Z @ ( image_option_a_a @ F @ A2 ) )
= ( ? [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_233_bex__imageD,axiom,
! [F: a > option_a,A2: set_a,P: option_a > $o] :
( ? [X4: option_a] :
( ( member_option_a @ X4 @ ( image_a_option_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_234_bex__imageD,axiom,
! [F: option_a > a,A2: set_option_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_option_a_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_235_image__cong,axiom,
! [M: set_option_a,N: set_option_a,F: option_a > a,G: option_a > a] :
( ( M = N )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_option_a_a @ F @ M )
= ( image_option_a_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_236_image__cong,axiom,
! [M: set_a,N: set_a,F: a > option_a,G: a > option_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_option_a @ F @ M )
= ( image_a_option_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_237_ball__imageD,axiom,
! [F: a > option_a,A2: set_a,P: option_a > $o] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ ( image_a_option_a @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_238_ball__imageD,axiom,
! [F: option_a > a,A2: set_option_a,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_option_a_a @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: option_a] :
( ( member_option_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_239_rev__image__eqI,axiom,
! [X: option_a,A2: set_option_a,B2: option_a,F: option_a > option_a] :
( ( member_option_a @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_option_a @ B2 @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_240_rev__image__eqI,axiom,
! [X: option_a,A2: set_option_a,B2: a,F: option_a > a] :
( ( member_option_a @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_option_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_241_rev__image__eqI,axiom,
! [X: a,A2: set_a,B2: option_a,F: a > option_a] :
( ( member_a @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_option_a @ B2 @ ( image_a_option_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_242_rev__image__eqI,axiom,
! [X: a,A2: set_a,B2: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_243_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_244_ex__in__conv,axiom,
! [A2: set_option_a] :
( ( ? [X3: option_a] : ( member_option_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_option_a ) ) ).
% ex_in_conv
thf(fact_245_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_246_equals0I,axiom,
! [A2: set_option_a] :
( ! [Y2: option_a] :
~ ( member_option_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_option_a ) ) ).
% equals0I
thf(fact_247_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_248_equals0D,axiom,
! [A2: set_option_a,A: option_a] :
( ( A2 = bot_bot_set_option_a )
=> ~ ( member_option_a @ A @ A2 ) ) ).
% equals0D
thf(fact_249_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_250_emptyE,axiom,
! [A: option_a] :
~ ( member_option_a @ A @ bot_bot_set_option_a ) ).
% emptyE
thf(fact_251_Collect__mono__iff,axiom,
! [P: option_a > $o,Q: option_a > $o] :
( ( ord_le1955136853071979460tion_a @ ( collect_option_a @ P ) @ ( collect_option_a @ Q ) )
= ( ! [X3: option_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_252_set__eq__subset,axiom,
( ( ^ [Y5: set_option_a,Z4: set_option_a] : ( Y5 = Z4 ) )
= ( ^ [A3: set_option_a,B3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A3 @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_253_subset__trans,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ord_le1955136853071979460tion_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_254_Collect__mono,axiom,
! [P: option_a > $o,Q: option_a > $o] :
( ! [X2: option_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1955136853071979460tion_a @ ( collect_option_a @ P ) @ ( collect_option_a @ Q ) ) ) ).
% Collect_mono
thf(fact_255_subset__refl,axiom,
! [A2: set_option_a] : ( ord_le1955136853071979460tion_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_256_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A3 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_257_subset__iff,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A3: set_option_a,B3: set_option_a] :
! [T2: option_a] :
( ( member_option_a @ T2 @ A3 )
=> ( member_option_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_258_equalityD2,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( A2 = B )
=> ( ord_le1955136853071979460tion_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_259_equalityD1,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( A2 = B )
=> ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_260_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_261_subset__eq,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A3: set_option_a,B3: set_option_a] :
! [X3: option_a] :
( ( member_option_a @ X3 @ A3 )
=> ( member_option_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_262_equalityE,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( A2 = B )
=> ~ ( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ~ ( ord_le1955136853071979460tion_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_263_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_264_subsetD,axiom,
! [A2: set_option_a,B: set_option_a,C: option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( member_option_a @ C @ A2 )
=> ( member_option_a @ C @ B ) ) ) ).
% subsetD
thf(fact_265_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_266_in__mono,axiom,
! [A2: set_option_a,B: set_option_a,X: option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( member_option_a @ X @ A2 )
=> ( member_option_a @ X @ B ) ) ) ).
% in_mono
thf(fact_267_mk__disjoint__insert,axiom,
! [A: option_a,A2: set_option_a] :
( ( member_option_a @ A @ A2 )
=> ? [B4: set_option_a] :
( ( A2
= ( insert_option_a @ A @ B4 ) )
& ~ ( member_option_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_268_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_269_insert__commute,axiom,
! [X: option_a,Y: option_a,A2: set_option_a] :
( ( insert_option_a @ X @ ( insert_option_a @ Y @ A2 ) )
= ( insert_option_a @ Y @ ( insert_option_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_270_insert__eq__iff,axiom,
! [A: option_a,A2: set_option_a,B2: option_a,B: set_option_a] :
( ~ ( member_option_a @ A @ A2 )
=> ( ~ ( member_option_a @ B2 @ B )
=> ( ( ( insert_option_a @ A @ A2 )
= ( insert_option_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_option_a] :
( ( A2
= ( insert_option_a @ B2 @ C3 ) )
& ~ ( member_option_a @ B2 @ C3 )
& ( B
= ( insert_option_a @ A @ C3 ) )
& ~ ( member_option_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_271_insert__eq__iff,axiom,
! [A: a,A2: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_272_insert__absorb,axiom,
! [A: option_a,A2: set_option_a] :
( ( member_option_a @ A @ A2 )
=> ( ( insert_option_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_273_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_274_insert__ident,axiom,
! [X: option_a,A2: set_option_a,B: set_option_a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ~ ( member_option_a @ X @ B )
=> ( ( ( insert_option_a @ X @ A2 )
= ( insert_option_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_275_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_276_Set_Oset__insert,axiom,
! [X: option_a,A2: set_option_a] :
( ( member_option_a @ X @ A2 )
=> ~ ! [B4: set_option_a] :
( ( A2
= ( insert_option_a @ X @ B4 ) )
=> ( member_option_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_277_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_278_insertI2,axiom,
! [A: option_a,B: set_option_a,B2: option_a] :
( ( member_option_a @ A @ B )
=> ( member_option_a @ A @ ( insert_option_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_279_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_280_insertI1,axiom,
! [A: option_a,B: set_option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ B ) ) ).
% insertI1
thf(fact_281_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_282_insertE,axiom,
! [A: option_a,B2: option_a,A2: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_option_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_283_insertE,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_284_DiffD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_285_DiffD2,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
=> ~ ( member_option_a @ C @ B ) ) ).
% DiffD2
thf(fact_286_DiffD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_287_DiffD1,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
=> ( member_option_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_288_DiffE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_289_DiffE,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
=> ~ ( ( member_option_a @ C @ A2 )
=> ( member_option_a @ C @ B ) ) ) ).
% DiffE
thf(fact_290_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > option_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) ) ).
% image_mono
thf(fact_291_image__mono,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_292_image__mono,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ).
% image_mono
thf(fact_293_image__subsetI,axiom,
! [A2: set_option_a,F: option_a > a,B: set_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_294_image__subsetI,axiom,
! [A2: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_295_image__subsetI,axiom,
! [A2: set_option_a,F: option_a > option_a,B: set_option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( member_option_a @ ( F @ X2 ) @ B ) )
=> ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_296_image__subsetI,axiom,
! [A2: set_a,F: a > option_a,B: set_option_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_option_a @ ( F @ X2 ) @ B ) )
=> ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_297_subset__imageE,axiom,
! [B: set_a,F: option_a > a,A2: set_option_a] :
( ( ord_less_eq_set_a @ B @ ( image_option_a_a @ F @ A2 ) )
=> ~ ! [C4: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C4 @ A2 )
=> ( B
!= ( image_option_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_298_subset__imageE,axiom,
! [B: set_option_a,F: a > option_a,A2: set_a] :
( ( ord_le1955136853071979460tion_a @ B @ ( image_a_option_a @ F @ A2 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( B
!= ( image_a_option_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_299_subset__imageE,axiom,
! [B: set_option_a,F: option_a > option_a,A2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ~ ! [C4: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C4 @ A2 )
=> ( B
!= ( image_7439109396645324421tion_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_300_image__subset__iff,axiom,
! [F: option_a > a,A2: set_option_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ B )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_301_image__subset__iff,axiom,
! [F: a > option_a,A2: set_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_option_a @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_302_subset__image__iff,axiom,
! [B: set_a,F: option_a > a,A2: set_option_a] :
( ( ord_less_eq_set_a @ B @ ( image_option_a_a @ F @ A2 ) )
= ( ? [AA: set_option_a] :
( ( ord_le1955136853071979460tion_a @ AA @ A2 )
& ( B
= ( image_option_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_303_subset__image__iff,axiom,
! [B: set_option_a,F: a > option_a,A2: set_a] :
( ( ord_le1955136853071979460tion_a @ B @ ( image_a_option_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_option_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_304_subset__image__iff,axiom,
! [B: set_option_a,F: option_a > option_a,A2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( ? [AA: set_option_a] :
( ( ord_le1955136853071979460tion_a @ AA @ A2 )
& ( B
= ( image_7439109396645324421tion_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_305_singleton__inject,axiom,
! [A: option_a,B2: option_a] :
( ( ( insert_option_a @ A @ bot_bot_set_option_a )
= ( insert_option_a @ B2 @ bot_bot_set_option_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_306_insert__not__empty,axiom,
! [A: option_a,A2: set_option_a] :
( ( insert_option_a @ A @ A2 )
!= bot_bot_set_option_a ) ).
% insert_not_empty
thf(fact_307_doubleton__eq__iff,axiom,
! [A: option_a,B2: option_a,C: option_a,D: option_a] :
( ( ( insert_option_a @ A @ ( insert_option_a @ B2 @ bot_bot_set_option_a ) )
= ( insert_option_a @ C @ ( insert_option_a @ D @ bot_bot_set_option_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_308_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_309_singleton__iff,axiom,
! [B2: option_a,A: option_a] :
( ( member_option_a @ B2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_310_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_311_singletonD,axiom,
! [B2: option_a,A: option_a] :
( ( member_option_a @ B2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_312_subset__insertI2,axiom,
! [A2: set_option_a,B: set_option_a,B2: option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_313_subset__insertI,axiom,
! [B: set_option_a,A: option_a] : ( ord_le1955136853071979460tion_a @ B @ ( insert_option_a @ A @ B ) ) ).
% subset_insertI
thf(fact_314_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_315_subset__insert,axiom,
! [X: option_a,A2: set_option_a,B: set_option_a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ X @ B ) )
= ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_316_insert__mono,axiom,
! [C2: set_option_a,D2: set_option_a,A: option_a] :
( ( ord_le1955136853071979460tion_a @ C2 @ D2 )
=> ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ C2 ) @ ( insert_option_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_317_double__diff,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ( minus_1574173051537231627tion_a @ B @ ( minus_1574173051537231627tion_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_318_Diff__subset,axiom,
! [A2: set_option_a,B: set_option_a] : ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_319_Diff__mono,axiom,
! [A2: set_option_a,C2: set_option_a,D2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ D2 @ B )
=> ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ ( minus_1574173051537231627tion_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_320_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_321_insert__Diff__if,axiom,
! [X: option_a,B: set_option_a,A2: set_option_a] :
( ( ( member_option_a @ X @ B )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A2 ) @ B )
= ( minus_1574173051537231627tion_a @ A2 @ B ) ) )
& ( ~ ( member_option_a @ X @ B )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A2 ) @ B )
= ( insert_option_a @ X @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_322_option_Odistinct_I1_J,axiom,
! [X22: a] :
( none_a
!= ( some_a @ X22 ) ) ).
% option.distinct(1)
thf(fact_323_option_OdiscI,axiom,
! [Option: option_a,X22: a] :
( ( Option
= ( some_a @ X22 ) )
=> ( Option != none_a ) ) ).
% option.discI
thf(fact_324_option_Oexhaust,axiom,
! [Y: option_a] :
( ( Y != none_a )
=> ~ ! [X23: a] :
( Y
!= ( some_a @ X23 ) ) ) ).
% option.exhaust
thf(fact_325_split__option__ex,axiom,
( ( ^ [P3: option_a > $o] :
? [X5: option_a] : ( P3 @ X5 ) )
= ( ^ [P2: option_a > $o] :
( ( P2 @ none_a )
| ? [X3: a] : ( P2 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_326_split__option__all,axiom,
( ( ^ [P3: option_a > $o] :
! [X5: option_a] : ( P3 @ X5 ) )
= ( ^ [P2: option_a > $o] :
( ( P2 @ none_a )
& ! [X3: a] : ( P2 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_327_combine__options__cases,axiom,
! [X: option_a,P: option_a > option_a > $o,Y: option_a] :
( ( ( X = none_a )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_a )
=> ( P @ X @ Y ) )
=> ( ! [A4: a,B5: a] :
( ( X
= ( some_a @ A4 ) )
=> ( ( Y
= ( some_a @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_328_option_Osel,axiom,
! [X22: a] :
( ( the_a @ ( some_a @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_329_option_Oexpand,axiom,
! [Option: option_a,Option2: option_a] :
( ( ( Option = none_a )
= ( Option2 = none_a ) )
=> ( ( ( Option != none_a )
=> ( ( Option2 != none_a )
=> ( ( the_a @ Option )
= ( the_a @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_330_subset__singleton__iff,axiom,
! [X6: set_option_a,A: option_a] :
( ( ord_le1955136853071979460tion_a @ X6 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( ( X6 = bot_bot_set_option_a )
| ( X6
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_331_subset__singletonD,axiom,
! [A2: set_option_a,X: option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
=> ( ( A2 = bot_bot_set_option_a )
| ( A2
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ).
% subset_singletonD
thf(fact_332_image__diff__subset,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_333_image__diff__subset,axiom,
! [F: a > option_a,A2: set_a,B: set_a] : ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) @ ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_334_image__diff__subset,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] : ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_335_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_336_Diff__insert__absorb,axiom,
! [X: option_a,A2: set_option_a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A2 ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_337_Diff__insert2,axiom,
! [A2: set_option_a,A: option_a,B: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ B ) )
= ( minus_1574173051537231627tion_a @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_338_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_339_insert__Diff,axiom,
! [A: option_a,A2: set_option_a] :
( ( member_option_a @ A @ A2 )
=> ( ( insert_option_a @ A @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_340_Diff__insert,axiom,
! [A2: set_option_a,A: option_a,B: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ B ) )
= ( minus_1574173051537231627tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ).
% Diff_insert
thf(fact_341_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_342_subset__Diff__insert,axiom,
! [A2: set_option_a,B: set_option_a,X: option_a,C2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ ( insert_option_a @ X @ C2 ) ) )
= ( ( ord_le1955136853071979460tion_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ C2 ) )
& ~ ( member_option_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_343_option_Oexhaust__sel,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( Option
= ( some_a @ ( the_a @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_344_None__notin__image__Some,axiom,
! [A2: set_a] :
~ ( member_option_a @ none_a @ ( image_a_option_a @ some_a @ A2 ) ) ).
% None_notin_image_Some
thf(fact_345_Diff__single__insert,axiom,
! [A2: set_option_a,X: option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B )
=> ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_346_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_347_subset__insert__iff,axiom,
! [A2: set_option_a,X: option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ X @ B ) )
= ( ( ( member_option_a @ X @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B ) )
& ( ~ ( member_option_a @ X @ A2 )
=> ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_348_lift__fun_Ofold__graph__finite,axiom,
! [Z: option_a,A2: set_option_a,Y: option_a] :
( ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 @ Y )
=> ( finite1674126218327898605tion_a @ A2 ) ) ).
% lift_fun.fold_graph_finite
thf(fact_349_lift__fun_Ofold__commute__f,axiom,
! [X: option_a,Xs: list_option_a,Y: option_a] :
( ( lift_f_total_a_a_a @ p @ f @ X @ ( foldr_8203685439773058855tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ Y ) )
= ( foldr_8203685439773058855tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ ( lift_f_total_a_a_a @ p @ f @ X @ Y ) ) ) ).
% lift_fun.fold_commute_f
thf(fact_350_lift__fun_Ofoldr__list__swap,axiom,
! [Xs: list_option_a,Ys: list_option_a,Y: option_a] :
( ( foldr_8203685439773058855tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ ( foldr_8203685439773058855tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Ys @ Y ) )
= ( foldr_8203685439773058855tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Ys @ ( foldr_8203685439773058855tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ Y ) ) ) ).
% lift_fun.foldr_list_swap
thf(fact_351_lift__fun_Ocomp__comp__fun__idem,axiom,
! [G: option_a > option_a] : ( finite6444408709487604428tion_a @ ( comp_o4936981127514773180tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ G ) ) ).
% lift_fun.comp_comp_fun_idem
thf(fact_352_lift__fun_Ocomp__comp__fun__commute,axiom,
! [G: option_a > option_a] : ( finite8922082839177200849tion_a @ ( comp_o4936981127514773180tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ G ) ) ).
% lift_fun.comp_comp_fun_commute
thf(fact_353_pred__closurep_Obase,axiom,
! [S2: a > $o,S: a] :
( ( S2 @ S )
=> ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ S ) ) ).
% pred_closurep.base
thf(fact_354_pred__closurep_Ocases,axiom,
! [S2: a > $o,A: a] :
( ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ A )
=> ( ~ ( S2 @ A )
=> ~ ! [S3: a,T: a] :
( ( A
= ( f @ S3 @ T ) )
=> ( ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ S3 )
=> ( ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ T )
=> ~ ( p @ S3 @ T ) ) ) ) ) ) ).
% pred_closurep.cases
thf(fact_355_pred__closurep_Osimps,axiom,
! [S2: a > $o,A: a] :
( ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ A )
= ( ? [S4: a] :
( ( A = S4 )
& ( S2 @ S4 ) )
| ? [S4: a,T2: a] :
( ( A
= ( f @ S4 @ T2 ) )
& ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ S4 )
& ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ T2 )
& ( p @ S4 @ T2 ) ) ) ) ).
% pred_closurep.simps
thf(fact_356_pred__closurep_Ostep,axiom,
! [S2: a > $o,S: a,T3: a] :
( ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ S )
=> ( ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ T3 )
=> ( ( p @ S @ T3 )
=> ( set_cl7081670816658834010urep_a @ p @ f @ S2 @ ( f @ S @ T3 ) ) ) ) ) ).
% pred_closurep.step
thf(fact_357_lift__fun_Ofinite__S__finite__closure,axiom,
! [S2: set_option_a] :
( ( finite1674126218327898605tion_a @ S2 )
=> ( finite1674126218327898605tion_a @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) ) ) ).
% lift_fun.finite_S_finite_closure
thf(fact_358_set__closure__partial__oprator_Opred__closurep_Ocong,axiom,
set_cl7081670816658834010urep_a = set_cl7081670816658834010urep_a ).
% set_closure_partial_oprator.pred_closurep.cong
thf(fact_359_comp__fun__commute_Ofold__commute__f,axiom,
! [F: option_a > option_a > option_a,X: option_a,Xs: list_option_a,Y: option_a] :
( ( finite8922082839177200849tion_a @ F )
=> ( ( F @ X @ ( foldr_8203685439773058855tion_a @ F @ Xs @ Y ) )
= ( foldr_8203685439773058855tion_a @ F @ Xs @ ( F @ X @ Y ) ) ) ) ).
% comp_fun_commute.fold_commute_f
thf(fact_360_comp__fun__commute_Ofoldr__list__swap,axiom,
! [F: option_a > option_a > option_a,Xs: list_option_a,Ys: list_option_a,Y: option_a] :
( ( finite8922082839177200849tion_a @ F )
=> ( ( foldr_8203685439773058855tion_a @ F @ Xs @ ( foldr_8203685439773058855tion_a @ F @ Ys @ Y ) )
= ( foldr_8203685439773058855tion_a @ F @ Ys @ ( foldr_8203685439773058855tion_a @ F @ Xs @ Y ) ) ) ) ).
% comp_fun_commute.foldr_list_swap
thf(fact_361_set__closure__partial__oprator_Opred__closurep_Obase,axiom,
! [P: a > a > $o,F: a > a > a,S2: a > $o,S: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( S2 @ S )
=> ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ S ) ) ) ).
% set_closure_partial_oprator.pred_closurep.base
thf(fact_362_set__closure__partial__oprator_Opred__closurep_Ostep,axiom,
! [P: a > a > $o,F: a > a > a,S2: a > $o,S: a,T3: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ S )
=> ( ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ T3 )
=> ( ( P @ S @ T3 )
=> ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ ( F @ S @ T3 ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closurep.step
thf(fact_363_set__closure__partial__oprator_Opred__closurep_Ocases,axiom,
! [P: a > a > $o,F: a > a > a,S2: a > $o,A: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ A )
=> ( ~ ( S2 @ A )
=> ~ ! [S3: a,T: a] :
( ( A
= ( F @ S3 @ T ) )
=> ( ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ S3 )
=> ( ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ T )
=> ~ ( P @ S3 @ T ) ) ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closurep.cases
thf(fact_364_set__closure__partial__oprator_Opred__closurep_Osimps,axiom,
! [P: a > a > $o,F: a > a > a,S2: a > $o,A: a] :
( ( set_cl4247418117474325792ator_a @ P @ F )
=> ( ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ A )
= ( ? [S4: a] :
( ( A = S4 )
& ( S2 @ S4 ) )
| ? [S4: a,T2: a] :
( ( A
= ( F @ S4 @ T2 ) )
& ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ S4 )
& ( set_cl7081670816658834010urep_a @ P @ F @ S2 @ T2 )
& ( P @ S4 @ T2 ) ) ) ) ) ).
% set_closure_partial_oprator.pred_closurep.simps
thf(fact_365_set__closure__oprator_Ofinite__S__finite__closure,axiom,
! [F: option_a > option_a > option_a,S2: set_option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( finite1674126218327898605tion_a @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ).
% set_closure_oprator.finite_S_finite_closure
thf(fact_366_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A2: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B )
=> ~ ( member_a @ X @ C4 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_367_in__image__insert__iff,axiom,
! [B: set_set_option_a,X: option_a,A2: set_option_a] :
( ! [C4: set_option_a] :
( ( member_set_option_a @ C4 @ B )
=> ~ ( member_option_a @ X @ C4 ) )
=> ( ( member_set_option_a @ A2 @ ( image_4886974585486332549tion_a @ ( insert_option_a @ X ) @ B ) )
= ( ( member_option_a @ X @ A2 )
& ( member_set_option_a @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_368_lift__fun_Ocomp__comp__fun__idem__on,axiom,
! [R: set_option_a,G: option_a > option_a] : ( finite9052942386770090443tion_a @ R @ ( comp_o4936981127514773180tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ G ) ) ).
% lift_fun.comp_comp_fun_idem_on
thf(fact_369_finite__Diff__insert,axiom,
! [A2: set_option_a,A: option_a,B: set_option_a] :
( ( finite1674126218327898605tion_a @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ B ) ) )
= ( finite1674126218327898605tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_370_lift__fun_Ocomp__comp__fun__commute__on,axiom,
! [R: set_option_a,G: option_a > option_a] : ( finite3844474811650721488tion_a @ R @ ( comp_o4936981127514773180tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ G ) ) ).
% lift_fun.comp_comp_fun_commute_on
thf(fact_371_lift__fun_Ofold__rec,axiom,
! [A2: set_option_a,X: option_a,Z: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( member_option_a @ X @ A2 )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 )
= ( lift_f_total_a_a_a @ p @ f @ X @ ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ) ) ).
% lift_fun.fold_rec
thf(fact_372_lift__fun_Ofold__insert__remove,axiom,
! [A2: set_option_a,Z: option_a,X: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( insert_option_a @ X @ A2 ) )
= ( lift_f_total_a_a_a @ p @ f @ X @ ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ) ).
% lift_fun.fold_insert_remove
thf(fact_373_finite__Diff,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( finite1674126218327898605tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_374_finite__Diff2,axiom,
! [B: set_option_a,A2: set_option_a] :
( ( finite1674126218327898605tion_a @ B )
=> ( ( finite1674126218327898605tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_375_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_376_remove__induct,axiom,
! [P: set_option_a > $o,B: set_option_a] :
( ( P @ bot_bot_set_option_a )
=> ( ( ~ ( finite1674126218327898605tion_a @ B )
=> ( P @ B ) )
=> ( ! [A5: set_option_a] :
( ( finite1674126218327898605tion_a @ A5 )
=> ( ( A5 != bot_bot_set_option_a )
=> ( ( ord_le1955136853071979460tion_a @ A5 @ B )
=> ( ! [X4: option_a] :
( ( member_option_a @ X4 @ A5 )
=> ( P @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_377_lift__fun_Ofold__fun__left__comm,axiom,
! [A2: set_option_a,X: option_a,Z: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( lift_f_total_a_a_a @ p @ f @ X @ ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 ) )
= ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( lift_f_total_a_a_a @ p @ f @ X @ Z ) @ A2 ) ) ) ).
% lift_fun.fold_fun_left_comm
thf(fact_378_lift__fun_Ofold__equality,axiom,
! [Z: option_a,A2: set_option_a,Y: option_a] :
( ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 @ Y )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 )
= Y ) ) ).
% lift_fun.fold_equality
thf(fact_379_lift__fun_Ofold__insert,axiom,
! [A2: set_option_a,X: option_a,Z: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ~ ( member_option_a @ X @ A2 )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( insert_option_a @ X @ A2 ) )
= ( lift_f_total_a_a_a @ p @ f @ X @ ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 ) ) ) ) ) ).
% lift_fun.fold_insert
thf(fact_380_lift__fun_Ofold__insert2,axiom,
! [A2: set_option_a,X: option_a,Z: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ~ ( member_option_a @ X @ A2 )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( insert_option_a @ X @ A2 ) )
= ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( lift_f_total_a_a_a @ p @ f @ X @ Z ) @ A2 ) ) ) ) ).
% lift_fun.fold_insert2
thf(fact_381_lift__fun_Ofold__insert__idem2,axiom,
! [A2: set_option_a,Z: option_a,X: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( insert_option_a @ X @ A2 ) )
= ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( lift_f_total_a_a_a @ p @ f @ X @ Z ) @ A2 ) ) ) ).
% lift_fun.fold_insert_idem2
thf(fact_382_finite__imageI,axiom,
! [F3: set_a,H: a > option_a] :
( ( finite_finite_a @ F3 )
=> ( finite1674126218327898605tion_a @ ( image_a_option_a @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_383_finite__imageI,axiom,
! [F3: set_option_a,H: option_a > a] :
( ( finite1674126218327898605tion_a @ F3 )
=> ( finite_finite_a @ ( image_option_a_a @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_384_finite__imageI,axiom,
! [F3: set_option_a,H: option_a > option_a] :
( ( finite1674126218327898605tion_a @ F3 )
=> ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_385_lift__fun_Ofold__graph__fold,axiom,
! [A2: set_option_a,Z: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( finite8508940333850120134tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 @ ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 ) ) ) ).
% lift_fun.fold_graph_fold
thf(fact_386_finite__insert,axiom,
! [A: option_a,A2: set_option_a] :
( ( finite1674126218327898605tion_a @ ( insert_option_a @ A @ A2 ) )
= ( finite1674126218327898605tion_a @ A2 ) ) ).
% finite_insert
thf(fact_387_fold__empty,axiom,
! [F: option_a > option_a > option_a,Z: option_a] :
( ( finite1758689504953034312tion_a @ F @ Z @ bot_bot_set_option_a )
= Z ) ).
% fold_empty
thf(fact_388_fold__infinite,axiom,
! [A2: set_option_a,F: option_a > option_a > option_a,Z: option_a] :
( ~ ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ A2 )
= Z ) ) ).
% fold_infinite
thf(fact_389_lift__fun_Ofold__insert__idem,axiom,
! [A2: set_option_a,Z: option_a,X: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( insert_option_a @ X @ A2 ) )
= ( lift_f_total_a_a_a @ p @ f @ X @ ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 ) ) ) ) ).
% lift_fun.fold_insert_idem
thf(fact_390_Finite__Set_Ofold__cong,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,G: option_a > option_a > option_a,A2: set_option_a,S: option_a,T3: option_a,B: set_option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( finite3844474811650721488tion_a @ S2 @ G )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( S = T3 )
=> ( ( A2 = B )
=> ( ( finite1758689504953034312tion_a @ F @ S @ A2 )
= ( finite1758689504953034312tion_a @ G @ T3 @ B ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_391_comp__fun__commute__on_Ointro,axiom,
! [S2: set_a,F: a > option_a > option_a] :
( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ S2 )
=> ( ( member_a @ Y2 @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y2 ) @ ( F @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( finite6006036441637571606tion_a @ S2 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_392_comp__fun__commute__on_Ointro,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a] :
( ! [X2: option_a,Y2: option_a] :
( ( member_option_a @ X2 @ S2 )
=> ( ( member_option_a @ Y2 @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y2 ) @ ( F @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( finite3844474811650721488tion_a @ S2 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_393_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S2: set_a,F: a > option_a > option_a,X: a,Y: a,G: option_a > option_a] :
( ( finite6006036441637571606tion_a @ S2 @ F )
=> ( ( member_a @ X @ S2 )
=> ( ( member_a @ Y @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y ) @ ( comp_o3154387707078715297tion_a @ ( F @ X ) @ G ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( comp_o3154387707078715297tion_a @ ( F @ Y ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_394_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,Y: option_a,G: option_a > option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( member_option_a @ X @ S2 )
=> ( ( member_option_a @ Y @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y ) @ ( comp_o3154387707078715297tion_a @ ( F @ X ) @ G ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( comp_o3154387707078715297tion_a @ ( F @ Y ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_395_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S2: set_a,F: a > option_a > option_a,X: a,Y: a] :
( ( finite6006036441637571606tion_a @ S2 @ F )
=> ( ( member_a @ X @ S2 )
=> ( ( member_a @ Y @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y ) @ ( F @ X ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_396_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,Y: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( member_option_a @ X @ S2 )
=> ( ( member_option_a @ Y @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y ) @ ( F @ X ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_397_comp__fun__commute__on__def,axiom,
( finite6006036441637571606tion_a
= ( ^ [S5: set_a,F2: a > option_a > option_a] :
! [X3: a,Y3: a] :
( ( member_a @ X3 @ S5 )
=> ( ( member_a @ Y3 @ S5 )
=> ( ( comp_o3154387707078715297tion_a @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
= ( comp_o3154387707078715297tion_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_398_comp__fun__commute__on__def,axiom,
( finite3844474811650721488tion_a
= ( ^ [S5: set_option_a,F2: option_a > option_a > option_a] :
! [X3: option_a,Y3: option_a] :
( ( member_option_a @ X3 @ S5 )
=> ( ( member_option_a @ Y3 @ S5 )
=> ( ( comp_o3154387707078715297tion_a @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
= ( comp_o3154387707078715297tion_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_399_fold__closed__eq,axiom,
! [A2: set_option_a,B: set_a,F: option_a > a > a,G: option_a > a > a,Z: a] :
( ! [A4: option_a,B5: a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: option_a,B5: a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( member_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_a @ Z @ B )
=> ( ( finite8724221345971246594on_a_a @ F @ Z @ A2 )
= ( finite8724221345971246594on_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_400_fold__closed__eq,axiom,
! [A2: set_a,B: set_option_a,F: a > option_a > option_a,G: a > option_a > option_a,Z: option_a] :
( ! [A4: a,B5: option_a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: option_a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( member_option_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_option_a @ Z @ B )
=> ( ( finite6501707464432451470tion_a @ F @ Z @ A2 )
= ( finite6501707464432451470tion_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_401_fold__closed__eq,axiom,
! [A2: set_a,B: set_a,F: a > a > a,G: a > a > a,Z: a] :
( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( member_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_a @ Z @ B )
=> ( ( finite_fold_a_a @ F @ Z @ A2 )
= ( finite_fold_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_402_fold__closed__eq,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > option_a > option_a,G: option_a > option_a > option_a,Z: option_a] :
( ! [A4: option_a,B5: option_a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: option_a,B5: option_a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( member_option_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_option_a @ Z @ B )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ A2 )
= ( finite1758689504953034312tion_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_403_comp__fun__commute__on_Ofun__left__comm,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,Y: option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( member_option_a @ X @ S2 )
=> ( ( member_option_a @ Y @ S2 )
=> ( ( F @ Y @ ( F @ X @ Z ) )
= ( F @ X @ ( F @ Y @ Z ) ) ) ) ) ) ).
% comp_fun_commute_on.fun_left_comm
thf(fact_404_comp__fun__idem__on_Ofun__left__idem,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,Z: option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( ( member_option_a @ X @ S2 )
=> ( ( F @ X @ ( F @ X @ Z ) )
= ( F @ X @ Z ) ) ) ) ).
% comp_fun_idem_on.fun_left_idem
thf(fact_405_comp__fun__idem__on_Oaxioms_I1_J,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( finite3844474811650721488tion_a @ S2 @ F ) ) ).
% comp_fun_idem_on.axioms(1)
thf(fact_406_comp__fun__commute__on_Ofold__equality,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,A2: set_option_a,Z: option_a,Y: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ S2 )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 @ Y )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ A2 )
= Y ) ) ) ) ).
% comp_fun_commute_on.fold_equality
thf(fact_407_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S2: set_a,F: a > option_a > option_a,X: a] :
( ( finite6148732787124368273tion_a @ S2 @ F )
=> ( ( member_a @ X @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_408_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( ( member_option_a @ X @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_409_comp__fun__commute__on_Ofold__insert,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,A2: set_option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ~ ( member_option_a @ X @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( insert_option_a @ X @ A2 ) )
= ( F @ X @ ( finite1758689504953034312tion_a @ F @ Z @ A2 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_410_comp__fun__commute__on_Ofold__insert2,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,A2: set_option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ~ ( member_option_a @ X @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( insert_option_a @ X @ A2 ) )
= ( finite1758689504953034312tion_a @ F @ ( F @ X @ Z ) @ A2 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_411_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,A2: set_option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ( F @ X @ ( finite1758689504953034312tion_a @ F @ Z @ A2 ) )
= ( finite1758689504953034312tion_a @ F @ ( F @ X @ Z ) @ A2 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_412_comp__fun__commute__on_Ofold__graph__fold,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,A2: set_option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( finite8508940333850120134tion_a @ F @ Z @ A2 @ ( finite1758689504953034312tion_a @ F @ Z @ A2 ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_fold
thf(fact_413_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,A2: set_option_a,Z: option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( insert_option_a @ X @ A2 ) )
= ( F @ X @ ( finite1758689504953034312tion_a @ F @ Z @ A2 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_414_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,A2: set_option_a,Z: option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( insert_option_a @ X @ A2 ) )
= ( finite1758689504953034312tion_a @ F @ ( F @ X @ Z ) @ A2 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_415_comp__fun__commute__on_Ofold__graph__finite,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,Z: option_a,A2: set_option_a,Y: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 @ Y )
=> ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% comp_fun_commute_on.fold_graph_finite
thf(fact_416_comp__fun__commute__on_Ofold__graph__determ,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,A2: set_option_a,Z: option_a,X: option_a,Y: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ S2 )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 @ X )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 @ Y )
=> ( Y = X ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_determ
thf(fact_417_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,A2: set_option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( insert_option_a @ X @ A2 ) )
= ( F @ X @ ( finite1758689504953034312tion_a @ F @ Z @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_418_comp__fun__commute__on_Ofold__rec,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,A2: set_option_a,X: option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ( member_option_a @ X @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ A2 )
= ( F @ X @ ( finite1758689504953034312tion_a @ F @ Z @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_419_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,X: option_a,A2: set_option_a,Z: option_a,V2: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A2 ) @ S2 )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ ( insert_option_a @ X @ A2 ) @ V2 )
=> ( ~ ( member_option_a @ X @ A2 )
=> ~ ! [Y2: option_a] :
( ( V2
= ( F @ X @ Y2 ) )
=> ~ ( finite8508940333850120134tion_a @ F @ Z @ A2 @ Y2 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_420_fold__graph__closed__lemma,axiom,
! [G: option_a > a > a,Z: a,A2: set_option_a,X: a,B: set_a,F: option_a > a > a] :
( ( finite4714158851117683840on_a_a @ G @ Z @ A2 @ X )
=> ( ! [A4: option_a,B5: a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: option_a,B5: a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( member_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_a @ Z @ B )
=> ( ( finite4714158851117683840on_a_a @ F @ Z @ A2 @ X )
& ( member_a @ X @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_421_fold__graph__closed__lemma,axiom,
! [G: a > option_a > option_a,Z: option_a,A2: set_a,X: option_a,B: set_option_a,F: a > option_a > option_a] :
( ( finite2491644969578888716tion_a @ G @ Z @ A2 @ X )
=> ( ! [A4: a,B5: option_a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: option_a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( member_option_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_option_a @ Z @ B )
=> ( ( finite2491644969578888716tion_a @ F @ Z @ A2 @ X )
& ( member_option_a @ X @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_422_fold__graph__closed__lemma,axiom,
! [G: a > a > a,Z: a,A2: set_a,X: a,B: set_a,F: a > a > a] :
( ( finite7874008084079289286ph_a_a @ G @ Z @ A2 @ X )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( member_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_a @ Z @ B )
=> ( ( finite7874008084079289286ph_a_a @ F @ Z @ A2 @ X )
& ( member_a @ X @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_423_fold__graph__closed__lemma,axiom,
! [G: option_a > option_a > option_a,Z: option_a,A2: set_option_a,X: option_a,B: set_option_a,F: option_a > option_a > option_a] :
( ( finite8508940333850120134tion_a @ G @ Z @ A2 @ X )
=> ( ! [A4: option_a,B5: option_a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: option_a,B5: option_a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( member_option_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_option_a @ Z @ B )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 @ X )
& ( member_option_a @ X @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_424_fold__graph__closed__eq,axiom,
! [A2: set_option_a,B: set_a,F: option_a > a > a,G: option_a > a > a,Z: a] :
( ! [A4: option_a,B5: a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: option_a,B5: a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( member_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_a @ Z @ B )
=> ( ( finite4714158851117683840on_a_a @ F @ Z @ A2 )
= ( finite4714158851117683840on_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_425_fold__graph__closed__eq,axiom,
! [A2: set_a,B: set_option_a,F: a > option_a > option_a,G: a > option_a > option_a,Z: option_a] :
( ! [A4: a,B5: option_a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: option_a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( member_option_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_option_a @ Z @ B )
=> ( ( finite2491644969578888716tion_a @ F @ Z @ A2 )
= ( finite2491644969578888716tion_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_426_fold__graph__closed__eq,axiom,
! [A2: set_a,B: set_a,F: a > a > a,G: a > a > a,Z: a] :
( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B )
=> ( member_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_a @ Z @ B )
=> ( ( finite7874008084079289286ph_a_a @ F @ Z @ A2 )
= ( finite7874008084079289286ph_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_427_fold__graph__closed__eq,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > option_a > option_a,G: option_a > option_a > option_a,Z: option_a] :
( ! [A4: option_a,B5: option_a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: option_a,B5: option_a] :
( ( member_option_a @ A4 @ A2 )
=> ( ( member_option_a @ B5 @ B )
=> ( member_option_a @ ( G @ A4 @ B5 ) @ B ) ) )
=> ( ( member_option_a @ Z @ B )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 )
= ( finite8508940333850120134tion_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_428_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,A2: set_option_a,Z: option_a,Y: option_a,A: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ S2 )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 @ Y )
=> ( ( member_option_a @ A @ A2 )
=> ? [Y4: option_a] :
( ( Y
= ( F @ A @ Y4 ) )
& ( finite8508940333850120134tion_a @ F @ Z @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ Y4 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_429_finite__has__minimal2,axiom,
! [A2: set_set_option_a,A: set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( member_set_option_a @ A @ A2 )
=> ? [X2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
& ( ord_le1955136853071979460tion_a @ X2 @ A )
& ! [Xa2: set_option_a] :
( ( member_set_option_a @ Xa2 @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_430_finite__has__maximal2,axiom,
! [A2: set_set_option_a,A: set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( member_set_option_a @ A @ A2 )
=> ? [X2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
& ( ord_le1955136853071979460tion_a @ A @ X2 )
& ! [Xa2: set_option_a] :
( ( member_set_option_a @ Xa2 @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_431_infinite__imp__nonempty,axiom,
! [S2: set_option_a] :
( ~ ( finite1674126218327898605tion_a @ S2 )
=> ( S2 != bot_bot_set_option_a ) ) ).
% infinite_imp_nonempty
thf(fact_432_finite_OemptyI,axiom,
finite1674126218327898605tion_a @ bot_bot_set_option_a ).
% finite.emptyI
thf(fact_433_all__subset__image,axiom,
! [F: option_a > a,A2: set_option_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_option_a_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B3 @ A2 )
=> ( P @ ( image_option_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_434_all__subset__image,axiom,
! [F: a > option_a,A2: set_a,P: set_option_a > $o] :
( ( ! [B3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B3 @ ( image_a_option_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_option_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_435_all__subset__image,axiom,
! [F: option_a > option_a,A2: set_option_a,P: set_option_a > $o] :
( ( ! [B3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B3 @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B3 @ A2 )
=> ( P @ ( image_7439109396645324421tion_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_436_rev__finite__subset,axiom,
! [B: set_option_a,A2: set_option_a] :
( ( finite1674126218327898605tion_a @ B )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_437_infinite__super,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( ord_le1955136853071979460tion_a @ S2 @ T4 )
=> ( ~ ( finite1674126218327898605tion_a @ S2 )
=> ~ ( finite1674126218327898605tion_a @ T4 ) ) ) ).
% infinite_super
thf(fact_438_finite__subset,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( finite1674126218327898605tion_a @ B )
=> ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% finite_subset
thf(fact_439_finite_OinsertI,axiom,
! [A2: set_option_a,A: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( finite1674126218327898605tion_a @ ( insert_option_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_440_Diff__infinite__finite,axiom,
! [T4: set_option_a,S2: set_option_a] :
( ( finite1674126218327898605tion_a @ T4 )
=> ( ~ ( finite1674126218327898605tion_a @ S2 )
=> ~ ( finite1674126218327898605tion_a @ ( minus_1574173051537231627tion_a @ S2 @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_441_empty__fold__graphE,axiom,
! [F: option_a > option_a > option_a,Z: option_a,X: option_a] :
( ( finite8508940333850120134tion_a @ F @ Z @ bot_bot_set_option_a @ X )
=> ( X = Z ) ) ).
% empty_fold_graphE
thf(fact_442_fold__graph_OemptyI,axiom,
! [F: option_a > option_a > option_a,Z: option_a] : ( finite8508940333850120134tion_a @ F @ Z @ bot_bot_set_option_a @ Z ) ).
% fold_graph.emptyI
thf(fact_443_finite__imp__fold__graph,axiom,
! [A2: set_option_a,F: option_a > option_a > option_a,Z: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ? [X_1: option_a] : ( finite8508940333850120134tion_a @ F @ Z @ A2 @ X_1 ) ) ).
% finite_imp_fold_graph
thf(fact_444_fold__graph_OinsertI,axiom,
! [X: option_a,A2: set_option_a,F: option_a > option_a > option_a,Z: option_a,Y: option_a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ A2 @ Y )
=> ( finite8508940333850120134tion_a @ F @ Z @ ( insert_option_a @ X @ A2 ) @ ( F @ X @ Y ) ) ) ) ).
% fold_graph.insertI
thf(fact_445_comp__fun__commute__def,axiom,
( finite8922082839177200849tion_a
= ( ^ [F2: option_a > option_a > option_a] :
! [Y3: option_a,X3: option_a] :
( ( comp_o3154387707078715297tion_a @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
= ( comp_o3154387707078715297tion_a @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_446_comp__fun__commute_Ocomp__fun__commute,axiom,
! [F: option_a > option_a > option_a,Y: option_a,X: option_a] :
( ( finite8922082839177200849tion_a @ F )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y ) @ ( F @ X ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_447_comp__fun__commute_Ointro,axiom,
! [F: option_a > option_a > option_a] :
( ! [Y2: option_a,X2: option_a] :
( ( comp_o3154387707078715297tion_a @ ( F @ Y2 ) @ ( F @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( finite8922082839177200849tion_a @ F ) ) ).
% comp_fun_commute.intro
thf(fact_448_comp__fun__idem_Ocomp__fun__idem,axiom,
! [F: option_a > option_a > option_a,X: option_a] :
( ( finite6444408709487604428tion_a @ F )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ).
% comp_fun_idem.comp_fun_idem
thf(fact_449_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [F: option_a > option_a > option_a,G: option_a > option_a] :
( ( finite8922082839177200849tion_a @ F )
=> ( finite8922082839177200849tion_a @ ( comp_o4936981127514773180tion_a @ F @ G ) ) ) ).
% comp_fun_commute.comp_comp_fun_commute
thf(fact_450_comp__fun__idem_Ocomp__comp__fun__idem,axiom,
! [F: option_a > option_a > option_a,G: option_a > option_a] :
( ( finite6444408709487604428tion_a @ F )
=> ( finite6444408709487604428tion_a @ ( comp_o4936981127514773180tion_a @ F @ G ) ) ) ).
% comp_fun_idem.comp_comp_fun_idem
thf(fact_451_comp__fun__idem_Oaxioms_I1_J,axiom,
! [F: option_a > option_a > option_a] :
( ( finite6444408709487604428tion_a @ F )
=> ( finite8922082839177200849tion_a @ F ) ) ).
% comp_fun_idem.axioms(1)
thf(fact_452_finite__has__minimal,axiom,
! [A2: set_set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( A2 != bot_bo651546871064636272tion_a )
=> ? [X2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
& ! [Xa2: set_option_a] :
( ( member_set_option_a @ Xa2 @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_453_finite__has__maximal,axiom,
! [A2: set_set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( A2 != bot_bo651546871064636272tion_a )
=> ? [X2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
& ! [Xa2: set_option_a] :
( ( member_set_option_a @ Xa2 @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_454_all__finite__subset__image,axiom,
! [F: option_a > a,A2: set_option_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_option_a_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_option_a] :
( ( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ A2 ) )
=> ( P @ ( image_option_a_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_455_all__finite__subset__image,axiom,
! [F: a > option_a,A2: set_a,P: set_option_a > $o] :
( ( ! [B3: set_option_a] :
( ( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ ( image_a_option_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) )
=> ( P @ ( image_a_option_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_456_all__finite__subset__image,axiom,
! [F: option_a > option_a,A2: set_option_a,P: set_option_a > $o] :
( ( ! [B3: set_option_a] :
( ( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ ( image_7439109396645324421tion_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_option_a] :
( ( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ A2 ) )
=> ( P @ ( image_7439109396645324421tion_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_457_ex__finite__subset__image,axiom,
! [F: option_a > a,A2: set_option_a,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_option_a_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_option_a] :
( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ A2 )
& ( P @ ( image_option_a_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_458_ex__finite__subset__image,axiom,
! [F: a > option_a,A2: set_a,P: set_option_a > $o] :
( ( ? [B3: set_option_a] :
( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ ( image_a_option_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 )
& ( P @ ( image_a_option_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_459_ex__finite__subset__image,axiom,
! [F: option_a > option_a,A2: set_option_a,P: set_option_a > $o] :
( ( ? [B3: set_option_a] :
( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ ( image_7439109396645324421tion_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_option_a] :
( ( finite1674126218327898605tion_a @ B3 )
& ( ord_le1955136853071979460tion_a @ B3 @ A2 )
& ( P @ ( image_7439109396645324421tion_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_460_finite__subset__image,axiom,
! [B: set_a,F: option_a > a,A2: set_option_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_option_a_a @ F @ A2 ) )
=> ? [C4: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C4 @ A2 )
& ( finite1674126218327898605tion_a @ C4 )
& ( B
= ( image_option_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_461_finite__subset__image,axiom,
! [B: set_option_a,F: a > option_a,A2: set_a] :
( ( finite1674126218327898605tion_a @ B )
=> ( ( ord_le1955136853071979460tion_a @ B @ ( image_a_option_a @ F @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B
= ( image_a_option_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_462_finite__subset__image,axiom,
! [B: set_option_a,F: option_a > option_a,A2: set_option_a] :
( ( finite1674126218327898605tion_a @ B )
=> ( ( ord_le1955136853071979460tion_a @ B @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ? [C4: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C4 @ A2 )
& ( finite1674126218327898605tion_a @ C4 )
& ( B
= ( image_7439109396645324421tion_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_463_finite__surj,axiom,
! [A2: set_option_a,B: set_a,F: option_a > a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_option_a_a @ F @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_464_finite__surj,axiom,
! [A2: set_a,B: set_option_a,F: a > option_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ ( image_a_option_a @ F @ A2 ) )
=> ( finite1674126218327898605tion_a @ B ) ) ) ).
% finite_surj
thf(fact_465_finite__surj,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ( finite1674126218327898605tion_a @ B ) ) ) ).
% finite_surj
thf(fact_466_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_467_infinite__finite__induct,axiom,
! [P: set_option_a > $o,A2: set_option_a] :
( ! [A5: set_option_a] :
( ~ ( finite1674126218327898605tion_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_option_a )
=> ( ! [X2: option_a,F4: set_option_a] :
( ( finite1674126218327898605tion_a @ F4 )
=> ( ~ ( member_option_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_option_a @ X2 @ F4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_468_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_469_finite__ne__induct,axiom,
! [F3: set_option_a,P: set_option_a > $o] :
( ( finite1674126218327898605tion_a @ F3 )
=> ( ( F3 != bot_bot_set_option_a )
=> ( ! [X2: option_a] : ( P @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) )
=> ( ! [X2: option_a,F4: set_option_a] :
( ( finite1674126218327898605tion_a @ F4 )
=> ( ( F4 != bot_bot_set_option_a )
=> ( ~ ( member_option_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_option_a @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_470_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_471_finite__induct,axiom,
! [F3: set_option_a,P: set_option_a > $o] :
( ( finite1674126218327898605tion_a @ F3 )
=> ( ( P @ bot_bot_set_option_a )
=> ( ! [X2: option_a,F4: set_option_a] :
( ( finite1674126218327898605tion_a @ F4 )
=> ( ~ ( member_option_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_option_a @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_472_finite_Osimps,axiom,
( finite1674126218327898605tion_a
= ( ^ [A6: set_option_a] :
( ( A6 = bot_bot_set_option_a )
| ? [A3: set_option_a,B6: option_a] :
( ( A6
= ( insert_option_a @ B6 @ A3 ) )
& ( finite1674126218327898605tion_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_473_finite_Ocases,axiom,
! [A: set_option_a] :
( ( finite1674126218327898605tion_a @ A )
=> ( ( A != bot_bot_set_option_a )
=> ~ ! [A5: set_option_a] :
( ? [A4: option_a] :
( A
= ( insert_option_a @ A4 @ A5 ) )
=> ~ ( finite1674126218327898605tion_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_474_fold__graph_Osimps,axiom,
( finite8508940333850120134tion_a
= ( ^ [F2: option_a > option_a > option_a,Z3: option_a,A1: set_option_a,A22: option_a] :
( ( ( A1 = bot_bot_set_option_a )
& ( A22 = Z3 ) )
| ? [X3: option_a,A3: set_option_a,Y3: option_a] :
( ( A1
= ( insert_option_a @ X3 @ A3 ) )
& ( A22
= ( F2 @ X3 @ Y3 ) )
& ~ ( member_option_a @ X3 @ A3 )
& ( finite8508940333850120134tion_a @ F2 @ Z3 @ A3 @ Y3 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_475_fold__graph_Ocases,axiom,
! [F: option_a > option_a > option_a,Z: option_a,A12: set_option_a,A23: option_a] :
( ( finite8508940333850120134tion_a @ F @ Z @ A12 @ A23 )
=> ( ( ( A12 = bot_bot_set_option_a )
=> ( A23 != Z ) )
=> ~ ! [X2: option_a,A5: set_option_a] :
( ( A12
= ( insert_option_a @ X2 @ A5 ) )
=> ! [Y2: option_a] :
( ( A23
= ( F @ X2 @ Y2 ) )
=> ( ~ ( member_option_a @ X2 @ A5 )
=> ~ ( finite8508940333850120134tion_a @ F @ Z @ A5 @ Y2 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_476_finite__subset__induct_H,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A4 @ A2 )
=> ( ( ord_less_eq_set_a @ F4 @ A2 )
=> ( ~ ( member_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_477_finite__subset__induct_H,axiom,
! [F3: set_option_a,A2: set_option_a,P: set_option_a > $o] :
( ( finite1674126218327898605tion_a @ F3 )
=> ( ( ord_le1955136853071979460tion_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_option_a )
=> ( ! [A4: option_a,F4: set_option_a] :
( ( finite1674126218327898605tion_a @ F4 )
=> ( ( member_option_a @ A4 @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ F4 @ A2 )
=> ( ~ ( member_option_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_option_a @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_478_finite__subset__induct,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A4 @ A2 )
=> ( ~ ( member_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_479_finite__subset__induct,axiom,
! [F3: set_option_a,A2: set_option_a,P: set_option_a > $o] :
( ( finite1674126218327898605tion_a @ F3 )
=> ( ( ord_le1955136853071979460tion_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_option_a )
=> ( ! [A4: option_a,F4: set_option_a] :
( ( finite1674126218327898605tion_a @ F4 )
=> ( ( member_option_a @ A4 @ A2 )
=> ( ~ ( member_option_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_option_a @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_480_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: a,A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_481_finite__empty__induct,axiom,
! [A2: set_option_a,P: set_option_a > $o] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: option_a,A5: set_option_a] :
( ( finite1674126218327898605tion_a @ A5 )
=> ( ( member_option_a @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A4 @ bot_bot_set_option_a ) ) ) ) ) )
=> ( P @ bot_bot_set_option_a ) ) ) ) ).
% finite_empty_induct
thf(fact_482_infinite__coinduct,axiom,
! [X6: set_option_a > $o,A2: set_option_a] :
( ( X6 @ A2 )
=> ( ! [A5: set_option_a] :
( ( X6 @ A5 )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ A5 )
& ( ( X6 @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) )
| ~ ( finite1674126218327898605tion_a @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) ) ) )
=> ~ ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_483_infinite__remove,axiom,
! [S2: set_option_a,A: option_a] :
( ~ ( finite1674126218327898605tion_a @ S2 )
=> ~ ( finite1674126218327898605tion_a @ ( minus_1574173051537231627tion_a @ S2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ).
% infinite_remove
thf(fact_484_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_485_finite__remove__induct,axiom,
! [B: set_option_a,P: set_option_a > $o] :
( ( finite1674126218327898605tion_a @ B )
=> ( ( P @ bot_bot_set_option_a )
=> ( ! [A5: set_option_a] :
( ( finite1674126218327898605tion_a @ A5 )
=> ( ( A5 != bot_bot_set_option_a )
=> ( ( ord_le1955136853071979460tion_a @ A5 @ B )
=> ( ! [X4: option_a] :
( ( member_option_a @ X4 @ A5 )
=> ( P @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_486_lift__fun_Ocomp__fun__commute__on__axioms,axiom,
finite3844474811650721488tion_a @ top_top_set_option_a @ ( lift_f_total_a_a_a @ p @ f ) ).
% lift_fun.comp_fun_commute_on_axioms
thf(fact_487_lift__fun_Ocomp__fun__idem__on__axioms,axiom,
finite9052942386770090443tion_a @ top_top_set_option_a @ ( lift_f_total_a_a_a @ p @ f ) ).
% lift_fun.comp_fun_idem_on_axioms
thf(fact_488_lift__fun_Ofoldr__to__fold,axiom,
! [Xs: list_option_a] :
( ( foldr_8203685439773058855tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs )
= ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs ) ) ).
% lift_fun.foldr_to_fold
thf(fact_489_diff__shunt__var,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( minus_1574173051537231627tion_a @ X @ Y )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_490_lift__fun_Ofold__list__swap,axiom,
! [Xs: list_option_a,Ys: list_option_a,Y: option_a] :
( ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Ys @ Y ) )
= ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Ys @ ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ Y ) ) ) ).
% lift_fun.fold_list_swap
thf(fact_491_lift__fun_Ocomp__fun__commute__on,axiom,
! [Y: option_a,X: option_a] :
( ( comp_o3154387707078715297tion_a @ ( lift_f_total_a_a_a @ p @ f @ Y ) @ ( lift_f_total_a_a_a @ p @ f @ X ) )
= ( comp_o3154387707078715297tion_a @ ( lift_f_total_a_a_a @ p @ f @ X ) @ ( lift_f_total_a_a_a @ p @ f @ Y ) ) ) ).
% lift_fun.comp_fun_commute_on
thf(fact_492_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_493_UNIV__I,axiom,
! [X: option_a] : ( member_option_a @ X @ top_top_set_option_a ) ).
% UNIV_I
thf(fact_494_lift__fun_Ocommute__left__comp,axiom,
! [Y: option_a,X: option_a,G: option_a > option_a] :
( ( comp_o3154387707078715297tion_a @ ( lift_f_total_a_a_a @ p @ f @ Y ) @ ( comp_o3154387707078715297tion_a @ ( lift_f_total_a_a_a @ p @ f @ X ) @ G ) )
= ( comp_o3154387707078715297tion_a @ ( lift_f_total_a_a_a @ p @ f @ X ) @ ( comp_o3154387707078715297tion_a @ ( lift_f_total_a_a_a @ p @ f @ Y ) @ G ) ) ) ).
% lift_fun.commute_left_comp
thf(fact_495_lift__fun_Ocomp__fun__idem__on,axiom,
! [X: option_a] :
( ( comp_o3154387707078715297tion_a @ ( lift_f_total_a_a_a @ p @ f @ X ) @ ( lift_f_total_a_a_a @ p @ f @ X ) )
= ( lift_f_total_a_a_a @ p @ f @ X ) ) ).
% lift_fun.comp_fun_idem_on
thf(fact_496_finite__Plus__UNIV__iff,axiom,
( ( finite2934087602147779004tion_a @ top_to5819210477232330763tion_a )
= ( ( finite1674126218327898605tion_a @ top_top_set_option_a )
& ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_497_Diff__UNIV,axiom,
! [A2: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ top_top_set_option_a )
= bot_bot_set_option_a ) ).
% Diff_UNIV
thf(fact_498_finite__option__UNIV,axiom,
( ( finite8114217219359860531tion_a @ top_to1659475022456381882tion_a )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% finite_option_UNIV
thf(fact_499_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_500_Finite__Set_Ofinite__set,axiom,
( ( finite3047895793355147469tion_a @ top_to7232205438743699540tion_a )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% Finite_Set.finite_set
thf(fact_501_finite__prod,axiom,
( ( finite5652798022079174736tion_a @ top_to31087889803499607tion_a )
= ( ( finite1674126218327898605tion_a @ top_top_set_option_a )
& ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ) ).
% finite_prod
thf(fact_502_finite__Prod__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ( ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ( finite5652798022079174736tion_a @ top_to31087889803499607tion_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_503_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_504_UNIV__witness,axiom,
? [X2: option_a] : ( member_option_a @ X2 @ top_top_set_option_a ) ).
% UNIV_witness
thf(fact_505_UNIV__eq__I,axiom,
! [A2: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A2 )
=> ( top_top_set_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_506_UNIV__eq__I,axiom,
! [A2: set_option_a] :
( ! [X2: option_a] : ( member_option_a @ X2 @ A2 )
=> ( top_top_set_option_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_507_bot__set__def,axiom,
( bot_bot_set_option_a
= ( collect_option_a @ bot_bot_option_a_o ) ) ).
% bot_set_def
thf(fact_508_range__eqI,axiom,
! [B2: option_a,F: a > option_a,X: a] :
( ( B2
= ( F @ X ) )
=> ( member_option_a @ B2 @ ( image_a_option_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_509_range__eqI,axiom,
! [B2: option_a,F: option_a > option_a,X: option_a] :
( ( B2
= ( F @ X ) )
=> ( member_option_a @ B2 @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) ) ) ).
% range_eqI
thf(fact_510_range__eqI,axiom,
! [B2: a,F: option_a > a,X: option_a] :
( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_option_a_a @ F @ top_top_set_option_a ) ) ) ).
% range_eqI
thf(fact_511_rangeI,axiom,
! [F: a > option_a,X: a] : ( member_option_a @ ( F @ X ) @ ( image_a_option_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_512_rangeI,axiom,
! [F: option_a > option_a,X: option_a] : ( member_option_a @ ( F @ X ) @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) ) ).
% rangeI
thf(fact_513_rangeI,axiom,
! [F: option_a > a,X: option_a] : ( member_a @ ( F @ X ) @ ( image_option_a_a @ F @ top_top_set_option_a ) ) ).
% rangeI
thf(fact_514_empty__not__UNIV,axiom,
bot_bot_set_option_a != top_top_set_option_a ).
% empty_not_UNIV
thf(fact_515_ex__new__if__finite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A2 )
=> ? [A4: a] :
~ ( member_a @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_516_ex__new__if__finite,axiom,
! [A2: set_option_a] :
( ~ ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ? [A4: option_a] :
~ ( member_option_a @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_517_subset__UNIV,axiom,
! [A2: set_option_a] : ( ord_le1955136853071979460tion_a @ A2 @ top_top_set_option_a ) ).
% subset_UNIV
thf(fact_518_insert__UNIV,axiom,
! [X: option_a] :
( ( insert_option_a @ X @ top_top_set_option_a )
= top_top_set_option_a ) ).
% insert_UNIV
thf(fact_519_comp__fun__commute_Ofold__list__swap,axiom,
! [F: option_a > option_a > option_a,Xs: list_option_a,Ys: list_option_a,Y: option_a] :
( ( finite8922082839177200849tion_a @ F )
=> ( ( fold_o675018875565802379tion_a @ F @ Xs @ ( fold_o675018875565802379tion_a @ F @ Ys @ Y ) )
= ( fold_o675018875565802379tion_a @ F @ Ys @ ( fold_o675018875565802379tion_a @ F @ Xs @ Y ) ) ) ) ).
% comp_fun_commute.fold_list_swap
thf(fact_520_comp__fun__idem__insert,axiom,
finite4778312610131166636tion_a @ insert_option_a ).
% comp_fun_idem_insert
thf(fact_521_range__subsetD,axiom,
! [F: option_a > a,B: set_a,I: option_a] :
( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ top_top_set_option_a ) @ B )
=> ( member_a @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_522_range__subsetD,axiom,
! [F: a > option_a,B: set_option_a,I: a] :
( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ top_top_set_a ) @ B )
=> ( member_option_a @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_523_range__subsetD,axiom,
! [F: option_a > option_a,B: set_option_a,I: option_a] :
( ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) @ B )
=> ( member_option_a @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_524_comp__fun__commute_Ofoldr__to__fold,axiom,
! [F: option_a > option_a > option_a,Xs: list_option_a] :
( ( finite8922082839177200849tion_a @ F )
=> ( ( foldr_8203685439773058855tion_a @ F @ Xs )
= ( fold_o675018875565802379tion_a @ F @ Xs ) ) ) ).
% comp_fun_commute.foldr_to_fold
thf(fact_525_UNIV__option__conv,axiom,
( top_to1659475022456381882tion_a
= ( insert605063979879581146tion_a @ none_option_a @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) ) ).
% UNIV_option_conv
thf(fact_526_UNIV__option__conv,axiom,
( top_top_set_option_a
= ( insert_option_a @ none_a @ ( image_a_option_a @ some_a @ top_top_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_527_comp__fun__commute__def_H,axiom,
( finite8922082839177200849tion_a
= ( finite3844474811650721488tion_a @ top_top_set_option_a ) ) ).
% comp_fun_commute_def'
thf(fact_528_comp__fun__idem__def_H,axiom,
( finite6444408709487604428tion_a
= ( finite9052942386770090443tion_a @ top_top_set_option_a ) ) ).
% comp_fun_idem_def'
thf(fact_529_range__eq__singletonD,axiom,
! [F: a > option_a,A: option_a,X: a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= ( insert_option_a @ A @ bot_bot_set_option_a ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_530_range__eq__singletonD,axiom,
! [F: option_a > a,A: a,X: option_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= ( insert_a @ A @ bot_bot_set_a ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_531_range__eq__singletonD,axiom,
! [F: option_a > option_a,A: option_a,X: option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= ( insert_option_a @ A @ bot_bot_set_option_a ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_532_notin__range__Some,axiom,
! [X: option_a] :
( ( ~ ( member_option_a @ X @ ( image_a_option_a @ some_a @ top_top_set_a ) ) )
= ( X = none_a ) ) ).
% notin_range_Some
thf(fact_533_notin__range__Some,axiom,
! [X: option_option_a] :
( ( ~ ( member5113800082084363315tion_a @ X @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) )
= ( X = none_option_a ) ) ).
% notin_range_Some
thf(fact_534_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [S2: set_a,F: a > option_a > option_a,G: option_a > a,R: set_option_a] :
( ( finite6006036441637571606tion_a @ S2 @ F )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ G @ top_top_set_option_a ) @ S2 )
=> ( finite3844474811650721488tion_a @ R @ ( comp_a700500124998732034tion_a @ F @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_535_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,G: a > option_a,R: set_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ G @ top_top_set_a ) @ S2 )
=> ( finite6006036441637571606tion_a @ R @ ( comp_o844531392756860022on_a_a @ F @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_536_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,G: option_a > option_a,R: set_option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ G @ top_top_set_option_a ) @ S2 )
=> ( finite3844474811650721488tion_a @ R @ ( comp_o4936981127514773180tion_a @ F @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_537_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [S2: set_a,F: a > option_a > option_a,G: option_a > a,R: set_option_a] :
( ( finite6148732787124368273tion_a @ S2 @ F )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ G @ top_top_set_option_a ) @ S2 )
=> ( finite9052942386770090443tion_a @ R @ ( comp_a700500124998732034tion_a @ F @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_538_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,G: a > option_a,R: set_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ G @ top_top_set_a ) @ S2 )
=> ( finite6148732787124368273tion_a @ R @ ( comp_o844531392756860022on_a_a @ F @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_539_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,G: option_a > option_a,R: set_option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ G @ top_top_set_option_a ) @ S2 )
=> ( finite9052942386770090443tion_a @ R @ ( comp_o4936981127514773180tion_a @ F @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_540_finite__range__Some,axiom,
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ some_a @ top_top_set_a ) )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_range_Some
thf(fact_541_finite__range__Some,axiom,
( ( finite8114217219359860531tion_a @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% finite_range_Some
thf(fact_542_lift__fun_Ofold__set__fold,axiom,
! [Y: option_a,Xs: list_option_a] :
( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Y @ ( set_option_a2 @ Xs ) )
= ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ Y ) ) ).
% lift_fun.fold_set_fold
thf(fact_543_fun_Oset__map,axiom,
! [F: option_a > a,V2: a > option_a] :
( ( image_a_a @ ( comp_option_a_a_a @ F @ V2 ) @ top_top_set_a )
= ( image_option_a_a @ F @ ( image_a_option_a @ V2 @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_544_fun_Oset__map,axiom,
! [F: option_a > option_a,V2: a > option_a] :
( ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ F @ V2 ) @ top_top_set_a )
= ( image_7439109396645324421tion_a @ F @ ( image_a_option_a @ V2 @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_545_fun_Oset__map,axiom,
! [F: a > option_a,V2: a > a] :
( ( image_a_option_a @ ( comp_a_option_a_a @ F @ V2 ) @ top_top_set_a )
= ( image_a_option_a @ F @ ( image_a_a @ V2 @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_546_fun_Oset__map,axiom,
! [F: option_a > option_a,V2: option_a > option_a] :
( ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ F @ V2 ) @ top_top_set_option_a )
= ( image_7439109396645324421tion_a @ F @ ( image_7439109396645324421tion_a @ V2 @ top_top_set_option_a ) ) ) ).
% fun.set_map
thf(fact_547_fun_Oset__map,axiom,
! [F: a > option_a,V2: option_a > a] :
( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ F @ V2 ) @ top_top_set_option_a )
= ( image_a_option_a @ F @ ( image_option_a_a @ V2 @ top_top_set_option_a ) ) ) ).
% fun.set_map
thf(fact_548_fun_Oset__map,axiom,
! [F: a > a,V2: option_a > a] :
( ( image_option_a_a @ ( comp_a_a_option_a @ F @ V2 ) @ top_top_set_option_a )
= ( image_a_a @ F @ ( image_option_a_a @ V2 @ top_top_set_option_a ) ) ) ).
% fun.set_map
thf(fact_549_fun_Oset__map,axiom,
! [F: option_a > a,V2: option_a > option_a] :
( ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ F @ V2 ) @ top_top_set_option_a )
= ( image_option_a_a @ F @ ( image_7439109396645324421tion_a @ V2 @ top_top_set_option_a ) ) ) ).
% fun.set_map
thf(fact_550_fun_Omap__cong,axiom,
! [X: option_a > option_a,Ya: option_a > option_a,F: option_a > option_a,G: option_a > option_a] :
( ( X = Ya )
=> ( ! [Z2: option_a] :
( ( member_option_a @ Z2 @ ( image_7439109396645324421tion_a @ Ya @ top_top_set_option_a ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp_o3154387707078715297tion_a @ F @ X )
= ( comp_o3154387707078715297tion_a @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_551_fun_Omap__cong0,axiom,
! [X: option_a > option_a,F: option_a > option_a,G: option_a > option_a] :
( ! [Z2: option_a] :
( ( member_option_a @ Z2 @ ( image_7439109396645324421tion_a @ X @ top_top_set_option_a ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp_o3154387707078715297tion_a @ F @ X )
= ( comp_o3154387707078715297tion_a @ G @ X ) ) ) ).
% fun.map_cong0
thf(fact_552_fun_Oinj__map__strong,axiom,
! [X: option_a > option_a,Xa: option_a > option_a,F: option_a > option_a,Fa: option_a > option_a] :
( ! [Z2: option_a,Za: option_a] :
( ( member_option_a @ Z2 @ ( image_7439109396645324421tion_a @ X @ top_top_set_option_a ) )
=> ( ( member_option_a @ Za @ ( image_7439109396645324421tion_a @ Xa @ top_top_set_option_a ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( comp_o3154387707078715297tion_a @ F @ X )
= ( comp_o3154387707078715297tion_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_553_comp__apply,axiom,
( comp_o3154387707078715297tion_a
= ( ^ [F2: option_a > option_a,G2: option_a > option_a,X3: option_a] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_554_comp__eq__dest__lhs,axiom,
! [A: option_a > option_a,B2: option_a > option_a,C: option_a > option_a,V2: option_a] :
( ( ( comp_o3154387707078715297tion_a @ A @ B2 )
= C )
=> ( ( A @ ( B2 @ V2 ) )
= ( C @ V2 ) ) ) ).
% comp_eq_dest_lhs
thf(fact_555_comp__eq__elim,axiom,
! [A: option_a > option_a,B2: option_a > option_a,C: option_a > option_a,D: option_a > option_a] :
( ( ( comp_o3154387707078715297tion_a @ A @ B2 )
= ( comp_o3154387707078715297tion_a @ C @ D ) )
=> ! [V3: option_a] :
( ( A @ ( B2 @ V3 ) )
= ( C @ ( D @ V3 ) ) ) ) ).
% comp_eq_elim
thf(fact_556_comp__eq__dest,axiom,
! [A: option_a > option_a,B2: option_a > option_a,C: option_a > option_a,D: option_a > option_a,V2: option_a] :
( ( ( comp_o3154387707078715297tion_a @ A @ B2 )
= ( comp_o3154387707078715297tion_a @ C @ D ) )
=> ( ( A @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_dest
thf(fact_557_comp__assoc,axiom,
! [F: option_a > option_a,G: option_a > option_a,H: option_a > option_a] :
( ( comp_o3154387707078715297tion_a @ ( comp_o3154387707078715297tion_a @ F @ G ) @ H )
= ( comp_o3154387707078715297tion_a @ F @ ( comp_o3154387707078715297tion_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_558_comp__def,axiom,
( comp_o3154387707078715297tion_a
= ( ^ [F2: option_a > option_a,G2: option_a > option_a,X3: option_a] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_559_fun_Omap__comp,axiom,
! [G: option_a > option_a,F: option_a > option_a,V2: option_a > option_a] :
( ( comp_o3154387707078715297tion_a @ G @ ( comp_o3154387707078715297tion_a @ F @ V2 ) )
= ( comp_o3154387707078715297tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ V2 ) ) ).
% fun.map_comp
thf(fact_560_surjD,axiom,
! [F: a > option_a,Y: option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ? [X2: a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_561_surjD,axiom,
! [F: option_a > a,Y: a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ? [X2: option_a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_562_surjD,axiom,
! [F: option_a > option_a,Y: option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ? [X2: option_a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_563_surjE,axiom,
! [F: a > option_a,Y: option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ~ ! [X2: a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_564_surjE,axiom,
! [F: option_a > a,Y: a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ~ ! [X2: option_a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_565_surjE,axiom,
! [F: option_a > option_a,Y: option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ~ ! [X2: option_a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_566_surjI,axiom,
! [G: a > option_a,F: option_a > a] :
( ! [X2: option_a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_option_a @ G @ top_top_set_a )
= top_top_set_option_a ) ) ).
% surjI
thf(fact_567_surjI,axiom,
! [G: option_a > a,F: a > option_a] :
( ! [X2: a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_option_a_a @ G @ top_top_set_option_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_568_surjI,axiom,
! [G: option_a > option_a,F: option_a > option_a] :
( ! [X2: option_a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
= top_top_set_option_a ) ) ).
% surjI
thf(fact_569_surj__def,axiom,
! [F: a > option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
= ( ! [Y3: option_a] :
? [X3: a] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_570_surj__def,axiom,
! [F: option_a > a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
= ( ! [Y3: a] :
? [X3: option_a] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_571_surj__def,axiom,
! [F: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
= ( ! [Y3: option_a] :
? [X3: option_a] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_572_image__eq__imp__comp,axiom,
! [F: option_a > option_a,A2: set_option_a,G: option_a > option_a,B: set_option_a,H: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ A2 )
= ( image_7439109396645324421tion_a @ G @ B ) )
=> ( ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ H @ F ) @ A2 )
= ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_573_image__eq__imp__comp,axiom,
! [F: option_a > option_a,A2: set_option_a,G: a > option_a,B: set_a,H: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ A2 )
= ( image_a_option_a @ G @ B ) )
=> ( ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ H @ F ) @ A2 )
= ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_574_image__eq__imp__comp,axiom,
! [F: option_a > option_a,A2: set_option_a,G: a > option_a,B: set_a,H: option_a > a] :
( ( ( image_7439109396645324421tion_a @ F @ A2 )
= ( image_a_option_a @ G @ B ) )
=> ( ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ H @ F ) @ A2 )
= ( image_a_a @ ( comp_option_a_a_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_575_image__eq__imp__comp,axiom,
! [F: a > a,A2: set_a,G: option_a > a,B: set_option_a,H: a > option_a] :
( ( ( image_a_a @ F @ A2 )
= ( image_option_a_a @ G @ B ) )
=> ( ( image_a_option_a @ ( comp_a_option_a_a @ H @ F ) @ A2 )
= ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_576_image__eq__imp__comp,axiom,
! [F: a > option_a,A2: set_a,G: option_a > option_a,B: set_option_a,H: option_a > a] :
( ( ( image_a_option_a @ F @ A2 )
= ( image_7439109396645324421tion_a @ G @ B ) )
=> ( ( image_a_a @ ( comp_option_a_a_a @ H @ F ) @ A2 )
= ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_577_image__eq__imp__comp,axiom,
! [F: a > option_a,A2: set_a,G: option_a > option_a,B: set_option_a,H: option_a > option_a] :
( ( ( image_a_option_a @ F @ A2 )
= ( image_7439109396645324421tion_a @ G @ B ) )
=> ( ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ H @ F ) @ A2 )
= ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_578_image__eq__imp__comp,axiom,
! [F: a > option_a,A2: set_a,G: a > option_a,B: set_a,H: option_a > option_a] :
( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ G @ B ) )
=> ( ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ H @ F ) @ A2 )
= ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_579_image__eq__imp__comp,axiom,
! [F: option_a > a,A2: set_option_a,G: a > a,B: set_a,H: a > option_a] :
( ( ( image_option_a_a @ F @ A2 )
= ( image_a_a @ G @ B ) )
=> ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ F ) @ A2 )
= ( image_a_option_a @ ( comp_a_option_a_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_580_image__eq__imp__comp,axiom,
! [F: option_a > a,A2: set_option_a,G: option_a > a,B: set_option_a,H: a > a] :
( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ G @ B ) )
=> ( ( image_option_a_a @ ( comp_a_a_option_a @ H @ F ) @ A2 )
= ( image_option_a_a @ ( comp_a_a_option_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_581_image__comp,axiom,
! [F: option_a > option_a,G: option_a > option_a,R2: set_option_a] :
( ( image_7439109396645324421tion_a @ F @ ( image_7439109396645324421tion_a @ G @ R2 ) )
= ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_582_image__comp,axiom,
! [F: option_a > option_a,G: a > option_a,R2: set_a] :
( ( image_7439109396645324421tion_a @ F @ ( image_a_option_a @ G @ R2 ) )
= ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_583_image__comp,axiom,
! [F: a > a,G: option_a > a,R2: set_option_a] :
( ( image_a_a @ F @ ( image_option_a_a @ G @ R2 ) )
= ( image_option_a_a @ ( comp_a_a_option_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_584_image__comp,axiom,
! [F: a > option_a,G: a > a,R2: set_a] :
( ( image_a_option_a @ F @ ( image_a_a @ G @ R2 ) )
= ( image_a_option_a @ ( comp_a_option_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_585_image__comp,axiom,
! [F: a > option_a,G: option_a > a,R2: set_option_a] :
( ( image_a_option_a @ F @ ( image_option_a_a @ G @ R2 ) )
= ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_586_image__comp,axiom,
! [F: option_a > a,G: option_a > option_a,R2: set_option_a] :
( ( image_option_a_a @ F @ ( image_7439109396645324421tion_a @ G @ R2 ) )
= ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_587_image__comp,axiom,
! [F: option_a > a,G: a > option_a,R2: set_a] :
( ( image_option_a_a @ F @ ( image_a_option_a @ G @ R2 ) )
= ( image_a_a @ ( comp_option_a_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_588_comp__surj,axiom,
! [F: a > a,G: a > option_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_option_a @ G @ top_top_set_a )
= top_top_set_option_a )
=> ( ( image_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_589_comp__surj,axiom,
! [F: a > option_a,G: option_a > a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( ( image_option_a_a @ G @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_option_a_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_590_comp__surj,axiom,
! [F: a > option_a,G: option_a > option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_591_comp__surj,axiom,
! [F: option_a > a,G: a > a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_option_a_a @ ( comp_a_a_option_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_592_comp__surj,axiom,
! [F: option_a > a,G: a > option_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ( image_a_option_a @ G @ top_top_set_a )
= top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_593_comp__surj,axiom,
! [F: option_a > option_a,G: option_a > a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( ( image_option_a_a @ G @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_594_comp__surj,axiom,
! [F: option_a > option_a,G: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_595_fun_Omap__ident__strong,axiom,
! [T3: a > option_a,F: option_a > option_a] :
( ! [Z2: option_a] :
( ( member_option_a @ Z2 @ ( image_a_option_a @ T3 @ top_top_set_a ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_o6087033147929006299on_a_a @ F @ T3 )
= T3 ) ) ).
% fun.map_ident_strong
thf(fact_596_fun_Omap__ident__strong,axiom,
! [T3: option_a > option_a,F: option_a > option_a] :
( ! [Z2: option_a] :
( ( member_option_a @ Z2 @ ( image_7439109396645324421tion_a @ T3 @ top_top_set_option_a ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_o3154387707078715297tion_a @ F @ T3 )
= T3 ) ) ).
% fun.map_ident_strong
thf(fact_597_fun_Omap__ident__strong,axiom,
! [T3: option_a > a,F: a > a] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( image_option_a_a @ T3 @ top_top_set_option_a ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_a_a_option_a @ F @ T3 )
= T3 ) ) ).
% fun.map_ident_strong
thf(fact_598_lift__fun_Ofold__set__fold__remdups,axiom,
! [Y: option_a,Xs: list_option_a] :
( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Y @ ( set_option_a2 @ Xs ) )
= ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( remdups_option_a @ Xs ) @ Y ) ) ).
% lift_fun.fold_set_fold_remdups
thf(fact_599_comp__fun__idem__on_Ofold__set__fold,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,Xs: list_option_a,Y: option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Xs ) @ S2 )
=> ( ( finite1758689504953034312tion_a @ F @ Y @ ( set_option_a2 @ Xs ) )
= ( fold_o675018875565802379tion_a @ F @ Xs @ Y ) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
thf(fact_600_List_Ofinite__set,axiom,
! [Xs: list_option_a] : ( finite1674126218327898605tion_a @ ( set_option_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_601_foldr__fold,axiom,
! [Xs: list_a,F: a > option_a > option_a] :
( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y2 ) @ ( F @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_a_option_a @ F @ Xs )
= ( fold_a_option_a @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_602_foldr__fold,axiom,
! [Xs: list_option_a,F: option_a > option_a > option_a] :
( ! [X2: option_a,Y2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
=> ( ( member_option_a @ Y2 @ ( set_option_a2 @ Xs ) )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ Y2 ) @ ( F @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr_8203685439773058855tion_a @ F @ Xs )
= ( fold_o675018875565802379tion_a @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_603_fold__commute,axiom,
! [Xs: list_a,H: option_a > option_a,G: a > option_a > option_a,F: a > option_a > option_a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( comp_o3154387707078715297tion_a @ H @ ( G @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ H ) ) )
=> ( ( comp_o3154387707078715297tion_a @ H @ ( fold_a_option_a @ G @ Xs ) )
= ( comp_o3154387707078715297tion_a @ ( fold_a_option_a @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_604_fold__commute,axiom,
! [Xs: list_option_a,H: option_a > option_a,G: option_a > option_a > option_a,F: option_a > option_a > option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
=> ( ( comp_o3154387707078715297tion_a @ H @ ( G @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ H ) ) )
=> ( ( comp_o3154387707078715297tion_a @ H @ ( fold_o675018875565802379tion_a @ G @ Xs ) )
= ( comp_o3154387707078715297tion_a @ ( fold_o675018875565802379tion_a @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_605_order__refl,axiom,
! [X: set_option_a] : ( ord_le1955136853071979460tion_a @ X @ X ) ).
% order_refl
thf(fact_606_dual__order_Orefl,axiom,
! [A: set_option_a] : ( ord_le1955136853071979460tion_a @ A @ A ) ).
% dual_order.refl
thf(fact_607_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_option_a,Z4: set_option_a] : ( Y5 = Z4 ) )
= ( ^ [X3: set_option_a,Y3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X3 @ Y3 )
& ( ord_le1955136853071979460tion_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_608_ord__eq__le__trans,axiom,
! [A: set_option_a,B2: set_option_a,C: set_option_a] :
( ( A = B2 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ C )
=> ( ord_le1955136853071979460tion_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_609_ord__le__eq__trans,axiom,
! [A: set_option_a,B2: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le1955136853071979460tion_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_610_order__antisym,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ Y )
=> ( ( ord_le1955136853071979460tion_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_611_order_Otrans,axiom,
! [A: set_option_a,B2: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ C )
=> ( ord_le1955136853071979460tion_a @ A @ C ) ) ) ).
% order.trans
thf(fact_612_order__trans,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ Y )
=> ( ( ord_le1955136853071979460tion_a @ Y @ Z )
=> ( ord_le1955136853071979460tion_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_613_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_option_a,Z4: set_option_a] : ( Y5 = Z4 ) )
= ( ^ [A6: set_option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B6 @ A6 )
& ( ord_le1955136853071979460tion_a @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_614_dual__order_Oantisym,axiom,
! [B2: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_615_dual__order_Otrans,axiom,
! [B2: set_option_a,A: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ( ( ord_le1955136853071979460tion_a @ C @ B2 )
=> ( ord_le1955136853071979460tion_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_616_antisym,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_617_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_option_a,Z4: set_option_a] : ( Y5 = Z4 ) )
= ( ^ [A6: set_option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A6 @ B6 )
& ( ord_le1955136853071979460tion_a @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_618_order__subst1,axiom,
! [A: set_option_a,F: set_option_a > set_option_a,B2: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ ( F @ B2 ) )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ C )
=> ( ! [X2: set_option_a,Y2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X2 @ Y2 )
=> ( ord_le1955136853071979460tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le1955136853071979460tion_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_619_order__subst2,axiom,
! [A: set_option_a,B2: set_option_a,F: set_option_a > set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( ord_le1955136853071979460tion_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_option_a,Y2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X2 @ Y2 )
=> ( ord_le1955136853071979460tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le1955136853071979460tion_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_620_order__eq__refl,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( X = Y )
=> ( ord_le1955136853071979460tion_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_621_ord__eq__le__subst,axiom,
! [A: set_option_a,F: set_option_a > set_option_a,B2: set_option_a,C: set_option_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ C )
=> ( ! [X2: set_option_a,Y2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X2 @ Y2 )
=> ( ord_le1955136853071979460tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le1955136853071979460tion_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_622_ord__le__eq__subst,axiom,
! [A: set_option_a,B2: set_option_a,F: set_option_a > set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_option_a,Y2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X2 @ Y2 )
=> ( ord_le1955136853071979460tion_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le1955136853071979460tion_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_623_order__antisym__conv,axiom,
! [Y: set_option_a,X: set_option_a] :
( ( ord_le1955136853071979460tion_a @ Y @ X )
=> ( ( ord_le1955136853071979460tion_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_624_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,Xs: list_option_a,Y: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Xs ) @ S2 )
=> ( ( finite1758689504953034312tion_a @ F @ Y @ ( set_option_a2 @ Xs ) )
= ( fold_o675018875565802379tion_a @ F @ ( remdups_option_a @ Xs ) @ Y ) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
thf(fact_625_bot_Oextremum,axiom,
! [A: set_option_a] : ( ord_le1955136853071979460tion_a @ bot_bot_set_option_a @ A ) ).
% bot.extremum
thf(fact_626_bot_Oextremum__unique,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ bot_bot_set_option_a )
= ( A = bot_bot_set_option_a ) ) ).
% bot.extremum_unique
thf(fact_627_bot_Oextremum__uniqueI,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ bot_bot_set_option_a )
=> ( A = bot_bot_set_option_a ) ) ).
% bot.extremum_uniqueI
thf(fact_628_top_Oextremum__uniqueI,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ top_top_set_option_a @ A )
=> ( A = top_top_set_option_a ) ) ).
% top.extremum_uniqueI
thf(fact_629_top_Oextremum__unique,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ top_top_set_option_a @ A )
= ( A = top_top_set_option_a ) ) ).
% top.extremum_unique
thf(fact_630_top__greatest,axiom,
! [A: set_option_a] : ( ord_le1955136853071979460tion_a @ A @ top_top_set_option_a ) ).
% top_greatest
thf(fact_631_finite__list,axiom,
! [A2: set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ? [Xs2: list_option_a] :
( ( set_option_a2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_632_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_633_subset__code_I1_J,axiom,
! [Xs: list_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Xs ) @ B )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
=> ( member_option_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_634_fold__invariant,axiom,
! [Xs: list_option_a,Q: option_a > $o,P: option_a > $o,S: option_a,F: option_a > option_a > option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
=> ( Q @ X2 ) )
=> ( ( P @ S )
=> ( ! [X2: option_a,S3: option_a] :
( ( Q @ X2 )
=> ( ( P @ S3 )
=> ( P @ ( F @ X2 @ S3 ) ) ) )
=> ( P @ ( fold_o675018875565802379tion_a @ F @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_635_List_Ofold__cong,axiom,
! [A: option_a,B2: option_a,Xs: list_option_a,Ys: list_option_a,F: option_a > option_a > option_a,G: option_a > option_a > option_a] :
( ( A = B2 )
=> ( ( Xs = Ys )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( fold_o675018875565802379tion_a @ F @ Xs @ A )
= ( fold_o675018875565802379tion_a @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_636_foldr__cong,axiom,
! [A: option_a,B2: option_a,L: list_option_a,K: list_option_a,F: option_a > option_a > option_a,G: option_a > option_a > option_a] :
( ( A = B2 )
=> ( ( L = K )
=> ( ! [A4: option_a,X2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ L ) )
=> ( ( F @ X2 @ A4 )
= ( G @ X2 @ A4 ) ) )
=> ( ( foldr_8203685439773058855tion_a @ F @ L @ A )
= ( foldr_8203685439773058855tion_a @ G @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_637_fold__commute__apply,axiom,
! [Xs: list_a,H: option_a > option_a,G: a > option_a > option_a,F: a > option_a > option_a,S: option_a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( comp_o3154387707078715297tion_a @ H @ ( G @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ H ) ) )
=> ( ( H @ ( fold_a_option_a @ G @ Xs @ S ) )
= ( fold_a_option_a @ F @ Xs @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_638_fold__commute__apply,axiom,
! [Xs: list_option_a,H: option_a > option_a,G: option_a > option_a > option_a,F: option_a > option_a > option_a,S: option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
=> ( ( comp_o3154387707078715297tion_a @ H @ ( G @ X2 ) )
= ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ H ) ) )
=> ( ( H @ ( fold_o675018875565802379tion_a @ G @ Xs @ S ) )
= ( fold_o675018875565802379tion_a @ F @ Xs @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_639_surj__fun__eq,axiom,
! [F: option_a > option_a,X6: set_option_a,G1: option_a > option_a,G22: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ X6 )
= top_top_set_option_a )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ X6 )
=> ( ( comp_o3154387707078715297tion_a @ G1 @ F @ X2 )
= ( comp_o3154387707078715297tion_a @ G22 @ F @ X2 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_640_the__elem__eq,axiom,
! [X: option_a] :
( ( the_elem_option_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= X ) ).
% the_elem_eq
thf(fact_641_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_642_iso__tuple__UNIV__I,axiom,
! [X: option_a] : ( member_option_a @ X @ top_top_set_option_a ) ).
% iso_tuple_UNIV_I
thf(fact_643_these__insert__Some,axiom,
! [X: option_a,A2: set_option_option_a] :
( ( these_option_a @ ( insert605063979879581146tion_a @ ( some_option_a @ X ) @ A2 ) )
= ( insert_option_a @ X @ ( these_option_a @ A2 ) ) ) ).
% these_insert_Some
thf(fact_644_these__insert__Some,axiom,
! [X: a,A2: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X ) @ A2 ) )
= ( insert_a @ X @ ( these_a @ A2 ) ) ) ).
% these_insert_Some
thf(fact_645_these__image__Some__eq,axiom,
! [A2: set_a] :
( ( these_a @ ( image_a_option_a @ some_a @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_646_these__empty,axiom,
( ( these_option_a @ bot_bo4163488203964334806tion_a )
= bot_bot_set_option_a ) ).
% these_empty
thf(fact_647_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_648_these__insert__None,axiom,
! [A2: set_option_a] :
( ( these_a @ ( insert_option_a @ none_a @ A2 ) )
= ( these_a @ A2 ) ) ).
% these_insert_None
thf(fact_649_top__set__def,axiom,
( top_top_set_option_a
= ( collect_option_a @ top_top_option_a_o ) ) ).
% top_set_def
thf(fact_650_in__these__eq,axiom,
! [X: option_a,A2: set_option_option_a] :
( ( member_option_a @ X @ ( these_option_a @ A2 ) )
= ( member5113800082084363315tion_a @ ( some_option_a @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_651_in__these__eq,axiom,
! [X: a,A2: set_option_a] :
( ( member_a @ X @ ( these_a @ A2 ) )
= ( member_option_a @ ( some_a @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_652_the__elem__image__unique,axiom,
! [A2: set_a,F: a > option_a,X: a] :
( ( A2 != bot_bot_set_a )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A2 )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_option_a @ ( image_a_option_a @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_653_the__elem__image__unique,axiom,
! [A2: set_option_a,F: option_a > a,X: option_a] :
( ( A2 != bot_bot_set_option_a )
=> ( ! [Y2: option_a] :
( ( member_option_a @ Y2 @ A2 )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_a @ ( image_option_a_a @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_654_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_655_subset__emptyI,axiom,
! [A2: set_option_a] :
( ! [X2: option_a] :
~ ( member_option_a @ X2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ A2 @ bot_bot_set_option_a ) ) ).
% subset_emptyI
thf(fact_656_insert__subsetI,axiom,
! [X: a,A2: set_a,X6: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X6 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_657_insert__subsetI,axiom,
! [X: option_a,A2: set_option_a,X6: set_option_a] :
( ( member_option_a @ X @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ X6 @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_658_these__not__empty__eq,axiom,
! [B: set_option_a] :
( ( ( these_a @ B )
!= bot_bot_set_a )
= ( ( B != bot_bot_set_option_a )
& ( B
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_659_these__not__empty__eq,axiom,
! [B: set_option_option_a] :
( ( ( these_option_a @ B )
!= bot_bot_set_option_a )
= ( ( B != bot_bo4163488203964334806tion_a )
& ( B
!= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_660_these__empty__eq,axiom,
! [B: set_option_a] :
( ( ( these_a @ B )
= bot_bot_set_a )
= ( ( B = bot_bot_set_option_a )
| ( B
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_661_these__empty__eq,axiom,
! [B: set_option_option_a] :
( ( ( these_option_a @ B )
= bot_bot_set_option_a )
= ( ( B = bot_bo4163488203964334806tion_a )
| ( B
= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_empty_eq
thf(fact_662_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_663_bot__empty__eq,axiom,
( bot_bot_option_a_o
= ( ^ [X3: option_a] : ( member_option_a @ X3 @ bot_bot_set_option_a ) ) ) ).
% bot_empty_eq
thf(fact_664_Collect__empty__eq__bot,axiom,
! [P: option_a > $o] :
( ( ( collect_option_a @ P )
= bot_bot_set_option_a )
= ( P = bot_bot_option_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_665_is__singleton__the__elem,axiom,
( is_sin3348965821858909752tion_a
= ( ^ [A3: set_option_a] :
( A3
= ( insert_option_a @ ( the_elem_option_a @ A3 ) @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_666_is__singletonI,axiom,
! [X: option_a] : ( is_sin3348965821858909752tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ).
% is_singletonI
thf(fact_667_set__removeAll,axiom,
! [X: option_a,Xs: list_option_a] :
( ( set_option_a2 @ ( removeAll_option_a @ X @ Xs ) )
= ( minus_1574173051537231627tion_a @ ( set_option_a2 @ Xs ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ).
% set_removeAll
thf(fact_668_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_669_is__singletonI_H,axiom,
! [A2: set_option_a] :
( ( A2 != bot_bot_set_option_a )
=> ( ! [X2: option_a,Y2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( ( member_option_a @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_sin3348965821858909752tion_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_670_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_671_top__empty__eq,axiom,
( top_top_option_a_o
= ( ^ [X3: option_a] : ( member_option_a @ X3 @ top_top_set_option_a ) ) ) ).
% top_empty_eq
thf(fact_672_is__singletonE,axiom,
! [A2: set_option_a] :
( ( is_sin3348965821858909752tion_a @ A2 )
=> ~ ! [X2: option_a] :
( A2
!= ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ).
% is_singletonE
thf(fact_673_is__singleton__def,axiom,
( is_sin3348965821858909752tion_a
= ( ^ [A3: set_option_a] :
? [X3: option_a] :
( A3
= ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_def
thf(fact_674_cofinite__bot,axiom,
( ( cofinite_option_a = bot_bo5190894291365496758tion_a )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% cofinite_bot
thf(fact_675_Inf_OINF__image,axiom,
! [Inf: set_option_a > option_a,G: option_a > option_a,F: option_a > option_a,A2: set_option_a] :
( ( Inf @ ( image_7439109396645324421tion_a @ G @ ( image_7439109396645324421tion_a @ F @ A2 ) ) )
= ( Inf @ ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_676_Inf_OINF__image,axiom,
! [Inf: set_option_a > option_a,G: option_a > option_a,F: a > option_a,A2: set_a] :
( ( Inf @ ( image_7439109396645324421tion_a @ G @ ( image_a_option_a @ F @ A2 ) ) )
= ( Inf @ ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_677_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: a > a,F: option_a > a,A2: set_option_a] :
( ( Inf @ ( image_a_a @ G @ ( image_option_a_a @ F @ A2 ) ) )
= ( Inf @ ( image_option_a_a @ ( comp_a_a_option_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_678_Inf_OINF__image,axiom,
! [Inf: set_option_a > option_a,G: a > option_a,F: a > a,A2: set_a] :
( ( Inf @ ( image_a_option_a @ G @ ( image_a_a @ F @ A2 ) ) )
= ( Inf @ ( image_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_679_Inf_OINF__image,axiom,
! [Inf: set_option_a > option_a,G: a > option_a,F: option_a > a,A2: set_option_a] :
( ( Inf @ ( image_a_option_a @ G @ ( image_option_a_a @ F @ A2 ) ) )
= ( Inf @ ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_680_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: option_a > a,F: option_a > option_a,A2: set_option_a] :
( ( Inf @ ( image_option_a_a @ G @ ( image_7439109396645324421tion_a @ F @ A2 ) ) )
= ( Inf @ ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_681_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: option_a > a,F: a > option_a,A2: set_a] :
( ( Inf @ ( image_option_a_a @ G @ ( image_a_option_a @ F @ A2 ) ) )
= ( Inf @ ( image_a_a @ ( comp_option_a_a_a @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_682_Sup_OSUP__image,axiom,
! [Sup: set_option_a > option_a,G: option_a > option_a,F: option_a > option_a,A2: set_option_a] :
( ( Sup @ ( image_7439109396645324421tion_a @ G @ ( image_7439109396645324421tion_a @ F @ A2 ) ) )
= ( Sup @ ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_683_Sup_OSUP__image,axiom,
! [Sup: set_option_a > option_a,G: option_a > option_a,F: a > option_a,A2: set_a] :
( ( Sup @ ( image_7439109396645324421tion_a @ G @ ( image_a_option_a @ F @ A2 ) ) )
= ( Sup @ ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_684_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: a > a,F: option_a > a,A2: set_option_a] :
( ( Sup @ ( image_a_a @ G @ ( image_option_a_a @ F @ A2 ) ) )
= ( Sup @ ( image_option_a_a @ ( comp_a_a_option_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_685_Sup_OSUP__image,axiom,
! [Sup: set_option_a > option_a,G: a > option_a,F: a > a,A2: set_a] :
( ( Sup @ ( image_a_option_a @ G @ ( image_a_a @ F @ A2 ) ) )
= ( Sup @ ( image_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_686_Sup_OSUP__image,axiom,
! [Sup: set_option_a > option_a,G: a > option_a,F: option_a > a,A2: set_option_a] :
( ( Sup @ ( image_a_option_a @ G @ ( image_option_a_a @ F @ A2 ) ) )
= ( Sup @ ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_687_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: option_a > a,F: option_a > option_a,A2: set_option_a] :
( ( Sup @ ( image_option_a_a @ G @ ( image_7439109396645324421tion_a @ F @ A2 ) ) )
= ( Sup @ ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_688_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: option_a > a,F: a > option_a,A2: set_a] :
( ( Sup @ ( image_option_a_a @ G @ ( image_a_option_a @ F @ A2 ) ) )
= ( Sup @ ( image_a_a @ ( comp_option_a_a_a @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_689_Inf_OINF__cong,axiom,
! [A2: set_option_a,B: set_option_a,C2: option_a > a,D2: option_a > a,Inf: set_a > a] :
( ( A2 = B )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_option_a_a @ C2 @ A2 ) )
= ( Inf @ ( image_option_a_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_690_Inf_OINF__cong,axiom,
! [A2: set_a,B: set_a,C2: a > option_a,D2: a > option_a,Inf: set_option_a > option_a] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_a_option_a @ C2 @ A2 ) )
= ( Inf @ ( image_a_option_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_691_Sup_OSUP__cong,axiom,
! [A2: set_option_a,B: set_option_a,C2: option_a > a,D2: option_a > a,Sup: set_a > a] :
( ( A2 = B )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_option_a_a @ C2 @ A2 ) )
= ( Sup @ ( image_option_a_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_692_Sup_OSUP__cong,axiom,
! [A2: set_a,B: set_a,C2: a > option_a,D2: a > option_a,Sup: set_option_a > option_a] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_a_option_a @ C2 @ A2 ) )
= ( Sup @ ( image_a_option_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_693_comp__fun__idem__on__def,axiom,
( finite9052942386770090443tion_a
= ( ^ [S5: set_option_a,F2: option_a > option_a > option_a] :
( ( finite3844474811650721488tion_a @ S5 @ F2 )
& ( finite3481171789878968558tion_a @ S5 @ F2 ) ) ) ) ).
% comp_fun_idem_on_def
thf(fact_694_comp__fun__idem__on_Ointro,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( finite3481171789878968558tion_a @ S2 @ F )
=> ( finite9052942386770090443tion_a @ S2 @ F ) ) ) ).
% comp_fun_idem_on.intro
thf(fact_695_comp__fun__idem__def,axiom,
( finite6444408709487604428tion_a
= ( ^ [F2: option_a > option_a > option_a] :
( ( finite8922082839177200849tion_a @ F2 )
& ( finite5854837498025318383tion_a @ F2 ) ) ) ) ).
% comp_fun_idem_def
thf(fact_696_comp__fun__idem_Ointro,axiom,
! [F: option_a > option_a > option_a] :
( ( finite8922082839177200849tion_a @ F )
=> ( ( finite5854837498025318383tion_a @ F )
=> ( finite6444408709487604428tion_a @ F ) ) ) ).
% comp_fun_idem.intro
thf(fact_697_remove__def,axiom,
( remove_option_a
= ( ^ [X3: option_a,A3: set_option_a] : ( minus_1574173051537231627tion_a @ A3 @ ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ).
% remove_def
thf(fact_698_member__remove,axiom,
! [X: option_a,Y: option_a,A2: set_option_a] :
( ( member_option_a @ X @ ( remove_option_a @ Y @ A2 ) )
= ( ( member_option_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_699_member__remove,axiom,
! [X: a,Y: a,A2: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_700_comp__fun__idem__on__axioms_Ointro,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( finite3481171789878968558tion_a @ S2 @ F ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_701_comp__fun__idem__on__axioms_Ointro,axiom,
! [S2: set_a,F: a > option_a > option_a] :
( ! [X2: a] :
( ( member_a @ X2 @ S2 )
=> ( ( comp_o3154387707078715297tion_a @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( finite1862468707813542708tion_a @ S2 @ F ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_702_comp__fun__idem__on__axioms__def,axiom,
( finite3481171789878968558tion_a
= ( ^ [S5: set_option_a,F2: option_a > option_a > option_a] :
! [X3: option_a] :
( ( member_option_a @ X3 @ S5 )
=> ( ( comp_o3154387707078715297tion_a @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_703_comp__fun__idem__on__axioms__def,axiom,
( finite1862468707813542708tion_a
= ( ^ [S5: set_a,F2: a > option_a > option_a] :
! [X3: a] :
( ( member_a @ X3 @ S5 )
=> ( ( comp_o3154387707078715297tion_a @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_704_comp__fun__idem_Oaxioms_I2_J,axiom,
! [F: option_a > option_a > option_a] :
( ( finite6444408709487604428tion_a @ F )
=> ( finite5854837498025318383tion_a @ F ) ) ).
% comp_fun_idem.axioms(2)
thf(fact_705_comp__fun__idem__on_Oaxioms_I2_J,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a] :
( ( finite9052942386770090443tion_a @ S2 @ F )
=> ( finite3481171789878968558tion_a @ S2 @ F ) ) ).
% comp_fun_idem_on.axioms(2)
thf(fact_706_minus__fold__remove,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( minus_1574173051537231627tion_a @ B @ A2 )
= ( finite3172252742789171496tion_a @ remove_option_a @ B @ A2 ) ) ) ).
% minus_fold_remove
thf(fact_707_minus__set__fold,axiom,
! [A2: set_option_a,Xs: list_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ ( set_option_a2 @ Xs ) )
= ( fold_o8770731556678408811tion_a @ remove_option_a @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_708_semilattice__closure__operator_Osupremum__in__closure,axiom,
! [F: a > a > a,Less_eq: a > a > $o,E: a,S2: set_a,R: set_a] :
( ( semila7314208447189752392ator_a @ F @ Less_eq @ E )
=> ( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ R @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ( ( R != bot_bot_set_a )
=> ( member_a @ ( finite_fold_a_a @ F @ E @ R ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ) ).
% semilattice_closure_operator.supremum_in_closure
thf(fact_709_semilattice__closure__operator_Osupremum__in__closure,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a,R: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( ( ord_le1955136853071979460tion_a @ R @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ( ( R != bot_bot_set_option_a )
=> ( member_option_a @ ( finite1758689504953034312tion_a @ F @ E @ R ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ) ).
% semilattice_closure_operator.supremum_in_closure
thf(fact_710_semilattice__closure__operator_Osupremum__neutral,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( ( ( finite1758689504953034312tion_a @ F @ E @ S2 )
= E )
=> ( ord_le1955136853071979460tion_a @ S2 @ ( insert_option_a @ E @ bot_bot_set_option_a ) ) ) ) ) ).
% semilattice_closure_operator.supremum_neutral
thf(fact_711_image__Fpow__mono,axiom,
! [F: option_a > a,A2: set_option_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_974160131257141311_set_a @ ( image_option_a_a @ F ) @ ( finite_Fpow_option_a @ A2 ) ) @ ( finite_Fpow_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_712_image__Fpow__mono,axiom,
! [F: a > option_a,A2: set_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ B )
=> ( ord_le7761976607055303332tion_a @ ( image_2200789291716222155tion_a @ ( image_a_option_a @ F ) @ ( finite_Fpow_a @ A2 ) ) @ ( finite_Fpow_option_a @ B ) ) ) ).
% image_Fpow_mono
thf(fact_713_lift__fun_Ofold__set__union__disj,axiom,
! [A2: set_option_a,B: set_option_a,Z: option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1674126218327898605tion_a @ B )
=> ( ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a )
=> ( ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( finite1758689504953034312tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Z @ A2 ) @ B ) ) ) ) ) ).
% lift_fun.fold_set_union_disj
thf(fact_714_IntI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_715_IntI,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ A2 )
=> ( ( member_option_a @ C @ B )
=> ( member_option_a @ C @ ( inf_inf_set_option_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_716_Int__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_717_Int__iff,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( ( member_option_a @ C @ A2 )
& ( member_option_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_718_UnCI,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_719_UnCI,axiom,
! [C: option_a,B: set_option_a,A2: set_option_a] :
( ( ~ ( member_option_a @ C @ B )
=> ( member_option_a @ C @ A2 ) )
=> ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_720_Un__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_721_Un__iff,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( ( member_option_a @ C @ A2 )
| ( member_option_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_722_le__inf__iff,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ ( inf_inf_set_option_a @ Y @ Z ) )
= ( ( ord_le1955136853071979460tion_a @ X @ Y )
& ( ord_le1955136853071979460tion_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_723_inf_Obounded__iff,axiom,
! [A: set_option_a,B2: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ ( inf_inf_set_option_a @ B2 @ C ) )
= ( ( ord_le1955136853071979460tion_a @ A @ B2 )
& ( ord_le1955136853071979460tion_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_724_le__sup__iff,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ X @ Y ) @ Z )
= ( ( ord_le1955136853071979460tion_a @ X @ Z )
& ( ord_le1955136853071979460tion_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_725_sup_Obounded__iff,axiom,
! [B2: set_option_a,C: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ B2 @ C ) @ A )
= ( ( ord_le1955136853071979460tion_a @ B2 @ A )
& ( ord_le1955136853071979460tion_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_726_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_727_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ bot_bot_set_option_a @ X )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_728_inf__bot__left,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ bot_bot_set_option_a @ X )
= bot_bot_set_option_a ) ).
% inf_bot_left
thf(fact_729_inf__bot__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% inf_bot_right
thf(fact_730_inf__top__left,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ top_top_set_option_a @ X )
= X ) ).
% inf_top_left
thf(fact_731_inf__top__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ top_top_set_option_a )
= X ) ).
% inf_top_right
thf(fact_732_inf__eq__top__iff,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( inf_inf_set_option_a @ X @ Y )
= top_top_set_option_a )
= ( ( X = top_top_set_option_a )
& ( Y = top_top_set_option_a ) ) ) ).
% inf_eq_top_iff
thf(fact_733_top__eq__inf__iff,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( top_top_set_option_a
= ( inf_inf_set_option_a @ X @ Y ) )
= ( ( X = top_top_set_option_a )
& ( Y = top_top_set_option_a ) ) ) ).
% top_eq_inf_iff
thf(fact_734_inf__top_Oeq__neutr__iff,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( ( inf_inf_set_option_a @ A @ B2 )
= top_top_set_option_a )
= ( ( A = top_top_set_option_a )
& ( B2 = top_top_set_option_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_735_inf__top_Oleft__neutral,axiom,
! [A: set_option_a] :
( ( inf_inf_set_option_a @ top_top_set_option_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_736_inf__top_Oneutr__eq__iff,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( top_top_set_option_a
= ( inf_inf_set_option_a @ A @ B2 ) )
= ( ( A = top_top_set_option_a )
& ( B2 = top_top_set_option_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_737_inf__top_Oright__neutral,axiom,
! [A: set_option_a] :
( ( inf_inf_set_option_a @ A @ top_top_set_option_a )
= A ) ).
% inf_top.right_neutral
thf(fact_738_sup__bot__left,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ bot_bot_set_option_a @ X )
= X ) ).
% sup_bot_left
thf(fact_739_sup__bot__right,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ X @ bot_bot_set_option_a )
= X ) ).
% sup_bot_right
thf(fact_740_bot__eq__sup__iff,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( bot_bot_set_option_a
= ( sup_sup_set_option_a @ X @ Y ) )
= ( ( X = bot_bot_set_option_a )
& ( Y = bot_bot_set_option_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_741_sup__eq__bot__iff,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( sup_sup_set_option_a @ X @ Y )
= bot_bot_set_option_a )
= ( ( X = bot_bot_set_option_a )
& ( Y = bot_bot_set_option_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_742_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( ( sup_sup_set_option_a @ A @ B2 )
= bot_bot_set_option_a )
= ( ( A = bot_bot_set_option_a )
& ( B2 = bot_bot_set_option_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_743_sup__bot_Oleft__neutral,axiom,
! [A: set_option_a] :
( ( sup_sup_set_option_a @ bot_bot_set_option_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_744_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( bot_bot_set_option_a
= ( sup_sup_set_option_a @ A @ B2 ) )
= ( ( A = bot_bot_set_option_a )
& ( B2 = bot_bot_set_option_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_745_sup__bot_Oright__neutral,axiom,
! [A: set_option_a] :
( ( sup_sup_set_option_a @ A @ bot_bot_set_option_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_746_boolean__algebra_Odisj__one__left,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ top_top_set_option_a @ X )
= top_top_set_option_a ) ).
% boolean_algebra.disj_one_left
thf(fact_747_boolean__algebra_Odisj__one__right,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ X @ top_top_set_option_a )
= top_top_set_option_a ) ).
% boolean_algebra.disj_one_right
thf(fact_748_sup__top__left,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ top_top_set_option_a @ X )
= top_top_set_option_a ) ).
% sup_top_left
thf(fact_749_sup__top__right,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ X @ top_top_set_option_a )
= top_top_set_option_a ) ).
% sup_top_right
thf(fact_750_Int__UNIV,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ( inf_inf_set_option_a @ A2 @ B )
= top_top_set_option_a )
= ( ( A2 = top_top_set_option_a )
& ( B = top_top_set_option_a ) ) ) ).
% Int_UNIV
thf(fact_751_finite__Int,axiom,
! [F3: set_option_a,G3: set_option_a] :
( ( ( finite1674126218327898605tion_a @ F3 )
| ( finite1674126218327898605tion_a @ G3 ) )
=> ( finite1674126218327898605tion_a @ ( inf_inf_set_option_a @ F3 @ G3 ) ) ) ).
% finite_Int
thf(fact_752_Un__empty,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ( sup_sup_set_option_a @ A2 @ B )
= bot_bot_set_option_a )
= ( ( A2 = bot_bot_set_option_a )
& ( B = bot_bot_set_option_a ) ) ) ).
% Un_empty
thf(fact_753_Int__subset__iff,axiom,
! [C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C2 @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( ( ord_le1955136853071979460tion_a @ C2 @ A2 )
& ( ord_le1955136853071979460tion_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_754_finite__Un,axiom,
! [F3: set_option_a,G3: set_option_a] :
( ( finite1674126218327898605tion_a @ ( sup_sup_set_option_a @ F3 @ G3 ) )
= ( ( finite1674126218327898605tion_a @ F3 )
& ( finite1674126218327898605tion_a @ G3 ) ) ) ).
% finite_Un
thf(fact_755_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_756_Int__insert__right__if1,axiom,
! [A: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ A @ A2 )
=> ( ( inf_inf_set_option_a @ A2 @ ( insert_option_a @ A @ B ) )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_757_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_758_Int__insert__right__if0,axiom,
! [A: option_a,A2: set_option_a,B: set_option_a] :
( ~ ( member_option_a @ A @ A2 )
=> ( ( inf_inf_set_option_a @ A2 @ ( insert_option_a @ A @ B ) )
= ( inf_inf_set_option_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_759_insert__inter__insert,axiom,
! [A: option_a,A2: set_option_a,B: set_option_a] :
( ( inf_inf_set_option_a @ ( insert_option_a @ A @ A2 ) @ ( insert_option_a @ A @ B ) )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_760_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_761_Int__insert__left__if1,axiom,
! [A: option_a,C2: set_option_a,B: set_option_a] :
( ( member_option_a @ A @ C2 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B ) @ C2 )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_762_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_763_Int__insert__left__if0,axiom,
! [A: option_a,C2: set_option_a,B: set_option_a] :
( ~ ( member_option_a @ A @ C2 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B ) @ C2 )
= ( inf_inf_set_option_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_764_Un__subset__iff,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ A2 @ B ) @ C2 )
= ( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
& ( ord_le1955136853071979460tion_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_765_Un__insert__right,axiom,
! [A2: set_option_a,A: option_a,B: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ A @ B ) )
= ( insert_option_a @ A @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_766_Un__insert__left,axiom,
! [A: option_a,B: set_option_a,C2: set_option_a] :
( ( sup_sup_set_option_a @ ( insert_option_a @ A @ B ) @ C2 )
= ( insert_option_a @ A @ ( sup_sup_set_option_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_767_Int__Un__eq_I4_J,axiom,
! [T4: set_option_a,S2: set_option_a] :
( ( sup_sup_set_option_a @ T4 @ ( inf_inf_set_option_a @ S2 @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_768_Int__Un__eq_I3_J,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( sup_sup_set_option_a @ S2 @ ( inf_inf_set_option_a @ S2 @ T4 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_769_Int__Un__eq_I2_J,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ S2 @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_770_Int__Un__eq_I1_J,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ S2 @ T4 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_771_Un__Int__eq_I4_J,axiom,
! [T4: set_option_a,S2: set_option_a] :
( ( inf_inf_set_option_a @ T4 @ ( sup_sup_set_option_a @ S2 @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_772_Un__Int__eq_I3_J,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( inf_inf_set_option_a @ S2 @ ( sup_sup_set_option_a @ S2 @ T4 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_773_Un__Int__eq_I2_J,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ S2 @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_774_Un__Int__eq_I1_J,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ S2 @ T4 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_775_Un__Diff__cancel2,axiom,
! [B: set_option_a,A2: set_option_a] :
( ( sup_sup_set_option_a @ ( minus_1574173051537231627tion_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_option_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_776_Un__Diff__cancel,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ A2 ) )
= ( sup_sup_set_option_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_777_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_778_insert__disjoint_I1_J,axiom,
! [A: option_a,A2: set_option_a,B: set_option_a] :
( ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ A2 ) @ B )
= bot_bot_set_option_a )
= ( ~ ( member_option_a @ A @ B )
& ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a ) ) ) ).
% insert_disjoint(1)
thf(fact_779_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_780_insert__disjoint_I2_J,axiom,
! [A: option_a,A2: set_option_a,B: set_option_a] :
( ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ ( insert_option_a @ A @ A2 ) @ B ) )
= ( ~ ( member_option_a @ A @ B )
& ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_781_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_782_disjoint__insert_I1_J,axiom,
! [B: set_option_a,A: option_a,A2: set_option_a] :
( ( ( inf_inf_set_option_a @ B @ ( insert_option_a @ A @ A2 ) )
= bot_bot_set_option_a )
= ( ~ ( member_option_a @ A @ B )
& ( ( inf_inf_set_option_a @ B @ A2 )
= bot_bot_set_option_a ) ) ) ).
% disjoint_insert(1)
thf(fact_783_disjoint__insert_I2_J,axiom,
! [A2: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_784_disjoint__insert_I2_J,axiom,
! [A2: set_option_a,B2: option_a,B: set_option_a] :
( ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ A2 @ ( insert_option_a @ B2 @ B ) ) )
= ( ~ ( member_option_a @ B2 @ A2 )
& ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_785_Diff__disjoint,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ A2 ) )
= bot_bot_set_option_a ) ).
% Diff_disjoint
thf(fact_786_boolean__algebra_Oconj__one__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ top_top_set_option_a )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_787_Un__UNIV__left,axiom,
! [B: set_option_a] :
( ( sup_sup_set_option_a @ top_top_set_option_a @ B )
= top_top_set_option_a ) ).
% Un_UNIV_left
thf(fact_788_Un__UNIV__right,axiom,
! [A2: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ top_top_set_option_a )
= top_top_set_option_a ) ).
% Un_UNIV_right
thf(fact_789_Int__UNIV__left,axiom,
! [B: set_option_a] :
( ( inf_inf_set_option_a @ top_top_set_option_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_790_Int__UNIV__right,axiom,
! [A2: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ top_top_set_option_a )
= A2 ) ).
% Int_UNIV_right
thf(fact_791_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_option_a,K: set_option_a,A: set_option_a,B2: set_option_a] :
( ( A2
= ( inf_inf_set_option_a @ K @ A ) )
=> ( ( inf_inf_set_option_a @ A2 @ B2 )
= ( inf_inf_set_option_a @ K @ ( inf_inf_set_option_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_792_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_option_a,K: set_option_a,B2: set_option_a,A: set_option_a] :
( ( B
= ( inf_inf_set_option_a @ K @ B2 ) )
=> ( ( inf_inf_set_option_a @ A @ B )
= ( inf_inf_set_option_a @ K @ ( inf_inf_set_option_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_793_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_option_a,K: set_option_a,A: set_option_a,B2: set_option_a] :
( ( A2
= ( sup_sup_set_option_a @ K @ A ) )
=> ( ( sup_sup_set_option_a @ A2 @ B2 )
= ( sup_sup_set_option_a @ K @ ( sup_sup_set_option_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_794_boolean__algebra__cancel_Osup2,axiom,
! [B: set_option_a,K: set_option_a,B2: set_option_a,A: set_option_a] :
( ( B
= ( sup_sup_set_option_a @ K @ B2 ) )
=> ( ( sup_sup_set_option_a @ A @ B )
= ( sup_sup_set_option_a @ K @ ( sup_sup_set_option_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_795_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( inf_inf_set_option_a @ X @ ( sup_sup_set_option_a @ Y @ Z ) )
= ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ X @ Y ) @ ( inf_inf_set_option_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_796_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( sup_sup_set_option_a @ X @ ( inf_inf_set_option_a @ Y @ Z ) )
= ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ X @ Y ) @ ( sup_sup_set_option_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_797_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_option_a,Z: set_option_a,X: set_option_a] :
( ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ Y @ Z ) @ X )
= ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ Y @ X ) @ ( inf_inf_set_option_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_798_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_option_a,Z: set_option_a,X: set_option_a] :
( ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ Y @ Z ) @ X )
= ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ Y @ X ) @ ( sup_sup_set_option_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_799_UnE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_800_UnE,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) )
=> ( ~ ( member_option_a @ C @ A2 )
=> ( member_option_a @ C @ B ) ) ) ).
% UnE
thf(fact_801_IntE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_802_IntE,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A2 @ B ) )
=> ~ ( ( member_option_a @ C @ A2 )
=> ~ ( member_option_a @ C @ B ) ) ) ).
% IntE
thf(fact_803_UnI1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_804_UnI1,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ A2 )
=> ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_805_UnI2,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_806_UnI2,axiom,
! [C: option_a,B: set_option_a,A2: set_option_a] :
( ( member_option_a @ C @ B )
=> ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_807_IntD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_808_IntD1,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A2 @ B ) )
=> ( member_option_a @ C @ A2 ) ) ).
% IntD1
thf(fact_809_IntD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_810_IntD2,axiom,
! [C: option_a,A2: set_option_a,B: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A2 @ B ) )
=> ( member_option_a @ C @ B ) ) ).
% IntD2
thf(fact_811_bex__Un,axiom,
! [A2: set_option_a,B: set_option_a,P: option_a > $o] :
( ( ? [X3: option_a] :
( ( member_option_a @ X3 @ ( sup_sup_set_option_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: option_a] :
( ( member_option_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_812_ball__Un,axiom,
! [A2: set_option_a,B: set_option_a,P: option_a > $o] :
( ( ! [X3: option_a] :
( ( member_option_a @ X3 @ ( sup_sup_set_option_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: option_a] :
( ( member_option_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_813_Un__assoc,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( sup_sup_set_option_a @ ( sup_sup_set_option_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_option_a @ A2 @ ( sup_sup_set_option_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_814_Int__assoc,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_option_a @ A2 @ ( inf_inf_set_option_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_815_Un__absorb,axiom,
! [A2: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_816_Int__absorb,axiom,
! [A2: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_817_Un__commute,axiom,
( sup_sup_set_option_a
= ( ^ [A3: set_option_a,B3: set_option_a] : ( sup_sup_set_option_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_818_Int__commute,axiom,
( inf_inf_set_option_a
= ( ^ [A3: set_option_a,B3: set_option_a] : ( inf_inf_set_option_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_819_Un__Int__crazy,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( sup_sup_set_option_a @ ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ A2 @ B ) @ ( inf_inf_set_option_a @ B @ C2 ) ) @ ( inf_inf_set_option_a @ C2 @ A2 ) )
= ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ A2 @ B ) @ ( sup_sup_set_option_a @ B @ C2 ) ) @ ( sup_sup_set_option_a @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_820_Int__Un__distrib,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ ( sup_sup_set_option_a @ B @ C2 ) )
= ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ A2 @ B ) @ ( inf_inf_set_option_a @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_821_Un__Int__distrib,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ ( inf_inf_set_option_a @ B @ C2 ) )
= ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ A2 @ B ) @ ( sup_sup_set_option_a @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_822_Un__left__absorb,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( sup_sup_set_option_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_823_Int__Un__distrib2,axiom,
! [B: set_option_a,C2: set_option_a,A2: set_option_a] :
( ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ B @ C2 ) @ A2 )
= ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ B @ A2 ) @ ( inf_inf_set_option_a @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_824_Int__left__absorb,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( inf_inf_set_option_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_825_Un__Int__distrib2,axiom,
! [B: set_option_a,C2: set_option_a,A2: set_option_a] :
( ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ B @ C2 ) @ A2 )
= ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ B @ A2 ) @ ( sup_sup_set_option_a @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_826_Un__left__commute,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ ( sup_sup_set_option_a @ B @ C2 ) )
= ( sup_sup_set_option_a @ B @ ( sup_sup_set_option_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_827_Int__left__commute,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ ( inf_inf_set_option_a @ B @ C2 ) )
= ( inf_inf_set_option_a @ B @ ( inf_inf_set_option_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_828_Int__Diff,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( minus_1574173051537231627tion_a @ ( inf_inf_set_option_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_option_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_829_Diff__Int2,axiom,
! [A2: set_option_a,C2: set_option_a,B: set_option_a] :
( ( minus_1574173051537231627tion_a @ ( inf_inf_set_option_a @ A2 @ C2 ) @ ( inf_inf_set_option_a @ B @ C2 ) )
= ( minus_1574173051537231627tion_a @ ( inf_inf_set_option_a @ A2 @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_830_Diff__Diff__Int,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( inf_inf_set_option_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_831_Diff__Int__distrib,axiom,
! [C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( inf_inf_set_option_a @ C2 @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( minus_1574173051537231627tion_a @ ( inf_inf_set_option_a @ C2 @ A2 ) @ ( inf_inf_set_option_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_832_Diff__Int__distrib2,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( inf_inf_set_option_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ C2 )
= ( minus_1574173051537231627tion_a @ ( inf_inf_set_option_a @ A2 @ C2 ) @ ( inf_inf_set_option_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_833_Un__Diff,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( minus_1574173051537231627tion_a @ ( sup_sup_set_option_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_option_a @ ( minus_1574173051537231627tion_a @ A2 @ C2 ) @ ( minus_1574173051537231627tion_a @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_834_Int__insert__left,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_835_Int__insert__left,axiom,
! [A: option_a,C2: set_option_a,B: set_option_a] :
( ( ( member_option_a @ A @ C2 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B ) @ C2 )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ B @ C2 ) ) ) )
& ( ~ ( member_option_a @ A @ C2 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B ) @ C2 )
= ( inf_inf_set_option_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_836_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_837_Int__insert__right,axiom,
! [A: option_a,A2: set_option_a,B: set_option_a] :
( ( ( member_option_a @ A @ A2 )
=> ( ( inf_inf_set_option_a @ A2 @ ( insert_option_a @ A @ B ) )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ A2 @ B ) ) ) )
& ( ~ ( member_option_a @ A @ A2 )
=> ( ( inf_inf_set_option_a @ A2 @ ( insert_option_a @ A @ B ) )
= ( inf_inf_set_option_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_838_Int__mono,axiom,
! [A2: set_option_a,C2: set_option_a,B: set_option_a,D2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ D2 )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A2 @ B ) @ ( inf_inf_set_option_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_839_Int__lower1,axiom,
! [A2: set_option_a,B: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_840_Int__lower2,axiom,
! [A2: set_option_a,B: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_841_Int__absorb1,axiom,
! [B: set_option_a,A2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B @ A2 )
=> ( ( inf_inf_set_option_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_842_Int__absorb2,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( inf_inf_set_option_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_843_Int__greatest,axiom,
! [C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C2 @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ C2 @ B )
=> ( ord_le1955136853071979460tion_a @ C2 @ ( inf_inf_set_option_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_844_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_845_Int__Collect__mono,axiom,
! [A2: set_option_a,B: set_option_a,P: option_a > $o,Q: option_a > $o] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A2 @ ( collect_option_a @ P ) ) @ ( inf_inf_set_option_a @ B @ ( collect_option_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_846_Un__mono,axiom,
! [A2: set_option_a,C2: set_option_a,B: set_option_a,D2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ D2 )
=> ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ A2 @ B ) @ ( sup_sup_set_option_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_847_Un__least,axiom,
! [A2: set_option_a,C2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_848_Un__upper1,axiom,
! [A2: set_option_a,B: set_option_a] : ( ord_le1955136853071979460tion_a @ A2 @ ( sup_sup_set_option_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_849_Un__upper2,axiom,
! [B: set_option_a,A2: set_option_a] : ( ord_le1955136853071979460tion_a @ B @ ( sup_sup_set_option_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_850_Un__absorb1,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( sup_sup_set_option_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_851_Un__absorb2,axiom,
! [B: set_option_a,A2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B @ A2 )
=> ( ( sup_sup_set_option_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_852_subset__UnE,axiom,
! [C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C2 @ ( sup_sup_set_option_a @ A2 @ B ) )
=> ~ ! [A7: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ A2 )
=> ! [B7: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B7 @ B )
=> ( C2
!= ( sup_sup_set_option_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_853_subset__Un__eq,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A3: set_option_a,B3: set_option_a] :
( ( sup_sup_set_option_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_854_infinite__Un,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ( ~ ( finite1674126218327898605tion_a @ ( sup_sup_set_option_a @ S2 @ T4 ) ) )
= ( ~ ( finite1674126218327898605tion_a @ S2 )
| ~ ( finite1674126218327898605tion_a @ T4 ) ) ) ).
% infinite_Un
thf(fact_855_Un__infinite,axiom,
! [S2: set_option_a,T4: set_option_a] :
( ~ ( finite1674126218327898605tion_a @ S2 )
=> ~ ( finite1674126218327898605tion_a @ ( sup_sup_set_option_a @ S2 @ T4 ) ) ) ).
% Un_infinite
thf(fact_856_finite__UnI,axiom,
! [F3: set_option_a,G3: set_option_a] :
( ( finite1674126218327898605tion_a @ F3 )
=> ( ( finite1674126218327898605tion_a @ G3 )
=> ( finite1674126218327898605tion_a @ ( sup_sup_set_option_a @ F3 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_857_image__Un,axiom,
! [F: a > option_a,A2: set_a,B: set_a] :
( ( image_a_option_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_option_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) ) ).
% image_Un
thf(fact_858_image__Un,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] :
( ( image_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) ) ).
% image_Un
thf(fact_859_image__Un,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] :
( ( image_7439109396645324421tion_a @ F @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( sup_sup_set_option_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ).
% image_Un
thf(fact_860_inf__sup__ord_I2_J,axiom,
! [X: set_option_a,Y: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_861_inf__sup__ord_I1_J,axiom,
! [X: set_option_a,Y: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_862_inf__le1,axiom,
! [X: set_option_a,Y: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_863_inf__le2,axiom,
! [X: set_option_a,Y: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_864_le__infE,axiom,
! [X: set_option_a,A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ ( inf_inf_set_option_a @ A @ B2 ) )
=> ~ ( ( ord_le1955136853071979460tion_a @ X @ A )
=> ~ ( ord_le1955136853071979460tion_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_865_le__infI,axiom,
! [X: set_option_a,A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ A )
=> ( ( ord_le1955136853071979460tion_a @ X @ B2 )
=> ( ord_le1955136853071979460tion_a @ X @ ( inf_inf_set_option_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_866_inf__mono,axiom,
! [A: set_option_a,C: set_option_a,B2: set_option_a,D: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ C )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ D )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A @ B2 ) @ ( inf_inf_set_option_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_867_le__infI1,axiom,
! [A: set_option_a,X: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ X )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_868_le__infI2,axiom,
! [B2: set_option_a,X: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ X )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_869_inf_OorderE,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( A
= ( inf_inf_set_option_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_870_inf_OorderI,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( A
= ( inf_inf_set_option_a @ A @ B2 ) )
=> ( ord_le1955136853071979460tion_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_871_inf__unique,axiom,
! [F: set_option_a > set_option_a > set_option_a,X: set_option_a,Y: set_option_a] :
( ! [X2: set_option_a,Y2: set_option_a] : ( ord_le1955136853071979460tion_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_option_a,Y2: set_option_a] : ( ord_le1955136853071979460tion_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_option_a,Y2: set_option_a,Z2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X2 @ Y2 )
=> ( ( ord_le1955136853071979460tion_a @ X2 @ Z2 )
=> ( ord_le1955136853071979460tion_a @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_set_option_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_872_le__iff__inf,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [X3: set_option_a,Y3: set_option_a] :
( ( inf_inf_set_option_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_873_inf_Oabsorb1,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( inf_inf_set_option_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_874_inf_Oabsorb2,axiom,
! [B2: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ( ( inf_inf_set_option_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_875_inf__absorb1,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ Y )
=> ( ( inf_inf_set_option_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_876_inf__absorb2,axiom,
! [Y: set_option_a,X: set_option_a] :
( ( ord_le1955136853071979460tion_a @ Y @ X )
=> ( ( inf_inf_set_option_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_877_inf_OboundedE,axiom,
! [A: set_option_a,B2: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ ( inf_inf_set_option_a @ B2 @ C ) )
=> ~ ( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ~ ( ord_le1955136853071979460tion_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_878_inf_OboundedI,axiom,
! [A: set_option_a,B2: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( ord_le1955136853071979460tion_a @ A @ C )
=> ( ord_le1955136853071979460tion_a @ A @ ( inf_inf_set_option_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_879_inf__greatest,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ Y )
=> ( ( ord_le1955136853071979460tion_a @ X @ Z )
=> ( ord_le1955136853071979460tion_a @ X @ ( inf_inf_set_option_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_880_inf_Oorder__iff,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A6: set_option_a,B6: set_option_a] :
( A6
= ( inf_inf_set_option_a @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_881_inf_Ocobounded1,axiom,
! [A: set_option_a,B2: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_882_inf_Ocobounded2,axiom,
! [A: set_option_a,B2: set_option_a] : ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_883_inf_Oabsorb__iff1,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A6: set_option_a,B6: set_option_a] :
( ( inf_inf_set_option_a @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_884_inf_Oabsorb__iff2,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [B6: set_option_a,A6: set_option_a] :
( ( inf_inf_set_option_a @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_885_inf_OcoboundedI1,axiom,
! [A: set_option_a,C: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ C )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_886_inf_OcoboundedI2,axiom,
! [B2: set_option_a,C: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ C )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_887_inf__sup__ord_I4_J,axiom,
! [Y: set_option_a,X: set_option_a] : ( ord_le1955136853071979460tion_a @ Y @ ( sup_sup_set_option_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_888_inf__sup__ord_I3_J,axiom,
! [X: set_option_a,Y: set_option_a] : ( ord_le1955136853071979460tion_a @ X @ ( sup_sup_set_option_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_889_le__supE,axiom,
! [A: set_option_a,B2: set_option_a,X: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ A @ B2 ) @ X )
=> ~ ( ( ord_le1955136853071979460tion_a @ A @ X )
=> ~ ( ord_le1955136853071979460tion_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_890_le__supI,axiom,
! [A: set_option_a,X: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ X )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ X )
=> ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_891_sup__ge1,axiom,
! [X: set_option_a,Y: set_option_a] : ( ord_le1955136853071979460tion_a @ X @ ( sup_sup_set_option_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_892_sup__ge2,axiom,
! [Y: set_option_a,X: set_option_a] : ( ord_le1955136853071979460tion_a @ Y @ ( sup_sup_set_option_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_893_le__supI1,axiom,
! [X: set_option_a,A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ A )
=> ( ord_le1955136853071979460tion_a @ X @ ( sup_sup_set_option_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_894_le__supI2,axiom,
! [X: set_option_a,B2: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ B2 )
=> ( ord_le1955136853071979460tion_a @ X @ ( sup_sup_set_option_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_895_sup_Omono,axiom,
! [C: set_option_a,A: set_option_a,D: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C @ A )
=> ( ( ord_le1955136853071979460tion_a @ D @ B2 )
=> ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ C @ D ) @ ( sup_sup_set_option_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_896_sup__mono,axiom,
! [A: set_option_a,C: set_option_a,B2: set_option_a,D: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ C )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ D )
=> ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ A @ B2 ) @ ( sup_sup_set_option_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_897_sup__least,axiom,
! [Y: set_option_a,X: set_option_a,Z: set_option_a] :
( ( ord_le1955136853071979460tion_a @ Y @ X )
=> ( ( ord_le1955136853071979460tion_a @ Z @ X )
=> ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_898_le__iff__sup,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [X3: set_option_a,Y3: set_option_a] :
( ( sup_sup_set_option_a @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_899_sup_OorderE,axiom,
! [B2: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ( A
= ( sup_sup_set_option_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_900_sup_OorderI,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( A
= ( sup_sup_set_option_a @ A @ B2 ) )
=> ( ord_le1955136853071979460tion_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_901_sup__unique,axiom,
! [F: set_option_a > set_option_a > set_option_a,X: set_option_a,Y: set_option_a] :
( ! [X2: set_option_a,Y2: set_option_a] : ( ord_le1955136853071979460tion_a @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_option_a,Y2: set_option_a] : ( ord_le1955136853071979460tion_a @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_option_a,Y2: set_option_a,Z2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ Y2 @ X2 )
=> ( ( ord_le1955136853071979460tion_a @ Z2 @ X2 )
=> ( ord_le1955136853071979460tion_a @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_set_option_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_902_sup_Oabsorb1,axiom,
! [B2: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ( ( sup_sup_set_option_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_903_sup_Oabsorb2,axiom,
! [A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ B2 )
=> ( ( sup_sup_set_option_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_904_sup__absorb1,axiom,
! [Y: set_option_a,X: set_option_a] :
( ( ord_le1955136853071979460tion_a @ Y @ X )
=> ( ( sup_sup_set_option_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_905_sup__absorb2,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ Y )
=> ( ( sup_sup_set_option_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_906_sup_OboundedE,axiom,
! [B2: set_option_a,C: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ B2 @ C ) @ A )
=> ~ ( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ~ ( ord_le1955136853071979460tion_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_907_sup_OboundedI,axiom,
! [B2: set_option_a,A: set_option_a,C: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ A )
=> ( ( ord_le1955136853071979460tion_a @ C @ A )
=> ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_908_sup_Oorder__iff,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [B6: set_option_a,A6: set_option_a] :
( A6
= ( sup_sup_set_option_a @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_909_sup_Ocobounded1,axiom,
! [A: set_option_a,B2: set_option_a] : ( ord_le1955136853071979460tion_a @ A @ ( sup_sup_set_option_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_910_sup_Ocobounded2,axiom,
! [B2: set_option_a,A: set_option_a] : ( ord_le1955136853071979460tion_a @ B2 @ ( sup_sup_set_option_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_911_sup_Oabsorb__iff1,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [B6: set_option_a,A6: set_option_a] :
( ( sup_sup_set_option_a @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_912_sup_Oabsorb__iff2,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A6: set_option_a,B6: set_option_a] :
( ( sup_sup_set_option_a @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_913_sup_OcoboundedI1,axiom,
! [C: set_option_a,A: set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C @ A )
=> ( ord_le1955136853071979460tion_a @ C @ ( sup_sup_set_option_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_914_sup_OcoboundedI2,axiom,
! [C: set_option_a,B2: set_option_a,A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ C @ B2 )
=> ( ord_le1955136853071979460tion_a @ C @ ( sup_sup_set_option_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_915_Un__Int__assoc__eq,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_option_a @ A2 @ ( sup_sup_set_option_a @ B @ C2 ) ) )
= ( ord_le1955136853071979460tion_a @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_916_Un__Diff__Int,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( sup_sup_set_option_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ ( inf_inf_set_option_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_917_Int__Diff__Un,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ A2 @ B ) @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_918_Diff__Int,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ ( inf_inf_set_option_a @ B @ C2 ) )
= ( sup_sup_set_option_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ ( minus_1574173051537231627tion_a @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_919_Diff__Un,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ ( sup_sup_set_option_a @ B @ C2 ) )
= ( inf_inf_set_option_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ ( minus_1574173051537231627tion_a @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_920_distrib__sup__le,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] : ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ X @ ( inf_inf_set_option_a @ Y @ Z ) ) @ ( inf_inf_set_option_a @ ( sup_sup_set_option_a @ X @ Y ) @ ( sup_sup_set_option_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_921_distrib__inf__le,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] : ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ ( inf_inf_set_option_a @ X @ Y ) @ ( inf_inf_set_option_a @ X @ Z ) ) @ ( inf_inf_set_option_a @ X @ ( sup_sup_set_option_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_922_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ X @ bot_bot_set_option_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_923_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_924_Int__emptyI,axiom,
! [A2: set_option_a,B: set_option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ~ ( member_option_a @ X2 @ B ) )
=> ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a ) ) ).
% Int_emptyI
thf(fact_925_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_926_disjoint__iff,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ~ ( member_option_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_927_Int__empty__left,axiom,
! [B: set_option_a] :
( ( inf_inf_set_option_a @ bot_bot_set_option_a @ B )
= bot_bot_set_option_a ) ).
% Int_empty_left
thf(fact_928_Int__empty__right,axiom,
! [A2: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% Int_empty_right
thf(fact_929_disjoint__iff__not__equal,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ! [Y3: option_a] :
( ( member_option_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_930_Un__empty__left,axiom,
! [B: set_option_a] :
( ( sup_sup_set_option_a @ bot_bot_set_option_a @ B )
= B ) ).
% Un_empty_left
thf(fact_931_Un__empty__right,axiom,
! [A2: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ bot_bot_set_option_a )
= A2 ) ).
% Un_empty_right
thf(fact_932_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_option_a,X: set_option_a,Y: set_option_a] :
( ( ( inf_inf_set_option_a @ A @ X )
= bot_bot_set_option_a )
=> ( ( ( sup_sup_set_option_a @ A @ X )
= top_top_set_option_a )
=> ( ( ( inf_inf_set_option_a @ A @ Y )
= bot_bot_set_option_a )
=> ( ( ( sup_sup_set_option_a @ A @ Y )
= top_top_set_option_a )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_933_comp__fun__idem__sup,axiom,
finite3263133667051808460tion_a @ sup_sup_set_option_a ).
% comp_fun_idem_sup
thf(fact_934_comp__fun__idem__inf,axiom,
finite3263133667051808460tion_a @ inf_inf_set_option_a ).
% comp_fun_idem_inf
thf(fact_935_semilattice__closure__operator_Oaxioms_I1_J,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( set_cl4772385950493004755tion_a @ F ) ) ).
% semilattice_closure_operator.axioms(1)
thf(fact_936_singleton__Un__iff,axiom,
! [X: option_a,A2: set_option_a,B: set_option_a] :
( ( ( insert_option_a @ X @ bot_bot_set_option_a )
= ( sup_sup_set_option_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_option_a )
& ( B
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
| ( ( A2
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B = bot_bot_set_option_a ) )
| ( ( A2
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_937_Un__singleton__iff,axiom,
! [A2: set_option_a,B: set_option_a,X: option_a] :
( ( ( sup_sup_set_option_a @ A2 @ B )
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
= ( ( ( A2 = bot_bot_set_option_a )
& ( B
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
| ( ( A2
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B = bot_bot_set_option_a ) )
| ( ( A2
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_938_insert__is__Un,axiom,
( insert_option_a
= ( ^ [A6: option_a] : ( sup_sup_set_option_a @ ( insert_option_a @ A6 @ bot_bot_set_option_a ) ) ) ) ).
% insert_is_Un
thf(fact_939_image__Int__subset,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] : ( ord_less_eq_set_a @ ( image_option_a_a @ F @ ( inf_inf_set_option_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_940_image__Int__subset,axiom,
! [F: a > option_a,A2: set_a,B: set_a] : ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_option_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_941_image__Int__subset,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] : ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ ( inf_inf_set_option_a @ A2 @ B ) ) @ ( inf_inf_set_option_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_942_union__fold__insert,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( sup_sup_set_option_a @ A2 @ B )
= ( finite3172252742789171496tion_a @ insert_option_a @ B @ A2 ) ) ) ).
% union_fold_insert
thf(fact_943_union__set__fold,axiom,
! [Xs: list_option_a,A2: set_option_a] :
( ( sup_sup_set_option_a @ ( set_option_a2 @ Xs ) @ A2 )
= ( fold_o8770731556678408811tion_a @ insert_option_a @ Xs @ A2 ) ) ).
% union_set_fold
thf(fact_944_Diff__subset__conv,axiom,
! [A2: set_option_a,B: set_option_a,C2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ C2 )
= ( ord_le1955136853071979460tion_a @ A2 @ ( sup_sup_set_option_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_945_Diff__partition,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( sup_sup_set_option_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_946_Int__Diff__disjoint,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ A2 @ B ) @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= bot_bot_set_option_a ) ).
% Int_Diff_disjoint
thf(fact_947_Diff__triv,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a )
=> ( ( minus_1574173051537231627tion_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_948_empty__in__Fpow,axiom,
! [A2: set_option_a] : ( member_set_option_a @ bot_bot_set_option_a @ ( finite_Fpow_option_a @ A2 ) ) ).
% empty_in_Fpow
thf(fact_949_Fpow__mono,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ord_le7761976607055303332tion_a @ ( finite_Fpow_option_a @ A2 ) @ ( finite_Fpow_option_a @ B ) ) ) ).
% Fpow_mono
thf(fact_950_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,A2: set_option_a,B: set_option_a,Z: option_a] :
( ( finite3844474811650721488tion_a @ S2 @ F )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ S2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ S2 )
=> ( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1674126218327898605tion_a @ B )
=> ( ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( finite1758689504953034312tion_a @ F @ ( finite1758689504953034312tion_a @ F @ Z @ A2 ) @ B ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_951_semilattice__closure__operator_Osupremum__sound,axiom,
! [F: a > a > a,Less_eq: a > a > $o,E: a,S2: set_a,T3: a] :
( ( semila7314208447189752392ator_a @ F @ Less_eq @ E )
=> ( ( finite_finite_a @ S2 )
=> ( ( member_a @ T3 @ S2 )
=> ( Less_eq @ T3 @ ( finite_fold_a_a @ F @ E @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.supremum_sound
thf(fact_952_semilattice__closure__operator_Osupremum__sound,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a,T3: option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( ( member_option_a @ T3 @ S2 )
=> ( Less_eq @ T3 @ ( finite1758689504953034312tion_a @ F @ E @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.supremum_sound
thf(fact_953_semilattice__closure__operator_Osupremum__subset__less__eq,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a,R: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( ( ord_le1955136853071979460tion_a @ R @ S2 )
=> ( Less_eq @ ( finite1758689504953034312tion_a @ F @ E @ R ) @ ( finite1758689504953034312tion_a @ F @ E @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.supremum_subset_less_eq
thf(fact_954_semilattice__closure__operator_Osupremum__in__smaller__closure,axiom,
! [F: a > a > a,Less_eq: a > a > $o,E: a,S2: set_a,X: a] :
( ( semila7314208447189752392ator_a @ F @ Less_eq @ E )
=> ( ( finite_finite_a @ S2 )
=> ( member_a @ ( finite_fold_a_a @ F @ E @ ( semila2432794410337176460bset_a @ Less_eq @ X @ S2 ) ) @ ( sup_sup_set_a @ ( insert_a @ E @ bot_bot_set_a ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.supremum_in_smaller_closure
thf(fact_955_semilattice__closure__operator_Osupremum__in__smaller__closure,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a,X: option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( member_option_a @ ( finite1758689504953034312tion_a @ F @ E @ ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) ) @ ( sup_sup_set_option_a @ ( insert_option_a @ E @ bot_bot_set_option_a ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.supremum_in_smaller_closure
thf(fact_956_inter__coset__fold,axiom,
! [A2: set_option_a,Xs: list_option_a] :
( ( inf_inf_set_option_a @ A2 @ ( coset_option_a @ Xs ) )
= ( fold_o8770731556678408811tion_a @ remove_option_a @ Xs @ A2 ) ) ).
% inter_coset_fold
thf(fact_957_semilattice__closure__operator_Osupremum__smaller__closure,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a,X: option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( ( finite1758689504953034312tion_a @ F @ E @ ( semila1830171625574221010tion_a @ Less_eq @ X @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) )
= ( finite1758689504953034312tion_a @ F @ E @ ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.supremum_smaller_closure
thf(fact_958_SUP__set__fold,axiom,
! [F: option_a > set_option_a,Xs: list_option_a] :
( ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ ( set_option_a2 @ Xs ) ) )
= ( fold_o8770731556678408811tion_a @ ( comp_s1392512783494491100tion_a @ sup_sup_set_option_a @ F ) @ Xs @ bot_bot_set_option_a ) ) ).
% SUP_set_fold
thf(fact_959_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_option_a] :
( ( bot_bot_set_option_a
= ( comple4629436989800923665tion_a @ A2 ) )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_option_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_960_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_option_a] :
( ( ( comple4629436989800923665tion_a @ A2 )
= bot_bot_set_option_a )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_option_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_961_finite__UN,axiom,
! [A2: set_option_a,B: option_a > set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1674126218327898605tion_a @ ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ B @ A2 ) ) )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( finite1674126218327898605tion_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_962_finite__Union,axiom,
! [A2: set_set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ! [M2: set_option_a] :
( ( member_set_option_a @ M2 @ A2 )
=> ( finite1674126218327898605tion_a @ M2 ) )
=> ( finite1674126218327898605tion_a @ ( comple4629436989800923665tion_a @ A2 ) ) ) ) ).
% finite_Union
thf(fact_963_Sup__empty,axiom,
( ( comple4629436989800923665tion_a @ bot_bo651546871064636272tion_a )
= bot_bot_set_option_a ) ).
% Sup_empty
thf(fact_964_Sup__UNIV,axiom,
( ( comple4629436989800923665tion_a @ top_to7232205438743699540tion_a )
= top_top_set_option_a ) ).
% Sup_UNIV
thf(fact_965_Sup__insert,axiom,
! [A: set_option_a,A2: set_set_option_a] :
( ( comple4629436989800923665tion_a @ ( insert_set_option_a @ A @ A2 ) )
= ( sup_sup_set_option_a @ A @ ( comple4629436989800923665tion_a @ A2 ) ) ) ).
% Sup_insert
thf(fact_966_Sup__upper2,axiom,
! [U: set_option_a,A2: set_set_option_a,V2: set_option_a] :
( ( member_set_option_a @ U @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ V2 @ U )
=> ( ord_le1955136853071979460tion_a @ V2 @ ( comple4629436989800923665tion_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_967_Sup__le__iff,axiom,
! [A2: set_set_option_a,B2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ A2 ) @ B2 )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( ord_le1955136853071979460tion_a @ X3 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_968_Sup__upper,axiom,
! [X: set_option_a,A2: set_set_option_a] :
( ( member_set_option_a @ X @ A2 )
=> ( ord_le1955136853071979460tion_a @ X @ ( comple4629436989800923665tion_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_969_Sup__least,axiom,
! [A2: set_set_option_a,Z: set_option_a] :
( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ X2 @ Z ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_970_Sup__mono,axiom,
! [A2: set_set_option_a,B: set_set_option_a] :
( ! [A4: set_option_a] :
( ( member_set_option_a @ A4 @ A2 )
=> ? [X4: set_option_a] :
( ( member_set_option_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ A4 @ X4 ) ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ A2 ) @ ( comple4629436989800923665tion_a @ B ) ) ) ).
% Sup_mono
thf(fact_971_Sup__eqI,axiom,
! [A2: set_set_option_a,X: set_option_a] :
( ! [Y2: set_option_a] :
( ( member_set_option_a @ Y2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ Y2 @ X ) )
=> ( ! [Y2: set_option_a] :
( ! [Z5: set_option_a] :
( ( member_set_option_a @ Z5 @ A2 )
=> ( ord_le1955136853071979460tion_a @ Z5 @ Y2 ) )
=> ( ord_le1955136853071979460tion_a @ X @ Y2 ) )
=> ( ( comple4629436989800923665tion_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_972_Union__subsetI,axiom,
! [A2: set_set_option_a,B: set_set_option_a] :
( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
=> ? [Y6: set_option_a] :
( ( member_set_option_a @ Y6 @ B )
& ( ord_le1955136853071979460tion_a @ X2 @ Y6 ) ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ A2 ) @ ( comple4629436989800923665tion_a @ B ) ) ) ).
% Union_subsetI
thf(fact_973_Union__upper,axiom,
! [B: set_option_a,A2: set_set_option_a] :
( ( member_set_option_a @ B @ A2 )
=> ( ord_le1955136853071979460tion_a @ B @ ( comple4629436989800923665tion_a @ A2 ) ) ) ).
% Union_upper
thf(fact_974_Union__least,axiom,
! [A2: set_set_option_a,C2: set_option_a] :
( ! [X7: set_option_a] :
( ( member_set_option_a @ X7 @ A2 )
=> ( ord_le1955136853071979460tion_a @ X7 @ C2 ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_975_Union__empty,axiom,
( ( comple4629436989800923665tion_a @ bot_bo651546871064636272tion_a )
= bot_bot_set_option_a ) ).
% Union_empty
thf(fact_976_Union__empty__conv,axiom,
! [A2: set_set_option_a] :
( ( ( comple4629436989800923665tion_a @ A2 )
= bot_bot_set_option_a )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_option_a ) ) ) ) ).
% Union_empty_conv
thf(fact_977_empty__Union__conv,axiom,
! [A2: set_set_option_a] :
( ( bot_bot_set_option_a
= ( comple4629436989800923665tion_a @ A2 ) )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = bot_bot_set_option_a ) ) ) ) ).
% empty_Union_conv
thf(fact_978_SUP__eq,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > set_option_a,G: option_a > set_option_a] :
( ! [I2: option_a] :
( ( member_option_a @ I2 @ A2 )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: option_a] :
( ( member_option_a @ J @ B )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) )
= ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_979_SUP__eq,axiom,
! [A2: set_option_a,B: set_a,F: option_a > set_option_a,G: a > set_option_a] :
( ! [I2: option_a] :
( ( member_option_a @ I2 @ A2 )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) )
= ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_980_SUP__eq,axiom,
! [A2: set_a,B: set_option_a,F: a > set_option_a,G: option_a > set_option_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: option_a] :
( ( member_option_a @ J @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ F @ A2 ) )
= ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_981_SUP__eq,axiom,
! [A2: set_a,B: set_a,F: a > set_option_a,G: a > set_option_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ F @ A2 ) )
= ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_982_less__eq__Sup,axiom,
! [A2: set_set_option_a,U: set_option_a] :
( ! [V: set_option_a] :
( ( member_set_option_a @ V @ A2 )
=> ( ord_le1955136853071979460tion_a @ U @ V ) )
=> ( ( A2 != bot_bo651546871064636272tion_a )
=> ( ord_le1955136853071979460tion_a @ U @ ( comple4629436989800923665tion_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_983_Sup__subset__mono,axiom,
! [A2: set_set_option_a,B: set_set_option_a] :
( ( ord_le7761976607055303332tion_a @ A2 @ B )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ A2 ) @ ( comple4629436989800923665tion_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_984_Union__disjoint,axiom,
! [C2: set_set_option_a,A2: set_option_a] :
( ( ( inf_inf_set_option_a @ ( comple4629436989800923665tion_a @ C2 ) @ A2 )
= bot_bot_set_option_a )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ C2 )
=> ( ( inf_inf_set_option_a @ X3 @ A2 )
= bot_bot_set_option_a ) ) ) ) ).
% Union_disjoint
thf(fact_985_insert__partition,axiom,
! [X: set_option_a,F3: set_set_option_a] :
( ~ ( member_set_option_a @ X @ F3 )
=> ( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ ( insert_set_option_a @ X @ F3 ) )
=> ! [Xa3: set_option_a] :
( ( member_set_option_a @ Xa3 @ ( insert_set_option_a @ X @ F3 ) )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf_set_option_a @ X2 @ Xa3 )
= bot_bot_set_option_a ) ) ) )
=> ( ( inf_inf_set_option_a @ X @ ( comple4629436989800923665tion_a @ F3 ) )
= bot_bot_set_option_a ) ) ) ).
% insert_partition
thf(fact_986_Union__Int__subset,axiom,
! [A2: set_set_option_a,B: set_set_option_a] : ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ ( inf_in4298045558108779506tion_a @ A2 @ B ) ) @ ( inf_inf_set_option_a @ ( comple4629436989800923665tion_a @ A2 ) @ ( comple4629436989800923665tion_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_987_Union__mono,axiom,
! [A2: set_set_option_a,B: set_set_option_a] :
( ( ord_le7761976607055303332tion_a @ A2 @ B )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ A2 ) @ ( comple4629436989800923665tion_a @ B ) ) ) ).
% Union_mono
thf(fact_988_Union__UNIV,axiom,
( ( comple4629436989800923665tion_a @ top_to7232205438743699540tion_a )
= top_top_set_option_a ) ).
% Union_UNIV
thf(fact_989_finite__UnionD,axiom,
! [A2: set_set_option_a] :
( ( finite1674126218327898605tion_a @ ( comple4629436989800923665tion_a @ A2 ) )
=> ( finite3047895793355147469tion_a @ A2 ) ) ).
% finite_UnionD
thf(fact_990_SUP__eq__iff,axiom,
! [I3: set_a,C: set_option_a,F: a > set_option_a] :
( ( I3 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I3 )
=> ( ord_le1955136853071979460tion_a @ C @ ( F @ I2 ) ) )
=> ( ( ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ F @ I3 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_991_SUP__eq__iff,axiom,
! [I3: set_option_a,C: set_option_a,F: option_a > set_option_a] :
( ( I3 != bot_bot_set_option_a )
=> ( ! [I2: option_a] :
( ( member_option_a @ I2 @ I3 )
=> ( ord_le1955136853071979460tion_a @ C @ ( F @ I2 ) ) )
=> ( ( ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ I3 ) )
= C )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ I3 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_992_Sup__inter__less__eq,axiom,
! [A2: set_set_option_a,B: set_set_option_a] : ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ ( inf_in4298045558108779506tion_a @ A2 @ B ) ) @ ( inf_inf_set_option_a @ ( comple4629436989800923665tion_a @ A2 ) @ ( comple4629436989800923665tion_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_993_finite__Sup__in,axiom,
! [A2: set_set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( A2 != bot_bo651546871064636272tion_a )
=> ( ! [X2: set_option_a,Y2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
=> ( ( member_set_option_a @ Y2 @ A2 )
=> ( member_set_option_a @ ( sup_sup_set_option_a @ X2 @ Y2 ) @ A2 ) ) )
=> ( member_set_option_a @ ( comple4629436989800923665tion_a @ A2 ) @ A2 ) ) ) ) ).
% finite_Sup_in
thf(fact_994_sup__Sup__fold__sup,axiom,
! [A2: set_set_option_a,B: set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( sup_sup_set_option_a @ ( comple4629436989800923665tion_a @ A2 ) @ B )
= ( finite8248554786851617352tion_a @ sup_sup_set_option_a @ B @ A2 ) ) ) ).
% sup_Sup_fold_sup
thf(fact_995_semilattice__closure__operator_Osmaller__subset__empty,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,X: option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( semila1830171625574221010tion_a @ Less_eq @ X @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ) ).
% semilattice_closure_operator.smaller_subset_empty
thf(fact_996_semilattice__closure__operator_Ofinite__smaller__subset,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a,X: option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( finite1674126218327898605tion_a @ ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) ) ) ) ).
% semilattice_closure_operator.finite_smaller_subset
thf(fact_997_semilattice__closure__operator_Osmaller__subset__mono,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,X: option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ord_le1955136853071979460tion_a @ ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) @ S2 ) ) ).
% semilattice_closure_operator.smaller_subset_mono
thf(fact_998_semilattice__closure__operator_Osmaller__subset__insert_I2_J,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,Y: option_a,X: option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ~ ( Less_eq @ Y @ X )
=> ( ( semila1830171625574221010tion_a @ Less_eq @ X @ ( insert_option_a @ Y @ S2 ) )
= ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) ) ) ) ).
% semilattice_closure_operator.smaller_subset_insert(2)
thf(fact_999_semilattice__closure__operator_Osmaller__subset__insert_I1_J,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,Y: option_a,X: option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( Less_eq @ Y @ X )
=> ( ( semila1830171625574221010tion_a @ Less_eq @ X @ ( insert_option_a @ Y @ S2 ) )
= ( insert_option_a @ Y @ ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.smaller_subset_insert(1)
thf(fact_1000_subset__code_I2_J,axiom,
! [A2: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A2 @ ( coset_a @ Ys ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_1001_subset__code_I2_J,axiom,
! [A2: set_option_a,Ys: list_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ ( coset_option_a @ Ys ) )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ ( set_option_a2 @ Ys ) )
=> ~ ( member_option_a @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_1002_semilattice__closure__operator_Opre__subset__eq__pos__subset,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,X: option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( semila1830171625574221010tion_a @ Less_eq @ X @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
= ( set_cl4093418192473505224tion_a @ F @ ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) ) ) ) ).
% semilattice_closure_operator.pre_subset_eq_pos_subset
thf(fact_1003_insert__code_I2_J,axiom,
! [X: option_a,Xs: list_option_a] :
( ( insert_option_a @ X @ ( coset_option_a @ Xs ) )
= ( coset_option_a @ ( removeAll_option_a @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1004_Sup__fold__sup,axiom,
! [A2: set_set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( comple4629436989800923665tion_a @ A2 )
= ( finite8248554786851617352tion_a @ sup_sup_set_option_a @ bot_bot_set_option_a @ A2 ) ) ) ).
% Sup_fold_sup
thf(fact_1005_Sup__set__fold,axiom,
! [Xs: list_set_option_a] :
( ( comple4629436989800923665tion_a @ ( set_set_option_a2 @ Xs ) )
= ( fold_s5742101689114034059tion_a @ sup_sup_set_option_a @ Xs @ bot_bot_set_option_a ) ) ).
% Sup_set_fold
thf(fact_1006_semilattice__closure__operator_Osupremum__smaller__subset,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S2: set_option_a,X: option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( finite1674126218327898605tion_a @ S2 )
=> ( Less_eq @ ( finite1758689504953034312tion_a @ F @ E @ ( semila1830171625574221010tion_a @ Less_eq @ X @ S2 ) ) @ X ) ) ) ).
% semilattice_closure_operator.supremum_smaller_subset
thf(fact_1007_sup__SUP__fold__sup,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( sup_sup_set_option_a @ B @ ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) ) )
= ( finite3172252742789171496tion_a @ ( comp_s1392512783494491100tion_a @ sup_sup_set_option_a @ F ) @ B @ A2 ) ) ) ).
% sup_SUP_fold_sup
thf(fact_1008_SUP__fold__sup,axiom,
! [A2: set_option_a,F: option_a > set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) )
= ( finite3172252742789171496tion_a @ ( comp_s1392512783494491100tion_a @ sup_sup_set_option_a @ F ) @ bot_bot_set_option_a @ A2 ) ) ) ).
% SUP_fold_sup
thf(fact_1009_finite__subset__Union,axiom,
! [A2: set_option_a,B8: set_set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ ( comple4629436989800923665tion_a @ B8 ) )
=> ~ ! [F5: set_set_option_a] :
( ( finite3047895793355147469tion_a @ F5 )
=> ( ( ord_le7761976607055303332tion_a @ F5 @ B8 )
=> ~ ( ord_le1955136853071979460tion_a @ A2 @ ( comple4629436989800923665tion_a @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1010_cSup__eq__maximum,axiom,
! [Z: set_option_a,X6: set_set_option_a] :
( ( member_set_option_a @ Z @ X6 )
=> ( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ X6 )
=> ( ord_le1955136853071979460tion_a @ X2 @ Z ) )
=> ( ( comple4629436989800923665tion_a @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1011_cSup__least,axiom,
! [X6: set_set_option_a,Z: set_option_a] :
( ( X6 != bot_bo651546871064636272tion_a )
=> ( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ X6 )
=> ( ord_le1955136853071979460tion_a @ X2 @ Z ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1012_cSup__eq__non__empty,axiom,
! [X6: set_set_option_a,A: set_option_a] :
( ( X6 != bot_bo651546871064636272tion_a )
=> ( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ X6 )
=> ( ord_le1955136853071979460tion_a @ X2 @ A ) )
=> ( ! [Y2: set_option_a] :
( ! [X4: set_option_a] :
( ( member_set_option_a @ X4 @ X6 )
=> ( ord_le1955136853071979460tion_a @ X4 @ Y2 ) )
=> ( ord_le1955136853071979460tion_a @ A @ Y2 ) )
=> ( ( comple4629436989800923665tion_a @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1013_le__cSup__finite,axiom,
! [X6: set_set_option_a,X: set_option_a] :
( ( finite3047895793355147469tion_a @ X6 )
=> ( ( member_set_option_a @ X @ X6 )
=> ( ord_le1955136853071979460tion_a @ X @ ( comple4629436989800923665tion_a @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_1014_inf__Sup,axiom,
! [A: set_option_a,B: set_set_option_a] :
( ( inf_inf_set_option_a @ A @ ( comple4629436989800923665tion_a @ B ) )
= ( comple4629436989800923665tion_a @ ( image_4886974585486332549tion_a @ ( inf_inf_set_option_a @ A ) @ B ) ) ) ).
% inf_Sup
thf(fact_1015_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_option_a,A: set_option_a] :
( ( ( inf_inf_set_option_a @ ( comple4629436989800923665tion_a @ B ) @ A )
= bot_bot_set_option_a )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ B )
=> ( ( inf_inf_set_option_a @ X3 @ A )
= bot_bot_set_option_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1016_cSUP__least,axiom,
! [A2: set_a,F: a > set_option_a,M: set_option_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( F @ X2 ) @ M ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1017_cSUP__least,axiom,
! [A2: set_option_a,F: option_a > set_option_a,M: set_option_a] :
( ( A2 != bot_bot_set_option_a )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( F @ X2 ) @ M ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1018_Union__image__insert,axiom,
! [F: option_a > set_option_a,A: option_a,B: set_option_a] :
( ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ ( insert_option_a @ A @ B ) ) )
= ( sup_sup_set_option_a @ ( F @ A ) @ ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ B ) ) ) ) ).
% Union_image_insert
thf(fact_1019_Union__image__empty,axiom,
! [A2: set_option_a,F: option_a > set_option_a] :
( ( sup_sup_set_option_a @ A2 @ ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ bot_bot_set_option_a ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_1020_type__copy__map__cong0,axiom,
! [M: option_a > option_a,G: option_a > option_a,X: option_a,N: option_a > option_a,H: option_a > option_a,F: option_a > option_a] :
( ( ( M @ ( G @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_o3154387707078715297tion_a @ ( comp_o3154387707078715297tion_a @ F @ M ) @ G @ X )
= ( comp_o3154387707078715297tion_a @ ( comp_o3154387707078715297tion_a @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_1021_UNION__fun__upd,axiom,
! [A2: a > set_option_a,I: a,B: set_option_a,J2: set_a] :
( ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ ( fun_up2298615654684665507tion_a @ A2 @ I @ B ) @ J2 ) )
= ( sup_sup_set_option_a @ ( comple4629436989800923665tion_a @ ( image_a_set_option_a @ A2 @ ( minus_minus_set_a @ J2 @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) @ ( if_set_option_a @ ( member_a @ I @ J2 ) @ B @ bot_bot_set_option_a ) ) ) ).
% UNION_fun_upd
thf(fact_1022_UNION__fun__upd,axiom,
! [A2: option_a > set_option_a,I: option_a,B: set_option_a,J2: set_option_a] :
( ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ ( fun_up3924216255703953885tion_a @ A2 @ I @ B ) @ J2 ) )
= ( sup_sup_set_option_a @ ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ A2 @ ( minus_1574173051537231627tion_a @ J2 @ ( insert_option_a @ I @ bot_bot_set_option_a ) ) ) ) @ ( if_set_option_a @ ( member_option_a @ I @ J2 ) @ B @ bot_bot_set_option_a ) ) ) ).
% UNION_fun_upd
thf(fact_1023_INF__set__fold,axiom,
! [F: option_a > set_option_a,Xs: list_option_a] :
( ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ ( set_option_a2 @ Xs ) ) )
= ( fold_o8770731556678408811tion_a @ ( comp_s1392512783494491100tion_a @ inf_inf_set_option_a @ F ) @ Xs @ top_top_set_option_a ) ) ).
% INF_set_fold
thf(fact_1024_Inter__UNIV__conv_I1_J,axiom,
! [A2: set_set_option_a] :
( ( ( comple4728569501423616811tion_a @ A2 )
= top_top_set_option_a )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = top_top_set_option_a ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_1025_Inter__UNIV__conv_I2_J,axiom,
! [A2: set_set_option_a] :
( ( top_top_set_option_a
= ( comple4728569501423616811tion_a @ A2 ) )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = top_top_set_option_a ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_1026_finite__Inter,axiom,
! [M: set_set_option_a] :
( ? [X4: set_option_a] :
( ( member_set_option_a @ X4 @ M )
& ( finite1674126218327898605tion_a @ X4 ) )
=> ( finite1674126218327898605tion_a @ ( comple4728569501423616811tion_a @ M ) ) ) ).
% finite_Inter
thf(fact_1027_Inf__top__conv_I1_J,axiom,
! [A2: set_set_option_a] :
( ( ( comple4728569501423616811tion_a @ A2 )
= top_top_set_option_a )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = top_top_set_option_a ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_1028_Inf__top__conv_I2_J,axiom,
! [A2: set_set_option_a] :
( ( top_top_set_option_a
= ( comple4728569501423616811tion_a @ A2 ) )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( X3 = top_top_set_option_a ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_1029_Inf__UNIV,axiom,
( ( comple4728569501423616811tion_a @ top_to7232205438743699540tion_a )
= bot_bot_set_option_a ) ).
% Inf_UNIV
thf(fact_1030_Inf__empty,axiom,
( ( comple4728569501423616811tion_a @ bot_bo651546871064636272tion_a )
= top_top_set_option_a ) ).
% Inf_empty
thf(fact_1031_Inf__insert,axiom,
! [A: set_option_a,A2: set_set_option_a] :
( ( comple4728569501423616811tion_a @ ( insert_set_option_a @ A @ A2 ) )
= ( inf_inf_set_option_a @ A @ ( comple4728569501423616811tion_a @ A2 ) ) ) ).
% Inf_insert
thf(fact_1032_Inf__sup__eq__top__iff,axiom,
! [B: set_set_option_a,A: set_option_a] :
( ( ( sup_sup_set_option_a @ ( comple4728569501423616811tion_a @ B ) @ A )
= top_top_set_option_a )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ B )
=> ( ( sup_sup_set_option_a @ X3 @ A )
= top_top_set_option_a ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_1033_sup__Inf,axiom,
! [A: set_option_a,B: set_set_option_a] :
( ( sup_sup_set_option_a @ A @ ( comple4728569501423616811tion_a @ B ) )
= ( comple4728569501423616811tion_a @ ( image_4886974585486332549tion_a @ ( sup_sup_set_option_a @ A ) @ B ) ) ) ).
% sup_Inf
thf(fact_1034_cInf__eq__non__empty,axiom,
! [X6: set_set_option_a,A: set_option_a] :
( ( X6 != bot_bo651546871064636272tion_a )
=> ( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ X6 )
=> ( ord_le1955136853071979460tion_a @ A @ X2 ) )
=> ( ! [Y2: set_option_a] :
( ! [X4: set_option_a] :
( ( member_set_option_a @ X4 @ X6 )
=> ( ord_le1955136853071979460tion_a @ Y2 @ X4 ) )
=> ( ord_le1955136853071979460tion_a @ Y2 @ A ) )
=> ( ( comple4728569501423616811tion_a @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1035_cInf__greatest,axiom,
! [X6: set_set_option_a,Z: set_option_a] :
( ( X6 != bot_bo651546871064636272tion_a )
=> ( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ X6 )
=> ( ord_le1955136853071979460tion_a @ Z @ X2 ) )
=> ( ord_le1955136853071979460tion_a @ Z @ ( comple4728569501423616811tion_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1036_INF__eq,axiom,
! [A2: set_option_a,B: set_option_a,G: option_a > set_option_a,F: option_a > set_option_a] :
( ! [I2: option_a] :
( ( member_option_a @ I2 @ A2 )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( G @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J: option_a] :
( ( member_option_a @ J @ B )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( F @ X4 ) @ ( G @ J ) ) ) )
=> ( ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) )
= ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1037_INF__eq,axiom,
! [A2: set_option_a,B: set_a,G: a > set_option_a,F: option_a > set_option_a] :
( ! [I2: option_a] :
( ( member_option_a @ I2 @ A2 )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( G @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( F @ X4 ) @ ( G @ J ) ) ) )
=> ( ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) )
= ( comple4728569501423616811tion_a @ ( image_a_set_option_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1038_INF__eq,axiom,
! [A2: set_a,B: set_option_a,G: option_a > set_option_a,F: a > set_option_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( G @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J: option_a] :
( ( member_option_a @ J @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( F @ X4 ) @ ( G @ J ) ) ) )
=> ( ( comple4728569501423616811tion_a @ ( image_a_set_option_a @ F @ A2 ) )
= ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1039_INF__eq,axiom,
! [A2: set_a,B: set_a,G: a > set_option_a,F: a > set_option_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A2 )
=> ? [X4: a] :
( ( member_a @ X4 @ B )
& ( ord_le1955136853071979460tion_a @ ( G @ X4 ) @ ( F @ I2 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( F @ X4 ) @ ( G @ J ) ) ) )
=> ( ( comple4728569501423616811tion_a @ ( image_a_set_option_a @ F @ A2 ) )
= ( comple4728569501423616811tion_a @ ( image_a_set_option_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_1040_cInf__eq__minimum,axiom,
! [Z: set_option_a,X6: set_set_option_a] :
( ( member_set_option_a @ Z @ X6 )
=> ( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ X6 )
=> ( ord_le1955136853071979460tion_a @ Z @ X2 ) )
=> ( ( comple4728569501423616811tion_a @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1041_cInf__le__finite,axiom,
! [X6: set_set_option_a,X: set_option_a] :
( ( finite3047895793355147469tion_a @ X6 )
=> ( ( member_set_option_a @ X @ X6 )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ X6 ) @ X ) ) ) ).
% cInf_le_finite
thf(fact_1042_Inter__anti__mono,axiom,
! [B: set_set_option_a,A2: set_set_option_a] :
( ( ord_le7761976607055303332tion_a @ B @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ ( comple4728569501423616811tion_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_1043_Inter__empty,axiom,
( ( comple4728569501423616811tion_a @ bot_bo651546871064636272tion_a )
= top_top_set_option_a ) ).
% Inter_empty
thf(fact_1044_Inter__subset,axiom,
! [A2: set_set_option_a,B: set_option_a] :
( ! [X7: set_option_a] :
( ( member_set_option_a @ X7 @ A2 )
=> ( ord_le1955136853071979460tion_a @ X7 @ B ) )
=> ( ( A2 != bot_bo651546871064636272tion_a )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ B ) ) ) ).
% Inter_subset
thf(fact_1045_Inter__UNIV,axiom,
( ( comple4728569501423616811tion_a @ top_to7232205438743699540tion_a )
= bot_bot_set_option_a ) ).
% Inter_UNIV
thf(fact_1046_Inf__greatest,axiom,
! [A2: set_set_option_a,Z: set_option_a] :
( ! [X2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ Z @ X2 ) )
=> ( ord_le1955136853071979460tion_a @ Z @ ( comple4728569501423616811tion_a @ A2 ) ) ) ).
% Inf_greatest
thf(fact_1047_le__Inf__iff,axiom,
! [B2: set_option_a,A2: set_set_option_a] :
( ( ord_le1955136853071979460tion_a @ B2 @ ( comple4728569501423616811tion_a @ A2 ) )
= ( ! [X3: set_option_a] :
( ( member_set_option_a @ X3 @ A2 )
=> ( ord_le1955136853071979460tion_a @ B2 @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_1048_Inf__lower2,axiom,
! [U: set_option_a,A2: set_set_option_a,V2: set_option_a] :
( ( member_set_option_a @ U @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ U @ V2 )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ V2 ) ) ) ).
% Inf_lower2
thf(fact_1049_Inf__lower,axiom,
! [X: set_option_a,A2: set_set_option_a] :
( ( member_set_option_a @ X @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ X ) ) ).
% Inf_lower
thf(fact_1050_Inf__mono,axiom,
! [B: set_set_option_a,A2: set_set_option_a] :
( ! [B5: set_option_a] :
( ( member_set_option_a @ B5 @ B )
=> ? [X4: set_option_a] :
( ( member_set_option_a @ X4 @ A2 )
& ( ord_le1955136853071979460tion_a @ X4 @ B5 ) ) )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ ( comple4728569501423616811tion_a @ B ) ) ) ).
% Inf_mono
thf(fact_1051_Inf__eqI,axiom,
! [A2: set_set_option_a,X: set_option_a] :
( ! [I2: set_option_a] :
( ( member_set_option_a @ I2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ X @ I2 ) )
=> ( ! [Y2: set_option_a] :
( ! [I4: set_option_a] :
( ( member_set_option_a @ I4 @ A2 )
=> ( ord_le1955136853071979460tion_a @ Y2 @ I4 ) )
=> ( ord_le1955136853071979460tion_a @ Y2 @ X ) )
=> ( ( comple4728569501423616811tion_a @ A2 )
= X ) ) ) ).
% Inf_eqI
thf(fact_1052_Inter__greatest,axiom,
! [A2: set_set_option_a,C2: set_option_a] :
( ! [X7: set_option_a] :
( ( member_set_option_a @ X7 @ A2 )
=> ( ord_le1955136853071979460tion_a @ C2 @ X7 ) )
=> ( ord_le1955136853071979460tion_a @ C2 @ ( comple4728569501423616811tion_a @ A2 ) ) ) ).
% Inter_greatest
thf(fact_1053_Inter__lower,axiom,
! [B: set_option_a,A2: set_set_option_a] :
( ( member_set_option_a @ B @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ B ) ) ).
% Inter_lower
thf(fact_1054_Inf__superset__mono,axiom,
! [B: set_set_option_a,A2: set_set_option_a] :
( ( ord_le7761976607055303332tion_a @ B @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ ( comple4728569501423616811tion_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_1055_Inf__less__eq,axiom,
! [A2: set_set_option_a,U: set_option_a] :
( ! [V: set_option_a] :
( ( member_set_option_a @ V @ A2 )
=> ( ord_le1955136853071979460tion_a @ V @ U ) )
=> ( ( A2 != bot_bo651546871064636272tion_a )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1056_fun__upd__comp,axiom,
! [F: option_a > option_a,G: option_a > option_a,X: option_a,Y: option_a] :
( ( comp_o3154387707078715297tion_a @ F @ ( fun_up1079276522633388797tion_a @ G @ X @ Y ) )
= ( fun_up1079276522633388797tion_a @ ( comp_o3154387707078715297tion_a @ F @ G ) @ X @ ( F @ Y ) ) ) ).
% fun_upd_comp
thf(fact_1057_INF__eq__iff,axiom,
! [I3: set_a,F: a > set_option_a,C: set_option_a] :
( ( I3 != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I3 )
=> ( ord_le1955136853071979460tion_a @ ( F @ I2 ) @ C ) )
=> ( ( ( comple4728569501423616811tion_a @ ( image_a_set_option_a @ F @ I3 ) )
= C )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1058_INF__eq__iff,axiom,
! [I3: set_option_a,F: option_a > set_option_a,C: set_option_a] :
( ( I3 != bot_bot_set_option_a )
=> ( ! [I2: option_a] :
( ( member_option_a @ I2 @ I3 )
=> ( ord_le1955136853071979460tion_a @ ( F @ I2 ) @ C ) )
=> ( ( ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ I3 ) )
= C )
= ( ! [X3: option_a] :
( ( member_option_a @ X3 @ I3 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1059_cINF__greatest,axiom,
! [A2: set_a,M3: set_option_a,F: a > set_option_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ M3 @ ( F @ X2 ) ) )
=> ( ord_le1955136853071979460tion_a @ M3 @ ( comple4728569501423616811tion_a @ ( image_a_set_option_a @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1060_cINF__greatest,axiom,
! [A2: set_option_a,M3: set_option_a,F: option_a > set_option_a] :
( ( A2 != bot_bot_set_option_a )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( ord_le1955136853071979460tion_a @ M3 @ ( F @ X2 ) ) )
=> ( ord_le1955136853071979460tion_a @ M3 @ ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1061_Inf__le__Sup,axiom,
! [A2: set_set_option_a] :
( ( A2 != bot_bo651546871064636272tion_a )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ A2 ) @ ( comple4629436989800923665tion_a @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1062_finite__Inf__in,axiom,
! [A2: set_set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( A2 != bot_bo651546871064636272tion_a )
=> ( ! [X2: set_option_a,Y2: set_option_a] :
( ( member_set_option_a @ X2 @ A2 )
=> ( ( member_set_option_a @ Y2 @ A2 )
=> ( member_set_option_a @ ( inf_inf_set_option_a @ X2 @ Y2 ) @ A2 ) ) )
=> ( member_set_option_a @ ( comple4728569501423616811tion_a @ A2 ) @ A2 ) ) ) ) ).
% finite_Inf_in
thf(fact_1063_less__eq__Inf__inter,axiom,
! [A2: set_set_option_a,B: set_set_option_a] : ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ ( comple4728569501423616811tion_a @ A2 ) @ ( comple4728569501423616811tion_a @ B ) ) @ ( comple4728569501423616811tion_a @ ( inf_in4298045558108779506tion_a @ A2 @ B ) ) ) ).
% less_eq_Inf_inter
thf(fact_1064_inf__Inf__fold__inf,axiom,
! [A2: set_set_option_a,B: set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( inf_inf_set_option_a @ ( comple4728569501423616811tion_a @ A2 ) @ B )
= ( finite8248554786851617352tion_a @ inf_inf_set_option_a @ B @ A2 ) ) ) ).
% inf_Inf_fold_inf
thf(fact_1065_Inter__Un__subset,axiom,
! [A2: set_set_option_a,B: set_set_option_a] : ( ord_le1955136853071979460tion_a @ ( sup_sup_set_option_a @ ( comple4728569501423616811tion_a @ A2 ) @ ( comple4728569501423616811tion_a @ B ) ) @ ( comple4728569501423616811tion_a @ ( inf_in4298045558108779506tion_a @ A2 @ B ) ) ) ).
% Inter_Un_subset
thf(fact_1066_INF__le__SUP,axiom,
! [A2: set_option_a,F: option_a > set_option_a] :
( ( A2 != bot_bot_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) ) @ ( comple4629436989800923665tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1067_Inf__fold__inf,axiom,
! [A2: set_set_option_a] :
( ( finite3047895793355147469tion_a @ A2 )
=> ( ( comple4728569501423616811tion_a @ A2 )
= ( finite8248554786851617352tion_a @ inf_inf_set_option_a @ top_top_set_option_a @ A2 ) ) ) ).
% Inf_fold_inf
thf(fact_1068_Inf__set__fold,axiom,
! [Xs: list_set_option_a] :
( ( comple4728569501423616811tion_a @ ( set_set_option_a2 @ Xs ) )
= ( fold_s5742101689114034059tion_a @ inf_inf_set_option_a @ Xs @ top_top_set_option_a ) ) ).
% Inf_set_fold
thf(fact_1069_fun__upd__image,axiom,
! [X: a,A2: set_a,F: a > option_a,Y: option_a] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_option_a @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A2 )
= ( insert_option_a @ Y @ ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_option_a @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A2 )
= ( image_a_option_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1070_fun__upd__image,axiom,
! [X: option_a,A2: set_option_a,F: option_a > a,Y: a] :
( ( ( member_option_a @ X @ A2 )
=> ( ( image_option_a_a @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X @ A2 )
=> ( ( image_option_a_a @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A2 )
= ( image_option_a_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1071_fun__upd__image,axiom,
! [X: option_a,A2: set_option_a,F: option_a > option_a,Y: option_a] :
( ( ( member_option_a @ X @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) @ A2 )
= ( insert_option_a @ Y @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) @ A2 )
= ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1072_inf__INF__fold__inf,axiom,
! [A2: set_option_a,B: set_option_a,F: option_a > set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( inf_inf_set_option_a @ B @ ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) ) )
= ( finite3172252742789171496tion_a @ ( comp_s1392512783494491100tion_a @ inf_inf_set_option_a @ F ) @ B @ A2 ) ) ) ).
% inf_INF_fold_inf
thf(fact_1073_INF__fold__inf,axiom,
! [A2: set_option_a,F: option_a > set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( comple4728569501423616811tion_a @ ( image_1157047679760422245tion_a @ F @ A2 ) )
= ( finite3172252742789171496tion_a @ ( comp_s1392512783494491100tion_a @ inf_inf_set_option_a @ F ) @ top_top_set_option_a @ A2 ) ) ) ).
% INF_fold_inf
thf(fact_1074_semilattice__closure__operator__def,axiom,
( semila6266762623024302478tion_a
= ( ^ [F2: option_a > option_a > option_a,Less_eq2: option_a > option_a > $o,E2: option_a] :
( ( set_cl4772385950493004755tion_a @ F2 )
& ( semila5752222179541100331tion_a @ F2 @ Less_eq2 @ E2 ) ) ) ) ).
% semilattice_closure_operator_def
thf(fact_1075_semilattice__closure__operator_Ointro,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( semila5752222179541100331tion_a @ F @ Less_eq @ E )
=> ( semila6266762623024302478tion_a @ F @ Less_eq @ E ) ) ) ).
% semilattice_closure_operator.intro
thf(fact_1076_image__map__upd,axiom,
! [X: option_a,A2: set_option_a,M3: option_a > option_a,Y: a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ M3 @ X @ ( some_a @ Y ) ) @ A2 )
= ( image_7439109396645324421tion_a @ M3 @ A2 ) ) ) ).
% image_map_upd
thf(fact_1077_image__map__upd,axiom,
! [X: a,A2: set_a,M3: a > option_a,Y: a] :
( ~ ( member_a @ X @ A2 )
=> ( ( image_a_option_a @ ( fun_upd_a_option_a @ M3 @ X @ ( some_a @ Y ) ) @ A2 )
= ( image_a_option_a @ M3 @ A2 ) ) ) ).
% image_map_upd
thf(fact_1078_finite__range__updI,axiom,
! [F: a > option_a,A: a,B2: a] :
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ F @ top_top_set_a ) )
=> ( finite1674126218327898605tion_a @ ( image_a_option_a @ ( fun_upd_a_option_a @ F @ A @ ( some_a @ B2 ) ) @ top_top_set_a ) ) ) ).
% finite_range_updI
thf(fact_1079_finite__range__updI,axiom,
! [F: option_a > option_a,A: option_a,B2: a] :
( ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) )
=> ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ A @ ( some_a @ B2 ) ) @ top_top_set_option_a ) ) ) ).
% finite_range_updI
thf(fact_1080_fun__upd__None__restrict,axiom,
! [X: a,D2: set_a,M3: a > option_a] :
( ( ( member_a @ X @ D2 )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M3 @ D2 ) @ X @ none_a )
= ( restrict_map_a_a @ M3 @ ( minus_minus_set_a @ D2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) )
& ( ~ ( member_a @ X @ D2 )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M3 @ D2 ) @ X @ none_a )
= ( restrict_map_a_a @ M3 @ D2 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_1081_fun__upd__None__restrict,axiom,
! [X: option_a,D2: set_option_a,M3: option_a > option_a] :
( ( ( member_option_a @ X @ D2 )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M3 @ D2 ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M3 @ ( minus_1574173051537231627tion_a @ D2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) )
& ( ~ ( member_option_a @ X @ D2 )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M3 @ D2 ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M3 @ D2 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_1082_restrict__out,axiom,
! [X: option_a,A2: set_option_a,M3: option_a > option_a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( restri3984065703976872170on_a_a @ M3 @ A2 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_1083_restrict__out,axiom,
! [X: a,A2: set_a,M3: a > option_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( restrict_map_a_a @ M3 @ A2 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_1084_restrict__map__to__empty,axiom,
! [M3: option_a > option_a] :
( ( restri3984065703976872170on_a_a @ M3 @ bot_bot_set_option_a )
= ( ^ [X3: option_a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_1085_restrict__map__def,axiom,
( restri3984065703976872170on_a_a
= ( ^ [M4: option_a > option_a,A3: set_option_a,X3: option_a] : ( if_option_a @ ( member_option_a @ X3 @ A3 ) @ ( M4 @ X3 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_1086_restrict__map__def,axiom,
( restrict_map_a_a
= ( ^ [M4: a > option_a,A3: set_a,X3: a] : ( if_option_a @ ( member_a @ X3 @ A3 ) @ ( M4 @ X3 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_1087_restrict__upd__same,axiom,
! [M3: option_a > option_a,X: option_a,Y: a] :
( ( restri3984065703976872170on_a_a @ ( fun_up1079276522633388797tion_a @ M3 @ X @ ( some_a @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( restri3984065703976872170on_a_a @ M3 @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ).
% restrict_upd_same
thf(fact_1088_dom__fun__upd,axiom,
! [Y: option_a,F: option_a > option_a,X: option_a] :
( ( ( Y = none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( insert_option_a @ X @ ( dom_option_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_1089_Compl__iff,axiom,
! [C: option_a,A2: set_option_a] :
( ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A2 ) )
= ( ~ ( member_option_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_1090_Compl__iff,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
= ( ~ ( member_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_1091_ComplI,axiom,
! [C: option_a,A2: set_option_a] :
( ~ ( member_option_a @ C @ A2 )
=> ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ).
% ComplI
thf(fact_1092_ComplI,axiom,
! [C: a,A2: set_a] :
( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% ComplI
thf(fact_1093_compl__le__compl__iff,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ X ) @ ( uminus6205308855922866075tion_a @ Y ) )
= ( ord_le1955136853071979460tion_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1094_Compl__anti__mono,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ B ) @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_1095_Compl__subset__Compl__iff,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ A2 ) @ ( uminus6205308855922866075tion_a @ B ) )
= ( ord_le1955136853071979460tion_a @ B @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1096_boolean__algebra_Ocompl__zero,axiom,
( ( uminus6205308855922866075tion_a @ bot_bot_set_option_a )
= top_top_set_option_a ) ).
% boolean_algebra.compl_zero
thf(fact_1097_boolean__algebra_Ocompl__one,axiom,
( ( uminus6205308855922866075tion_a @ top_top_set_option_a )
= bot_bot_set_option_a ) ).
% boolean_algebra.compl_one
thf(fact_1098_inf__compl__bot__left1,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ ( inf_inf_set_option_a @ X @ Y ) )
= bot_bot_set_option_a ) ).
% inf_compl_bot_left1
thf(fact_1099_inf__compl__bot__left2,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( inf_inf_set_option_a @ X @ ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ Y ) )
= bot_bot_set_option_a ) ).
% inf_compl_bot_left2
thf(fact_1100_inf__compl__bot__right,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( inf_inf_set_option_a @ X @ ( inf_inf_set_option_a @ Y @ ( uminus6205308855922866075tion_a @ X ) ) )
= bot_bot_set_option_a ) ).
% inf_compl_bot_right
thf(fact_1101_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ X )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1102_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ ( uminus6205308855922866075tion_a @ X ) )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1103_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ X @ ( uminus6205308855922866075tion_a @ X ) )
= top_top_set_option_a ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1104_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ X )
= top_top_set_option_a ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1105_sup__compl__top__left2,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( sup_sup_set_option_a @ X @ ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ Y ) )
= top_top_set_option_a ) ).
% sup_compl_top_left2
thf(fact_1106_sup__compl__top__left1,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ ( sup_sup_set_option_a @ X @ Y ) )
= top_top_set_option_a ) ).
% sup_compl_top_left1
thf(fact_1107_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( inf_inf_set_option_a @ X @ Y ) )
= ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ ( uminus6205308855922866075tion_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_1108_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( sup_sup_set_option_a @ X @ Y ) )
= ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ ( uminus6205308855922866075tion_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_1109_finite__compl,axiom,
! [A2: set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1674126218327898605tion_a @ ( uminus6205308855922866075tion_a @ A2 ) )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ) ).
% finite_compl
thf(fact_1110_Compl__disjoint,axiom,
! [A2: set_option_a] :
( ( inf_inf_set_option_a @ A2 @ ( uminus6205308855922866075tion_a @ A2 ) )
= bot_bot_set_option_a ) ).
% Compl_disjoint
thf(fact_1111_Compl__disjoint2,axiom,
! [A2: set_option_a] :
( ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ A2 ) @ A2 )
= bot_bot_set_option_a ) ).
% Compl_disjoint2
thf(fact_1112_dom__eq__empty__conv,axiom,
! [F: option_a > option_a] :
( ( ( dom_option_a_a @ F )
= bot_bot_set_option_a )
= ( F
= ( ^ [X3: option_a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_1113_Diff__Compl,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( minus_1574173051537231627tion_a @ A2 @ ( uminus6205308855922866075tion_a @ B ) )
= ( inf_inf_set_option_a @ A2 @ B ) ) ).
% Diff_Compl
thf(fact_1114_Compl__Diff__eq,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ A2 ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_1115_fun__upd__None__if__notin__dom,axiom,
! [K: option_a,M3: option_a > option_a] :
( ~ ( member_option_a @ K @ ( dom_option_a_a @ M3 ) )
=> ( ( fun_up1079276522633388797tion_a @ M3 @ K @ none_a )
= M3 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_1116_fun__upd__None__if__notin__dom,axiom,
! [K: a,M3: a > option_a] :
( ~ ( member_a @ K @ ( dom_a_a @ M3 ) )
=> ( ( fun_upd_a_option_a @ M3 @ K @ none_a )
= M3 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_1117_subset__Compl__singleton,axiom,
! [A2: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1118_subset__Compl__singleton,axiom,
! [A2: set_option_a,B2: option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ B2 @ bot_bot_set_option_a ) ) )
= ( ~ ( member_option_a @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_1119_finite__ran,axiom,
! [P4: option_a > option_option_a] :
( ( finite1674126218327898605tion_a @ ( dom_op4724496951392727122tion_a @ P4 ) )
=> ( finite1674126218327898605tion_a @ ( ran_op6317565877353657455tion_a @ P4 ) ) ) ).
% finite_ran
thf(fact_1120_compl__mono,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ord_le1955136853071979460tion_a @ X @ Y )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ Y ) @ ( uminus6205308855922866075tion_a @ X ) ) ) ).
% compl_mono
thf(fact_1121_compl__le__swap1,axiom,
! [Y: set_option_a,X: set_option_a] :
( ( ord_le1955136853071979460tion_a @ Y @ ( uminus6205308855922866075tion_a @ X ) )
=> ( ord_le1955136853071979460tion_a @ X @ ( uminus6205308855922866075tion_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1122_compl__le__swap2,axiom,
! [Y: set_option_a,X: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ Y ) @ X )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1123_subset__Compl__self__eq,axiom,
! [A2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ ( uminus6205308855922866075tion_a @ A2 ) )
= ( A2 = bot_bot_set_option_a ) ) ).
% subset_Compl_self_eq
thf(fact_1124_Compl__empty__eq,axiom,
( ( uminus6205308855922866075tion_a @ bot_bot_set_option_a )
= top_top_set_option_a ) ).
% Compl_empty_eq
thf(fact_1125_Compl__UNIV__eq,axiom,
( ( uminus6205308855922866075tion_a @ top_top_set_option_a )
= bot_bot_set_option_a ) ).
% Compl_UNIV_eq
thf(fact_1126_inf__cancel__left1,axiom,
! [X: set_option_a,A: set_option_a,B2: set_option_a] :
( ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ X @ A ) @ ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ B2 ) )
= bot_bot_set_option_a ) ).
% inf_cancel_left1
thf(fact_1127_inf__cancel__left2,axiom,
! [X: set_option_a,A: set_option_a,B2: set_option_a] :
( ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ A ) @ ( inf_inf_set_option_a @ X @ B2 ) )
= bot_bot_set_option_a ) ).
% inf_cancel_left2
thf(fact_1128_diff__eq,axiom,
( minus_1574173051537231627tion_a
= ( ^ [X3: set_option_a,Y3: set_option_a] : ( inf_inf_set_option_a @ X3 @ ( uminus6205308855922866075tion_a @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1129_sup__cancel__left2,axiom,
! [X: set_option_a,A: set_option_a,B2: set_option_a] :
( ( sup_sup_set_option_a @ ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ A ) @ ( sup_sup_set_option_a @ X @ B2 ) )
= top_top_set_option_a ) ).
% sup_cancel_left2
thf(fact_1130_sup__cancel__left1,axiom,
! [X: set_option_a,A: set_option_a,B2: set_option_a] :
( ( sup_sup_set_option_a @ ( sup_sup_set_option_a @ X @ A ) @ ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ B2 ) )
= top_top_set_option_a ) ).
% sup_cancel_left1
thf(fact_1131_Compl__partition,axiom,
! [A2: set_option_a] :
( ( sup_sup_set_option_a @ A2 @ ( uminus6205308855922866075tion_a @ A2 ) )
= top_top_set_option_a ) ).
% Compl_partition
thf(fact_1132_Compl__partition2,axiom,
! [A2: set_option_a] :
( ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ A2 ) @ A2 )
= top_top_set_option_a ) ).
% Compl_partition2
thf(fact_1133_Compl__eq__Diff__UNIV,axiom,
( uminus6205308855922866075tion_a
= ( minus_1574173051537231627tion_a @ top_top_set_option_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1134_Compl__Un,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ A2 ) @ ( uminus6205308855922866075tion_a @ B ) ) ) ).
% Compl_Un
thf(fact_1135_Compl__Int,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ A2 ) @ ( uminus6205308855922866075tion_a @ B ) ) ) ).
% Compl_Int
thf(fact_1136_Diff__eq,axiom,
( minus_1574173051537231627tion_a
= ( ^ [A3: set_option_a,B3: set_option_a] : ( inf_inf_set_option_a @ A3 @ ( uminus6205308855922866075tion_a @ B3 ) ) ) ) ).
% Diff_eq
thf(fact_1137_ComplD,axiom,
! [C: option_a,A2: set_option_a] :
( ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A2 ) )
=> ~ ( member_option_a @ C @ A2 ) ) ).
% ComplD
thf(fact_1138_ComplD,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
=> ~ ( member_a @ C @ A2 ) ) ).
% ComplD
thf(fact_1139_domI,axiom,
! [M3: option_a > option_a,A: option_a,B2: a] :
( ( ( M3 @ A )
= ( some_a @ B2 ) )
=> ( member_option_a @ A @ ( dom_option_a_a @ M3 ) ) ) ).
% domI
thf(fact_1140_domI,axiom,
! [M3: a > option_a,A: a,B2: a] :
( ( ( M3 @ A )
= ( some_a @ B2 ) )
=> ( member_a @ A @ ( dom_a_a @ M3 ) ) ) ).
% domI
thf(fact_1141_domD,axiom,
! [A: option_a,M3: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M3 ) )
=> ? [B5: a] :
( ( M3 @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_1142_domD,axiom,
! [A: a,M3: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M3 ) )
=> ? [B5: a] :
( ( M3 @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_1143_domIff,axiom,
! [A: option_a,M3: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M3 ) )
= ( ( M3 @ A )
!= none_a ) ) ).
% domIff
thf(fact_1144_domIff,axiom,
! [A: a,M3: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M3 ) )
= ( ( M3 @ A )
!= none_a ) ) ).
% domIff
thf(fact_1145_insert__dom,axiom,
! [F: option_a > option_a,X: option_a,Y: a] :
( ( ( F @ X )
= ( some_a @ Y ) )
=> ( ( insert_option_a @ X @ ( dom_option_a_a @ F ) )
= ( dom_option_a_a @ F ) ) ) ).
% insert_dom
thf(fact_1146_finite__map__freshness,axiom,
! [F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ F ) )
=> ( ~ ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ? [X2: option_a] :
( ( F @ X2 )
= none_a ) ) ) ).
% finite_map_freshness
thf(fact_1147_dom__minus,axiom,
! [F: option_a > option_a,X: option_a,A2: set_option_a] :
( ( ( F @ X )
= none_a )
=> ( ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ A2 ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ A2 ) ) ) ).
% dom_minus
thf(fact_1148_inf__shunt,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( inf_inf_set_option_a @ X @ Y )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ X @ ( uminus6205308855922866075tion_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1149_sup__shunt,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( sup_sup_set_option_a @ X @ Y )
= top_top_set_option_a )
= ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_1150_shunt1,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ X @ Y ) @ Z )
= ( ord_le1955136853071979460tion_a @ X @ ( sup_sup_set_option_a @ ( uminus6205308855922866075tion_a @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_1151_shunt2,axiom,
! [X: set_option_a,Y: set_option_a,Z: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ X @ ( uminus6205308855922866075tion_a @ Y ) ) @ Z )
= ( ord_le1955136853071979460tion_a @ X @ ( sup_sup_set_option_a @ Y @ Z ) ) ) ).
% shunt2
thf(fact_1152_sup__neg__inf,axiom,
! [P4: set_option_a,Q2: set_option_a,R2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ P4 @ ( sup_sup_set_option_a @ Q2 @ R2 ) )
= ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ P4 @ ( uminus6205308855922866075tion_a @ Q2 ) ) @ R2 ) ) ).
% sup_neg_inf
thf(fact_1153_surj__Compl__image__subset,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_option_a_a @ F @ A2 ) ) @ ( image_option_a_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1154_surj__Compl__image__subset,axiom,
! [F: a > option_a,A2: set_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ ( image_a_option_a @ F @ A2 ) ) @ ( image_a_option_a @ F @ ( uminus_uminus_set_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1155_surj__Compl__image__subset,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) ) @ ( image_7439109396645324421tion_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1156_disjoint__eq__subset__Compl,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ( inf_inf_set_option_a @ A2 @ B )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ A2 @ ( uminus6205308855922866075tion_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1157_Compl__insert,axiom,
! [X: option_a,A2: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ A2 ) )
= ( minus_1574173051537231627tion_a @ ( uminus6205308855922866075tion_a @ A2 ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ).
% Compl_insert
thf(fact_1158_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( inf_inf_set_option_a @ X @ Y )
= bot_bot_set_option_a )
=> ( ( ( sup_sup_set_option_a @ X @ Y )
= top_top_set_option_a )
=> ( ( uminus6205308855922866075tion_a @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1159_restrict__complement__singleton__eq,axiom,
! [F: option_a > option_a,X: option_a] :
( ( restri3984065703976872170on_a_a @ F @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( fun_up1079276522633388797tion_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_1160_ran__map__add,axiom,
! [M1: option_a > option_option_a,M22: option_a > option_option_a] :
( ( ( inf_inf_set_option_a @ ( dom_op4724496951392727122tion_a @ M1 ) @ ( dom_op4724496951392727122tion_a @ M22 ) )
= bot_bot_set_option_a )
=> ( ( ran_op6317565877353657455tion_a @ ( map_ad299661227452494261tion_a @ M1 @ M22 ) )
= ( sup_sup_set_option_a @ ( ran_op6317565877353657455tion_a @ M1 ) @ ( ran_op6317565877353657455tion_a @ M22 ) ) ) ) ).
% ran_map_add
thf(fact_1161_inj__on__insert,axiom,
! [F: a > option_a,A: a,A2: set_a] :
( ( inj_on_a_option_a @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on_a_option_a @ F @ A2 )
& ~ ( member_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1162_inj__on__insert,axiom,
! [F: option_a > option_a,A: option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ ( insert_option_a @ A @ A2 ) )
= ( ( inj_on8559383841115902449tion_a @ F @ A2 )
& ~ ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1163_inj__on__insert,axiom,
! [F: option_a > a,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ ( insert_option_a @ A @ A2 ) )
= ( ( inj_on_option_a_a @ F @ A2 )
& ~ ( member_a @ ( F @ A ) @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1164_inj__on__fun__updI,axiom,
! [F: a > option_a,A2: set_a,Y: option_a,X: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ~ ( member_option_a @ Y @ ( image_a_option_a @ F @ A2 ) )
=> ( inj_on_a_option_a @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_1165_inj__on__fun__updI,axiom,
! [F: option_a > a,A2: set_option_a,Y: a,X: option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ~ ( member_a @ Y @ ( image_option_a_a @ F @ A2 ) )
=> ( inj_on_option_a_a @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_1166_inj__compose,axiom,
! [F: option_a > option_a,G: option_a > option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( inj_on8559383841115902449tion_a @ G @ top_top_set_option_a )
=> ( inj_on8559383841115902449tion_a @ ( comp_o3154387707078715297tion_a @ F @ G ) @ top_top_set_option_a ) ) ) ).
% inj_compose
thf(fact_1167_inj__Some,axiom,
! [A2: set_a] : ( inj_on_a_option_a @ some_a @ A2 ) ).
% inj_Some
thf(fact_1168_inj__on__imageI2,axiom,
! [F6: option_a > option_a,F: option_a > option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ ( comp_o3154387707078715297tion_a @ F6 @ F ) @ A2 )
=> ( inj_on8559383841115902449tion_a @ F @ A2 ) ) ).
% inj_on_imageI2
thf(fact_1169_fun_Oinj__map,axiom,
! [F: option_a > option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( inj_on5338052465248252989tion_a @ ( comp_o3154387707078715297tion_a @ F ) @ top_to6364489741703166539tion_a ) ) ).
% fun.inj_map
thf(fact_1170_inj__on__image__Fpow,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( inj_on478642083548346935tion_a @ ( image_a_option_a @ F ) @ ( finite_Fpow_a @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_1171_inj__on__image__Fpow,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( inj_on8475384959944041899_set_a @ ( image_option_a_a @ F ) @ ( finite_Fpow_option_a @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_1172_inj__on__Un__image__eq__iff,axiom,
! [F: a > option_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_1173_inj__on__Un__image__eq__iff,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ B ) )
=> ( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_1174_comp__inj__on__iff,axiom,
! [F: option_a > option_a,A2: set_option_a,F6: option_a > option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( inj_on8559383841115902449tion_a @ F6 @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( inj_on8559383841115902449tion_a @ ( comp_o3154387707078715297tion_a @ F6 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_1175_inj__on__imageI,axiom,
! [G: option_a > option_a,F: option_a > option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ A2 )
=> ( inj_on8559383841115902449tion_a @ G @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_1176_comp__inj__on,axiom,
! [F: option_a > option_a,A2: set_option_a,G: option_a > option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( inj_on8559383841115902449tion_a @ G @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ( inj_on8559383841115902449tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_1177_inj__img__insertE,axiom,
! [F: option_a > option_a,A2: set_option_a,X: option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ~ ( member_option_a @ X @ B )
=> ( ( ( insert_option_a @ X @ B )
= ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ~ ! [X8: option_a,A7: set_option_a] :
( ~ ( member_option_a @ X8 @ A7 )
=> ( ( A2
= ( insert_option_a @ X8 @ A7 ) )
=> ( ( X
= ( F @ X8 ) )
=> ( B
!= ( image_7439109396645324421tion_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1178_inj__img__insertE,axiom,
! [F: a > option_a,A2: set_a,X: option_a,B: set_option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ~ ( member_option_a @ X @ B )
=> ( ( ( insert_option_a @ X @ B )
= ( image_a_option_a @ F @ A2 ) )
=> ~ ! [X8: a,A7: set_a] :
( ~ ( member_a @ X8 @ A7 )
=> ( ( A2
= ( insert_a @ X8 @ A7 ) )
=> ( ( X
= ( F @ X8 ) )
=> ( B
!= ( image_a_option_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1179_inj__img__insertE,axiom,
! [F: option_a > a,A2: set_option_a,X: a,B: set_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ B )
= ( image_option_a_a @ F @ A2 ) )
=> ~ ! [X8: option_a,A7: set_option_a] :
( ~ ( member_option_a @ X8 @ A7 )
=> ( ( A2
= ( insert_option_a @ X8 @ A7 ) )
=> ( ( X
= ( F @ X8 ) )
=> ( B
!= ( image_option_a_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1180_inj__img__insertE,axiom,
! [F: a > a,A2: set_a,X: a,B: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ B )
= ( image_a_a @ F @ A2 ) )
=> ~ ! [X8: a,A7: set_a] :
( ~ ( member_a @ X8 @ A7 )
=> ( ( A2
= ( insert_a @ X8 @ A7 ) )
=> ( ( X
= ( F @ X8 ) )
=> ( B
!= ( image_a_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1181_inj__image__mem__iff,axiom,
! [F: a > option_a,A: a,A2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( member_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_1182_inj__image__mem__iff,axiom,
! [F: a > a,A: a,A2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_1183_inj__image__mem__iff,axiom,
! [F: option_a > option_a,A: option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_1184_inj__image__mem__iff,axiom,
! [F: option_a > a,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( member_a @ ( F @ A ) @ ( image_option_a_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_1185_inj__image__eq__iff,axiom,
! [F: a > option_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_image_eq_iff
thf(fact_1186_inj__image__eq__iff,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ F @ B ) )
= ( A2 = B ) ) ) ).
% inj_image_eq_iff
thf(fact_1187_range__ex1__eq,axiom,
! [F: a > option_a,B2: option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( member_option_a @ B2 @ ( image_a_option_a @ F @ top_top_set_a ) )
= ( ? [X3: a] :
( ( B2
= ( F @ X3 ) )
& ! [Y3: a] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1188_range__ex1__eq,axiom,
! [F: option_a > option_a,B2: option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( member_option_a @ B2 @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) )
= ( ? [X3: option_a] :
( ( B2
= ( F @ X3 ) )
& ! [Y3: option_a] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1189_range__ex1__eq,axiom,
! [F: option_a > a,B2: a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( member_a @ B2 @ ( image_option_a_a @ F @ top_top_set_option_a ) )
= ( ? [X3: option_a] :
( ( B2
= ( F @ X3 ) )
& ! [Y3: option_a] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1190_inj__on__image,axiom,
! [F: a > option_a,A2: set_set_a] :
( ( inj_on_a_option_a @ F @ ( comple2307003609928055243_set_a @ A2 ) )
=> ( inj_on478642083548346935tion_a @ ( image_a_option_a @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_1191_inj__on__image,axiom,
! [F: option_a > a,A2: set_set_option_a] :
( ( inj_on_option_a_a @ F @ ( comple4629436989800923665tion_a @ A2 ) )
=> ( inj_on8475384959944041899_set_a @ ( image_option_a_a @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_1192_finite__imageD,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( finite_finite_a @ ( image_option_a_a @ F @ A2 ) )
=> ( ( inj_on_option_a_a @ F @ A2 )
=> ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% finite_imageD
thf(fact_1193_finite__imageD,axiom,
! [F: a > option_a,A2: set_a] :
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ F @ A2 ) )
=> ( ( inj_on_a_option_a @ F @ A2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_imageD
thf(fact_1194_finite__imageD,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% finite_imageD
thf(fact_1195_finite__image__iff,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( finite_finite_a @ ( image_option_a_a @ F @ A2 ) )
= ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% finite_image_iff
thf(fact_1196_finite__image__iff,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( finite1674126218327898605tion_a @ ( image_a_option_a @ F @ A2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_image_iff
thf(fact_1197_finite__image__iff,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( finite1674126218327898605tion_a @ A2 ) ) ) ).
% finite_image_iff
thf(fact_1198_inj__on__image__eq__iff,axiom,
! [F: a > option_a,C2: set_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ F @ B ) )
= ( A2 = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1199_inj__on__image__eq__iff,axiom,
! [F: option_a > a,C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ F @ B ) )
= ( A2 = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1200_inj__on__image__mem__iff,axiom,
! [F: a > option_a,B: set_a,A: a,A2: set_a] :
( ( inj_on_a_option_a @ F @ B )
=> ( ( member_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1201_inj__on__image__mem__iff,axiom,
! [F: a > a,B: set_a,A: a,A2: set_a] :
( ( inj_on_a_a @ F @ B )
=> ( ( member_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1202_inj__on__image__mem__iff,axiom,
! [F: option_a > option_a,B: set_option_a,A: option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ B )
=> ( ( member_option_a @ A @ B )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1203_inj__on__image__mem__iff,axiom,
! [F: option_a > a,B: set_option_a,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ B )
=> ( ( member_option_a @ A @ B )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( ( member_a @ ( F @ A ) @ ( image_option_a_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1204_subset__image__inj,axiom,
! [S2: set_a,F: option_a > a,T4: set_option_a] :
( ( ord_less_eq_set_a @ S2 @ ( image_option_a_a @ F @ T4 ) )
= ( ? [U2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ U2 @ T4 )
& ( inj_on_option_a_a @ F @ U2 )
& ( S2
= ( image_option_a_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1205_subset__image__inj,axiom,
! [S2: set_option_a,F: a > option_a,T4: set_a] :
( ( ord_le1955136853071979460tion_a @ S2 @ ( image_a_option_a @ F @ T4 ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T4 )
& ( inj_on_a_option_a @ F @ U2 )
& ( S2
= ( image_a_option_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1206_subset__image__inj,axiom,
! [S2: set_option_a,F: option_a > option_a,T4: set_option_a] :
( ( ord_le1955136853071979460tion_a @ S2 @ ( image_7439109396645324421tion_a @ F @ T4 ) )
= ( ? [U2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ U2 @ T4 )
& ( inj_on8559383841115902449tion_a @ F @ U2 )
& ( S2
= ( image_7439109396645324421tion_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1207_finite__UNIV__inj__surj,axiom,
! [F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1208_finite__UNIV__surj__inj,axiom,
! [F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1209_inj__image__subset__iff,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) )
= ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1210_inj__image__subset__iff,axiom,
! [F: a > option_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1211_inj__image__subset__iff,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) )
= ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1212_inj__on__iff__surj,axiom,
! [A2: set_option_a,A8: set_a] :
( ( A2 != bot_bot_set_option_a )
=> ( ( ? [F2: option_a > a] :
( ( inj_on_option_a_a @ F2 @ A2 )
& ( ord_less_eq_set_a @ ( image_option_a_a @ F2 @ A2 ) @ A8 ) ) )
= ( ? [G2: a > option_a] :
( ( image_a_option_a @ G2 @ A8 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1213_inj__on__iff__surj,axiom,
! [A2: set_a,A8: set_option_a] :
( ( A2 != bot_bot_set_a )
=> ( ( ? [F2: a > option_a] :
( ( inj_on_a_option_a @ F2 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F2 @ A2 ) @ A8 ) ) )
= ( ? [G2: option_a > a] :
( ( image_option_a_a @ G2 @ A8 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1214_inj__on__iff__surj,axiom,
! [A2: set_option_a,A8: set_option_a] :
( ( A2 != bot_bot_set_option_a )
=> ( ( ? [F2: option_a > option_a] :
( ( inj_on8559383841115902449tion_a @ F2 @ A2 )
& ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F2 @ A2 ) @ A8 ) ) )
= ( ? [G2: option_a > option_a] :
( ( image_7439109396645324421tion_a @ G2 @ A8 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1215_finite__surj__inj,axiom,
! [A2: set_option_a,F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ( inj_on8559383841115902449tion_a @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1216_inj__on__finite,axiom,
! [F: option_a > a,A2: set_option_a,B: set_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ B )
=> ( ( finite_finite_a @ B )
=> ( finite1674126218327898605tion_a @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1217_inj__on__finite,axiom,
! [F: a > option_a,A2: set_a,B: set_option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ B )
=> ( ( finite1674126218327898605tion_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1218_inj__on__finite,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ B )
=> ( ( finite1674126218327898605tion_a @ B )
=> ( finite1674126218327898605tion_a @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1219_endo__inj__surj,axiom,
! [A2: set_option_a,F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ A2 )
=> ( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( image_7439109396645324421tion_a @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1220_image__Int,axiom,
! [F: a > option_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( image_a_option_a @ F @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_option_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) ) ) ).
% image_Int
thf(fact_1221_image__Int,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( image_option_a_a @ F @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( inf_inf_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) ) ) ).
% image_Int
thf(fact_1222_image__Int,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ F @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( inf_inf_set_option_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ) ).
% image_Int
thf(fact_1223_image__set__diff,axiom,
! [F: a > option_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_1574173051537231627tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_1224_image__set__diff,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_1225_image__set__diff,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( minus_1574173051537231627tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ) ).
% image_set_diff
thf(fact_1226_inj__on__image__Int,axiom,
! [F: a > option_a,C2: set_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( image_a_option_a @ F @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_option_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1227_inj__on__image__Int,axiom,
! [F: option_a > a,C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ( image_option_a_a @ F @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( inf_inf_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1228_inj__on__image__Int,axiom,
! [F: option_a > option_a,C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ( image_7439109396645324421tion_a @ F @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( inf_inf_set_option_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1229_inj__on__image__set__diff,axiom,
! [F: a > option_a,C2: set_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_1574173051537231627tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1230_inj__on__image__set__diff,axiom,
! [F: option_a > a,C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1231_inj__on__image__set__diff,axiom,
! [F: option_a > option_a,C2: set_option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B ) @ C2 )
=> ( ( ord_le1955136853071979460tion_a @ B @ C2 )
=> ( ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) )
= ( minus_1574173051537231627tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1232_fold__image,axiom,
! [G: option_a > a,A2: set_option_a,F: a > option_a > option_a,Z: option_a] :
( ( inj_on_option_a_a @ G @ A2 )
=> ( ( finite6501707464432451470tion_a @ F @ Z @ ( image_option_a_a @ G @ A2 ) )
= ( finite1758689504953034312tion_a @ ( comp_a700500124998732034tion_a @ F @ G ) @ Z @ A2 ) ) ) ).
% fold_image
thf(fact_1233_fold__image,axiom,
! [G: a > option_a,A2: set_a,F: option_a > option_a > option_a,Z: option_a] :
( ( inj_on_a_option_a @ G @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( image_a_option_a @ G @ A2 ) )
= ( finite6501707464432451470tion_a @ ( comp_o844531392756860022on_a_a @ F @ G ) @ Z @ A2 ) ) ) ).
% fold_image
thf(fact_1234_fold__image,axiom,
! [G: option_a > option_a,A2: set_option_a,F: option_a > option_a > option_a,Z: option_a] :
( ( inj_on8559383841115902449tion_a @ G @ A2 )
=> ( ( finite1758689504953034312tion_a @ F @ Z @ ( image_7439109396645324421tion_a @ G @ A2 ) )
= ( finite1758689504953034312tion_a @ ( comp_o4936981127514773180tion_a @ F @ G ) @ Z @ A2 ) ) ) ).
% fold_image
thf(fact_1235_fold__graph__image,axiom,
! [G: option_a > a,A2: set_option_a,F: a > option_a > option_a,Z: option_a] :
( ( inj_on_option_a_a @ G @ A2 )
=> ( ( finite2491644969578888716tion_a @ F @ Z @ ( image_option_a_a @ G @ A2 ) )
= ( finite8508940333850120134tion_a @ ( comp_a700500124998732034tion_a @ F @ G ) @ Z @ A2 ) ) ) ).
% fold_graph_image
thf(fact_1236_fold__graph__image,axiom,
! [G: a > option_a,A2: set_a,F: option_a > option_a > option_a,Z: option_a] :
( ( inj_on_a_option_a @ G @ A2 )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ ( image_a_option_a @ G @ A2 ) )
= ( finite2491644969578888716tion_a @ ( comp_o844531392756860022on_a_a @ F @ G ) @ Z @ A2 ) ) ) ).
% fold_graph_image
thf(fact_1237_fold__graph__image,axiom,
! [G: option_a > option_a,A2: set_option_a,F: option_a > option_a > option_a,Z: option_a] :
( ( inj_on8559383841115902449tion_a @ G @ A2 )
=> ( ( finite8508940333850120134tion_a @ F @ Z @ ( image_7439109396645324421tion_a @ G @ A2 ) )
= ( finite8508940333850120134tion_a @ ( comp_o4936981127514773180tion_a @ F @ G ) @ Z @ A2 ) ) ) ).
% fold_graph_image
thf(fact_1238_map__add__upd__left,axiom,
! [M3: option_a,E22: option_a > option_a,E1: option_a > option_a,U1: a] :
( ~ ( member_option_a @ M3 @ ( dom_option_a_a @ E22 ) )
=> ( ( map_add_option_a_a @ ( fun_up1079276522633388797tion_a @ E1 @ M3 @ ( some_a @ U1 ) ) @ E22 )
= ( fun_up1079276522633388797tion_a @ ( map_add_option_a_a @ E1 @ E22 ) @ M3 @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_1239_map__add__upd__left,axiom,
! [M3: a,E22: a > option_a,E1: a > option_a,U1: a] :
( ~ ( member_a @ M3 @ ( dom_a_a @ E22 ) )
=> ( ( map_add_a_a @ ( fun_upd_a_option_a @ E1 @ M3 @ ( some_a @ U1 ) ) @ E22 )
= ( fun_upd_a_option_a @ ( map_add_a_a @ E1 @ E22 ) @ M3 @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_1240_inj__image__Compl__subset,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ord_less_eq_set_a @ ( image_option_a_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) @ ( uminus_uminus_set_a @ ( image_option_a_a @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1241_inj__image__Compl__subset,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ ( uminus_uminus_set_a @ A2 ) ) @ ( uminus6205308855922866075tion_a @ ( image_a_option_a @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1242_inj__image__Compl__subset,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) @ ( uminus6205308855922866075tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1243_inj__on__Un,axiom,
! [F: a > option_a,A2: set_a,B: set_a] :
( ( inj_on_a_option_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( inj_on_a_option_a @ F @ A2 )
& ( inj_on_a_option_a @ F @ B )
& ( ( inf_inf_set_option_a @ ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B ) ) @ ( image_a_option_a @ F @ ( minus_minus_set_a @ B @ A2 ) ) )
= bot_bot_set_option_a ) ) ) ).
% inj_on_Un
thf(fact_1244_inj__on__Un,axiom,
! [F: option_a > a,A2: set_option_a,B: set_option_a] :
( ( inj_on_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( ( inj_on_option_a_a @ F @ A2 )
& ( inj_on_option_a_a @ F @ B )
& ( ( inf_inf_set_a @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ B @ A2 ) ) )
= bot_bot_set_a ) ) ) ).
% inj_on_Un
thf(fact_1245_inj__on__Un,axiom,
! [F: option_a > option_a,A2: set_option_a,B: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ ( sup_sup_set_option_a @ A2 @ B ) )
= ( ( inj_on8559383841115902449tion_a @ F @ A2 )
& ( inj_on8559383841115902449tion_a @ F @ B )
& ( ( inf_inf_set_option_a @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ B @ A2 ) ) )
= bot_bot_set_option_a ) ) ) ).
% inj_on_Un
thf(fact_1246_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea3266301835125955738tion_a @ inf_inf_set_option_a @ sup_sup_set_option_a @ uminus6205308855922866075tion_a @ bot_bot_set_option_a @ top_top_set_option_a ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1247_semilattice__closure__operator_Osound__fold,axiom,
! [F: a > a > a,Less_eq: a > a > $o,E: a,Ss: list_a,S2: set_a] :
( ( semila7314208447189752392ator_a @ F @ Less_eq @ E )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ss ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ( ( Ss != nil_a )
=> ( member_a @ ( fold_a_a @ F @ Ss @ E ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.sound_fold
thf(fact_1248_semilattice__closure__operator_Osound__fold,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,Ss: list_option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Ss ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ( ( Ss != nil_option_a )
=> ( member_option_a @ ( fold_o675018875565802379tion_a @ F @ Ss @ E ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.sound_fold
thf(fact_1249_set__empty,axiom,
! [Xs: list_option_a] :
( ( ( set_option_a2 @ Xs )
= bot_bot_set_option_a )
= ( Xs = nil_option_a ) ) ).
% set_empty
thf(fact_1250_set__empty2,axiom,
! [Xs: list_option_a] :
( ( bot_bot_set_option_a
= ( set_option_a2 @ Xs ) )
= ( Xs = nil_option_a ) ) ).
% set_empty2
thf(fact_1251_infinite__iff__countable__subset,axiom,
! [S2: set_option_a] :
( ( ~ ( finite1674126218327898605tion_a @ S2 ) )
= ( ? [F2: nat > option_a] :
( ( inj_on_nat_option_a @ F2 @ top_top_set_nat )
& ( ord_le1955136853071979460tion_a @ ( image_nat_option_a @ F2 @ top_top_set_nat ) @ S2 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1252_infinite__countable__subset,axiom,
! [S2: set_option_a] :
( ~ ( finite1674126218327898605tion_a @ S2 )
=> ? [F7: nat > option_a] :
( ( inj_on_nat_option_a @ F7 @ top_top_set_nat )
& ( ord_le1955136853071979460tion_a @ ( image_nat_option_a @ F7 @ top_top_set_nat ) @ S2 ) ) ) ).
% infinite_countable_subset
thf(fact_1253_UNIV__coset,axiom,
( top_top_set_option_a
= ( coset_option_a @ nil_option_a ) ) ).
% UNIV_coset
thf(fact_1254_empty__set,axiom,
( bot_bot_set_option_a
= ( set_option_a2 @ nil_option_a ) ) ).
% empty_set
thf(fact_1255_fold__simps_I1_J,axiom,
! [F: option_a > option_a > option_a,S: option_a] :
( ( fold_o675018875565802379tion_a @ F @ nil_option_a @ S )
= S ) ).
% fold_simps(1)
thf(fact_1256_subset__code_I3_J,axiom,
~ ( ord_le1955136853071979460tion_a @ ( coset_option_a @ nil_option_a ) @ ( set_option_a2 @ nil_option_a ) ) ).
% subset_code(3)
thf(fact_1257_semilattice__closure__operator_Osound__set__fold,axiom,
! [F: a > a > a,Less_eq: a > a > $o,E: a,Ss: list_a,S2: set_a] :
( ( semila7314208447189752392ator_a @ F @ Less_eq @ E )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ss ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ( ( Ss != nil_a )
=> ( member_a @ ( finite_fold_a_a @ F @ E @ ( set_a2 @ Ss ) ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.sound_set_fold
thf(fact_1258_semilattice__closure__operator_Osound__set__fold,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,Ss: list_option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Ss ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ( ( Ss != nil_option_a )
=> ( member_option_a @ ( finite1758689504953034312tion_a @ F @ E @ ( set_option_a2 @ Ss ) ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.sound_set_fold
thf(fact_1259_semilattice__closure__operator_Oclosure__to__cons__listE,axiom,
! [F: a > a > a,Less_eq: a > a > $o,E: a,S: a,S2: set_a] :
( ( semila7314208447189752392ator_a @ F @ Less_eq @ E )
=> ( ( member_a @ S @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ~ ! [Ss2: list_a] :
( ( Ss2 != nil_a )
=> ( ( ( fold_a_a @ F @ Ss2 @ E )
= S )
=> ~ ( ord_less_eq_set_a @ ( set_a2 @ Ss2 ) @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.closure_to_cons_listE
thf(fact_1260_semilattice__closure__operator_Oclosure__to__cons__listE,axiom,
! [F: option_a > option_a > option_a,Less_eq: option_a > option_a > $o,E: option_a,S: option_a,S2: set_option_a] :
( ( semila6266762623024302478tion_a @ F @ Less_eq @ E )
=> ( ( member_option_a @ S @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ~ ! [Ss2: list_option_a] :
( ( Ss2 != nil_option_a )
=> ( ( ( fold_o675018875565802379tion_a @ F @ Ss2 @ E )
= S )
=> ~ ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Ss2 ) @ S2 ) ) ) ) ) ).
% semilattice_closure_operator.closure_to_cons_listE
thf(fact_1261_lift__fun_Osound__fold,axiom,
! [Ss: list_option_a,S2: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Ss ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) )
=> ( ( Ss != nil_option_a )
=> ( member_option_a @ ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( tl_option_a @ Ss ) @ ( hd_option_a @ Ss ) ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ S2 ) ) ) ) ).
% lift_fun.sound_fold
thf(fact_1262_Pow__set_I1_J,axiom,
( ( pow_option_a @ ( set_option_a2 @ nil_option_a ) )
= ( insert_set_option_a @ bot_bot_set_option_a @ bot_bo651546871064636272tion_a ) ) ).
% Pow_set(1)
thf(fact_1263_Pow__iff,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( member_set_option_a @ A2 @ ( pow_option_a @ B ) )
= ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ).
% Pow_iff
thf(fact_1264_PowI,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A2 @ B )
=> ( member_set_option_a @ A2 @ ( pow_option_a @ B ) ) ) ).
% PowI
thf(fact_1265_lift__fun_Ofold__dist,axiom,
! [Xs: list_option_a,T3: option_a] :
( ( Xs != nil_option_a )
=> ( ( lift_f_total_a_a_a @ p @ f @ ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( tl_option_a @ Xs ) @ ( hd_option_a @ Xs ) ) @ T3 )
= ( fold_o675018875565802379tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ Xs @ T3 ) ) ) ).
% lift_fun.fold_dist
thf(fact_1266_Pow__UNIV,axiom,
( ( pow_option_a @ top_top_set_option_a )
= top_to7232205438743699540tion_a ) ).
% Pow_UNIV
thf(fact_1267_finite__Pow__iff,axiom,
! [A2: set_option_a] :
( ( finite3047895793355147469tion_a @ ( pow_option_a @ A2 ) )
= ( finite1674126218327898605tion_a @ A2 ) ) ).
% finite_Pow_iff
thf(fact_1268_Pow__Int__eq,axiom,
! [A2: set_option_a,B: set_option_a] :
( ( pow_option_a @ ( inf_inf_set_option_a @ A2 @ B ) )
= ( inf_in4298045558108779506tion_a @ ( pow_option_a @ A2 ) @ ( pow_option_a @ B ) ) ) ).
% Pow_Int_eq
thf(fact_1269_Pow__empty,axiom,
( ( pow_option_a @ bot_bot_set_option_a )
= ( insert_set_option_a @ bot_bot_set_option_a @ bot_bo651546871064636272tion_a ) ) ).
% Pow_empty
thf(fact_1270_Pow__singleton__iff,axiom,
! [X6: set_option_a,Y7: set_option_a] :
( ( ( pow_option_a @ X6 )
= ( insert_set_option_a @ Y7 @ bot_bo651546871064636272tion_a ) )
= ( ( X6 = bot_bot_set_option_a )
& ( Y7 = bot_bot_set_option_a ) ) ) ).
% Pow_singleton_iff
thf(fact_1271_set__closure__oprator_Osound__fold,axiom,
! [F: a > a > a,Ss: list_a,S2: set_a] :
( ( set_cl1870446787657330701ator_a @ F )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ss ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) )
=> ( ( Ss != nil_a )
=> ( member_a @ ( fold_a_a @ F @ ( tl_a @ Ss ) @ ( hd_a @ Ss ) ) @ ( set_cl2401587249382337922sure_a @ F @ S2 ) ) ) ) ) ).
% set_closure_oprator.sound_fold
thf(fact_1272_set__closure__oprator_Osound__fold,axiom,
! [F: option_a > option_a > option_a,Ss: list_option_a,S2: set_option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Ss ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) )
=> ( ( Ss != nil_option_a )
=> ( member_option_a @ ( fold_o675018875565802379tion_a @ F @ ( tl_option_a @ Ss ) @ ( hd_option_a @ Ss ) ) @ ( set_cl4093418192473505224tion_a @ F @ S2 ) ) ) ) ) ).
% set_closure_oprator.sound_fold
thf(fact_1273_closure__sound,axiom,
! [S2: set_a,F: a > a > a,Ss: list_a] :
( ! [S3: a,T: a] :
( ( member_a @ S3 @ S2 )
=> ( ( member_a @ T @ S2 )
=> ( member_a @ ( F @ S3 @ T ) @ S2 ) ) )
=> ( ! [X2: a,Y2: a] :
( ( F @ X2 @ Y2 )
= ( F @ Y2 @ X2 ) )
=> ( ! [X2: a,Y2: a,Z2: a] :
( ( F @ X2 @ ( F @ Y2 @ Z2 ) )
= ( F @ ( F @ X2 @ Y2 ) @ Z2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ss ) @ S2 )
=> ( ( Ss != nil_a )
=> ( member_a @ ( fold_a_a @ F @ ( tl_a @ Ss ) @ ( hd_a @ Ss ) ) @ S2 ) ) ) ) ) ) ).
% closure_sound
thf(fact_1274_closure__sound,axiom,
! [S2: set_option_a,F: option_a > option_a > option_a,Ss: list_option_a] :
( ! [S3: option_a,T: option_a] :
( ( member_option_a @ S3 @ S2 )
=> ( ( member_option_a @ T @ S2 )
=> ( member_option_a @ ( F @ S3 @ T ) @ S2 ) ) )
=> ( ! [X2: option_a,Y2: option_a] :
( ( F @ X2 @ Y2 )
= ( F @ Y2 @ X2 ) )
=> ( ! [X2: option_a,Y2: option_a,Z2: option_a] :
( ( F @ X2 @ ( F @ Y2 @ Z2 ) )
= ( F @ ( F @ X2 @ Y2 ) @ Z2 ) )
=> ( ( ord_le1955136853071979460tion_a @ ( set_option_a2 @ Ss ) @ S2 )
=> ( ( Ss != nil_option_a )
=> ( member_option_a @ ( fold_o675018875565802379tion_a @ F @ ( tl_option_a @ Ss ) @ ( hd_option_a @ Ss ) ) @ S2 ) ) ) ) ) ) ).
% closure_sound
thf(fact_1275_set__closure__oprator_Ofold__dist,axiom,
! [F: option_a > option_a > option_a,Xs: list_option_a,T3: option_a] :
( ( set_cl4772385950493004755tion_a @ F )
=> ( ( Xs != nil_option_a )
=> ( ( F @ ( fold_o675018875565802379tion_a @ F @ ( tl_option_a @ Xs ) @ ( hd_option_a @ Xs ) ) @ T3 )
= ( fold_o675018875565802379tion_a @ F @ Xs @ T3 ) ) ) ) ).
% set_closure_oprator.fold_dist
thf(fact_1276_Pow__bottom,axiom,
! [B: set_option_a] : ( member_set_option_a @ bot_bot_set_option_a @ ( pow_option_a @ B ) ) ).
% Pow_bottom
% Helper facts (5)
thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_T,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( if_set_option_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_T,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( if_set_option_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_option_a @ ( some_a @ s2 ) @ ( set_cl4093418192473505224tion_a @ ( lift_f_total_a_a_a @ p @ f ) @ ( image_a_option_a @ some_a @ s ) ) ).
%------------------------------------------------------------------------------