TPTP Problem File: SLH0430^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Combinable_Wands/0002_PartialHeapSA/prob_00174_005161__7246502_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1657 ( 501 unt; 372 typ; 0 def)
% Number of atoms : 3651 (1807 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 12329 ( 455 ~; 36 |; 227 &;9956 @)
% ( 0 <=>;1655 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 38 ( 37 usr)
% Number of type conns : 1845 (1845 >; 0 *; 0 +; 0 <<)
% Number of symbols : 338 ( 335 usr; 33 con; 0-4 aty)
% Number of variables : 3772 ( 147 ^;3487 !; 138 ?;3772 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:08:31.993
%------------------------------------------------------------------------------
% Could-be-implicit typings (37)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Mask__Oval_J_J,type,
list_P375024181684815772ar_val: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Mask__Oval_J_J,type,
set_Pr7730668076155855478ar_val: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Mask__Oval_J,type,
produc4494260595660908054ar_val: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__Mask__Oval_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
set_Pr2339170832469995719on_val: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
set_op9137726805559506488t_char: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Mask__Oval_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
produc5122021904355189735on_val: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
produc4953844613479565601on_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
option2736423997234022232t_char: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
set_Pr2507339680178222706t_char: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_op4508134149509766951at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc7819656566062154093et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
produc1457211279475724562t_char: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
option4927543243414619207at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Option__Ooption_It__Mask__Oval_J_J_J,type,
set_op1974636978870893421on_val: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Mask__Oval_J_J,type,
set_Pr5907909515674339164at_val: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
set_op6961666426309957030on_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_option_set_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Option__Ooption_It__Mask__Oval_J_J,type,
option_option_val: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Mask__Oval_J,type,
product_prod_nat_val: $tType ).
thf(ty_n_t__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J,type,
option_option_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Mask__Oval_J_J,type,
set_option_val: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
option_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
set_option_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Option__Ooption_It__Mask__Oval_J,type,
option_val: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Mask__Oval_J,type,
set_val: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Option__Ooption_I_Eo_J,type,
option_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Mask__Oval,type,
val: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (335)
thf(sy_c_BNF__Wellorder__Embedding_Ocompat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_We5309077942755642313at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_We2139224379931913761at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_BNF__Wellorder__Embedding_Oembed_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_We650101353357441270at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_We2525705117968701890at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc_001t__Nat__Onat,type,
bNF_We6243639482952820783uc_nat: set_Pr1261947904930325089at_nat > set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
condit1738341127787009408ow_nat: set_nat > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001t__Nat__Onat,type,
equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Equiv__Relations_Oproj_001t__Nat__Onat_001t__Nat__Onat,type,
equiv_proj_nat_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
thf(sy_c_Equiv__Relations_Oquotient_001t__Nat__Onat,type,
equiv_quotient_nat: set_nat > set_Pr1261947904930325089at_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Mask__Oval,type,
finite_finite_val: set_val > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Mask__Oval_J,type,
finite4212117559080419824on_val: set_option_val > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
finite5523153139673422903on_nat: set_option_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
finite3262071724885184827t_char: set_Pr2507339680178222706t_char > $o ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
finite5529483035118572448et_nat: ( nat > set_nat > set_nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite4178521680790401110et_nat: ( nat > set_set_nat > set_set_nat ) > set_set_nat > set_nat > set_set_nat ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be3438014552859920132et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Option__Ooption_It__Mask__Oval_J,type,
fun_upd_o_option_val: ( $o > option_val ) > $o > option_val > $o > option_val ).
thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Option__Ooption_It__Nat__Onat_J,type,
fun_upd_o_option_nat: ( $o > option_nat ) > $o > option_nat > $o > option_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Mask__Oval_001t__Option__Ooption_It__Mask__Oval_J,type,
fun_up735531404784515971on_val: ( val > option_val ) > val > option_val > val > option_val ).
thf(sy_c_Fun_Ofun__upd_001t__Mask__Oval_001t__Option__Ooption_It__Nat__Onat_J,type,
fun_up7213561917443673572on_nat: ( val > option_nat ) > val > option_nat > val > option_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_It__Mask__Oval_J,type,
fun_up810121132392276792on_val: ( nat > option_val ) > nat > option_val > nat > option_val ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
fun_up1493157387958331631on_nat: ( nat > option_nat ) > nat > option_nat > nat > option_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fun_upd_nat_set_nat: ( nat > set_nat ) > nat > set_nat > nat > set_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
fun_up8761876891205509693on_val: ( option_val > option_val ) > option_val > option_val > option_val > option_val ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Nat__Onat_J,type,
fun_up1535178098986571946on_nat: ( option_val > option_nat ) > option_val > option_nat > option_val > option_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Option__Ooption_It__Mask__Oval_J_J,type,
fun_up1682854050439270531on_val: ( option_val > option_option_val ) > option_val > option_option_val > option_val > option_option_val ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Mask__Oval_J,type,
fun_up2035529056102068968on_val: ( option_nat > option_val ) > option_nat > option_val > option_nat > option_val ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
fun_up5972625598298123583on_nat: ( option_nat > option_nat ) > option_nat > option_nat > option_nat > option_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
fun_up5735975307950539954on_val: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val > produc1457211279475724562t_char > option_val ).
thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Nat__Onat_J,type,
fun_up1015184567166388469on_nat: ( produc1457211279475724562t_char > option_nat ) > produc1457211279475724562t_char > option_nat > produc1457211279475724562t_char > option_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Nat__Onat_J_001t__Option__Ooption_It__Mask__Oval_J,type,
fun_up1611807965235150338on_val: ( set_nat > option_val ) > set_nat > option_val > set_nat > option_val ).
thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Nat__Onat_J_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
fun_up7668491525253874779et_nat: ( set_nat > option_set_nat ) > set_nat > option_set_nat > set_nat > option_set_nat ).
thf(sy_c_Fun_Oinj__on_001t__Mask__Oval_001t__Option__Ooption_It__Mask__Oval_J,type,
inj_on4074814566786472055on_val: ( val > option_val ) > set_val > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
inj_on4474484569326248358on_val: ( produc1457211279475724562t_char > option_val ) > set_Pr2507339680178222706t_char > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Ooverride__on_001t__Nat__Onat_001t__Nat__Onat,type,
override_on_nat_nat: ( nat > nat ) > ( nat > nat ) > set_nat > nat > nat ).
thf(sy_c_Fun_Ooverride__on_001t__Nat__Onat_001t__Option__Ooption_It__Mask__Oval_J,type,
overri87716735465204535on_val: ( nat > option_val ) > ( nat > option_val ) > set_nat > nat > option_val ).
thf(sy_c_Fun_Ooverride__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
overri358428573850456982et_nat: ( nat > set_nat ) > ( nat > set_nat ) > set_nat > nat > set_nat ).
thf(sy_c_Fun_Ooverride__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
overri6326584771168529523on_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > set_Pr2507339680178222706t_char > produc1457211279475724562t_char > option_val ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Mask__Oval_J,type,
minus_minus_set_val: set_val > set_val > set_val ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_It__Mask__Oval_J_J,type,
minus_3256168188512067470on_val: set_option_val > set_option_val > set_option_val ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
minus_5999362281193037231on_nat: set_option_nat > set_option_nat > set_option_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Nat__Onat,type,
hilber5277034221543178913on_nat: ( nat > nat ) > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
hilber3633877196798814958at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Option__Ooption_It__Mask__Oval_J,type,
if_option_val: $o > option_val > option_val > option_val ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
inf_in3544766959038609504t_char: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Mask__Oval_J,type,
sup_sup_set_val: set_val > set_val > set_val ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
sup_su7899656352754981958t_char: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Mask__Oval_J_J,type,
sup_su5754804325903363146ar_val: set_Pr7730668076155855478ar_val > set_Pr7730668076155855478ar_val > set_Pr7730668076155855478ar_val ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Map_Odom_001_Eo_001t__Mask__Oval,type,
dom_o_val: ( $o > option_val ) > set_o ).
thf(sy_c_Map_Odom_001_Eo_001t__Nat__Onat,type,
dom_o_nat: ( $o > option_nat ) > set_o ).
thf(sy_c_Map_Odom_001t__Mask__Oval_001t__Mask__Oval,type,
dom_val_val: ( val > option_val ) > set_val ).
thf(sy_c_Map_Odom_001t__Mask__Oval_001t__Nat__Onat,type,
dom_val_nat: ( val > option_nat ) > set_val ).
thf(sy_c_Map_Odom_001t__Nat__Onat_001t__Mask__Oval,type,
dom_nat_val: ( nat > option_val ) > set_nat ).
thf(sy_c_Map_Odom_001t__Nat__Onat_001t__Nat__Onat,type,
dom_nat_nat: ( nat > option_nat ) > set_nat ).
thf(sy_c_Map_Odom_001t__Option__Ooption_It__Mask__Oval_J_001t__Mask__Oval,type,
dom_option_val_val: ( option_val > option_val ) > set_option_val ).
thf(sy_c_Map_Odom_001t__Option__Ooption_It__Mask__Oval_J_001t__Nat__Onat,type,
dom_option_val_nat: ( option_val > option_nat ) > set_option_val ).
thf(sy_c_Map_Odom_001t__Option__Ooption_It__Nat__Onat_J_001t__Mask__Oval,type,
dom_option_nat_val: ( option_nat > option_val ) > set_option_nat ).
thf(sy_c_Map_Odom_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
dom_option_nat_nat: ( option_nat > option_nat ) > set_option_nat ).
thf(sy_c_Map_Odom_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
dom_Pr8989479155164304065ar_val: ( produc1457211279475724562t_char > option_val ) > set_Pr2507339680178222706t_char ).
thf(sy_c_Map_Odom_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Nat__Onat,type,
dom_Pr3443001725208114576ar_nat: ( produc1457211279475724562t_char > option_nat ) > set_Pr2507339680178222706t_char ).
thf(sy_c_Map_Odom_001t__Set__Oset_It__Nat__Onat_J_001t__Mask__Oval,type,
dom_set_nat_val: ( set_nat > option_val ) > set_set_nat ).
thf(sy_c_Map_Odom_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
dom_set_nat_nat: ( set_nat > option_nat ) > set_set_nat ).
thf(sy_c_Map_Ograph_001t__Nat__Onat_001t__Mask__Oval,type,
graph_nat_val: ( nat > option_val ) > set_Pr5907909515674339164at_val ).
thf(sy_c_Map_Ograph_001t__Nat__Onat_001t__Nat__Onat,type,
graph_nat_nat: ( nat > option_nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Map_Ograph_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
graph_7684138220035623358on_val: ( option_val > option_option_val ) > set_Pr2339170832469995719on_val ).
thf(sy_c_Map_Ograph_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
graph_1733029056113894637ar_val: ( produc1457211279475724562t_char > option_val ) > set_Pr7730668076155855478ar_val ).
thf(sy_c_Map_Ograph_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
graph_8728433485795580044et_nat: ( set_nat > option_set_nat ) > set_Pr5488025237498180813et_nat ).
thf(sy_c_Map_Omap__add_001_Eo_001t__Mask__Oval,type,
map_add_o_val: ( $o > option_val ) > ( $o > option_val ) > $o > option_val ).
thf(sy_c_Map_Omap__add_001_Eo_001t__Nat__Onat,type,
map_add_o_nat: ( $o > option_nat ) > ( $o > option_nat ) > $o > option_nat ).
thf(sy_c_Map_Omap__add_001t__Mask__Oval_001t__Mask__Oval,type,
map_add_val_val: ( val > option_val ) > ( val > option_val ) > val > option_val ).
thf(sy_c_Map_Omap__add_001t__Mask__Oval_001t__Nat__Onat,type,
map_add_val_nat: ( val > option_nat ) > ( val > option_nat ) > val > option_nat ).
thf(sy_c_Map_Omap__add_001t__Nat__Onat_001t__Mask__Oval,type,
map_add_nat_val: ( nat > option_val ) > ( nat > option_val ) > nat > option_val ).
thf(sy_c_Map_Omap__add_001t__Nat__Onat_001t__Nat__Onat,type,
map_add_nat_nat: ( nat > option_nat ) > ( nat > option_nat ) > nat > option_nat ).
thf(sy_c_Map_Omap__add_001t__Option__Ooption_It__Mask__Oval_J_001t__Mask__Oval,type,
map_ad2791809220819668783al_val: ( option_val > option_val ) > ( option_val > option_val ) > option_val > option_val ).
thf(sy_c_Map_Omap__add_001t__Option__Ooption_It__Nat__Onat_J_001t__Mask__Oval,type,
map_ad3338430187718117610at_val: ( option_nat > option_val ) > ( option_nat > option_val ) > option_nat > option_val ).
thf(sy_c_Map_Omap__add_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
map_ad2908576761495157028ar_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val ).
thf(sy_c_Map_Omap__add_001t__Set__Oset_It__Nat__Onat_J_001t__Mask__Oval,type,
map_add_set_nat_val: ( set_nat > option_val ) > ( set_nat > option_val ) > set_nat > option_val ).
thf(sy_c_Map_Omap__le_001t__Nat__Onat_001t__Mask__Oval,type,
map_le_nat_val: ( nat > option_val ) > ( nat > option_val ) > $o ).
thf(sy_c_Map_Omap__le_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
map_le3432596473071271956ar_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).
thf(sy_c_Map_Omap__of_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
map_of6844234013878170066ar_val: list_P375024181684815772ar_val > produc1457211279475724562t_char > option_val ).
thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Mask__Oval,type,
ran_nat_val: ( nat > option_val ) > set_val ).
thf(sy_c_Map_Oran_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
ran_Pr786515303459910110ar_val: ( produc1457211279475724562t_char > option_val ) > set_val ).
thf(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__Mask__Oval,type,
restrict_map_nat_val: ( nat > option_val ) > set_nat > nat > option_val ).
thf(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_map_nat_nat: ( nat > option_nat ) > set_nat > nat > option_nat ).
thf(sy_c_Map_Orestrict__map_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
restri7220945691392438384on_val: ( option_val > option_option_val ) > set_option_val > option_val > option_option_val ).
thf(sy_c_Map_Orestrict__map_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
restri4256959289591943583ar_val: ( produc1457211279475724562t_char > option_val ) > set_Pr2507339680178222706t_char > produc1457211279475724562t_char > option_val ).
thf(sy_c_Map_Orestrict__map_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
restri1233848060703379006et_nat: ( set_nat > option_set_nat ) > set_set_nat > set_nat > option_set_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Mask__Oval_J,type,
size_size_option_val: option_val > nat ).
thf(sy_c_Option_Obind_001t__Mask__Oval_001t__Mask__Oval,type,
bind_val_val: option_val > ( val > option_val ) > option_val ).
thf(sy_c_Option_Obind_001t__Mask__Oval_001t__Nat__Onat,type,
bind_val_nat: option_val > ( val > option_nat ) > option_nat ).
thf(sy_c_Option_Obind_001t__Nat__Onat_001t__Mask__Oval,type,
bind_nat_val: option_nat > ( nat > option_val ) > option_val ).
thf(sy_c_Option_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: option_nat > ( nat > option_nat ) > option_nat ).
thf(sy_c_Option_Ocombine__options_001t__Mask__Oval,type,
combine_options_val: ( val > val > val ) > option_val > option_val > option_val ).
thf(sy_c_Option_Ocombine__options_001t__Nat__Onat,type,
combine_options_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(sy_c_Option_Ois__none_001t__Mask__Oval,type,
is_none_val: option_val > $o ).
thf(sy_c_Option_Ois__none_001t__Nat__Onat,type,
is_none_nat: option_nat > $o ).
thf(sy_c_Option_Ooption_ONone_001_Eo,type,
none_o: option_o ).
thf(sy_c_Option_Ooption_ONone_001t__Mask__Oval,type,
none_val: option_val ).
thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
none_nat: option_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_It__Mask__Oval_J,type,
none_option_val: option_option_val ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_It__Nat__Onat_J,type,
none_option_nat: option_option_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
none_P5486107665612286067t_char: option2736423997234022232t_char ).
thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
none_set_nat: option_set_nat ).
thf(sy_c_Option_Ooption_OSome_001_Eo,type,
some_o: $o > option_o ).
thf(sy_c_Option_Ooption_OSome_001t__Mask__Oval,type,
some_val: val > option_val ).
thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
some_nat: nat > option_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_It__Mask__Oval_J,type,
some_option_val: option_val > option_option_val ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_It__Nat__Onat_J,type,
some_option_nat: option_nat > option_option_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
some_P6851457240968045559t_char: produc1457211279475724562t_char > option2736423997234022232t_char ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
some_set_nat: set_nat > option_set_nat ).
thf(sy_c_Option_Ooption_Opred__option_001_Eo,type,
pred_option_o: ( $o > $o ) > option_o > $o ).
thf(sy_c_Option_Ooption_Opred__option_001t__Mask__Oval,type,
pred_option_val: ( val > $o ) > option_val > $o ).
thf(sy_c_Option_Ooption_Opred__option_001t__Nat__Onat,type,
pred_option_nat: ( nat > $o ) > option_nat > $o ).
thf(sy_c_Option_Ooption_Opred__option_001t__Option__Ooption_It__Mask__Oval_J,type,
pred_o8905241415863828810on_val: ( option_val > $o ) > option_option_val > $o ).
thf(sy_c_Option_Ooption_Opred__option_001t__Option__Ooption_It__Nat__Onat_J,type,
pred_o8911740758570919517on_nat: ( option_nat > $o ) > option_option_nat > $o ).
thf(sy_c_Option_Ooption_Opred__option_001t__Set__Oset_It__Nat__Onat_J,type,
pred_option_set_nat: ( set_nat > $o ) > option_set_nat > $o ).
thf(sy_c_Option_Ooption_Oset__option_001_Eo,type,
set_option_o2: option_o > set_o ).
thf(sy_c_Option_Ooption_Oset__option_001t__Mask__Oval,type,
set_option_val2: option_val > set_val ).
thf(sy_c_Option_Ooption_Oset__option_001t__Nat__Onat,type,
set_option_nat2: option_nat > set_nat ).
thf(sy_c_Option_Ooption_Oset__option_001t__Option__Ooption_It__Mask__Oval_J,type,
set_op911010586042427349on_val: option_option_val > set_option_val ).
thf(sy_c_Option_Ooption_Oset__option_001t__Option__Ooption_It__Nat__Onat_J,type,
set_op3360498428384587026on_nat: option_option_nat > set_option_nat ).
thf(sy_c_Option_Ooption_Oset__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
set_op7440730737645789856t_char: option2736423997234022232t_char > set_Pr2507339680178222706t_char ).
thf(sy_c_Option_Ooption_Oset__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_op5809825150163292047at_nat: option4927543243414619207at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Option_Ooption_Oset__option_001t__Set__Oset_It__Nat__Onat_J,type,
set_option_set_nat2: option_set_nat > set_set_nat ).
thf(sy_c_Option_Ooption_Osize__option_001t__Mask__Oval,type,
size_option_val: ( val > nat ) > option_val > nat ).
thf(sy_c_Option_Ooption_Othe_001_Eo,type,
the_o: option_o > $o ).
thf(sy_c_Option_Ooption_Othe_001t__Mask__Oval,type,
the_val: option_val > val ).
thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
the_nat: option_nat > nat ).
thf(sy_c_Option_Ooption_Othe_001t__Option__Ooption_It__Mask__Oval_J,type,
the_option_val: option_option_val > option_val ).
thf(sy_c_Option_Ooption_Othe_001t__Option__Ooption_It__Nat__Onat_J,type,
the_option_nat: option_option_nat > option_nat ).
thf(sy_c_Option_Ooption_Othe_001t__Set__Oset_It__Nat__Onat_J,type,
the_set_nat: option_set_nat > set_nat ).
thf(sy_c_Option_Othese_001_Eo,type,
these_o: set_option_o > set_o ).
thf(sy_c_Option_Othese_001t__Mask__Oval,type,
these_val: set_option_val > set_val ).
thf(sy_c_Option_Othese_001t__Nat__Onat,type,
these_nat: set_option_nat > set_nat ).
thf(sy_c_Option_Othese_001t__Option__Ooption_It__Mask__Oval_J,type,
these_option_val: set_op1974636978870893421on_val > set_option_val ).
thf(sy_c_Option_Othese_001t__Option__Ooption_It__Nat__Onat_J,type,
these_option_nat: set_op6961666426309957030on_nat > set_option_nat ).
thf(sy_c_Option_Othese_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
these_8061402112538453438at_nat: set_op4508134149509766951at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Option_Othese_001t__Set__Oset_It__Nat__Onat_J,type,
these_set_nat: set_option_set_nat > set_set_nat ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Nat__Onat,type,
order_4473980167227706203on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Order__Relation_Oofilter_001t__Nat__Onat,type,
order_ofilter_nat: set_Pr1261947904930325089at_nat > set_nat > $o ).
thf(sy_c_Order__Relation_Opartial__order__on_001t__Nat__Onat,type,
order_5251275573222108571on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Nat__Onat,type,
order_4861654808422542329on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
thf(sy_c_Order__Relation_Ounder_001t__Nat__Onat,type,
order_under_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
thf(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
order_2888998067076097458on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Mask__Oval_J,type,
bot_bot_set_val: set_val ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Mask__Oval_J_J,type,
bot_bo5906790151302476563on_val: set_option_val ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
bot_bo5009843511495006442on_nat: set_option_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
bot_bo1414094081888755934t_char: set_Pr2507339680178222706t_char ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Mask__Oval_J,type,
top_top_set_val: set_val ).
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__Mask__Oval_J_J,type,
top_to853123400021307383on_val: set_option_val ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_It__Mask__Oval_J_J_J,type,
top_to6510155901028473021on_val: set_op1974636978870893421on_val ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
top_to2947398533425636438on_nat: set_op6961666426309957030on_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
top_to8912972517910612872t_char: set_op9137726805559506488t_char ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
top_to1250346684925019074t_char: set_Pr2507339680178222706t_char ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_PackageLogic_Opackage__logic__axioms_001t__Mask__Oval,type,
packag6139878888743580961ms_val: ( val > val > option_val ) > val > ( val > $o ) > $o ).
thf(sy_c_PackageLogic_Opackage__logic__axioms_001t__Nat__Onat,type,
packag4637937461133392688ms_nat: ( nat > nat > option_nat ) > nat > ( nat > $o ) > $o ).
thf(sy_c_PartialHeapSA_Ocompatible__heaps,type,
compatible_heaps: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).
thf(sy_c_PartialHeapSA_Ocompatible__options_001t__Mask__Oval,type,
compat7209469420100717861ns_val: option_val > option_val > $o ).
thf(sy_c_PartialHeapSA_Ocompatible__options_001t__Nat__Onat,type,
compat6420650667825766956ns_nat: option_nat > option_nat > $o ).
thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__Mask__Oval,type,
compat8294034345482142370el_val: produc5122021904355189735on_val > produc5122021904355189735on_val > $o ).
thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__Nat__Onat,type,
compat6933303284653735279el_nat: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ).
thf(sy_c_PartialHeapSA_Olarger__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
larger1532547405891354081ar_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Mask__Oval,type,
product_Pair_nat_val: nat > val > product_prod_nat_val ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
produc904852209897547607on_val: option_val > option_val > produc5122021904355189735on_val ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
produc5138639219294217606ar_val: produc1457211279475724562t_char > val > produc4494260595660908054ar_val ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
thf(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
produc1310100445399344235_nat_o: ( product_prod_nat_nat > $o ) > nat > nat > $o ).
thf(sy_c_Product__Type_Ocurry_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J_001_Eo,type,
produc3993299840032831053_val_o: ( produc5122021904355189735on_val > $o ) > option_val > option_val > $o ).
thf(sy_c_Product__Type_Ocurry_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
produc8481458855017059839_nat_o: ( produc7819656566062154093et_nat > $o ) > set_nat > set_nat > $o ).
thf(sy_c_Relation_OField_001t__Nat__Onat,type,
field_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(sy_c_Relation_OImage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: set_Pr1261947904930325089at_nat > set_nat > set_nat ).
thf(sy_c_Relation_Oantisym__on_001t__Nat__Onat,type,
antisym_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Relation_Orefl__on_001t__Nat__Onat,type,
refl_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Relation_Otrans__on_001t__Nat__Onat,type,
trans_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Mask__Oval,type,
collect_val: ( val > $o ) > set_val ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Mask__Oval_J,type,
collect_option_val: ( option_val > $o ) > set_option_val ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Nat__Onat_J,type,
collect_option_nat: ( option_nat > $o ) > set_option_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Odisjnt_001t__Nat__Onat,type,
disjnt_nat: set_nat > set_nat > $o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Mask__Oval,type,
image_o_val: ( $o > val ) > set_o > set_val ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Option__Ooption_It__Mask__Oval_J,type,
image_o_option_val: ( $o > option_val ) > set_o > set_option_val ).
thf(sy_c_Set_Oimage_001_Eo_001t__Option__Ooption_It__Nat__Onat_J,type,
image_o_option_nat: ( $o > option_nat ) > set_o > set_option_nat ).
thf(sy_c_Set_Oimage_001t__Mask__Oval_001_Eo,type,
image_val_o: ( val > $o ) > set_val > set_o ).
thf(sy_c_Set_Oimage_001t__Mask__Oval_001t__Mask__Oval,type,
image_val_val: ( val > val ) > set_val > set_val ).
thf(sy_c_Set_Oimage_001t__Mask__Oval_001t__Nat__Onat,type,
image_val_nat: ( val > nat ) > set_val > set_nat ).
thf(sy_c_Set_Oimage_001t__Mask__Oval_001t__Option__Ooption_It__Mask__Oval_J,type,
image_val_option_val: ( val > option_val ) > set_val > set_option_val ).
thf(sy_c_Set_Oimage_001t__Mask__Oval_001t__Option__Ooption_It__Nat__Onat_J,type,
image_val_option_nat: ( val > option_nat ) > set_val > set_option_nat ).
thf(sy_c_Set_Oimage_001t__Mask__Oval_001t__Set__Oset_It__Nat__Onat_J,type,
image_val_set_nat: ( val > set_nat ) > set_val > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Mask__Oval,type,
image_nat_val: ( nat > val ) > set_nat > set_val ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat2: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Option__Ooption_It__Mask__Oval_J,type,
image_nat_option_val: ( nat > option_val ) > set_nat > set_option_val ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
image_nat_option_nat: ( nat > option_nat ) > set_nat > set_option_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Mask__Oval_J_001t__Mask__Oval,type,
image_option_val_val: ( option_val > val ) > set_option_val > set_val ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Mask__Oval_J_001t__Nat__Onat,type,
image_option_val_nat: ( option_val > nat ) > set_option_val > set_nat ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
image_5994708315261978053on_val: ( option_val > option_val ) > set_option_val > set_option_val ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Nat__Onat_J,type,
image_1875112342138737442on_nat: ( option_val > option_nat ) > set_option_val > set_option_nat ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Option__Ooption_It__Mask__Oval_J_J,type,
image_3464815402931832843on_val: ( option_val > option_option_val ) > set_option_val > set_op1974636978870893421on_val ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Nat__Onat_J_001t__Mask__Oval,type,
image_option_nat_val: ( option_nat > val ) > set_option_nat > set_val ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
image_option_nat_nat: ( option_nat > nat ) > set_option_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Mask__Oval_J,type,
image_2375463299254234464on_val: ( option_nat > option_val ) > set_option_nat > set_option_val ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
image_2533357264035992775on_nat: ( option_nat > option_nat ) > set_option_nat > set_option_nat ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J,type,
image_5486090377411570711on_nat: ( option_nat > option_option_nat ) > set_option_nat > set_op6961666426309957030on_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Mask__Oval_J,type,
image_4269386950610408506on_val: ( produc1457211279475724562t_char > option_val ) > set_Pr2507339680178222706t_char > set_option_val ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Nat__Onat_J,type,
image_4937545321503807341on_nat: ( produc1457211279475724562t_char > option_nat ) > set_Pr2507339680178222706t_char > set_option_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
image_912670848803670091t_char: ( produc1457211279475724562t_char > option2736423997234022232t_char ) > set_Pr2507339680178222706t_char > set_op9137726805559506488t_char ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Mask__Oval,type,
image_set_nat_val: ( set_nat > val ) > set_set_nat > set_val ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Option__Ooption_It__Mask__Oval_J,type,
image_4570703338782665850on_val: ( set_nat > option_val ) > set_set_nat > set_option_val ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Mask__Oval,type,
insert_val: val > set_val > set_val ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Mask__Oval_J,type,
insert_option_val: option_val > set_option_val > set_option_val ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Nat__Onat_J,type,
insert_option_nat: option_nat > set_option_nat > set_option_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Option__Ooption_It__Mask__Oval_J_J,type,
insert7563226391266834525on_val: option_option_val > set_op1974636978870893421on_val > set_op1974636978870893421on_val ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J,type,
insert504548404241883424on_nat: option_option_nat > set_op6961666426309957030on_nat > set_op6961666426309957030on_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
insert213255421748673832t_char: option2736423997234022232t_char > set_op9137726805559506488t_char > set_op9137726805559506488t_char ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
insert6842972033726296599at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > set_op4508134149509766951at_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
insert6281397740311767046et_nat: option_set_nat > set_option_set_nat > set_option_set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
insert3372640069787401954t_char: produc1457211279475724562t_char > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Mask__Oval_J,type,
insert8107552998908906966at_val: product_prod_nat_val > set_Pr5907909515674339164at_val > set_Pr5907909515674339164at_val ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Option__Ooption_It__Mask__Oval_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
insert7026822616767984055on_val: produc5122021904355189735on_val > set_Pr2339170832469995719on_val > set_Pr2339170832469995719on_val ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Mask__Oval_J,type,
insert8754101819412505574ar_val: produc4494260595660908054ar_val > set_Pr7730668076155855478ar_val > set_Pr7730668076155855478ar_val ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
insert3810226134351308605et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Opairwise_001t__Nat__Onat,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Nat__Onat,type,
type_d6250493948777748686at_nat: ( nat > nat ) > ( nat > nat ) > set_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Mask__Oval_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
accp_P4269981079083053040on_val: ( produc5122021904355189735on_val > produc5122021904355189735on_val > $o ) > produc5122021904355189735on_val > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
accp_P8646395344606611882on_nat: ( produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ) > produc4953844613479565601on_nat > $o ).
thf(sy_c_Wellfounded_Omax__ext_001t__Nat__Onat,type,
max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Nat__Onat,type,
pred_chain_nat: set_nat > ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Mask__Oval,type,
member_val: val > set_val > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_I_Eo_J,type,
member_option_o: option_o > set_option_o > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Mask__Oval_J,type,
member_option_val: option_val > set_option_val > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
member_option_nat: option_nat > set_option_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_It__Mask__Oval_J_J,type,
member8465289789052943542on_val: option_option_val > set_op1974636978870893421on_val > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_It__Nat__Onat_J_J,type,
member6603469285749945991on_nat: option_option_nat > set_op6961666426309957030on_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
member1900516542280067841t_char: option2736423997234022232t_char > set_op9137726805559506488t_char > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
member8989860449721436141et_nat: option_set_nat > set_option_set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
member484235747325421115t_char: produc1457211279475724562t_char > set_Pr2507339680178222706t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Mask__Oval_J,type,
member4983101843562193725at_val: product_prod_nat_val > set_Pr5907909515674339164at_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Mask__Oval_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
member7947933638605243152on_val: produc5122021904355189735on_val > set_Pr2339170832469995719on_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Mask__Oval_J,type,
member8657528774786783295ar_val: produc4494260595660908054ar_val > set_Pr7730668076155855478ar_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_a,type,
a: produc1457211279475724562t_char > option_val ).
thf(sy_v_aa____,type,
aa: val ).
thf(sy_v_b,type,
b: produc1457211279475724562t_char > option_val ).
thf(sy_v_ba____,type,
ba: val ).
thf(sy_v_c,type,
c: produc1457211279475724562t_char > option_val ).
thf(sy_v_hl____,type,
hl: produc1457211279475724562t_char ).
% Relevant facts (1277)
thf(fact_0_assms_I2_J,axiom,
compatible_heaps @ a @ c ).
% assms(2)
thf(fact_1__092_060open_062b_Ahl_A_061_ASome_Aa_092_060close_062,axiom,
( ( b @ hl )
= ( some_val @ aa ) ) ).
% \<open>b hl = Some a\<close>
thf(fact_2__092_060open_062c_Ahl_A_061_ASome_Aba_092_060close_062,axiom,
( ( c @ hl )
= ( some_val @ ba ) ) ).
% \<open>c hl = Some ba\<close>
thf(fact_3_assms_I1_J,axiom,
larger1532547405891354081ar_val @ a @ b ).
% assms(1)
thf(fact_4_larger__heap__anti,axiom,
! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A @ B )
=> ( ( larger1532547405891354081ar_val @ B @ A )
=> ( A = B ) ) ) ).
% larger_heap_anti
thf(fact_5_larger__heap__refl,axiom,
! [H: produc1457211279475724562t_char > option_val] : ( larger1532547405891354081ar_val @ H @ H ) ).
% larger_heap_refl
thf(fact_6_larger__heap__trans,axiom,
! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A @ B )
=> ( ( larger1532547405891354081ar_val @ B @ C )
=> ( larger1532547405891354081ar_val @ A @ C ) ) ) ).
% larger_heap_trans
thf(fact_7_larger__heapI,axiom,
! [H: produc1457211279475724562t_char > option_val,H2: produc1457211279475724562t_char > option_val] :
( ! [Hl: produc1457211279475724562t_char,X: val] :
( ( ( H @ Hl )
= ( some_val @ X ) )
=> ( ( H2 @ Hl )
= ( some_val @ X ) ) )
=> ( larger1532547405891354081ar_val @ H2 @ H ) ) ).
% larger_heapI
thf(fact_8_larger__heap__def,axiom,
( larger1532547405891354081ar_val
= ( ^ [H3: produc1457211279475724562t_char > option_val,H4: produc1457211279475724562t_char > option_val] :
! [Hl2: produc1457211279475724562t_char,X2: val] :
( ( ( H4 @ Hl2 )
= ( some_val @ X2 ) )
=> ( ( H3 @ Hl2 )
= ( some_val @ X2 ) ) ) ) ) ).
% larger_heap_def
thf(fact_9_compatible__heapsI,axiom,
! [H: produc1457211279475724562t_char > option_val,H2: produc1457211279475724562t_char > option_val] :
( ! [Hl: produc1457211279475724562t_char,A2: val,B2: val] :
( ( ( H @ Hl )
= ( some_val @ A2 ) )
=> ( ( ( H2 @ Hl )
= ( some_val @ B2 ) )
=> ( A2 = B2 ) ) )
=> ( compatible_heaps @ H @ H2 ) ) ).
% compatible_heapsI
thf(fact_10_option_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( some_nat @ X22 )
= ( some_nat @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_11_option_Oinject,axiom,
! [X22: val,Y2: val] :
( ( ( some_val @ X22 )
= ( some_val @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_12_larger__heaps__sum__ineq,axiom,
! [A3: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > option_val,B3: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A3 @ A )
=> ( ( larger1532547405891354081ar_val @ B3 @ B )
=> ( ( compatible_heaps @ A3 @ B3 )
=> ( larger1532547405891354081ar_val @ ( map_ad2908576761495157028ar_val @ A3 @ B3 ) @ ( map_ad2908576761495157028ar_val @ A @ B ) ) ) ) ) ).
% larger_heaps_sum_ineq
thf(fact_13_compatible__heaps__def,axiom,
( compatible_heaps
= ( ^ [H4: produc1457211279475724562t_char > option_val,H3: produc1457211279475724562t_char > option_val] :
! [Hl2: produc1457211279475724562t_char] : ( compat7209469420100717861ns_val @ ( H4 @ Hl2 ) @ ( H3 @ Hl2 ) ) ) ) ).
% compatible_heaps_def
thf(fact_14_compatible__options_Oelims_I3_J,axiom,
! [X3: option_nat,Xa: option_nat] :
( ~ ( compat6420650667825766956ns_nat @ X3 @ Xa )
=> ~ ! [A2: nat] :
( ( X3
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xa
= ( some_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% compatible_options.elims(3)
thf(fact_15_compatible__options_Oelims_I3_J,axiom,
! [X3: option_val,Xa: option_val] :
( ~ ( compat7209469420100717861ns_val @ X3 @ Xa )
=> ~ ! [A2: val] :
( ( X3
= ( some_val @ A2 ) )
=> ! [B2: val] :
( ( Xa
= ( some_val @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% compatible_options.elims(3)
thf(fact_16_compatible__options_Osimps_I1_J,axiom,
! [A: nat,B: nat] :
( ( compat6420650667825766956ns_nat @ ( some_nat @ A ) @ ( some_nat @ B ) )
= ( A = B ) ) ).
% compatible_options.simps(1)
thf(fact_17_compatible__options_Osimps_I1_J,axiom,
! [A: val,B: val] :
( ( compat7209469420100717861ns_val @ ( some_val @ A ) @ ( some_val @ B ) )
= ( A = B ) ) ).
% compatible_options.simps(1)
thf(fact_18_compatible__heaps__comm,axiom,
! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
( ( compatible_heaps @ A @ B )
=> ( ( map_ad2908576761495157028ar_val @ A @ B )
= ( map_ad2908576761495157028ar_val @ B @ A ) ) ) ).
% compatible_heaps_comm
thf(fact_19_package__logic__axioms_Ointro,axiom,
! [Plus: nat > nat > option_nat,Unit: nat,Stable: nat > $o] :
( ! [A2: nat] :
( ( some_nat @ A2 )
= ( Plus @ A2 @ Unit ) )
=> ( ! [A2: nat,B2: nat,X: nat] :
( ( Stable @ A2 )
=> ( ( Stable @ B2 )
=> ( ( ( some_nat @ X )
= ( Plus @ A2 @ B2 ) )
=> ( Stable @ X ) ) ) )
=> ( ( Stable @ Unit )
=> ( packag4637937461133392688ms_nat @ Plus @ Unit @ Stable ) ) ) ) ).
% package_logic_axioms.intro
thf(fact_20_package__logic__axioms_Ointro,axiom,
! [Plus: val > val > option_val,Unit: val,Stable: val > $o] :
( ! [A2: val] :
( ( some_val @ A2 )
= ( Plus @ A2 @ Unit ) )
=> ( ! [A2: val,B2: val,X: val] :
( ( Stable @ A2 )
=> ( ( Stable @ B2 )
=> ( ( ( some_val @ X )
= ( Plus @ A2 @ B2 ) )
=> ( Stable @ X ) ) ) )
=> ( ( Stable @ Unit )
=> ( packag6139878888743580961ms_val @ Plus @ Unit @ Stable ) ) ) ) ).
% package_logic_axioms.intro
thf(fact_21_package__logic__axioms__def,axiom,
( packag4637937461133392688ms_nat
= ( ^ [Plus2: nat > nat > option_nat,Unit2: nat,Stable2: nat > $o] :
( ! [A4: nat] :
( ( some_nat @ A4 )
= ( Plus2 @ A4 @ Unit2 ) )
& ! [A4: nat,B4: nat,X2: nat] :
( ( Stable2 @ A4 )
=> ( ( Stable2 @ B4 )
=> ( ( ( some_nat @ X2 )
= ( Plus2 @ A4 @ B4 ) )
=> ( Stable2 @ X2 ) ) ) )
& ( Stable2 @ Unit2 ) ) ) ) ).
% package_logic_axioms_def
thf(fact_22_package__logic__axioms__def,axiom,
( packag6139878888743580961ms_val
= ( ^ [Plus2: val > val > option_val,Unit2: val,Stable2: val > $o] :
( ! [A4: val] :
( ( some_val @ A4 )
= ( Plus2 @ A4 @ Unit2 ) )
& ! [A4: val,B4: val,X2: val] :
( ( Stable2 @ A4 )
=> ( ( Stable2 @ B4 )
=> ( ( ( some_val @ X2 )
= ( Plus2 @ A4 @ B4 ) )
=> ( Stable2 @ X2 ) ) ) )
& ( Stable2 @ Unit2 ) ) ) ) ).
% package_logic_axioms_def
thf(fact_23_combine__options__simps_I3_J,axiom,
! [F: nat > nat > nat,A: nat,B: nat] :
( ( combine_options_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
= ( some_nat @ ( F @ A @ B ) ) ) ).
% combine_options_simps(3)
thf(fact_24_combine__options__simps_I3_J,axiom,
! [F: val > val > val,A: val,B: val] :
( ( combine_options_val @ F @ ( some_val @ A ) @ ( some_val @ B ) )
= ( some_val @ ( F @ A @ B ) ) ) ).
% combine_options_simps(3)
thf(fact_25_map__add__assoc,axiom,
! [M1: nat > option_val,M2: nat > option_val,M3: nat > option_val] :
( ( map_add_nat_val @ M1 @ ( map_add_nat_val @ M2 @ M3 ) )
= ( map_add_nat_val @ ( map_add_nat_val @ M1 @ M2 ) @ M3 ) ) ).
% map_add_assoc
thf(fact_26_map__add__assoc,axiom,
! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val,M3: produc1457211279475724562t_char > option_val] :
( ( map_ad2908576761495157028ar_val @ M1 @ ( map_ad2908576761495157028ar_val @ M2 @ M3 ) )
= ( map_ad2908576761495157028ar_val @ ( map_ad2908576761495157028ar_val @ M1 @ M2 ) @ M3 ) ) ).
% map_add_assoc
thf(fact_27_not__Some__eq,axiom,
! [X3: option_nat] :
( ( ! [Y: nat] :
( X3
!= ( some_nat @ Y ) ) )
= ( X3 = none_nat ) ) ).
% not_Some_eq
thf(fact_28_not__Some__eq,axiom,
! [X3: option_val] :
( ( ! [Y: val] :
( X3
!= ( some_val @ Y ) ) )
= ( X3 = none_val ) ) ).
% not_Some_eq
thf(fact_29_not__None__eq,axiom,
! [X3: option_nat] :
( ( X3 != none_nat )
= ( ? [Y: nat] :
( X3
= ( some_nat @ Y ) ) ) ) ).
% not_None_eq
thf(fact_30_not__None__eq,axiom,
! [X3: option_val] :
( ( X3 != none_val )
= ( ? [Y: val] :
( X3
= ( some_val @ Y ) ) ) ) ).
% not_None_eq
thf(fact_31_map__add__find__right,axiom,
! [N: nat > option_val,K: nat,Xx: val,M: nat > option_val] :
( ( ( N @ K )
= ( some_val @ Xx ) )
=> ( ( map_add_nat_val @ M @ N @ K )
= ( some_val @ Xx ) ) ) ).
% map_add_find_right
thf(fact_32_map__add__find__right,axiom,
! [N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,Xx: val,M: produc1457211279475724562t_char > option_val] :
( ( ( N @ K )
= ( some_val @ Xx ) )
=> ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
= ( some_val @ Xx ) ) ) ).
% map_add_find_right
thf(fact_33_map__add__eq__empty__iff,axiom,
! [F: nat > option_val,G: nat > option_val] :
( ( ( map_add_nat_val @ F @ G )
= ( ^ [X2: nat] : none_val ) )
= ( ( F
= ( ^ [X2: nat] : none_val ) )
& ( G
= ( ^ [X2: nat] : none_val ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_34_map__add__eq__empty__iff,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
( ( ( map_ad2908576761495157028ar_val @ F @ G )
= ( ^ [X2: produc1457211279475724562t_char] : none_val ) )
= ( ( F
= ( ^ [X2: produc1457211279475724562t_char] : none_val ) )
& ( G
= ( ^ [X2: produc1457211279475724562t_char] : none_val ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_35_map__add__None,axiom,
! [M: nat > option_val,N: nat > option_val,K: nat] :
( ( ( map_add_nat_val @ M @ N @ K )
= none_val )
= ( ( ( N @ K )
= none_val )
& ( ( M @ K )
= none_val ) ) ) ).
% map_add_None
thf(fact_36_map__add__None,axiom,
! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char] :
( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
= none_val )
= ( ( ( N @ K )
= none_val )
& ( ( M @ K )
= none_val ) ) ) ).
% map_add_None
thf(fact_37_combine__options__simps_I1_J,axiom,
! [F: nat > nat > nat,Y3: option_nat] :
( ( combine_options_nat @ F @ none_nat @ Y3 )
= Y3 ) ).
% combine_options_simps(1)
thf(fact_38_combine__options__simps_I1_J,axiom,
! [F: val > val > val,Y3: option_val] :
( ( combine_options_val @ F @ none_val @ Y3 )
= Y3 ) ).
% combine_options_simps(1)
thf(fact_39_combine__options__simps_I2_J,axiom,
! [F: nat > nat > nat,X3: option_nat] :
( ( combine_options_nat @ F @ X3 @ none_nat )
= X3 ) ).
% combine_options_simps(2)
thf(fact_40_combine__options__simps_I2_J,axiom,
! [F: val > val > val,X3: option_val] :
( ( combine_options_val @ F @ X3 @ none_val )
= X3 ) ).
% combine_options_simps(2)
thf(fact_41_map__add__SomeD,axiom,
! [M: nat > option_val,N: nat > option_val,K: nat,X3: val] :
( ( ( map_add_nat_val @ M @ N @ K )
= ( some_val @ X3 ) )
=> ( ( ( N @ K )
= ( some_val @ X3 ) )
| ( ( ( N @ K )
= none_val )
& ( ( M @ K )
= ( some_val @ X3 ) ) ) ) ) ).
% map_add_SomeD
thf(fact_42_map__add__SomeD,axiom,
! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
= ( some_val @ X3 ) )
=> ( ( ( N @ K )
= ( some_val @ X3 ) )
| ( ( ( N @ K )
= none_val )
& ( ( M @ K )
= ( some_val @ X3 ) ) ) ) ) ).
% map_add_SomeD
thf(fact_43_map__add__Some__iff,axiom,
! [M: nat > option_val,N: nat > option_val,K: nat,X3: val] :
( ( ( map_add_nat_val @ M @ N @ K )
= ( some_val @ X3 ) )
= ( ( ( N @ K )
= ( some_val @ X3 ) )
| ( ( ( N @ K )
= none_val )
& ( ( M @ K )
= ( some_val @ X3 ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_44_map__add__Some__iff,axiom,
! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
= ( some_val @ X3 ) )
= ( ( ( N @ K )
= ( some_val @ X3 ) )
| ( ( ( N @ K )
= none_val )
& ( ( M @ K )
= ( some_val @ X3 ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_45_combine__options__left__commute,axiom,
! [F: val > val > val,Y3: option_val,X3: option_val,Z: option_val] :
( ! [X: val,Y4: val] :
( ( F @ X @ Y4 )
= ( F @ Y4 @ X ) )
=> ( ! [X: val,Y4: val,Z2: val] :
( ( F @ ( F @ X @ Y4 ) @ Z2 )
= ( F @ X @ ( F @ Y4 @ Z2 ) ) )
=> ( ( combine_options_val @ F @ Y3 @ ( combine_options_val @ F @ X3 @ Z ) )
= ( combine_options_val @ F @ X3 @ ( combine_options_val @ F @ Y3 @ Z ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_46_combine__options__commute,axiom,
! [F: val > val > val,X3: option_val,Y3: option_val] :
( ! [X: val,Y4: val] :
( ( F @ X @ Y4 )
= ( F @ Y4 @ X ) )
=> ( ( combine_options_val @ F @ X3 @ Y3 )
= ( combine_options_val @ F @ Y3 @ X3 ) ) ) ).
% combine_options_commute
thf(fact_47_combine__options__assoc,axiom,
! [F: val > val > val,X3: option_val,Y3: option_val,Z: option_val] :
( ! [X: val,Y4: val,Z2: val] :
( ( F @ ( F @ X @ Y4 ) @ Z2 )
= ( F @ X @ ( F @ Y4 @ Z2 ) ) )
=> ( ( combine_options_val @ F @ ( combine_options_val @ F @ X3 @ Y3 ) @ Z )
= ( combine_options_val @ F @ X3 @ ( combine_options_val @ F @ Y3 @ Z ) ) ) ) ).
% combine_options_assoc
thf(fact_48_compatible__options_Osimps_I2_J,axiom,
! [Uv: option_nat] : ( compat6420650667825766956ns_nat @ none_nat @ Uv ) ).
% compatible_options.simps(2)
thf(fact_49_compatible__options_Osimps_I2_J,axiom,
! [Uv: option_val] : ( compat7209469420100717861ns_val @ none_val @ Uv ) ).
% compatible_options.simps(2)
thf(fact_50_compatible__options_Osimps_I3_J,axiom,
! [Uu: option_nat] : ( compat6420650667825766956ns_nat @ Uu @ none_nat ) ).
% compatible_options.simps(3)
thf(fact_51_compatible__options_Osimps_I3_J,axiom,
! [Uu: option_val] : ( compat7209469420100717861ns_val @ Uu @ none_val ) ).
% compatible_options.simps(3)
thf(fact_52_combine__options__cases,axiom,
! [X3: option_val,P: option_val > option_nat > $o,Y3: option_nat] :
( ( ( X3 = none_val )
=> ( P @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_nat )
=> ( P @ X3 @ Y3 ) )
=> ( ! [A2: val,B2: nat] :
( ( X3
= ( some_val @ A2 ) )
=> ( ( Y3
= ( some_nat @ B2 ) )
=> ( P @ X3 @ Y3 ) ) )
=> ( P @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_53_combine__options__cases,axiom,
! [X3: option_nat,P: option_nat > option_val > $o,Y3: option_val] :
( ( ( X3 = none_nat )
=> ( P @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_val )
=> ( P @ X3 @ Y3 ) )
=> ( ! [A2: nat,B2: val] :
( ( X3
= ( some_nat @ A2 ) )
=> ( ( Y3
= ( some_val @ B2 ) )
=> ( P @ X3 @ Y3 ) ) )
=> ( P @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_54_combine__options__cases,axiom,
! [X3: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
( ( ( X3 = none_nat )
=> ( P @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_nat )
=> ( P @ X3 @ Y3 ) )
=> ( ! [A2: nat,B2: nat] :
( ( X3
= ( some_nat @ A2 ) )
=> ( ( Y3
= ( some_nat @ B2 ) )
=> ( P @ X3 @ Y3 ) ) )
=> ( P @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_55_combine__options__cases,axiom,
! [X3: option_val,P: option_val > option_val > $o,Y3: option_val] :
( ( ( X3 = none_val )
=> ( P @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_val )
=> ( P @ X3 @ Y3 ) )
=> ( ! [A2: val,B2: val] :
( ( X3
= ( some_val @ A2 ) )
=> ( ( Y3
= ( some_val @ B2 ) )
=> ( P @ X3 @ Y3 ) ) )
=> ( P @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_56_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_57_split__option__all,axiom,
( ( ^ [P2: option_val > $o] :
! [X4: option_val] : ( P2 @ X4 ) )
= ( ^ [P3: option_val > $o] :
( ( P3 @ none_val )
& ! [X2: val] : ( P3 @ ( some_val @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_58_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_59_split__option__ex,axiom,
( ( ^ [P2: option_val > $o] :
? [X4: option_val] : ( P2 @ X4 ) )
= ( ^ [P3: option_val > $o] :
( ( P3 @ none_val )
| ? [X2: val] : ( P3 @ ( some_val @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_60_mem__Collect__eq,axiom,
! [A: option_val,P: option_val > $o] :
( ( member_option_val @ A @ ( collect_option_val @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: option_nat,P: option_nat > $o] :
( ( member_option_nat @ A @ ( collect_option_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: val,P: val > $o] :
( ( member_val @ A @ ( collect_val @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A5: set_option_val] :
( ( collect_option_val
@ ^ [X2: option_val] : ( member_option_val @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A5: set_option_nat] :
( ( collect_option_nat
@ ^ [X2: option_nat] : ( member_option_nat @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A5: set_val] :
( ( collect_val
@ ^ [X2: val] : ( member_val @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A5: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A5: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A5: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_72_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_73_option_Oexhaust,axiom,
! [Y3: option_nat] :
( ( Y3 != none_nat )
=> ~ ! [X23: nat] :
( Y3
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_74_option_Oexhaust,axiom,
! [Y3: option_val] :
( ( Y3 != none_val )
=> ~ ! [X23: val] :
( Y3
!= ( some_val @ X23 ) ) ) ).
% option.exhaust
thf(fact_75_option_OdiscI,axiom,
! [Option: option_nat,X22: nat] :
( ( Option
= ( some_nat @ X22 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_76_option_OdiscI,axiom,
! [Option: option_val,X22: val] :
( ( Option
= ( some_val @ X22 ) )
=> ( Option != none_val ) ) ).
% option.discI
thf(fact_77_option_Odistinct_I1_J,axiom,
! [X22: nat] :
( none_nat
!= ( some_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_78_option_Odistinct_I1_J,axiom,
! [X22: val] :
( none_val
!= ( some_val @ X22 ) ) ).
% option.distinct(1)
thf(fact_79_compatible__options_Oelims_I1_J,axiom,
! [X3: option_nat,Xa: option_nat,Y3: $o] :
( ( ( compat6420650667825766956ns_nat @ X3 @ Xa )
= Y3 )
=> ( ! [A2: nat] :
( ( X3
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xa
= ( some_nat @ B2 ) )
=> ( Y3
= ( A2 != B2 ) ) ) )
=> ( ( ( X3 = none_nat )
=> ~ Y3 )
=> ~ ( ( Xa = none_nat )
=> ~ Y3 ) ) ) ) ).
% compatible_options.elims(1)
thf(fact_80_compatible__options_Oelims_I1_J,axiom,
! [X3: option_val,Xa: option_val,Y3: $o] :
( ( ( compat7209469420100717861ns_val @ X3 @ Xa )
= Y3 )
=> ( ! [A2: val] :
( ( X3
= ( some_val @ A2 ) )
=> ! [B2: val] :
( ( Xa
= ( some_val @ B2 ) )
=> ( Y3
= ( A2 != B2 ) ) ) )
=> ( ( ( X3 = none_val )
=> ~ Y3 )
=> ~ ( ( Xa = none_val )
=> ~ Y3 ) ) ) ) ).
% compatible_options.elims(1)
thf(fact_81_compatible__options_Oelims_I2_J,axiom,
! [X3: option_nat,Xa: option_nat] :
( ( compat6420650667825766956ns_nat @ X3 @ Xa )
=> ( ! [A2: nat] :
( ( X3
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xa
= ( some_nat @ B2 ) )
=> ( A2 != B2 ) ) )
=> ( ( X3 != none_nat )
=> ( Xa = none_nat ) ) ) ) ).
% compatible_options.elims(2)
thf(fact_82_compatible__options_Oelims_I2_J,axiom,
! [X3: option_val,Xa: option_val] :
( ( compat7209469420100717861ns_val @ X3 @ Xa )
=> ( ! [A2: val] :
( ( X3
= ( some_val @ A2 ) )
=> ! [B2: val] :
( ( Xa
= ( some_val @ B2 ) )
=> ( A2 != B2 ) ) )
=> ( ( X3 != none_val )
=> ( Xa = none_val ) ) ) ) ).
% compatible_options.elims(2)
thf(fact_83_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( ( some_nat @ ( the_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_84_option_Ocollapse,axiom,
! [Option: option_val] :
( ( Option != none_val )
=> ( ( some_val @ ( the_val @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_85_bind__split,axiom,
! [P: option_val > $o,M: option_val,F: val > option_val] :
( ( P @ ( bind_val_val @ M @ F ) )
= ( ( ( M = none_val )
=> ( P @ none_val ) )
& ! [V: val] :
( ( M
= ( some_val @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_86_bind__split,axiom,
! [P: option_nat > $o,M: option_val,F: val > option_nat] :
( ( P @ ( bind_val_nat @ M @ F ) )
= ( ( ( M = none_val )
=> ( P @ none_nat ) )
& ! [V: val] :
( ( M
= ( some_val @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_87_bind__split,axiom,
! [P: option_val > $o,M: option_nat,F: nat > option_val] :
( ( P @ ( bind_nat_val @ M @ F ) )
= ( ( ( M = none_nat )
=> ( P @ none_val ) )
& ! [V: nat] :
( ( M
= ( some_nat @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_88_bind__split,axiom,
! [P: option_nat > $o,M: option_nat,F: nat > option_nat] :
( ( P @ ( bind_nat_nat @ M @ F ) )
= ( ( ( M = none_nat )
=> ( P @ none_nat ) )
& ! [V: nat] :
( ( M
= ( some_nat @ V ) )
=> ( P @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_89_bind__split__asm,axiom,
! [P: option_val > $o,M: option_val,F: val > option_val] :
( ( P @ ( bind_val_val @ M @ F ) )
= ( ~ ( ( ( M = none_val )
& ~ ( P @ none_val ) )
| ? [X2: val] :
( ( M
= ( some_val @ X2 ) )
& ~ ( P @ ( F @ X2 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_90_bind__split__asm,axiom,
! [P: option_nat > $o,M: option_val,F: val > option_nat] :
( ( P @ ( bind_val_nat @ M @ F ) )
= ( ~ ( ( ( M = none_val )
& ~ ( P @ none_nat ) )
| ? [X2: val] :
( ( M
= ( some_val @ X2 ) )
& ~ ( P @ ( F @ X2 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_91_bind__split__asm,axiom,
! [P: option_val > $o,M: option_nat,F: nat > option_val] :
( ( P @ ( bind_nat_val @ M @ F ) )
= ( ~ ( ( ( M = none_nat )
& ~ ( P @ none_val ) )
| ? [X2: nat] :
( ( M
= ( some_nat @ X2 ) )
& ~ ( P @ ( F @ X2 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_92_bind__split__asm,axiom,
! [P: option_nat > $o,M: option_nat,F: nat > option_nat] :
( ( P @ ( bind_nat_nat @ M @ F ) )
= ( ~ ( ( ( M = none_nat )
& ~ ( P @ none_nat ) )
| ? [X2: nat] :
( ( M
= ( some_nat @ X2 ) )
& ~ ( P @ ( F @ X2 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_93_map__add__upd,axiom,
! [F: nat > option_val,G: nat > option_val,X3: nat,Y3: val] :
( ( map_add_nat_val @ F @ ( fun_up810121132392276792on_val @ G @ X3 @ ( some_val @ Y3 ) ) )
= ( fun_up810121132392276792on_val @ ( map_add_nat_val @ F @ G ) @ X3 @ ( some_val @ Y3 ) ) ) ).
% map_add_upd
thf(fact_94_map__add__upd,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: val] :
( ( map_ad2908576761495157028ar_val @ F @ ( fun_up5735975307950539954on_val @ G @ X3 @ ( some_val @ Y3 ) ) )
= ( fun_up5735975307950539954on_val @ ( map_ad2908576761495157028ar_val @ F @ G ) @ X3 @ ( some_val @ Y3 ) ) ) ).
% map_add_upd
thf(fact_95_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option
= ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_96_option_Oexhaust__sel,axiom,
! [Option: option_val] :
( ( Option != none_val )
=> ( Option
= ( some_val @ ( the_val @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_97_is__none__code_I1_J,axiom,
is_none_val @ none_val ).
% is_none_code(1)
thf(fact_98_is__none__code_I1_J,axiom,
is_none_nat @ none_nat ).
% is_none_code(1)
thf(fact_99_is__none__code_I2_J,axiom,
! [X3: nat] :
~ ( is_none_nat @ ( some_nat @ X3 ) ) ).
% is_none_code(2)
thf(fact_100_is__none__code_I2_J,axiom,
! [X3: val] :
~ ( is_none_val @ ( some_val @ X3 ) ) ).
% is_none_code(2)
thf(fact_101_compatible__options_Ocases,axiom,
! [X3: produc4953844613479565601on_nat] :
( ! [A2: nat,B2: nat] :
( X3
!= ( produc5098337634421038937on_nat @ ( some_nat @ A2 ) @ ( some_nat @ B2 ) ) )
=> ( ! [Uv2: option_nat] :
( X3
!= ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) )
=> ~ ! [Uu2: option_nat] :
( X3
!= ( produc5098337634421038937on_nat @ Uu2 @ none_nat ) ) ) ) ).
% compatible_options.cases
thf(fact_102_compatible__options_Ocases,axiom,
! [X3: produc5122021904355189735on_val] :
( ! [A2: val,B2: val] :
( X3
!= ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) )
=> ( ! [Uv2: option_val] :
( X3
!= ( produc904852209897547607on_val @ none_val @ Uv2 ) )
=> ~ ! [Uu2: option_val] :
( X3
!= ( produc904852209897547607on_val @ Uu2 @ none_val ) ) ) ) ).
% compatible_options.cases
thf(fact_103_in__these__eq,axiom,
! [X3: option_val,A5: set_op1974636978870893421on_val] :
( ( member_option_val @ X3 @ ( these_option_val @ A5 ) )
= ( member8465289789052943542on_val @ ( some_option_val @ X3 ) @ A5 ) ) ).
% in_these_eq
thf(fact_104_in__these__eq,axiom,
! [X3: option_nat,A5: set_op6961666426309957030on_nat] :
( ( member_option_nat @ X3 @ ( these_option_nat @ A5 ) )
= ( member6603469285749945991on_nat @ ( some_option_nat @ X3 ) @ A5 ) ) ).
% in_these_eq
thf(fact_105_in__these__eq,axiom,
! [X3: $o,A5: set_option_o] :
( ( member_o @ X3 @ ( these_o @ A5 ) )
= ( member_option_o @ ( some_o @ X3 ) @ A5 ) ) ).
% in_these_eq
thf(fact_106_in__these__eq,axiom,
! [X3: set_nat,A5: set_option_set_nat] :
( ( member_set_nat @ X3 @ ( these_set_nat @ A5 ) )
= ( member8989860449721436141et_nat @ ( some_set_nat @ X3 ) @ A5 ) ) ).
% in_these_eq
thf(fact_107_in__these__eq,axiom,
! [X3: nat,A5: set_option_nat] :
( ( member_nat @ X3 @ ( these_nat @ A5 ) )
= ( member_option_nat @ ( some_nat @ X3 ) @ A5 ) ) ).
% in_these_eq
thf(fact_108_in__these__eq,axiom,
! [X3: val,A5: set_option_val] :
( ( member_val @ X3 @ ( these_val @ A5 ) )
= ( member_option_val @ ( some_val @ X3 ) @ A5 ) ) ).
% in_these_eq
thf(fact_109_option_Opred__inject_I2_J,axiom,
! [P: nat > $o,A: nat] :
( ( pred_option_nat @ P @ ( some_nat @ A ) )
= ( P @ A ) ) ).
% option.pred_inject(2)
thf(fact_110_option_Opred__inject_I2_J,axiom,
! [P: val > $o,A: val] :
( ( pred_option_val @ P @ ( some_val @ A ) )
= ( P @ A ) ) ).
% option.pred_inject(2)
thf(fact_111_bind__runit,axiom,
! [X3: option_nat] :
( ( bind_nat_nat @ X3 @ some_nat )
= X3 ) ).
% bind_runit
thf(fact_112_bind__runit,axiom,
! [X3: option_val] :
( ( bind_val_val @ X3 @ some_val )
= X3 ) ).
% bind_runit
thf(fact_113_bind__option__cong__code,axiom,
! [X3: option_val,Y3: option_val,F: val > option_val] :
( ( X3 = Y3 )
=> ( ( bind_val_val @ X3 @ F )
= ( bind_val_val @ Y3 @ F ) ) ) ).
% bind_option_cong_code
thf(fact_114_is__none__bind,axiom,
! [F: option_val,G: val > option_val] :
( ( is_none_val @ ( bind_val_val @ F @ G ) )
= ( ( is_none_val @ F )
| ( is_none_val @ ( G @ ( the_val @ F ) ) ) ) ) ).
% is_none_bind
thf(fact_115_is__none__bind,axiom,
! [F: option_nat,G: nat > option_val] :
( ( is_none_val @ ( bind_nat_val @ F @ G ) )
= ( ( is_none_nat @ F )
| ( is_none_val @ ( G @ ( the_nat @ F ) ) ) ) ) ).
% is_none_bind
thf(fact_116_bind__eq__None__conv,axiom,
! [A: option_val,F: val > option_val] :
( ( ( bind_val_val @ A @ F )
= none_val )
= ( ( A = none_val )
| ( ( F @ ( the_val @ A ) )
= none_val ) ) ) ).
% bind_eq_None_conv
thf(fact_117_bind__eq__None__conv,axiom,
! [A: option_nat,F: nat > option_val] :
( ( ( bind_nat_val @ A @ F )
= none_val )
= ( ( A = none_nat )
| ( ( F @ ( the_nat @ A ) )
= none_val ) ) ) ).
% bind_eq_None_conv
thf(fact_118_bind__eq__None__conv,axiom,
! [A: option_val,F: val > option_nat] :
( ( ( bind_val_nat @ A @ F )
= none_nat )
= ( ( A = none_val )
| ( ( F @ ( the_val @ A ) )
= none_nat ) ) ) ).
% bind_eq_None_conv
thf(fact_119_bind__eq__None__conv,axiom,
! [A: option_nat,F: nat > option_nat] :
( ( ( bind_nat_nat @ A @ F )
= none_nat )
= ( ( A = none_nat )
| ( ( F @ ( the_nat @ A ) )
= none_nat ) ) ) ).
% bind_eq_None_conv
thf(fact_120_map__upd__eqD1,axiom,
! [M: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char,X3: val,N: produc1457211279475724562t_char > option_val,Y3: val] :
( ( ( fun_up5735975307950539954on_val @ M @ A @ ( some_val @ X3 ) )
= ( fun_up5735975307950539954on_val @ N @ A @ ( some_val @ Y3 ) ) )
=> ( X3 = Y3 ) ) ).
% map_upd_eqD1
thf(fact_121_map__upd__eqD1,axiom,
! [M: nat > option_val,A: nat,X3: val,N: nat > option_val,Y3: val] :
( ( ( fun_up810121132392276792on_val @ M @ A @ ( some_val @ X3 ) )
= ( fun_up810121132392276792on_val @ N @ A @ ( some_val @ Y3 ) ) )
=> ( X3 = Y3 ) ) ).
% map_upd_eqD1
thf(fact_122_map__upd__triv,axiom,
! [T: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
( ( ( T @ K )
= ( some_val @ X3 ) )
=> ( ( fun_up5735975307950539954on_val @ T @ K @ ( some_val @ X3 ) )
= T ) ) ).
% map_upd_triv
thf(fact_123_map__upd__triv,axiom,
! [T: nat > option_val,K: nat,X3: val] :
( ( ( T @ K )
= ( some_val @ X3 ) )
=> ( ( fun_up810121132392276792on_val @ T @ K @ ( some_val @ X3 ) )
= T ) ) ).
% map_upd_triv
thf(fact_124_map__upd__Some__unfold,axiom,
! [M: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char,B: val,X3: produc1457211279475724562t_char,Y3: val] :
( ( ( fun_up5735975307950539954on_val @ M @ A @ ( some_val @ B ) @ X3 )
= ( some_val @ Y3 ) )
= ( ( ( X3 = A )
& ( B = Y3 ) )
| ( ( X3 != A )
& ( ( M @ X3 )
= ( some_val @ Y3 ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_125_map__upd__Some__unfold,axiom,
! [M: nat > option_val,A: nat,B: val,X3: nat,Y3: val] :
( ( ( fun_up810121132392276792on_val @ M @ A @ ( some_val @ B ) @ X3 )
= ( some_val @ Y3 ) )
= ( ( ( X3 = A )
& ( B = Y3 ) )
| ( ( X3 != A )
& ( ( M @ X3 )
= ( some_val @ Y3 ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_126_option_Osel,axiom,
! [X22: nat] :
( ( the_nat @ ( some_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_127_option_Osel,axiom,
! [X22: val] :
( ( the_val @ ( some_val @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_128_option_Oexpand,axiom,
! [Option: option_val,Option2: option_val] :
( ( ( Option = none_val )
= ( Option2 = none_val ) )
=> ( ( ( Option != none_val )
=> ( ( Option2 != none_val )
=> ( ( the_val @ Option )
= ( the_val @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_129_option_Oexpand,axiom,
! [Option: option_nat,Option2: option_nat] :
( ( ( Option = none_nat )
= ( Option2 = none_nat ) )
=> ( ( ( Option != none_nat )
=> ( ( Option2 != none_nat )
=> ( ( the_nat @ Option )
= ( the_nat @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_130_bind_Obind__lunit,axiom,
! [X3: val,F: val > option_val] :
( ( bind_val_val @ ( some_val @ X3 ) @ F )
= ( F @ X3 ) ) ).
% bind.bind_lunit
thf(fact_131_Option_Obind__cong,axiom,
! [X3: option_val,Y3: option_val,F: val > option_val,G: val > option_val] :
( ( X3 = Y3 )
=> ( ! [A2: val] :
( ( Y3
= ( some_val @ A2 ) )
=> ( ( F @ A2 )
= ( G @ A2 ) ) )
=> ( ( bind_val_val @ X3 @ F )
= ( bind_val_val @ Y3 @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_132_bind__eq__Some__conv,axiom,
! [F: option_nat,G: nat > option_val,X3: val] :
( ( ( bind_nat_val @ F @ G )
= ( some_val @ X3 ) )
= ( ? [Y: nat] :
( ( F
= ( some_nat @ Y ) )
& ( ( G @ Y )
= ( some_val @ X3 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_133_bind__eq__Some__conv,axiom,
! [F: option_val,G: val > option_nat,X3: nat] :
( ( ( bind_val_nat @ F @ G )
= ( some_nat @ X3 ) )
= ( ? [Y: val] :
( ( F
= ( some_val @ Y ) )
& ( ( G @ Y )
= ( some_nat @ X3 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_134_bind__eq__Some__conv,axiom,
! [F: option_nat,G: nat > option_nat,X3: nat] :
( ( ( bind_nat_nat @ F @ G )
= ( some_nat @ X3 ) )
= ( ? [Y: nat] :
( ( F
= ( some_nat @ Y ) )
& ( ( G @ Y )
= ( some_nat @ X3 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_135_bind__eq__Some__conv,axiom,
! [F: option_val,G: val > option_val,X3: val] :
( ( ( bind_val_val @ F @ G )
= ( some_val @ X3 ) )
= ( ? [Y: val] :
( ( F
= ( some_val @ Y ) )
& ( ( G @ Y )
= ( some_val @ X3 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_136_bind_Obind__lzero,axiom,
! [F: val > option_val] :
( ( bind_val_val @ none_val @ F )
= none_val ) ).
% bind.bind_lzero
thf(fact_137_bind_Obind__lzero,axiom,
! [F: val > option_nat] :
( ( bind_val_nat @ none_val @ F )
= none_nat ) ).
% bind.bind_lzero
thf(fact_138_bind_Obind__lzero,axiom,
! [F: nat > option_val] :
( ( bind_nat_val @ none_nat @ F )
= none_val ) ).
% bind.bind_lzero
thf(fact_139_bind_Obind__lzero,axiom,
! [F: nat > option_nat] :
( ( bind_nat_nat @ none_nat @ F )
= none_nat ) ).
% bind.bind_lzero
thf(fact_140_option_Opred__inject_I1_J,axiom,
! [P: val > $o] : ( pred_option_val @ P @ none_val ) ).
% option.pred_inject(1)
thf(fact_141_option_Opred__inject_I1_J,axiom,
! [P: nat > $o] : ( pred_option_nat @ P @ none_nat ) ).
% option.pred_inject(1)
thf(fact_142_is__none__simps_I2_J,axiom,
! [X3: nat] :
~ ( is_none_nat @ ( some_nat @ X3 ) ) ).
% is_none_simps(2)
thf(fact_143_is__none__simps_I2_J,axiom,
! [X3: val] :
~ ( is_none_val @ ( some_val @ X3 ) ) ).
% is_none_simps(2)
thf(fact_144_is__none__simps_I1_J,axiom,
is_none_val @ none_val ).
% is_none_simps(1)
thf(fact_145_is__none__simps_I1_J,axiom,
is_none_nat @ none_nat ).
% is_none_simps(1)
thf(fact_146_Option_Ois__none__def,axiom,
( is_none_val
= ( ^ [X2: option_val] : ( X2 = none_val ) ) ) ).
% Option.is_none_def
thf(fact_147_Option_Ois__none__def,axiom,
( is_none_nat
= ( ^ [X2: option_nat] : ( X2 = none_nat ) ) ) ).
% Option.is_none_def
thf(fact_148_map__upd__nonempty,axiom,
! [T: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
( ( fun_up5735975307950539954on_val @ T @ K @ ( some_val @ X3 ) )
!= ( ^ [X2: produc1457211279475724562t_char] : none_val ) ) ).
% map_upd_nonempty
thf(fact_149_map__upd__nonempty,axiom,
! [T: nat > option_val,K: nat,X3: val] :
( ( fun_up810121132392276792on_val @ T @ K @ ( some_val @ X3 ) )
!= ( ^ [X2: nat] : none_val ) ) ).
% map_upd_nonempty
thf(fact_150_fun__upd__upd,axiom,
! [F: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: option_val,Z: option_val] :
( ( fun_up5735975307950539954on_val @ ( fun_up5735975307950539954on_val @ F @ X3 @ Y3 ) @ X3 @ Z )
= ( fun_up5735975307950539954on_val @ F @ X3 @ Z ) ) ).
% fun_upd_upd
thf(fact_151_fun__upd__upd,axiom,
! [F: nat > option_val,X3: nat,Y3: option_val,Z: option_val] :
( ( fun_up810121132392276792on_val @ ( fun_up810121132392276792on_val @ F @ X3 @ Y3 ) @ X3 @ Z )
= ( fun_up810121132392276792on_val @ F @ X3 @ Z ) ) ).
% fun_upd_upd
thf(fact_152_fun__upd__upd,axiom,
! [F: nat > nat,X3: nat,Y3: nat,Z: nat] :
( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ F @ X3 @ Y3 ) @ X3 @ Z )
= ( fun_upd_nat_nat @ F @ X3 @ Z ) ) ).
% fun_upd_upd
thf(fact_153_fun__upd__upd,axiom,
! [F: nat > set_nat,X3: nat,Y3: set_nat,Z: set_nat] :
( ( fun_upd_nat_set_nat @ ( fun_upd_nat_set_nat @ F @ X3 @ Y3 ) @ X3 @ Z )
= ( fun_upd_nat_set_nat @ F @ X3 @ Z ) ) ).
% fun_upd_upd
thf(fact_154_fun__upd__triv,axiom,
! [F: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char] :
( ( fun_up5735975307950539954on_val @ F @ X3 @ ( F @ X3 ) )
= F ) ).
% fun_upd_triv
thf(fact_155_fun__upd__triv,axiom,
! [F: nat > option_val,X3: nat] :
( ( fun_up810121132392276792on_val @ F @ X3 @ ( F @ X3 ) )
= F ) ).
% fun_upd_triv
thf(fact_156_fun__upd__triv,axiom,
! [F: nat > nat,X3: nat] :
( ( fun_upd_nat_nat @ F @ X3 @ ( F @ X3 ) )
= F ) ).
% fun_upd_triv
thf(fact_157_fun__upd__triv,axiom,
! [F: nat > set_nat,X3: nat] :
( ( fun_upd_nat_set_nat @ F @ X3 @ ( F @ X3 ) )
= F ) ).
% fun_upd_triv
thf(fact_158_fun__upd__apply,axiom,
( fun_up5735975307950539954on_val
= ( ^ [F2: produc1457211279475724562t_char > option_val,X2: produc1457211279475724562t_char,Y: option_val,Z3: produc1457211279475724562t_char] : ( if_option_val @ ( Z3 = X2 ) @ Y @ ( F2 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_159_fun__upd__apply,axiom,
( fun_up810121132392276792on_val
= ( ^ [F2: nat > option_val,X2: nat,Y: option_val,Z3: nat] : ( if_option_val @ ( Z3 = X2 ) @ Y @ ( F2 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_160_fun__upd__apply,axiom,
( fun_upd_nat_nat
= ( ^ [F2: nat > nat,X2: nat,Y: nat,Z3: nat] : ( if_nat @ ( Z3 = X2 ) @ Y @ ( F2 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_161_fun__upd__apply,axiom,
( fun_upd_nat_set_nat
= ( ^ [F2: nat > set_nat,X2: nat,Y: set_nat,Z3: nat] : ( if_set_nat @ ( Z3 = X2 ) @ Y @ ( F2 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_162_prod_Oinject,axiom,
! [X1: option_val,X22: option_val,Y1: option_val,Y2: option_val] :
( ( ( produc904852209897547607on_val @ X1 @ X22 )
= ( produc904852209897547607on_val @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_163_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_164_prod_Oinject,axiom,
! [X1: set_nat,X22: set_nat,Y1: set_nat,Y2: set_nat] :
( ( ( produc4532415448927165861et_nat @ X1 @ X22 )
= ( produc4532415448927165861et_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_165_old_Oprod_Oinject,axiom,
! [A: option_val,B: option_val,A3: option_val,B3: option_val] :
( ( ( produc904852209897547607on_val @ A @ B )
= ( produc904852209897547607on_val @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_166_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_167_old_Oprod_Oinject,axiom,
! [A: set_nat,B: set_nat,A3: set_nat,B3: set_nat] :
( ( ( produc4532415448927165861et_nat @ A @ B )
= ( produc4532415448927165861et_nat @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_168_map__le__imp__upd__le,axiom,
! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: val] :
( ( map_le3432596473071271956ar_val @ M1 @ M2 )
=> ( map_le3432596473071271956ar_val @ ( fun_up5735975307950539954on_val @ M1 @ X3 @ none_val ) @ ( fun_up5735975307950539954on_val @ M2 @ X3 @ ( some_val @ Y3 ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_169_map__le__imp__upd__le,axiom,
! [M1: nat > option_val,M2: nat > option_val,X3: nat,Y3: val] :
( ( map_le_nat_val @ M1 @ M2 )
=> ( map_le_nat_val @ ( fun_up810121132392276792on_val @ M1 @ X3 @ none_val ) @ ( fun_up810121132392276792on_val @ M2 @ X3 @ ( some_val @ Y3 ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_170_fun__upd__def,axiom,
( fun_up5735975307950539954on_val
= ( ^ [F2: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char,B4: option_val,X2: produc1457211279475724562t_char] : ( if_option_val @ ( X2 = A4 ) @ B4 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_171_fun__upd__def,axiom,
( fun_up810121132392276792on_val
= ( ^ [F2: nat > option_val,A4: nat,B4: option_val,X2: nat] : ( if_option_val @ ( X2 = A4 ) @ B4 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_172_fun__upd__def,axiom,
( fun_upd_nat_nat
= ( ^ [F2: nat > nat,A4: nat,B4: nat,X2: nat] : ( if_nat @ ( X2 = A4 ) @ B4 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_173_fun__upd__def,axiom,
( fun_upd_nat_set_nat
= ( ^ [F2: nat > set_nat,A4: nat,B4: set_nat,X2: nat] : ( if_set_nat @ ( X2 = A4 ) @ B4 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_174_fun__upd__eqD,axiom,
! [F: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: option_val,G: produc1457211279475724562t_char > option_val,Z: option_val] :
( ( ( fun_up5735975307950539954on_val @ F @ X3 @ Y3 )
= ( fun_up5735975307950539954on_val @ G @ X3 @ Z ) )
=> ( Y3 = Z ) ) ).
% fun_upd_eqD
thf(fact_175_fun__upd__eqD,axiom,
! [F: nat > option_val,X3: nat,Y3: option_val,G: nat > option_val,Z: option_val] :
( ( ( fun_up810121132392276792on_val @ F @ X3 @ Y3 )
= ( fun_up810121132392276792on_val @ G @ X3 @ Z ) )
=> ( Y3 = Z ) ) ).
% fun_upd_eqD
thf(fact_176_fun__upd__eqD,axiom,
! [F: nat > nat,X3: nat,Y3: nat,G: nat > nat,Z: nat] :
( ( ( fun_upd_nat_nat @ F @ X3 @ Y3 )
= ( fun_upd_nat_nat @ G @ X3 @ Z ) )
=> ( Y3 = Z ) ) ).
% fun_upd_eqD
thf(fact_177_fun__upd__eqD,axiom,
! [F: nat > set_nat,X3: nat,Y3: set_nat,G: nat > set_nat,Z: set_nat] :
( ( ( fun_upd_nat_set_nat @ F @ X3 @ Y3 )
= ( fun_upd_nat_set_nat @ G @ X3 @ Z ) )
=> ( Y3 = Z ) ) ).
% fun_upd_eqD
thf(fact_178_map__le__refl,axiom,
! [F: produc1457211279475724562t_char > option_val] : ( map_le3432596473071271956ar_val @ F @ F ) ).
% map_le_refl
thf(fact_179_map__le__trans,axiom,
! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val,M3: produc1457211279475724562t_char > option_val] :
( ( map_le3432596473071271956ar_val @ M1 @ M2 )
=> ( ( map_le3432596473071271956ar_val @ M2 @ M3 )
=> ( map_le3432596473071271956ar_val @ M1 @ M3 ) ) ) ).
% map_le_trans
thf(fact_180_map__le__antisym,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
( ( map_le3432596473071271956ar_val @ F @ G )
=> ( ( map_le3432596473071271956ar_val @ G @ F )
=> ( F = G ) ) ) ).
% map_le_antisym
thf(fact_181_map__le__upd,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char,B: option_val] :
( ( map_le3432596473071271956ar_val @ F @ G )
=> ( map_le3432596473071271956ar_val @ ( fun_up5735975307950539954on_val @ F @ A @ B ) @ ( fun_up5735975307950539954on_val @ G @ A @ B ) ) ) ).
% map_le_upd
thf(fact_182_map__le__upd,axiom,
! [F: nat > option_val,G: nat > option_val,A: nat,B: option_val] :
( ( map_le_nat_val @ F @ G )
=> ( map_le_nat_val @ ( fun_up810121132392276792on_val @ F @ A @ B ) @ ( fun_up810121132392276792on_val @ G @ A @ B ) ) ) ).
% map_le_upd
thf(fact_183_map__le__iff__map__add__commute,axiom,
! [F: nat > option_val,G: nat > option_val] :
( ( map_le_nat_val @ F @ ( map_add_nat_val @ F @ G ) )
= ( ( map_add_nat_val @ F @ G )
= ( map_add_nat_val @ G @ F ) ) ) ).
% map_le_iff_map_add_commute
thf(fact_184_map__le__iff__map__add__commute,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
( ( map_le3432596473071271956ar_val @ F @ ( map_ad2908576761495157028ar_val @ F @ G ) )
= ( ( map_ad2908576761495157028ar_val @ F @ G )
= ( map_ad2908576761495157028ar_val @ G @ F ) ) ) ).
% map_le_iff_map_add_commute
thf(fact_185_map__add__subsumed2,axiom,
! [F: nat > option_val,G: nat > option_val] :
( ( map_le_nat_val @ F @ G )
=> ( ( map_add_nat_val @ G @ F )
= G ) ) ).
% map_add_subsumed2
thf(fact_186_map__add__subsumed2,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
( ( map_le3432596473071271956ar_val @ F @ G )
=> ( ( map_ad2908576761495157028ar_val @ G @ F )
= G ) ) ).
% map_add_subsumed2
thf(fact_187_map__add__subsumed1,axiom,
! [F: nat > option_val,G: nat > option_val] :
( ( map_le_nat_val @ F @ G )
=> ( ( map_add_nat_val @ F @ G )
= G ) ) ).
% map_add_subsumed1
thf(fact_188_map__add__subsumed1,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
( ( map_le3432596473071271956ar_val @ F @ G )
=> ( ( map_ad2908576761495157028ar_val @ F @ G )
= G ) ) ).
% map_add_subsumed1
thf(fact_189_map__add__le__mapI,axiom,
! [F: nat > option_val,H: nat > option_val,G: nat > option_val] :
( ( map_le_nat_val @ F @ H )
=> ( ( map_le_nat_val @ G @ H )
=> ( map_le_nat_val @ ( map_add_nat_val @ F @ G ) @ H ) ) ) ).
% map_add_le_mapI
thf(fact_190_map__add__le__mapI,axiom,
! [F: produc1457211279475724562t_char > option_val,H: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
( ( map_le3432596473071271956ar_val @ F @ H )
=> ( ( map_le3432596473071271956ar_val @ G @ H )
=> ( map_le3432596473071271956ar_val @ ( map_ad2908576761495157028ar_val @ F @ G ) @ H ) ) ) ).
% map_add_le_mapI
thf(fact_191_map__add__le__mapE,axiom,
! [F: nat > option_val,G: nat > option_val,H: nat > option_val] :
( ( map_le_nat_val @ ( map_add_nat_val @ F @ G ) @ H )
=> ( map_le_nat_val @ G @ H ) ) ).
% map_add_le_mapE
thf(fact_192_map__add__le__mapE,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val,H: produc1457211279475724562t_char > option_val] :
( ( map_le3432596473071271956ar_val @ ( map_ad2908576761495157028ar_val @ F @ G ) @ H )
=> ( map_le3432596473071271956ar_val @ G @ H ) ) ).
% map_add_le_mapE
thf(fact_193_map__le__map__add,axiom,
! [F: nat > option_val,G: nat > option_val] : ( map_le_nat_val @ F @ ( map_add_nat_val @ G @ F ) ) ).
% map_le_map_add
thf(fact_194_map__le__map__add,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] : ( map_le3432596473071271956ar_val @ F @ ( map_ad2908576761495157028ar_val @ G @ F ) ) ).
% map_le_map_add
thf(fact_195_upd__None__map__le,axiom,
! [F: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char] : ( map_le3432596473071271956ar_val @ ( fun_up5735975307950539954on_val @ F @ X3 @ none_val ) @ F ) ).
% upd_None_map_le
thf(fact_196_upd__None__map__le,axiom,
! [F: nat > option_val,X3: nat] : ( map_le_nat_val @ ( fun_up810121132392276792on_val @ F @ X3 @ none_val ) @ F ) ).
% upd_None_map_le
thf(fact_197_Pair__inject,axiom,
! [A: option_val,B: option_val,A3: option_val,B3: option_val] :
( ( ( produc904852209897547607on_val @ A @ B )
= ( produc904852209897547607on_val @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_198_Pair__inject,axiom,
! [A: nat,B: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_199_Pair__inject,axiom,
! [A: set_nat,B: set_nat,A3: set_nat,B3: set_nat] :
( ( ( produc4532415448927165861et_nat @ A @ B )
= ( produc4532415448927165861et_nat @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_200_prod__cases,axiom,
! [P: produc5122021904355189735on_val > $o,P4: produc5122021904355189735on_val] :
( ! [A2: option_val,B2: option_val] : ( P @ ( produc904852209897547607on_val @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_201_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
( ! [A2: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_202_prod__cases,axiom,
! [P: produc7819656566062154093et_nat > $o,P4: produc7819656566062154093et_nat] :
( ! [A2: set_nat,B2: set_nat] : ( P @ ( produc4532415448927165861et_nat @ A2 @ B2 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_203_surj__pair,axiom,
! [P4: produc5122021904355189735on_val] :
? [X: option_val,Y4: option_val] :
( P4
= ( produc904852209897547607on_val @ X @ Y4 ) ) ).
% surj_pair
thf(fact_204_surj__pair,axiom,
! [P4: product_prod_nat_nat] :
? [X: nat,Y4: nat] :
( P4
= ( product_Pair_nat_nat @ X @ Y4 ) ) ).
% surj_pair
thf(fact_205_surj__pair,axiom,
! [P4: produc7819656566062154093et_nat] :
? [X: set_nat,Y4: set_nat] :
( P4
= ( produc4532415448927165861et_nat @ X @ Y4 ) ) ).
% surj_pair
thf(fact_206_old_Oprod_Oexhaust,axiom,
! [Y3: produc5122021904355189735on_val] :
~ ! [A2: option_val,B2: option_val] :
( Y3
!= ( produc904852209897547607on_val @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_207_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [A2: nat,B2: nat] :
( Y3
!= ( product_Pair_nat_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_208_old_Oprod_Oexhaust,axiom,
! [Y3: produc7819656566062154093et_nat] :
~ ! [A2: set_nat,B2: set_nat] :
( Y3
!= ( produc4532415448927165861et_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_209_fun__upd__idem__iff,axiom,
! [F: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: option_val] :
( ( ( fun_up5735975307950539954on_val @ F @ X3 @ Y3 )
= F )
= ( ( F @ X3 )
= Y3 ) ) ).
% fun_upd_idem_iff
thf(fact_210_fun__upd__idem__iff,axiom,
! [F: nat > option_val,X3: nat,Y3: option_val] :
( ( ( fun_up810121132392276792on_val @ F @ X3 @ Y3 )
= F )
= ( ( F @ X3 )
= Y3 ) ) ).
% fun_upd_idem_iff
thf(fact_211_fun__upd__idem__iff,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( ( fun_upd_nat_nat @ F @ X3 @ Y3 )
= F )
= ( ( F @ X3 )
= Y3 ) ) ).
% fun_upd_idem_iff
thf(fact_212_fun__upd__idem__iff,axiom,
! [F: nat > set_nat,X3: nat,Y3: set_nat] :
( ( ( fun_upd_nat_set_nat @ F @ X3 @ Y3 )
= F )
= ( ( F @ X3 )
= Y3 ) ) ).
% fun_upd_idem_iff
thf(fact_213_fun__upd__twist,axiom,
! [A: produc1457211279475724562t_char,C: produc1457211279475724562t_char,M: produc1457211279475724562t_char > option_val,B: option_val,D: option_val] :
( ( A != C )
=> ( ( fun_up5735975307950539954on_val @ ( fun_up5735975307950539954on_val @ M @ A @ B ) @ C @ D )
= ( fun_up5735975307950539954on_val @ ( fun_up5735975307950539954on_val @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_214_fun__upd__twist,axiom,
! [A: nat,C: nat,M: nat > option_val,B: option_val,D: option_val] :
( ( A != C )
=> ( ( fun_up810121132392276792on_val @ ( fun_up810121132392276792on_val @ M @ A @ B ) @ C @ D )
= ( fun_up810121132392276792on_val @ ( fun_up810121132392276792on_val @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_215_fun__upd__twist,axiom,
! [A: nat,C: nat,M: nat > nat,B: nat,D: nat] :
( ( A != C )
=> ( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ A @ B ) @ C @ D )
= ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_216_fun__upd__twist,axiom,
! [A: nat,C: nat,M: nat > set_nat,B: set_nat,D: set_nat] :
( ( A != C )
=> ( ( fun_upd_nat_set_nat @ ( fun_upd_nat_set_nat @ M @ A @ B ) @ C @ D )
= ( fun_upd_nat_set_nat @ ( fun_upd_nat_set_nat @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_217_fun__upd__other,axiom,
! [Z: produc1457211279475724562t_char,X3: produc1457211279475724562t_char,F: produc1457211279475724562t_char > option_val,Y3: option_val] :
( ( Z != X3 )
=> ( ( fun_up5735975307950539954on_val @ F @ X3 @ Y3 @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_218_fun__upd__other,axiom,
! [Z: nat,X3: nat,F: nat > option_val,Y3: option_val] :
( ( Z != X3 )
=> ( ( fun_up810121132392276792on_val @ F @ X3 @ Y3 @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_219_fun__upd__other,axiom,
! [Z: nat,X3: nat,F: nat > nat,Y3: nat] :
( ( Z != X3 )
=> ( ( fun_upd_nat_nat @ F @ X3 @ Y3 @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_220_fun__upd__other,axiom,
! [Z: nat,X3: nat,F: nat > set_nat,Y3: set_nat] :
( ( Z != X3 )
=> ( ( fun_upd_nat_set_nat @ F @ X3 @ Y3 @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_221_fun__upd__same,axiom,
! [F: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: option_val] :
( ( fun_up5735975307950539954on_val @ F @ X3 @ Y3 @ X3 )
= Y3 ) ).
% fun_upd_same
thf(fact_222_fun__upd__same,axiom,
! [F: nat > option_val,X3: nat,Y3: option_val] :
( ( fun_up810121132392276792on_val @ F @ X3 @ Y3 @ X3 )
= Y3 ) ).
% fun_upd_same
thf(fact_223_fun__upd__same,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( fun_upd_nat_nat @ F @ X3 @ Y3 @ X3 )
= Y3 ) ).
% fun_upd_same
thf(fact_224_fun__upd__same,axiom,
! [F: nat > set_nat,X3: nat,Y3: set_nat] :
( ( fun_upd_nat_set_nat @ F @ X3 @ Y3 @ X3 )
= Y3 ) ).
% fun_upd_same
thf(fact_225_fun__upd__idem,axiom,
! [F: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,Y3: option_val] :
( ( ( F @ X3 )
= Y3 )
=> ( ( fun_up5735975307950539954on_val @ F @ X3 @ Y3 )
= F ) ) ).
% fun_upd_idem
thf(fact_226_fun__upd__idem,axiom,
! [F: nat > option_val,X3: nat,Y3: option_val] :
( ( ( F @ X3 )
= Y3 )
=> ( ( fun_up810121132392276792on_val @ F @ X3 @ Y3 )
= F ) ) ).
% fun_upd_idem
thf(fact_227_fun__upd__idem,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( ( F @ X3 )
= Y3 )
=> ( ( fun_upd_nat_nat @ F @ X3 @ Y3 )
= F ) ) ).
% fun_upd_idem
thf(fact_228_fun__upd__idem,axiom,
! [F: nat > set_nat,X3: nat,Y3: set_nat] :
( ( ( F @ X3 )
= Y3 )
=> ( ( fun_upd_nat_set_nat @ F @ X3 @ Y3 )
= F ) ) ).
% fun_upd_idem
thf(fact_229_map__add__upd__left,axiom,
! [M: nat,E2: nat > option_val,E1: nat > option_val,U1: val] :
( ~ ( member_nat @ M @ ( dom_nat_val @ E2 ) )
=> ( ( map_add_nat_val @ ( fun_up810121132392276792on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
= ( fun_up810121132392276792on_val @ ( map_add_nat_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_230_map__add__upd__left,axiom,
! [M: produc1457211279475724562t_char,E2: produc1457211279475724562t_char > option_val,E1: produc1457211279475724562t_char > option_val,U1: val] :
( ~ ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ E2 ) )
=> ( ( map_ad2908576761495157028ar_val @ ( fun_up5735975307950539954on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
= ( fun_up5735975307950539954on_val @ ( map_ad2908576761495157028ar_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_231_map__add__upd__left,axiom,
! [M: val,E2: val > option_val,E1: val > option_val,U1: val] :
( ~ ( member_val @ M @ ( dom_val_val @ E2 ) )
=> ( ( map_add_val_val @ ( fun_up735531404784515971on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
= ( fun_up735531404784515971on_val @ ( map_add_val_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_232_map__add__upd__left,axiom,
! [M: $o,E2: $o > option_val,E1: $o > option_val,U1: val] :
( ~ ( member_o @ M @ ( dom_o_val @ E2 ) )
=> ( ( map_add_o_val @ ( fun_upd_o_option_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
= ( fun_upd_o_option_val @ ( map_add_o_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_233_map__add__upd__left,axiom,
! [M: nat,E2: nat > option_nat,E1: nat > option_nat,U1: nat] :
( ~ ( member_nat @ M @ ( dom_nat_nat @ E2 ) )
=> ( ( map_add_nat_nat @ ( fun_up1493157387958331631on_nat @ E1 @ M @ ( some_nat @ U1 ) ) @ E2 )
= ( fun_up1493157387958331631on_nat @ ( map_add_nat_nat @ E1 @ E2 ) @ M @ ( some_nat @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_234_map__add__upd__left,axiom,
! [M: val,E2: val > option_nat,E1: val > option_nat,U1: nat] :
( ~ ( member_val @ M @ ( dom_val_nat @ E2 ) )
=> ( ( map_add_val_nat @ ( fun_up7213561917443673572on_nat @ E1 @ M @ ( some_nat @ U1 ) ) @ E2 )
= ( fun_up7213561917443673572on_nat @ ( map_add_val_nat @ E1 @ E2 ) @ M @ ( some_nat @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_235_map__add__upd__left,axiom,
! [M: $o,E2: $o > option_nat,E1: $o > option_nat,U1: nat] :
( ~ ( member_o @ M @ ( dom_o_nat @ E2 ) )
=> ( ( map_add_o_nat @ ( fun_upd_o_option_nat @ E1 @ M @ ( some_nat @ U1 ) ) @ E2 )
= ( fun_upd_o_option_nat @ ( map_add_o_nat @ E1 @ E2 ) @ M @ ( some_nat @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_236_map__add__upd__left,axiom,
! [M: option_val,E2: option_val > option_val,E1: option_val > option_val,U1: val] :
( ~ ( member_option_val @ M @ ( dom_option_val_val @ E2 ) )
=> ( ( map_ad2791809220819668783al_val @ ( fun_up8761876891205509693on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
= ( fun_up8761876891205509693on_val @ ( map_ad2791809220819668783al_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_237_map__add__upd__left,axiom,
! [M: option_nat,E2: option_nat > option_val,E1: option_nat > option_val,U1: val] :
( ~ ( member_option_nat @ M @ ( dom_option_nat_val @ E2 ) )
=> ( ( map_ad3338430187718117610at_val @ ( fun_up2035529056102068968on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
= ( fun_up2035529056102068968on_val @ ( map_ad3338430187718117610at_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_238_map__add__upd__left,axiom,
! [M: set_nat,E2: set_nat > option_val,E1: set_nat > option_val,U1: val] :
( ~ ( member_set_nat @ M @ ( dom_set_nat_val @ E2 ) )
=> ( ( map_add_set_nat_val @ ( fun_up1611807965235150338on_val @ E1 @ M @ ( some_val @ U1 ) ) @ E2 )
= ( fun_up1611807965235150338on_val @ ( map_add_set_nat_val @ E1 @ E2 ) @ M @ ( some_val @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_239_ssubst__Pair__rhs,axiom,
! [R: option_val,S: option_val,R2: set_Pr2339170832469995719on_val,S2: option_val] :
( ( member7947933638605243152on_val @ ( produc904852209897547607on_val @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member7947933638605243152on_val @ ( produc904852209897547607on_val @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_240_ssubst__Pair__rhs,axiom,
! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_241_ssubst__Pair__rhs,axiom,
! [R: set_nat,S: set_nat,R2: set_Pr5488025237498180813et_nat,S2: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_242_these__insert__None,axiom,
! [A5: set_option_val] :
( ( these_val @ ( insert_option_val @ none_val @ A5 ) )
= ( these_val @ A5 ) ) ).
% these_insert_None
thf(fact_243_these__insert__None,axiom,
! [A5: set_option_nat] :
( ( these_nat @ ( insert_option_nat @ none_nat @ A5 ) )
= ( these_nat @ A5 ) ) ).
% these_insert_None
thf(fact_244_compatible__options_Opelims_I2_J,axiom,
! [X3: option_nat,Xa: option_nat] :
( ( compat6420650667825766956ns_nat @ X3 @ Xa )
=> ( ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ X3 @ Xa ) )
=> ( ! [A2: nat] :
( ( X3
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xa
= ( some_nat @ B2 ) )
=> ( ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ ( some_nat @ A2 ) @ ( some_nat @ B2 ) ) )
=> ( A2 != B2 ) ) ) )
=> ( ( ( X3 = none_nat )
=> ~ ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ none_nat @ Xa ) ) )
=> ~ ( ( Xa = none_nat )
=> ~ ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ X3 @ none_nat ) ) ) ) ) ) ) ).
% compatible_options.pelims(2)
thf(fact_245_compatible__options_Opelims_I2_J,axiom,
! [X3: option_val,Xa: option_val] :
( ( compat7209469420100717861ns_val @ X3 @ Xa )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa ) )
=> ( ! [A2: val] :
( ( X3
= ( some_val @ A2 ) )
=> ! [B2: val] :
( ( Xa
= ( some_val @ B2 ) )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) )
=> ( A2 != B2 ) ) ) )
=> ( ( ( X3 = none_val )
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) )
=> ~ ( ( Xa = none_val )
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ none_val ) ) ) ) ) ) ) ).
% compatible_options.pelims(2)
thf(fact_246_compatible__options_Opelims_I1_J,axiom,
! [X3: option_nat,Xa: option_nat,Y3: $o] :
( ( ( compat6420650667825766956ns_nat @ X3 @ Xa )
= Y3 )
=> ( ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ X3 @ Xa ) )
=> ( ! [A2: nat] :
( ( X3
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xa
= ( some_nat @ B2 ) )
=> ( ( Y3
= ( A2 = B2 ) )
=> ~ ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ ( some_nat @ A2 ) @ ( some_nat @ B2 ) ) ) ) ) )
=> ( ( ( X3 = none_nat )
=> ( Y3
=> ~ ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ none_nat @ Xa ) ) ) )
=> ~ ( ( Xa = none_nat )
=> ( Y3
=> ~ ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ X3 @ none_nat ) ) ) ) ) ) ) ) ).
% compatible_options.pelims(1)
thf(fact_247_compatible__options_Opelims_I1_J,axiom,
! [X3: option_val,Xa: option_val,Y3: $o] :
( ( ( compat7209469420100717861ns_val @ X3 @ Xa )
= Y3 )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa ) )
=> ( ! [A2: val] :
( ( X3
= ( some_val @ A2 ) )
=> ! [B2: val] :
( ( Xa
= ( some_val @ B2 ) )
=> ( ( Y3
= ( A2 = B2 ) )
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) ) ) ) )
=> ( ( ( X3 = none_val )
=> ( Y3
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) ) )
=> ~ ( ( Xa = none_val )
=> ( Y3
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ none_val ) ) ) ) ) ) ) ) ).
% compatible_options.pelims(1)
thf(fact_248_these__image__Some__eq,axiom,
! [A5: set_nat] :
( ( these_nat @ ( image_nat_option_nat @ some_nat @ A5 ) )
= A5 ) ).
% these_image_Some_eq
thf(fact_249_these__image__Some__eq,axiom,
! [A5: set_val] :
( ( these_val @ ( image_val_option_val @ some_val @ A5 ) )
= A5 ) ).
% these_image_Some_eq
thf(fact_250_option_Oset__sel,axiom,
! [A: option_option_val] :
( ( A != none_option_val )
=> ( member_option_val @ ( the_option_val @ A ) @ ( set_op911010586042427349on_val @ A ) ) ) ).
% option.set_sel
thf(fact_251_option_Oset__sel,axiom,
! [A: option_option_nat] :
( ( A != none_option_nat )
=> ( member_option_nat @ ( the_option_nat @ A ) @ ( set_op3360498428384587026on_nat @ A ) ) ) ).
% option.set_sel
thf(fact_252_option_Oset__sel,axiom,
! [A: option_o] :
( ( A != none_o )
=> ( member_o @ ( the_o @ A ) @ ( set_option_o2 @ A ) ) ) ).
% option.set_sel
thf(fact_253_option_Oset__sel,axiom,
! [A: option_set_nat] :
( ( A != none_set_nat )
=> ( member_set_nat @ ( the_set_nat @ A ) @ ( set_option_set_nat2 @ A ) ) ) ).
% option.set_sel
thf(fact_254_option_Oset__sel,axiom,
! [A: option_val] :
( ( A != none_val )
=> ( member_val @ ( the_val @ A ) @ ( set_option_val2 @ A ) ) ) ).
% option.set_sel
thf(fact_255_option_Oset__sel,axiom,
! [A: option_nat] :
( ( A != none_nat )
=> ( member_nat @ ( the_nat @ A ) @ ( set_option_nat2 @ A ) ) ) ).
% option.set_sel
thf(fact_256_curryI,axiom,
! [F: produc5122021904355189735on_val > $o,A: option_val,B: option_val] :
( ( F @ ( produc904852209897547607on_val @ A @ B ) )
=> ( produc3993299840032831053_val_o @ F @ A @ B ) ) ).
% curryI
thf(fact_257_curryI,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( F @ ( product_Pair_nat_nat @ A @ B ) )
=> ( produc1310100445399344235_nat_o @ F @ A @ B ) ) ).
% curryI
thf(fact_258_curryI,axiom,
! [F: produc7819656566062154093et_nat > $o,A: set_nat,B: set_nat] :
( ( F @ ( produc4532415448927165861et_nat @ A @ B ) )
=> ( produc8481458855017059839_nat_o @ F @ A @ B ) ) ).
% curryI
thf(fact_259_elem__set,axiom,
! [X3: option_val,Xo: option_option_val] :
( ( member_option_val @ X3 @ ( set_op911010586042427349on_val @ Xo ) )
= ( Xo
= ( some_option_val @ X3 ) ) ) ).
% elem_set
thf(fact_260_elem__set,axiom,
! [X3: option_nat,Xo: option_option_nat] :
( ( member_option_nat @ X3 @ ( set_op3360498428384587026on_nat @ Xo ) )
= ( Xo
= ( some_option_nat @ X3 ) ) ) ).
% elem_set
thf(fact_261_elem__set,axiom,
! [X3: $o,Xo: option_o] :
( ( member_o @ X3 @ ( set_option_o2 @ Xo ) )
= ( Xo
= ( some_o @ X3 ) ) ) ).
% elem_set
thf(fact_262_elem__set,axiom,
! [X3: set_nat,Xo: option_set_nat] :
( ( member_set_nat @ X3 @ ( set_option_set_nat2 @ Xo ) )
= ( Xo
= ( some_set_nat @ X3 ) ) ) ).
% elem_set
thf(fact_263_elem__set,axiom,
! [X3: nat,Xo: option_nat] :
( ( member_nat @ X3 @ ( set_option_nat2 @ Xo ) )
= ( Xo
= ( some_nat @ X3 ) ) ) ).
% elem_set
thf(fact_264_elem__set,axiom,
! [X3: val,Xo: option_val] :
( ( member_val @ X3 @ ( set_option_val2 @ Xo ) )
= ( Xo
= ( some_val @ X3 ) ) ) ).
% elem_set
thf(fact_265_fun__upd__None__if__notin__dom,axiom,
! [K: val,M: val > option_val] :
( ~ ( member_val @ K @ ( dom_val_val @ M ) )
=> ( ( fun_up735531404784515971on_val @ M @ K @ none_val )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_266_fun__upd__None__if__notin__dom,axiom,
! [K: $o,M: $o > option_val] :
( ~ ( member_o @ K @ ( dom_o_val @ M ) )
=> ( ( fun_upd_o_option_val @ M @ K @ none_val )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_267_fun__upd__None__if__notin__dom,axiom,
! [K: nat,M: nat > option_val] :
( ~ ( member_nat @ K @ ( dom_nat_val @ M ) )
=> ( ( fun_up810121132392276792on_val @ M @ K @ none_val )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_268_fun__upd__None__if__notin__dom,axiom,
! [K: nat,M: nat > option_nat] :
( ~ ( member_nat @ K @ ( dom_nat_nat @ M ) )
=> ( ( fun_up1493157387958331631on_nat @ M @ K @ none_nat )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_269_fun__upd__None__if__notin__dom,axiom,
! [K: val,M: val > option_nat] :
( ~ ( member_val @ K @ ( dom_val_nat @ M ) )
=> ( ( fun_up7213561917443673572on_nat @ M @ K @ none_nat )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_270_fun__upd__None__if__notin__dom,axiom,
! [K: $o,M: $o > option_nat] :
( ~ ( member_o @ K @ ( dom_o_nat @ M ) )
=> ( ( fun_upd_o_option_nat @ M @ K @ none_nat )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_271_fun__upd__None__if__notin__dom,axiom,
! [K: option_val,M: option_val > option_val] :
( ~ ( member_option_val @ K @ ( dom_option_val_val @ M ) )
=> ( ( fun_up8761876891205509693on_val @ M @ K @ none_val )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_272_fun__upd__None__if__notin__dom,axiom,
! [K: option_nat,M: option_nat > option_val] :
( ~ ( member_option_nat @ K @ ( dom_option_nat_val @ M ) )
=> ( ( fun_up2035529056102068968on_val @ M @ K @ none_val )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_273_fun__upd__None__if__notin__dom,axiom,
! [K: set_nat,M: set_nat > option_val] :
( ~ ( member_set_nat @ K @ ( dom_set_nat_val @ M ) )
=> ( ( fun_up1611807965235150338on_val @ M @ K @ none_val )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_274_fun__upd__None__if__notin__dom,axiom,
! [K: option_val,M: option_val > option_nat] :
( ~ ( member_option_val @ K @ ( dom_option_val_nat @ M ) )
=> ( ( fun_up1535178098986571946on_nat @ M @ K @ none_nat )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_275_image__map__upd,axiom,
! [X3: nat,A5: set_nat,M: nat > option_val,Y3: val] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( image_nat_option_val @ ( fun_up810121132392276792on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A5 )
= ( image_nat_option_val @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_276_image__map__upd,axiom,
! [X3: val,A5: set_val,M: val > option_val,Y3: val] :
( ~ ( member_val @ X3 @ A5 )
=> ( ( image_val_option_val @ ( fun_up735531404784515971on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A5 )
= ( image_val_option_val @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_277_image__map__upd,axiom,
! [X3: $o,A5: set_o,M: $o > option_val,Y3: val] :
( ~ ( member_o @ X3 @ A5 )
=> ( ( image_o_option_val @ ( fun_upd_o_option_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A5 )
= ( image_o_option_val @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_278_image__map__upd,axiom,
! [X3: nat,A5: set_nat,M: nat > option_nat,Y3: nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( image_nat_option_nat @ ( fun_up1493157387958331631on_nat @ M @ X3 @ ( some_nat @ Y3 ) ) @ A5 )
= ( image_nat_option_nat @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_279_image__map__upd,axiom,
! [X3: val,A5: set_val,M: val > option_nat,Y3: nat] :
( ~ ( member_val @ X3 @ A5 )
=> ( ( image_val_option_nat @ ( fun_up7213561917443673572on_nat @ M @ X3 @ ( some_nat @ Y3 ) ) @ A5 )
= ( image_val_option_nat @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_280_image__map__upd,axiom,
! [X3: $o,A5: set_o,M: $o > option_nat,Y3: nat] :
( ~ ( member_o @ X3 @ A5 )
=> ( ( image_o_option_nat @ ( fun_upd_o_option_nat @ M @ X3 @ ( some_nat @ Y3 ) ) @ A5 )
= ( image_o_option_nat @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_281_image__map__upd,axiom,
! [X3: option_val,A5: set_option_val,M: option_val > option_val,Y3: val] :
( ~ ( member_option_val @ X3 @ A5 )
=> ( ( image_5994708315261978053on_val @ ( fun_up8761876891205509693on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A5 )
= ( image_5994708315261978053on_val @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_282_image__map__upd,axiom,
! [X3: option_nat,A5: set_option_nat,M: option_nat > option_val,Y3: val] :
( ~ ( member_option_nat @ X3 @ A5 )
=> ( ( image_2375463299254234464on_val @ ( fun_up2035529056102068968on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A5 )
= ( image_2375463299254234464on_val @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_283_image__map__upd,axiom,
! [X3: set_nat,A5: set_set_nat,M: set_nat > option_val,Y3: val] :
( ~ ( member_set_nat @ X3 @ A5 )
=> ( ( image_4570703338782665850on_val @ ( fun_up1611807965235150338on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ A5 )
= ( image_4570703338782665850on_val @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_284_image__map__upd,axiom,
! [X3: option_val,A5: set_option_val,M: option_val > option_nat,Y3: nat] :
( ~ ( member_option_val @ X3 @ A5 )
=> ( ( image_1875112342138737442on_nat @ ( fun_up1535178098986571946on_nat @ M @ X3 @ ( some_nat @ Y3 ) ) @ A5 )
= ( image_1875112342138737442on_nat @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_285_these__insert__Some,axiom,
! [X3: option_val,A5: set_op1974636978870893421on_val] :
( ( these_option_val @ ( insert7563226391266834525on_val @ ( some_option_val @ X3 ) @ A5 ) )
= ( insert_option_val @ X3 @ ( these_option_val @ A5 ) ) ) ).
% these_insert_Some
thf(fact_286_these__insert__Some,axiom,
! [X3: option_nat,A5: set_op6961666426309957030on_nat] :
( ( these_option_nat @ ( insert504548404241883424on_nat @ ( some_option_nat @ X3 ) @ A5 ) )
= ( insert_option_nat @ X3 @ ( these_option_nat @ A5 ) ) ) ).
% these_insert_Some
thf(fact_287_these__insert__Some,axiom,
! [X3: set_nat,A5: set_option_set_nat] :
( ( these_set_nat @ ( insert6281397740311767046et_nat @ ( some_set_nat @ X3 ) @ A5 ) )
= ( insert_set_nat @ X3 @ ( these_set_nat @ A5 ) ) ) ).
% these_insert_Some
thf(fact_288_these__insert__Some,axiom,
! [X3: product_prod_nat_nat,A5: set_op4508134149509766951at_nat] :
( ( these_8061402112538453438at_nat @ ( insert6842972033726296599at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ A5 ) )
= ( insert8211810215607154385at_nat @ X3 @ ( these_8061402112538453438at_nat @ A5 ) ) ) ).
% these_insert_Some
thf(fact_289_these__insert__Some,axiom,
! [X3: nat,A5: set_option_nat] :
( ( these_nat @ ( insert_option_nat @ ( some_nat @ X3 ) @ A5 ) )
= ( insert_nat @ X3 @ ( these_nat @ A5 ) ) ) ).
% these_insert_Some
thf(fact_290_these__insert__Some,axiom,
! [X3: val,A5: set_option_val] :
( ( these_val @ ( insert_option_val @ ( some_val @ X3 ) @ A5 ) )
= ( insert_val @ X3 @ ( these_val @ A5 ) ) ) ).
% these_insert_Some
thf(fact_291_insert__dom,axiom,
! [F: nat > option_val,X3: nat,Y3: val] :
( ( ( F @ X3 )
= ( some_val @ Y3 ) )
=> ( ( insert_nat @ X3 @ ( dom_nat_val @ F ) )
= ( dom_nat_val @ F ) ) ) ).
% insert_dom
thf(fact_292_insert__dom,axiom,
! [F: val > option_val,X3: val,Y3: val] :
( ( ( F @ X3 )
= ( some_val @ Y3 ) )
=> ( ( insert_val @ X3 @ ( dom_val_val @ F ) )
= ( dom_val_val @ F ) ) ) ).
% insert_dom
thf(fact_293_insert__dom,axiom,
! [F: nat > option_nat,X3: nat,Y3: nat] :
( ( ( F @ X3 )
= ( some_nat @ Y3 ) )
=> ( ( insert_nat @ X3 @ ( dom_nat_nat @ F ) )
= ( dom_nat_nat @ F ) ) ) ).
% insert_dom
thf(fact_294_insert__dom,axiom,
! [F: val > option_nat,X3: val,Y3: nat] :
( ( ( F @ X3 )
= ( some_nat @ Y3 ) )
=> ( ( insert_val @ X3 @ ( dom_val_nat @ F ) )
= ( dom_val_nat @ F ) ) ) ).
% insert_dom
thf(fact_295_insert__dom,axiom,
! [F: option_val > option_val,X3: option_val,Y3: val] :
( ( ( F @ X3 )
= ( some_val @ Y3 ) )
=> ( ( insert_option_val @ X3 @ ( dom_option_val_val @ F ) )
= ( dom_option_val_val @ F ) ) ) ).
% insert_dom
thf(fact_296_insert__dom,axiom,
! [F: option_nat > option_val,X3: option_nat,Y3: val] :
( ( ( F @ X3 )
= ( some_val @ Y3 ) )
=> ( ( insert_option_nat @ X3 @ ( dom_option_nat_val @ F ) )
= ( dom_option_nat_val @ F ) ) ) ).
% insert_dom
thf(fact_297_insert__dom,axiom,
! [F: set_nat > option_val,X3: set_nat,Y3: val] :
( ( ( F @ X3 )
= ( some_val @ Y3 ) )
=> ( ( insert_set_nat @ X3 @ ( dom_set_nat_val @ F ) )
= ( dom_set_nat_val @ F ) ) ) ).
% insert_dom
thf(fact_298_insert__dom,axiom,
! [F: option_val > option_nat,X3: option_val,Y3: nat] :
( ( ( F @ X3 )
= ( some_nat @ Y3 ) )
=> ( ( insert_option_val @ X3 @ ( dom_option_val_nat @ F ) )
= ( dom_option_val_nat @ F ) ) ) ).
% insert_dom
thf(fact_299_insert__dom,axiom,
! [F: option_nat > option_nat,X3: option_nat,Y3: nat] :
( ( ( F @ X3 )
= ( some_nat @ Y3 ) )
=> ( ( insert_option_nat @ X3 @ ( dom_option_nat_nat @ F ) )
= ( dom_option_nat_nat @ F ) ) ) ).
% insert_dom
thf(fact_300_insert__dom,axiom,
! [F: set_nat > option_nat,X3: set_nat,Y3: nat] :
( ( ( F @ X3 )
= ( some_nat @ Y3 ) )
=> ( ( insert_set_nat @ X3 @ ( dom_set_nat_nat @ F ) )
= ( dom_set_nat_nat @ F ) ) ) ).
% insert_dom
thf(fact_301_domD,axiom,
! [A: nat,M: nat > option_val] :
( ( member_nat @ A @ ( dom_nat_val @ M ) )
=> ? [B2: val] :
( ( M @ A )
= ( some_val @ B2 ) ) ) ).
% domD
thf(fact_302_domD,axiom,
! [A: val,M: val > option_val] :
( ( member_val @ A @ ( dom_val_val @ M ) )
=> ? [B2: val] :
( ( M @ A )
= ( some_val @ B2 ) ) ) ).
% domD
thf(fact_303_domD,axiom,
! [A: $o,M: $o > option_val] :
( ( member_o @ A @ ( dom_o_val @ M ) )
=> ? [B2: val] :
( ( M @ A )
= ( some_val @ B2 ) ) ) ).
% domD
thf(fact_304_domD,axiom,
! [A: nat,M: nat > option_nat] :
( ( member_nat @ A @ ( dom_nat_nat @ M ) )
=> ? [B2: nat] :
( ( M @ A )
= ( some_nat @ B2 ) ) ) ).
% domD
thf(fact_305_domD,axiom,
! [A: val,M: val > option_nat] :
( ( member_val @ A @ ( dom_val_nat @ M ) )
=> ? [B2: nat] :
( ( M @ A )
= ( some_nat @ B2 ) ) ) ).
% domD
thf(fact_306_domD,axiom,
! [A: $o,M: $o > option_nat] :
( ( member_o @ A @ ( dom_o_nat @ M ) )
=> ? [B2: nat] :
( ( M @ A )
= ( some_nat @ B2 ) ) ) ).
% domD
thf(fact_307_domD,axiom,
! [A: option_val,M: option_val > option_val] :
( ( member_option_val @ A @ ( dom_option_val_val @ M ) )
=> ? [B2: val] :
( ( M @ A )
= ( some_val @ B2 ) ) ) ).
% domD
thf(fact_308_domD,axiom,
! [A: option_nat,M: option_nat > option_val] :
( ( member_option_nat @ A @ ( dom_option_nat_val @ M ) )
=> ? [B2: val] :
( ( M @ A )
= ( some_val @ B2 ) ) ) ).
% domD
thf(fact_309_domD,axiom,
! [A: set_nat,M: set_nat > option_val] :
( ( member_set_nat @ A @ ( dom_set_nat_val @ M ) )
=> ? [B2: val] :
( ( M @ A )
= ( some_val @ B2 ) ) ) ).
% domD
thf(fact_310_domD,axiom,
! [A: option_val,M: option_val > option_nat] :
( ( member_option_val @ A @ ( dom_option_val_nat @ M ) )
=> ? [B2: nat] :
( ( M @ A )
= ( some_nat @ B2 ) ) ) ).
% domD
thf(fact_311_domI,axiom,
! [M: nat > option_val,A: nat,B: val] :
( ( ( M @ A )
= ( some_val @ B ) )
=> ( member_nat @ A @ ( dom_nat_val @ M ) ) ) ).
% domI
thf(fact_312_domI,axiom,
! [M: val > option_val,A: val,B: val] :
( ( ( M @ A )
= ( some_val @ B ) )
=> ( member_val @ A @ ( dom_val_val @ M ) ) ) ).
% domI
thf(fact_313_domI,axiom,
! [M: $o > option_val,A: $o,B: val] :
( ( ( M @ A )
= ( some_val @ B ) )
=> ( member_o @ A @ ( dom_o_val @ M ) ) ) ).
% domI
thf(fact_314_domI,axiom,
! [M: nat > option_nat,A: nat,B: nat] :
( ( ( M @ A )
= ( some_nat @ B ) )
=> ( member_nat @ A @ ( dom_nat_nat @ M ) ) ) ).
% domI
thf(fact_315_domI,axiom,
! [M: val > option_nat,A: val,B: nat] :
( ( ( M @ A )
= ( some_nat @ B ) )
=> ( member_val @ A @ ( dom_val_nat @ M ) ) ) ).
% domI
thf(fact_316_domI,axiom,
! [M: $o > option_nat,A: $o,B: nat] :
( ( ( M @ A )
= ( some_nat @ B ) )
=> ( member_o @ A @ ( dom_o_nat @ M ) ) ) ).
% domI
thf(fact_317_domI,axiom,
! [M: option_val > option_val,A: option_val,B: val] :
( ( ( M @ A )
= ( some_val @ B ) )
=> ( member_option_val @ A @ ( dom_option_val_val @ M ) ) ) ).
% domI
thf(fact_318_domI,axiom,
! [M: option_nat > option_val,A: option_nat,B: val] :
( ( ( M @ A )
= ( some_val @ B ) )
=> ( member_option_nat @ A @ ( dom_option_nat_val @ M ) ) ) ).
% domI
thf(fact_319_domI,axiom,
! [M: set_nat > option_val,A: set_nat,B: val] :
( ( ( M @ A )
= ( some_val @ B ) )
=> ( member_set_nat @ A @ ( dom_set_nat_val @ M ) ) ) ).
% domI
thf(fact_320_domI,axiom,
! [M: option_val > option_nat,A: option_val,B: nat] :
( ( ( M @ A )
= ( some_nat @ B ) )
=> ( member_option_val @ A @ ( dom_option_val_nat @ M ) ) ) ).
% domI
thf(fact_321_domIff,axiom,
! [A: nat,M: nat > option_val] :
( ( member_nat @ A @ ( dom_nat_val @ M ) )
= ( ( M @ A )
!= none_val ) ) ).
% domIff
thf(fact_322_domIff,axiom,
! [A: val,M: val > option_val] :
( ( member_val @ A @ ( dom_val_val @ M ) )
= ( ( M @ A )
!= none_val ) ) ).
% domIff
thf(fact_323_domIff,axiom,
! [A: $o,M: $o > option_val] :
( ( member_o @ A @ ( dom_o_val @ M ) )
= ( ( M @ A )
!= none_val ) ) ).
% domIff
thf(fact_324_domIff,axiom,
! [A: nat,M: nat > option_nat] :
( ( member_nat @ A @ ( dom_nat_nat @ M ) )
= ( ( M @ A )
!= none_nat ) ) ).
% domIff
thf(fact_325_domIff,axiom,
! [A: val,M: val > option_nat] :
( ( member_val @ A @ ( dom_val_nat @ M ) )
= ( ( M @ A )
!= none_nat ) ) ).
% domIff
thf(fact_326_domIff,axiom,
! [A: $o,M: $o > option_nat] :
( ( member_o @ A @ ( dom_o_nat @ M ) )
= ( ( M @ A )
!= none_nat ) ) ).
% domIff
thf(fact_327_domIff,axiom,
! [A: option_val,M: option_val > option_val] :
( ( member_option_val @ A @ ( dom_option_val_val @ M ) )
= ( ( M @ A )
!= none_val ) ) ).
% domIff
thf(fact_328_domIff,axiom,
! [A: option_nat,M: option_nat > option_val] :
( ( member_option_nat @ A @ ( dom_option_nat_val @ M ) )
= ( ( M @ A )
!= none_val ) ) ).
% domIff
thf(fact_329_domIff,axiom,
! [A: set_nat,M: set_nat > option_val] :
( ( member_set_nat @ A @ ( dom_set_nat_val @ M ) )
= ( ( M @ A )
!= none_val ) ) ).
% domIff
thf(fact_330_domIff,axiom,
! [A: option_val,M: option_val > option_nat] :
( ( member_option_val @ A @ ( dom_option_val_nat @ M ) )
= ( ( M @ A )
!= none_nat ) ) ).
% domIff
thf(fact_331_map__add__dom__app__simps_I3_J,axiom,
! [M: nat,L2: nat > option_val,L1: nat > option_val] :
( ~ ( member_nat @ M @ ( dom_nat_val @ L2 ) )
=> ( ( map_add_nat_val @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_332_map__add__dom__app__simps_I3_J,axiom,
! [M: produc1457211279475724562t_char,L2: produc1457211279475724562t_char > option_val,L1: produc1457211279475724562t_char > option_val] :
( ~ ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ L2 ) )
=> ( ( map_ad2908576761495157028ar_val @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_333_map__add__dom__app__simps_I2_J,axiom,
! [M: nat,L1: nat > option_val,L2: nat > option_val] :
( ~ ( member_nat @ M @ ( dom_nat_val @ L1 ) )
=> ( ( map_add_nat_val @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_334_map__add__dom__app__simps_I2_J,axiom,
! [M: produc1457211279475724562t_char,L1: produc1457211279475724562t_char > option_val,L2: produc1457211279475724562t_char > option_val] :
( ~ ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ L1 ) )
=> ( ( map_ad2908576761495157028ar_val @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_335_map__add__dom__app__simps_I1_J,axiom,
! [M: nat,L2: nat > option_val,L1: nat > option_val] :
( ( member_nat @ M @ ( dom_nat_val @ L2 ) )
=> ( ( map_add_nat_val @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_336_map__add__dom__app__simps_I1_J,axiom,
! [M: produc1457211279475724562t_char,L2: produc1457211279475724562t_char > option_val,L1: produc1457211279475724562t_char > option_val] :
( ( member484235747325421115t_char @ M @ ( dom_Pr8989479155164304065ar_val @ L2 ) )
=> ( ( map_ad2908576761495157028ar_val @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_337_map__le__def,axiom,
( map_le_nat_val
= ( ^ [M_1: nat > option_val,M_2: nat > option_val] :
! [X2: nat] :
( ( member_nat @ X2 @ ( dom_nat_val @ M_1 ) )
=> ( ( M_1 @ X2 )
= ( M_2 @ X2 ) ) ) ) ) ).
% map_le_def
thf(fact_338_map__le__def,axiom,
( map_le3432596473071271956ar_val
= ( ^ [M_1: produc1457211279475724562t_char > option_val,M_2: produc1457211279475724562t_char > option_val] :
! [X2: produc1457211279475724562t_char] :
( ( member484235747325421115t_char @ X2 @ ( dom_Pr8989479155164304065ar_val @ M_1 ) )
=> ( ( M_1 @ X2 )
= ( M_2 @ X2 ) ) ) ) ) ).
% map_le_def
thf(fact_339_option_Oset__cases,axiom,
! [E: option_val,A: option_option_val] :
( ( member_option_val @ E @ ( set_op911010586042427349on_val @ A ) )
=> ( A
= ( some_option_val @ E ) ) ) ).
% option.set_cases
thf(fact_340_option_Oset__cases,axiom,
! [E: option_nat,A: option_option_nat] :
( ( member_option_nat @ E @ ( set_op3360498428384587026on_nat @ A ) )
=> ( A
= ( some_option_nat @ E ) ) ) ).
% option.set_cases
thf(fact_341_option_Oset__cases,axiom,
! [E: $o,A: option_o] :
( ( member_o @ E @ ( set_option_o2 @ A ) )
=> ( A
= ( some_o @ E ) ) ) ).
% option.set_cases
thf(fact_342_option_Oset__cases,axiom,
! [E: set_nat,A: option_set_nat] :
( ( member_set_nat @ E @ ( set_option_set_nat2 @ A ) )
=> ( A
= ( some_set_nat @ E ) ) ) ).
% option.set_cases
thf(fact_343_option_Oset__cases,axiom,
! [E: nat,A: option_nat] :
( ( member_nat @ E @ ( set_option_nat2 @ A ) )
=> ( A
= ( some_nat @ E ) ) ) ).
% option.set_cases
thf(fact_344_option_Oset__cases,axiom,
! [E: val,A: option_val] :
( ( member_val @ E @ ( set_option_val2 @ A ) )
=> ( A
= ( some_val @ E ) ) ) ).
% option.set_cases
thf(fact_345_option_Oset__intros,axiom,
! [X22: option_val] : ( member_option_val @ X22 @ ( set_op911010586042427349on_val @ ( some_option_val @ X22 ) ) ) ).
% option.set_intros
thf(fact_346_option_Oset__intros,axiom,
! [X22: option_nat] : ( member_option_nat @ X22 @ ( set_op3360498428384587026on_nat @ ( some_option_nat @ X22 ) ) ) ).
% option.set_intros
thf(fact_347_option_Oset__intros,axiom,
! [X22: $o] : ( member_o @ X22 @ ( set_option_o2 @ ( some_o @ X22 ) ) ) ).
% option.set_intros
thf(fact_348_option_Oset__intros,axiom,
! [X22: set_nat] : ( member_set_nat @ X22 @ ( set_option_set_nat2 @ ( some_set_nat @ X22 ) ) ) ).
% option.set_intros
thf(fact_349_option_Oset__intros,axiom,
! [X22: nat] : ( member_nat @ X22 @ ( set_option_nat2 @ ( some_nat @ X22 ) ) ) ).
% option.set_intros
thf(fact_350_option_Oset__intros,axiom,
! [X22: val] : ( member_val @ X22 @ ( set_option_val2 @ ( some_val @ X22 ) ) ) ).
% option.set_intros
thf(fact_351_ospec,axiom,
! [A5: option_nat,P: nat > $o,X3: nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_option_nat2 @ A5 ) )
=> ( P @ X ) )
=> ( ( A5
= ( some_nat @ X3 ) )
=> ( P @ X3 ) ) ) ).
% ospec
thf(fact_352_ospec,axiom,
! [A5: option_val,P: val > $o,X3: val] :
( ! [X: val] :
( ( member_val @ X @ ( set_option_val2 @ A5 ) )
=> ( P @ X ) )
=> ( ( A5
= ( some_val @ X3 ) )
=> ( P @ X3 ) ) ) ).
% ospec
thf(fact_353_bind__option__cong,axiom,
! [X3: option_val,Y3: option_val,F: val > option_val,G: val > option_val] :
( ( X3 = Y3 )
=> ( ! [Z2: val] :
( ( member_val @ Z2 @ ( set_option_val2 @ Y3 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( bind_val_val @ X3 @ F )
= ( bind_val_val @ Y3 @ G ) ) ) ) ).
% bind_option_cong
thf(fact_354_option_Opred__mono__strong,axiom,
! [P: option_val > $o,X3: option_option_val,Pa: option_val > $o] :
( ( pred_o8905241415863828810on_val @ P @ X3 )
=> ( ! [Z2: option_val] :
( ( member_option_val @ Z2 @ ( set_op911010586042427349on_val @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( pred_o8905241415863828810on_val @ Pa @ X3 ) ) ) ).
% option.pred_mono_strong
thf(fact_355_option_Opred__mono__strong,axiom,
! [P: option_nat > $o,X3: option_option_nat,Pa: option_nat > $o] :
( ( pred_o8911740758570919517on_nat @ P @ X3 )
=> ( ! [Z2: option_nat] :
( ( member_option_nat @ Z2 @ ( set_op3360498428384587026on_nat @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( pred_o8911740758570919517on_nat @ Pa @ X3 ) ) ) ).
% option.pred_mono_strong
thf(fact_356_option_Opred__mono__strong,axiom,
! [P: $o > $o,X3: option_o,Pa: $o > $o] :
( ( pred_option_o @ P @ X3 )
=> ( ! [Z2: $o] :
( ( member_o @ Z2 @ ( set_option_o2 @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( pred_option_o @ Pa @ X3 ) ) ) ).
% option.pred_mono_strong
thf(fact_357_option_Opred__mono__strong,axiom,
! [P: set_nat > $o,X3: option_set_nat,Pa: set_nat > $o] :
( ( pred_option_set_nat @ P @ X3 )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( set_option_set_nat2 @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( pred_option_set_nat @ Pa @ X3 ) ) ) ).
% option.pred_mono_strong
thf(fact_358_option_Opred__mono__strong,axiom,
! [P: val > $o,X3: option_val,Pa: val > $o] :
( ( pred_option_val @ P @ X3 )
=> ( ! [Z2: val] :
( ( member_val @ Z2 @ ( set_option_val2 @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( pred_option_val @ Pa @ X3 ) ) ) ).
% option.pred_mono_strong
thf(fact_359_option_Opred__mono__strong,axiom,
! [P: nat > $o,X3: option_nat,Pa: nat > $o] :
( ( pred_option_nat @ P @ X3 )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_option_nat2 @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( pred_option_nat @ Pa @ X3 ) ) ) ).
% option.pred_mono_strong
thf(fact_360_option_Opred__cong,axiom,
! [X3: option_option_val,Ya: option_option_val,P: option_val > $o,Pa: option_val > $o] :
( ( X3 = Ya )
=> ( ! [Z2: option_val] :
( ( member_option_val @ Z2 @ ( set_op911010586042427349on_val @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( pred_o8905241415863828810on_val @ P @ X3 )
= ( pred_o8905241415863828810on_val @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_361_option_Opred__cong,axiom,
! [X3: option_option_nat,Ya: option_option_nat,P: option_nat > $o,Pa: option_nat > $o] :
( ( X3 = Ya )
=> ( ! [Z2: option_nat] :
( ( member_option_nat @ Z2 @ ( set_op3360498428384587026on_nat @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( pred_o8911740758570919517on_nat @ P @ X3 )
= ( pred_o8911740758570919517on_nat @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_362_option_Opred__cong,axiom,
! [X3: option_o,Ya: option_o,P: $o > $o,Pa: $o > $o] :
( ( X3 = Ya )
=> ( ! [Z2: $o] :
( ( member_o @ Z2 @ ( set_option_o2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( pred_option_o @ P @ X3 )
= ( pred_option_o @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_363_option_Opred__cong,axiom,
! [X3: option_set_nat,Ya: option_set_nat,P: set_nat > $o,Pa: set_nat > $o] :
( ( X3 = Ya )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( set_option_set_nat2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( pred_option_set_nat @ P @ X3 )
= ( pred_option_set_nat @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_364_option_Opred__cong,axiom,
! [X3: option_val,Ya: option_val,P: val > $o,Pa: val > $o] :
( ( X3 = Ya )
=> ( ! [Z2: val] :
( ( member_val @ Z2 @ ( set_option_val2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( pred_option_val @ P @ X3 )
= ( pred_option_val @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_365_option_Opred__cong,axiom,
! [X3: option_nat,Ya: option_nat,P: nat > $o,Pa: nat > $o] :
( ( X3 = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_option_nat2 @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( pred_option_nat @ P @ X3 )
= ( pred_option_nat @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_366_curryD,axiom,
! [F: produc5122021904355189735on_val > $o,A: option_val,B: option_val] :
( ( produc3993299840032831053_val_o @ F @ A @ B )
=> ( F @ ( produc904852209897547607on_val @ A @ B ) ) ) ).
% curryD
thf(fact_367_curryD,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryD
thf(fact_368_curryD,axiom,
! [F: produc7819656566062154093et_nat > $o,A: set_nat,B: set_nat] :
( ( produc8481458855017059839_nat_o @ F @ A @ B )
=> ( F @ ( produc4532415448927165861et_nat @ A @ B ) ) ) ).
% curryD
thf(fact_369_curryE,axiom,
! [F: produc5122021904355189735on_val > $o,A: option_val,B: option_val] :
( ( produc3993299840032831053_val_o @ F @ A @ B )
=> ( F @ ( produc904852209897547607on_val @ A @ B ) ) ) ).
% curryE
thf(fact_370_curryE,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryE
thf(fact_371_curryE,axiom,
! [F: produc7819656566062154093et_nat > $o,A: set_nat,B: set_nat] :
( ( produc8481458855017059839_nat_o @ F @ A @ B )
=> ( F @ ( produc4532415448927165861et_nat @ A @ B ) ) ) ).
% curryE
thf(fact_372_None__notin__image__Some,axiom,
! [A5: set_nat] :
~ ( member_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ A5 ) ) ).
% None_notin_image_Some
thf(fact_373_None__notin__image__Some,axiom,
! [A5: set_val] :
~ ( member_option_val @ none_val @ ( image_val_option_val @ some_val @ A5 ) ) ).
% None_notin_image_Some
thf(fact_374_compatible__options_Opelims_I3_J,axiom,
! [X3: option_nat,Xa: option_nat] :
( ~ ( compat6420650667825766956ns_nat @ X3 @ Xa )
=> ( ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ X3 @ Xa ) )
=> ~ ! [A2: nat] :
( ( X3
= ( some_nat @ A2 ) )
=> ! [B2: nat] :
( ( Xa
= ( some_nat @ B2 ) )
=> ( ( accp_P8646395344606611882on_nat @ compat6933303284653735279el_nat @ ( produc5098337634421038937on_nat @ ( some_nat @ A2 ) @ ( some_nat @ B2 ) ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% compatible_options.pelims(3)
thf(fact_375_compatible__options_Opelims_I3_J,axiom,
! [X3: option_val,Xa: option_val] :
( ~ ( compat7209469420100717861ns_val @ X3 @ Xa )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa ) )
=> ~ ! [A2: val] :
( ( X3
= ( some_val @ A2 ) )
=> ! [B2: val] :
( ( Xa
= ( some_val @ B2 ) )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% compatible_options.pelims(3)
thf(fact_376_image__insert,axiom,
! [F: nat > nat,A: nat,B5: set_nat] :
( ( image_nat_nat2 @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ).
% image_insert
thf(fact_377_image__insert,axiom,
! [F: nat > $o,A: nat,B5: set_nat] :
( ( image_nat_o @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_o @ ( F @ A ) @ ( image_nat_o @ F @ B5 ) ) ) ).
% image_insert
thf(fact_378_image__insert,axiom,
! [F: nat > val,A: nat,B5: set_nat] :
( ( image_nat_val @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_val @ ( F @ A ) @ ( image_nat_val @ F @ B5 ) ) ) ).
% image_insert
thf(fact_379_image__insert,axiom,
! [F: val > nat,A: val,B5: set_val] :
( ( image_val_nat @ F @ ( insert_val @ A @ B5 ) )
= ( insert_nat @ ( F @ A ) @ ( image_val_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_380_image__insert,axiom,
! [F: val > val,A: val,B5: set_val] :
( ( image_val_val @ F @ ( insert_val @ A @ B5 ) )
= ( insert_val @ ( F @ A ) @ ( image_val_val @ F @ B5 ) ) ) ).
% image_insert
thf(fact_381_image__insert,axiom,
! [F: nat > option_val,A: nat,B5: set_nat] :
( ( image_nat_option_val @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_option_val @ ( F @ A ) @ ( image_nat_option_val @ F @ B5 ) ) ) ).
% image_insert
thf(fact_382_image__insert,axiom,
! [F: nat > option_nat,A: nat,B5: set_nat] :
( ( image_nat_option_nat @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_option_nat @ ( F @ A ) @ ( image_nat_option_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_383_image__insert,axiom,
! [F: nat > set_nat,A: nat,B5: set_nat] :
( ( image_nat_set_nat @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_384_image__insert,axiom,
! [F: option_val > nat,A: option_val,B5: set_option_val] :
( ( image_option_val_nat @ F @ ( insert_option_val @ A @ B5 ) )
= ( insert_nat @ ( F @ A ) @ ( image_option_val_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_385_image__insert,axiom,
! [F: option_val > val,A: option_val,B5: set_option_val] :
( ( image_option_val_val @ F @ ( insert_option_val @ A @ B5 ) )
= ( insert_val @ ( F @ A ) @ ( image_option_val_val @ F @ B5 ) ) ) ).
% image_insert
thf(fact_386_insert__image,axiom,
! [X3: nat,A5: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_nat_nat2 @ F @ A5 ) )
= ( image_nat_nat2 @ F @ A5 ) ) ) ).
% insert_image
thf(fact_387_insert__image,axiom,
! [X3: nat,A5: set_nat,F: nat > $o] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_o @ ( F @ X3 ) @ ( image_nat_o @ F @ A5 ) )
= ( image_nat_o @ F @ A5 ) ) ) ).
% insert_image
thf(fact_388_insert__image,axiom,
! [X3: nat,A5: set_nat,F: nat > val] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_val @ ( F @ X3 ) @ ( image_nat_val @ F @ A5 ) )
= ( image_nat_val @ F @ A5 ) ) ) ).
% insert_image
thf(fact_389_insert__image,axiom,
! [X3: val,A5: set_val,F: val > nat] :
( ( member_val @ X3 @ A5 )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_val_nat @ F @ A5 ) )
= ( image_val_nat @ F @ A5 ) ) ) ).
% insert_image
thf(fact_390_insert__image,axiom,
! [X3: val,A5: set_val,F: val > val] :
( ( member_val @ X3 @ A5 )
=> ( ( insert_val @ ( F @ X3 ) @ ( image_val_val @ F @ A5 ) )
= ( image_val_val @ F @ A5 ) ) ) ).
% insert_image
thf(fact_391_insert__image,axiom,
! [X3: $o,A5: set_o,F: $o > nat] :
( ( member_o @ X3 @ A5 )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_o_nat @ F @ A5 ) )
= ( image_o_nat @ F @ A5 ) ) ) ).
% insert_image
thf(fact_392_insert__image,axiom,
! [X3: $o,A5: set_o,F: $o > val] :
( ( member_o @ X3 @ A5 )
=> ( ( insert_val @ ( F @ X3 ) @ ( image_o_val @ F @ A5 ) )
= ( image_o_val @ F @ A5 ) ) ) ).
% insert_image
thf(fact_393_insert__image,axiom,
! [X3: nat,A5: set_nat,F: nat > option_val] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_option_val @ ( F @ X3 ) @ ( image_nat_option_val @ F @ A5 ) )
= ( image_nat_option_val @ F @ A5 ) ) ) ).
% insert_image
thf(fact_394_insert__image,axiom,
! [X3: nat,A5: set_nat,F: nat > option_nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_option_nat @ ( F @ X3 ) @ ( image_nat_option_nat @ F @ A5 ) )
= ( image_nat_option_nat @ F @ A5 ) ) ) ).
% insert_image
thf(fact_395_insert__image,axiom,
! [X3: nat,A5: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_set_nat @ ( F @ X3 ) @ ( image_nat_set_nat @ F @ A5 ) )
= ( image_nat_set_nat @ F @ A5 ) ) ) ).
% insert_image
thf(fact_396_insertCI,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ~ ( member8440522571783428010at_nat @ A @ B5 )
=> ( A = B ) )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_397_insertCI,axiom,
! [A: option_val,B5: set_option_val,B: option_val] :
( ( ~ ( member_option_val @ A @ B5 )
=> ( A = B ) )
=> ( member_option_val @ A @ ( insert_option_val @ B @ B5 ) ) ) ).
% insertCI
thf(fact_398_insertCI,axiom,
! [A: option_nat,B5: set_option_nat,B: option_nat] :
( ( ~ ( member_option_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_option_nat @ A @ ( insert_option_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_399_insertCI,axiom,
! [A: val,B5: set_val,B: val] :
( ( ~ ( member_val @ A @ B5 )
=> ( A = B ) )
=> ( member_val @ A @ ( insert_val @ B @ B5 ) ) ) ).
% insertCI
thf(fact_400_insertCI,axiom,
! [A: $o,B5: set_o,B: $o] :
( ( ~ ( member_o @ A @ B5 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).
% insertCI
thf(fact_401_insertCI,axiom,
! [A: set_nat,B5: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_402_insertCI,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_403_insert__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A5 ) )
= ( ( A = B )
| ( member8440522571783428010at_nat @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_404_insert__iff,axiom,
! [A: option_val,B: option_val,A5: set_option_val] :
( ( member_option_val @ A @ ( insert_option_val @ B @ A5 ) )
= ( ( A = B )
| ( member_option_val @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_405_insert__iff,axiom,
! [A: option_nat,B: option_nat,A5: set_option_nat] :
( ( member_option_nat @ A @ ( insert_option_nat @ B @ A5 ) )
= ( ( A = B )
| ( member_option_nat @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_406_insert__iff,axiom,
! [A: val,B: val,A5: set_val] :
( ( member_val @ A @ ( insert_val @ B @ A5 ) )
= ( ( A = B )
| ( member_val @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_407_insert__iff,axiom,
! [A: $o,B: $o,A5: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A5 ) )
= ( ( A = B )
| ( member_o @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_408_insert__iff,axiom,
! [A: set_nat,B: set_nat,A5: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A5 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_409_insert__iff,axiom,
! [A: nat,B: nat,A5: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A5 ) )
= ( ( A = B )
| ( member_nat @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_410_insert__absorb2,axiom,
! [X3: option_val,A5: set_option_val] :
( ( insert_option_val @ X3 @ ( insert_option_val @ X3 @ A5 ) )
= ( insert_option_val @ X3 @ A5 ) ) ).
% insert_absorb2
thf(fact_411_insert__absorb2,axiom,
! [X3: val,A5: set_val] :
( ( insert_val @ X3 @ ( insert_val @ X3 @ A5 ) )
= ( insert_val @ X3 @ A5 ) ) ).
% insert_absorb2
thf(fact_412_insert__absorb2,axiom,
! [X3: option_nat,A5: set_option_nat] :
( ( insert_option_nat @ X3 @ ( insert_option_nat @ X3 @ A5 ) )
= ( insert_option_nat @ X3 @ A5 ) ) ).
% insert_absorb2
thf(fact_413_insert__absorb2,axiom,
! [X3: set_nat,A5: set_set_nat] :
( ( insert_set_nat @ X3 @ ( insert_set_nat @ X3 @ A5 ) )
= ( insert_set_nat @ X3 @ A5 ) ) ).
% insert_absorb2
thf(fact_414_insert__absorb2,axiom,
! [X3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ X3 @ A5 ) )
= ( insert8211810215607154385at_nat @ X3 @ A5 ) ) ).
% insert_absorb2
thf(fact_415_insert__absorb2,axiom,
! [X3: nat,A5: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A5 ) )
= ( insert_nat @ X3 @ A5 ) ) ).
% insert_absorb2
thf(fact_416_image__eqI,axiom,
! [B: nat,F: nat > nat,X3: nat,A5: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member_nat @ B @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_417_image__eqI,axiom,
! [B: val,F: nat > val,X3: nat,A5: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member_val @ B @ ( image_nat_val @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_418_image__eqI,axiom,
! [B: $o,F: nat > $o,X3: nat,A5: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member_o @ B @ ( image_nat_o @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_419_image__eqI,axiom,
! [B: nat,F: val > nat,X3: val,A5: set_val] :
( ( B
= ( F @ X3 ) )
=> ( ( member_val @ X3 @ A5 )
=> ( member_nat @ B @ ( image_val_nat @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_420_image__eqI,axiom,
! [B: val,F: val > val,X3: val,A5: set_val] :
( ( B
= ( F @ X3 ) )
=> ( ( member_val @ X3 @ A5 )
=> ( member_val @ B @ ( image_val_val @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_421_image__eqI,axiom,
! [B: $o,F: val > $o,X3: val,A5: set_val] :
( ( B
= ( F @ X3 ) )
=> ( ( member_val @ X3 @ A5 )
=> ( member_o @ B @ ( image_val_o @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_422_image__eqI,axiom,
! [B: nat,F: $o > nat,X3: $o,A5: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A5 )
=> ( member_nat @ B @ ( image_o_nat @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_423_image__eqI,axiom,
! [B: val,F: $o > val,X3: $o,A5: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A5 )
=> ( member_val @ B @ ( image_o_val @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_424_image__eqI,axiom,
! [B: $o,F: $o > $o,X3: $o,A5: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A5 )
=> ( member_o @ B @ ( image_o_o @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_425_image__eqI,axiom,
! [B: option_val,F: nat > option_val,X3: nat,A5: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member_option_val @ B @ ( image_nat_option_val @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_426_ran__map__upd,axiom,
! [M: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char,B: val] :
( ( ( M @ A )
= none_val )
=> ( ( ran_Pr786515303459910110ar_val @ ( fun_up5735975307950539954on_val @ M @ A @ ( some_val @ B ) ) )
= ( insert_val @ B @ ( ran_Pr786515303459910110ar_val @ M ) ) ) ) ).
% ran_map_upd
thf(fact_427_ran__map__upd,axiom,
! [M: nat > option_val,A: nat,B: val] :
( ( ( M @ A )
= none_val )
=> ( ( ran_nat_val @ ( fun_up810121132392276792on_val @ M @ A @ ( some_val @ B ) ) )
= ( insert_val @ B @ ( ran_nat_val @ M ) ) ) ) ).
% ran_map_upd
thf(fact_428_override__on__insert_H,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,X5: set_Pr2507339680178222706t_char] :
( ( overri6326584771168529523on_val @ F @ G @ ( insert3372640069787401954t_char @ X3 @ X5 ) )
= ( overri6326584771168529523on_val @ ( fun_up5735975307950539954on_val @ F @ X3 @ ( G @ X3 ) ) @ G @ X5 ) ) ).
% override_on_insert'
thf(fact_429_override__on__insert_H,axiom,
! [F: nat > option_val,G: nat > option_val,X3: nat,X5: set_nat] :
( ( overri87716735465204535on_val @ F @ G @ ( insert_nat @ X3 @ X5 ) )
= ( overri87716735465204535on_val @ ( fun_up810121132392276792on_val @ F @ X3 @ ( G @ X3 ) ) @ G @ X5 ) ) ).
% override_on_insert'
thf(fact_430_override__on__insert_H,axiom,
! [F: nat > nat,G: nat > nat,X3: nat,X5: set_nat] :
( ( override_on_nat_nat @ F @ G @ ( insert_nat @ X3 @ X5 ) )
= ( override_on_nat_nat @ ( fun_upd_nat_nat @ F @ X3 @ ( G @ X3 ) ) @ G @ X5 ) ) ).
% override_on_insert'
thf(fact_431_override__on__insert_H,axiom,
! [F: nat > set_nat,G: nat > set_nat,X3: nat,X5: set_nat] :
( ( overri358428573850456982et_nat @ F @ G @ ( insert_nat @ X3 @ X5 ) )
= ( overri358428573850456982et_nat @ ( fun_upd_nat_set_nat @ F @ X3 @ ( G @ X3 ) ) @ G @ X5 ) ) ).
% override_on_insert'
thf(fact_432_override__on__insert,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char,X5: set_Pr2507339680178222706t_char] :
( ( overri6326584771168529523on_val @ F @ G @ ( insert3372640069787401954t_char @ X3 @ X5 ) )
= ( fun_up5735975307950539954on_val @ ( overri6326584771168529523on_val @ F @ G @ X5 ) @ X3 @ ( G @ X3 ) ) ) ).
% override_on_insert
thf(fact_433_override__on__insert,axiom,
! [F: nat > option_val,G: nat > option_val,X3: nat,X5: set_nat] :
( ( overri87716735465204535on_val @ F @ G @ ( insert_nat @ X3 @ X5 ) )
= ( fun_up810121132392276792on_val @ ( overri87716735465204535on_val @ F @ G @ X5 ) @ X3 @ ( G @ X3 ) ) ) ).
% override_on_insert
thf(fact_434_override__on__insert,axiom,
! [F: nat > nat,G: nat > nat,X3: nat,X5: set_nat] :
( ( override_on_nat_nat @ F @ G @ ( insert_nat @ X3 @ X5 ) )
= ( fun_upd_nat_nat @ ( override_on_nat_nat @ F @ G @ X5 ) @ X3 @ ( G @ X3 ) ) ) ).
% override_on_insert
thf(fact_435_override__on__insert,axiom,
! [F: nat > set_nat,G: nat > set_nat,X3: nat,X5: set_nat] :
( ( overri358428573850456982et_nat @ F @ G @ ( insert_nat @ X3 @ X5 ) )
= ( fun_upd_nat_set_nat @ ( overri358428573850456982et_nat @ F @ G @ X5 ) @ X3 @ ( G @ X3 ) ) ) ).
% override_on_insert
thf(fact_436_ranI,axiom,
! [M: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char,B: val] :
( ( ( M @ A )
= ( some_val @ B ) )
=> ( member_val @ B @ ( ran_Pr786515303459910110ar_val @ M ) ) ) ).
% ranI
thf(fact_437_rev__image__eqI,axiom,
! [X3: nat,A5: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_438_rev__image__eqI,axiom,
! [X3: nat,A5: set_nat,B: val,F: nat > val] :
( ( member_nat @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_val @ B @ ( image_nat_val @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_439_rev__image__eqI,axiom,
! [X3: nat,A5: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_440_rev__image__eqI,axiom,
! [X3: val,A5: set_val,B: nat,F: val > nat] :
( ( member_val @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_val_nat @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_441_rev__image__eqI,axiom,
! [X3: val,A5: set_val,B: val,F: val > val] :
( ( member_val @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_val @ B @ ( image_val_val @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_442_rev__image__eqI,axiom,
! [X3: val,A5: set_val,B: $o,F: val > $o] :
( ( member_val @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_val_o @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_443_rev__image__eqI,axiom,
! [X3: $o,A5: set_o,B: nat,F: $o > nat] :
( ( member_o @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_444_rev__image__eqI,axiom,
! [X3: $o,A5: set_o,B: val,F: $o > val] :
( ( member_o @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_val @ B @ ( image_o_val @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_445_rev__image__eqI,axiom,
! [X3: $o,A5: set_o,B: $o,F: $o > $o] :
( ( member_o @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_o_o @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_446_rev__image__eqI,axiom,
! [X3: nat,A5: set_nat,B: option_val,F: nat > option_val] :
( ( member_nat @ X3 @ A5 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_option_val @ B @ ( image_nat_option_val @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_447_ball__imageD,axiom,
! [F: nat > option_val,A5: set_nat,P: option_val > $o] :
( ! [X: option_val] :
( ( member_option_val @ X @ ( image_nat_option_val @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A5 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_448_ball__imageD,axiom,
! [F: nat > option_nat,A5: set_nat,P: option_nat > $o] :
( ! [X: option_nat] :
( ( member_option_nat @ X @ ( image_nat_option_nat @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A5 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_449_ball__imageD,axiom,
! [F: nat > $o,A5: set_nat,P: $o > $o] :
( ! [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A5 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_450_ball__imageD,axiom,
! [F: nat > set_nat,A5: set_nat,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A5 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_451_ball__imageD,axiom,
! [F: ( nat > $o ) > set_nat,A5: set_nat_o,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_o_set_nat @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X6: nat > $o] :
( ( member_nat_o @ X6 @ A5 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_452_ball__imageD,axiom,
! [F: nat > nat,A5: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat2 @ F @ A5 ) )
=> ( P @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A5 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_453_image__cong,axiom,
! [M4: set_nat_o,N2: set_nat_o,F: ( nat > $o ) > set_nat,G: ( nat > $o ) > set_nat] :
( ( M4 = N2 )
=> ( ! [X: nat > $o] :
( ( member_nat_o @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_o_set_nat @ F @ M4 )
= ( image_nat_o_set_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_454_image__cong,axiom,
! [M4: set_nat,N2: set_nat,F: nat > option_val,G: nat > option_val] :
( ( M4 = N2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_option_val @ F @ M4 )
= ( image_nat_option_val @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_455_image__cong,axiom,
! [M4: set_nat,N2: set_nat,F: nat > option_nat,G: nat > option_nat] :
( ( M4 = N2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_option_nat @ F @ M4 )
= ( image_nat_option_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_456_image__cong,axiom,
! [M4: set_nat,N2: set_nat,F: nat > $o,G: nat > $o] :
( ( M4 = N2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_o @ F @ M4 )
= ( image_nat_o @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_457_image__cong,axiom,
! [M4: set_nat,N2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M4 = N2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_set_nat @ F @ M4 )
= ( image_nat_set_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_458_image__cong,axiom,
! [M4: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M4 = N2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat2 @ F @ M4 )
= ( image_nat_nat2 @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_459_bex__imageD,axiom,
! [F: nat > option_val,A5: set_nat,P: option_val > $o] :
( ? [X6: option_val] :
( ( member_option_val @ X6 @ ( image_nat_option_val @ F @ A5 ) )
& ( P @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_460_bex__imageD,axiom,
! [F: nat > option_nat,A5: set_nat,P: option_nat > $o] :
( ? [X6: option_nat] :
( ( member_option_nat @ X6 @ ( image_nat_option_nat @ F @ A5 ) )
& ( P @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_461_bex__imageD,axiom,
! [F: nat > $o,A5: set_nat,P: $o > $o] :
( ? [X6: $o] :
( ( member_o @ X6 @ ( image_nat_o @ F @ A5 ) )
& ( P @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_462_bex__imageD,axiom,
! [F: nat > set_nat,A5: set_nat,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_nat_set_nat @ F @ A5 ) )
& ( P @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_463_bex__imageD,axiom,
! [F: ( nat > $o ) > set_nat,A5: set_nat_o,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_nat_o_set_nat @ F @ A5 ) )
& ( P @ X6 ) )
=> ? [X: nat > $o] :
( ( member_nat_o @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_464_bex__imageD,axiom,
! [F: nat > nat,A5: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat2 @ F @ A5 ) )
& ( P @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_465_image__iff,axiom,
! [Z: option_val,F: nat > option_val,A5: set_nat] :
( ( member_option_val @ Z @ ( image_nat_option_val @ F @ A5 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_466_image__iff,axiom,
! [Z: option_nat,F: nat > option_nat,A5: set_nat] :
( ( member_option_nat @ Z @ ( image_nat_option_nat @ F @ A5 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_467_image__iff,axiom,
! [Z: $o,F: nat > $o,A5: set_nat] :
( ( member_o @ Z @ ( image_nat_o @ F @ A5 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_468_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A5: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A5 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_469_image__iff,axiom,
! [Z: set_nat,F: ( nat > $o ) > set_nat,A5: set_nat_o] :
( ( member_set_nat @ Z @ ( image_nat_o_set_nat @ F @ A5 ) )
= ( ? [X2: nat > $o] :
( ( member_nat_o @ X2 @ A5 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_470_image__iff,axiom,
! [Z: nat,F: nat > nat,A5: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat2 @ F @ A5 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_471_imageI,axiom,
! [X3: nat,A5: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat2 @ F @ A5 ) ) ) ).
% imageI
thf(fact_472_imageI,axiom,
! [X3: nat,A5: set_nat,F: nat > val] :
( ( member_nat @ X3 @ A5 )
=> ( member_val @ ( F @ X3 ) @ ( image_nat_val @ F @ A5 ) ) ) ).
% imageI
thf(fact_473_imageI,axiom,
! [X3: nat,A5: set_nat,F: nat > $o] :
( ( member_nat @ X3 @ A5 )
=> ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ A5 ) ) ) ).
% imageI
thf(fact_474_imageI,axiom,
! [X3: val,A5: set_val,F: val > nat] :
( ( member_val @ X3 @ A5 )
=> ( member_nat @ ( F @ X3 ) @ ( image_val_nat @ F @ A5 ) ) ) ).
% imageI
thf(fact_475_imageI,axiom,
! [X3: val,A5: set_val,F: val > val] :
( ( member_val @ X3 @ A5 )
=> ( member_val @ ( F @ X3 ) @ ( image_val_val @ F @ A5 ) ) ) ).
% imageI
thf(fact_476_imageI,axiom,
! [X3: val,A5: set_val,F: val > $o] :
( ( member_val @ X3 @ A5 )
=> ( member_o @ ( F @ X3 ) @ ( image_val_o @ F @ A5 ) ) ) ).
% imageI
thf(fact_477_imageI,axiom,
! [X3: $o,A5: set_o,F: $o > nat] :
( ( member_o @ X3 @ A5 )
=> ( member_nat @ ( F @ X3 ) @ ( image_o_nat @ F @ A5 ) ) ) ).
% imageI
thf(fact_478_imageI,axiom,
! [X3: $o,A5: set_o,F: $o > val] :
( ( member_o @ X3 @ A5 )
=> ( member_val @ ( F @ X3 ) @ ( image_o_val @ F @ A5 ) ) ) ).
% imageI
thf(fact_479_imageI,axiom,
! [X3: $o,A5: set_o,F: $o > $o] :
( ( member_o @ X3 @ A5 )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A5 ) ) ) ).
% imageI
thf(fact_480_imageI,axiom,
! [X3: nat,A5: set_nat,F: nat > option_val] :
( ( member_nat @ X3 @ A5 )
=> ( member_option_val @ ( F @ X3 ) @ ( image_nat_option_val @ F @ A5 ) ) ) ).
% imageI
thf(fact_481_mk__disjoint__insert,axiom,
! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A5 )
=> ? [B6: set_Pr1261947904930325089at_nat] :
( ( A5
= ( insert8211810215607154385at_nat @ A @ B6 ) )
& ~ ( member8440522571783428010at_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_482_mk__disjoint__insert,axiom,
! [A: option_val,A5: set_option_val] :
( ( member_option_val @ A @ A5 )
=> ? [B6: set_option_val] :
( ( A5
= ( insert_option_val @ A @ B6 ) )
& ~ ( member_option_val @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_483_mk__disjoint__insert,axiom,
! [A: option_nat,A5: set_option_nat] :
( ( member_option_nat @ A @ A5 )
=> ? [B6: set_option_nat] :
( ( A5
= ( insert_option_nat @ A @ B6 ) )
& ~ ( member_option_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_484_mk__disjoint__insert,axiom,
! [A: val,A5: set_val] :
( ( member_val @ A @ A5 )
=> ? [B6: set_val] :
( ( A5
= ( insert_val @ A @ B6 ) )
& ~ ( member_val @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_485_mk__disjoint__insert,axiom,
! [A: $o,A5: set_o] :
( ( member_o @ A @ A5 )
=> ? [B6: set_o] :
( ( A5
= ( insert_o @ A @ B6 ) )
& ~ ( member_o @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_486_mk__disjoint__insert,axiom,
! [A: set_nat,A5: set_set_nat] :
( ( member_set_nat @ A @ A5 )
=> ? [B6: set_set_nat] :
( ( A5
= ( insert_set_nat @ A @ B6 ) )
& ~ ( member_set_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_487_mk__disjoint__insert,axiom,
! [A: nat,A5: set_nat] :
( ( member_nat @ A @ A5 )
=> ? [B6: set_nat] :
( ( A5
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_488_insert__commute,axiom,
! [X3: option_val,Y3: option_val,A5: set_option_val] :
( ( insert_option_val @ X3 @ ( insert_option_val @ Y3 @ A5 ) )
= ( insert_option_val @ Y3 @ ( insert_option_val @ X3 @ A5 ) ) ) ).
% insert_commute
thf(fact_489_insert__commute,axiom,
! [X3: val,Y3: val,A5: set_val] :
( ( insert_val @ X3 @ ( insert_val @ Y3 @ A5 ) )
= ( insert_val @ Y3 @ ( insert_val @ X3 @ A5 ) ) ) ).
% insert_commute
thf(fact_490_insert__commute,axiom,
! [X3: option_nat,Y3: option_nat,A5: set_option_nat] :
( ( insert_option_nat @ X3 @ ( insert_option_nat @ Y3 @ A5 ) )
= ( insert_option_nat @ Y3 @ ( insert_option_nat @ X3 @ A5 ) ) ) ).
% insert_commute
thf(fact_491_insert__commute,axiom,
! [X3: set_nat,Y3: set_nat,A5: set_set_nat] :
( ( insert_set_nat @ X3 @ ( insert_set_nat @ Y3 @ A5 ) )
= ( insert_set_nat @ Y3 @ ( insert_set_nat @ X3 @ A5 ) ) ) ).
% insert_commute
thf(fact_492_insert__commute,axiom,
! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y3 @ A5 ) )
= ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X3 @ A5 ) ) ) ).
% insert_commute
thf(fact_493_insert__commute,axiom,
! [X3: nat,Y3: nat,A5: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ Y3 @ A5 ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X3 @ A5 ) ) ) ).
% insert_commute
thf(fact_494_insert__eq__iff,axiom,
! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ A5 )
=> ( ~ ( member8440522571783428010at_nat @ B @ B5 )
=> ( ( ( insert8211810215607154385at_nat @ A @ A5 )
= ( insert8211810215607154385at_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C2: set_Pr1261947904930325089at_nat] :
( ( A5
= ( insert8211810215607154385at_nat @ B @ C2 ) )
& ~ ( member8440522571783428010at_nat @ B @ C2 )
& ( B5
= ( insert8211810215607154385at_nat @ A @ C2 ) )
& ~ ( member8440522571783428010at_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_495_insert__eq__iff,axiom,
! [A: option_val,A5: set_option_val,B: option_val,B5: set_option_val] :
( ~ ( member_option_val @ A @ A5 )
=> ( ~ ( member_option_val @ B @ B5 )
=> ( ( ( insert_option_val @ A @ A5 )
= ( insert_option_val @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C2: set_option_val] :
( ( A5
= ( insert_option_val @ B @ C2 ) )
& ~ ( member_option_val @ B @ C2 )
& ( B5
= ( insert_option_val @ A @ C2 ) )
& ~ ( member_option_val @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_496_insert__eq__iff,axiom,
! [A: option_nat,A5: set_option_nat,B: option_nat,B5: set_option_nat] :
( ~ ( member_option_nat @ A @ A5 )
=> ( ~ ( member_option_nat @ B @ B5 )
=> ( ( ( insert_option_nat @ A @ A5 )
= ( insert_option_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C2: set_option_nat] :
( ( A5
= ( insert_option_nat @ B @ C2 ) )
& ~ ( member_option_nat @ B @ C2 )
& ( B5
= ( insert_option_nat @ A @ C2 ) )
& ~ ( member_option_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_497_insert__eq__iff,axiom,
! [A: val,A5: set_val,B: val,B5: set_val] :
( ~ ( member_val @ A @ A5 )
=> ( ~ ( member_val @ B @ B5 )
=> ( ( ( insert_val @ A @ A5 )
= ( insert_val @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C2: set_val] :
( ( A5
= ( insert_val @ B @ C2 ) )
& ~ ( member_val @ B @ C2 )
& ( B5
= ( insert_val @ A @ C2 ) )
& ~ ( member_val @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_498_insert__eq__iff,axiom,
! [A: $o,A5: set_o,B: $o,B5: set_o] :
( ~ ( member_o @ A @ A5 )
=> ( ~ ( member_o @ B @ B5 )
=> ( ( ( insert_o @ A @ A5 )
= ( insert_o @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A = ~ B )
=> ? [C2: set_o] :
( ( A5
= ( insert_o @ B @ C2 ) )
& ~ ( member_o @ B @ C2 )
& ( B5
= ( insert_o @ A @ C2 ) )
& ~ ( member_o @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_499_insert__eq__iff,axiom,
! [A: set_nat,A5: set_set_nat,B: set_nat,B5: set_set_nat] :
( ~ ( member_set_nat @ A @ A5 )
=> ( ~ ( member_set_nat @ B @ B5 )
=> ( ( ( insert_set_nat @ A @ A5 )
= ( insert_set_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C2: set_set_nat] :
( ( A5
= ( insert_set_nat @ B @ C2 ) )
& ~ ( member_set_nat @ B @ C2 )
& ( B5
= ( insert_set_nat @ A @ C2 ) )
& ~ ( member_set_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_500_insert__eq__iff,axiom,
! [A: nat,A5: set_nat,B: nat,B5: set_nat] :
( ~ ( member_nat @ A @ A5 )
=> ( ~ ( member_nat @ B @ B5 )
=> ( ( ( insert_nat @ A @ A5 )
= ( insert_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C2: set_nat] :
( ( A5
= ( insert_nat @ B @ C2 ) )
& ~ ( member_nat @ B @ C2 )
& ( B5
= ( insert_nat @ A @ C2 ) )
& ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_501_insert__absorb,axiom,
! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A5 )
=> ( ( insert8211810215607154385at_nat @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_502_insert__absorb,axiom,
! [A: option_val,A5: set_option_val] :
( ( member_option_val @ A @ A5 )
=> ( ( insert_option_val @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_503_insert__absorb,axiom,
! [A: option_nat,A5: set_option_nat] :
( ( member_option_nat @ A @ A5 )
=> ( ( insert_option_nat @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_504_insert__absorb,axiom,
! [A: val,A5: set_val] :
( ( member_val @ A @ A5 )
=> ( ( insert_val @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_505_insert__absorb,axiom,
! [A: $o,A5: set_o] :
( ( member_o @ A @ A5 )
=> ( ( insert_o @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_506_insert__absorb,axiom,
! [A: set_nat,A5: set_set_nat] :
( ( member_set_nat @ A @ A5 )
=> ( ( insert_set_nat @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_507_insert__absorb,axiom,
! [A: nat,A5: set_nat] :
( ( member_nat @ A @ A5 )
=> ( ( insert_nat @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_508_insert__ident,axiom,
! [X3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X3 @ A5 )
=> ( ~ ( member8440522571783428010at_nat @ X3 @ B5 )
=> ( ( ( insert8211810215607154385at_nat @ X3 @ A5 )
= ( insert8211810215607154385at_nat @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_509_insert__ident,axiom,
! [X3: option_val,A5: set_option_val,B5: set_option_val] :
( ~ ( member_option_val @ X3 @ A5 )
=> ( ~ ( member_option_val @ X3 @ B5 )
=> ( ( ( insert_option_val @ X3 @ A5 )
= ( insert_option_val @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_510_insert__ident,axiom,
! [X3: option_nat,A5: set_option_nat,B5: set_option_nat] :
( ~ ( member_option_nat @ X3 @ A5 )
=> ( ~ ( member_option_nat @ X3 @ B5 )
=> ( ( ( insert_option_nat @ X3 @ A5 )
= ( insert_option_nat @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_511_insert__ident,axiom,
! [X3: val,A5: set_val,B5: set_val] :
( ~ ( member_val @ X3 @ A5 )
=> ( ~ ( member_val @ X3 @ B5 )
=> ( ( ( insert_val @ X3 @ A5 )
= ( insert_val @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_512_insert__ident,axiom,
! [X3: $o,A5: set_o,B5: set_o] :
( ~ ( member_o @ X3 @ A5 )
=> ( ~ ( member_o @ X3 @ B5 )
=> ( ( ( insert_o @ X3 @ A5 )
= ( insert_o @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_513_insert__ident,axiom,
! [X3: set_nat,A5: set_set_nat,B5: set_set_nat] :
( ~ ( member_set_nat @ X3 @ A5 )
=> ( ~ ( member_set_nat @ X3 @ B5 )
=> ( ( ( insert_set_nat @ X3 @ A5 )
= ( insert_set_nat @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_514_insert__ident,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ~ ( member_nat @ X3 @ B5 )
=> ( ( ( insert_nat @ X3 @ A5 )
= ( insert_nat @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_515_Set_Oset__insert,axiom,
! [X3: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X3 @ A5 )
=> ~ ! [B6: set_Pr1261947904930325089at_nat] :
( ( A5
= ( insert8211810215607154385at_nat @ X3 @ B6 ) )
=> ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_516_Set_Oset__insert,axiom,
! [X3: option_val,A5: set_option_val] :
( ( member_option_val @ X3 @ A5 )
=> ~ ! [B6: set_option_val] :
( ( A5
= ( insert_option_val @ X3 @ B6 ) )
=> ( member_option_val @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_517_Set_Oset__insert,axiom,
! [X3: option_nat,A5: set_option_nat] :
( ( member_option_nat @ X3 @ A5 )
=> ~ ! [B6: set_option_nat] :
( ( A5
= ( insert_option_nat @ X3 @ B6 ) )
=> ( member_option_nat @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_518_Set_Oset__insert,axiom,
! [X3: val,A5: set_val] :
( ( member_val @ X3 @ A5 )
=> ~ ! [B6: set_val] :
( ( A5
= ( insert_val @ X3 @ B6 ) )
=> ( member_val @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_519_Set_Oset__insert,axiom,
! [X3: $o,A5: set_o] :
( ( member_o @ X3 @ A5 )
=> ~ ! [B6: set_o] :
( ( A5
= ( insert_o @ X3 @ B6 ) )
=> ( member_o @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_520_Set_Oset__insert,axiom,
! [X3: set_nat,A5: set_set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ~ ! [B6: set_set_nat] :
( ( A5
= ( insert_set_nat @ X3 @ B6 ) )
=> ( member_set_nat @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_521_Set_Oset__insert,axiom,
! [X3: nat,A5: set_nat] :
( ( member_nat @ X3 @ A5 )
=> ~ ! [B6: set_nat] :
( ( A5
= ( insert_nat @ X3 @ B6 ) )
=> ( member_nat @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_522_insertI2,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A @ B5 )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_523_insertI2,axiom,
! [A: option_val,B5: set_option_val,B: option_val] :
( ( member_option_val @ A @ B5 )
=> ( member_option_val @ A @ ( insert_option_val @ B @ B5 ) ) ) ).
% insertI2
thf(fact_524_insertI2,axiom,
! [A: option_nat,B5: set_option_nat,B: option_nat] :
( ( member_option_nat @ A @ B5 )
=> ( member_option_nat @ A @ ( insert_option_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_525_insertI2,axiom,
! [A: val,B5: set_val,B: val] :
( ( member_val @ A @ B5 )
=> ( member_val @ A @ ( insert_val @ B @ B5 ) ) ) ).
% insertI2
thf(fact_526_insertI2,axiom,
! [A: $o,B5: set_o,B: $o] :
( ( member_o @ A @ B5 )
=> ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).
% insertI2
thf(fact_527_insertI2,axiom,
! [A: set_nat,B5: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ B5 )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_528_insertI2,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( member_nat @ A @ B5 )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_529_insertI1,axiom,
! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_530_insertI1,axiom,
! [A: option_val,B5: set_option_val] : ( member_option_val @ A @ ( insert_option_val @ A @ B5 ) ) ).
% insertI1
thf(fact_531_insertI1,axiom,
! [A: option_nat,B5: set_option_nat] : ( member_option_nat @ A @ ( insert_option_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_532_insertI1,axiom,
! [A: val,B5: set_val] : ( member_val @ A @ ( insert_val @ A @ B5 ) ) ).
% insertI1
thf(fact_533_insertI1,axiom,
! [A: $o,B5: set_o] : ( member_o @ A @ ( insert_o @ A @ B5 ) ) ).
% insertI1
thf(fact_534_insertI1,axiom,
! [A: set_nat,B5: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_535_insertI1,axiom,
! [A: nat,B5: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_536_insertE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A5 ) )
=> ( ( A != B )
=> ( member8440522571783428010at_nat @ A @ A5 ) ) ) ).
% insertE
thf(fact_537_insertE,axiom,
! [A: option_val,B: option_val,A5: set_option_val] :
( ( member_option_val @ A @ ( insert_option_val @ B @ A5 ) )
=> ( ( A != B )
=> ( member_option_val @ A @ A5 ) ) ) ).
% insertE
thf(fact_538_insertE,axiom,
! [A: option_nat,B: option_nat,A5: set_option_nat] :
( ( member_option_nat @ A @ ( insert_option_nat @ B @ A5 ) )
=> ( ( A != B )
=> ( member_option_nat @ A @ A5 ) ) ) ).
% insertE
thf(fact_539_insertE,axiom,
! [A: val,B: val,A5: set_val] :
( ( member_val @ A @ ( insert_val @ B @ A5 ) )
=> ( ( A != B )
=> ( member_val @ A @ A5 ) ) ) ).
% insertE
thf(fact_540_insertE,axiom,
! [A: $o,B: $o,A5: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A5 ) )
=> ( ( A = ~ B )
=> ( member_o @ A @ A5 ) ) ) ).
% insertE
thf(fact_541_insertE,axiom,
! [A: set_nat,B: set_nat,A5: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A5 ) )
=> ( ( A != B )
=> ( member_set_nat @ A @ A5 ) ) ) ).
% insertE
thf(fact_542_insertE,axiom,
! [A: nat,B: nat,A5: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A5 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A5 ) ) ) ).
% insertE
thf(fact_543_graph__map__upd,axiom,
! [M: option_val > option_option_val,K: option_val,V2: option_val] :
( ( graph_7684138220035623358on_val @ ( fun_up1682854050439270531on_val @ M @ K @ ( some_option_val @ V2 ) ) )
= ( insert7026822616767984055on_val @ ( produc904852209897547607on_val @ K @ V2 ) @ ( graph_7684138220035623358on_val @ ( fun_up1682854050439270531on_val @ M @ K @ none_option_val ) ) ) ) ).
% graph_map_upd
thf(fact_544_graph__map__upd,axiom,
! [M: set_nat > option_set_nat,K: set_nat,V2: set_nat] :
( ( graph_8728433485795580044et_nat @ ( fun_up7668491525253874779et_nat @ M @ K @ ( some_set_nat @ V2 ) ) )
= ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ K @ V2 ) @ ( graph_8728433485795580044et_nat @ ( fun_up7668491525253874779et_nat @ M @ K @ none_set_nat ) ) ) ) ).
% graph_map_upd
thf(fact_545_graph__map__upd,axiom,
! [M: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,V2: val] :
( ( graph_1733029056113894637ar_val @ ( fun_up5735975307950539954on_val @ M @ K @ ( some_val @ V2 ) ) )
= ( insert8754101819412505574ar_val @ ( produc5138639219294217606ar_val @ K @ V2 ) @ ( graph_1733029056113894637ar_val @ ( fun_up5735975307950539954on_val @ M @ K @ none_val ) ) ) ) ).
% graph_map_upd
thf(fact_546_graph__map__upd,axiom,
! [M: nat > option_val,K: nat,V2: val] :
( ( graph_nat_val @ ( fun_up810121132392276792on_val @ M @ K @ ( some_val @ V2 ) ) )
= ( insert8107552998908906966at_val @ ( product_Pair_nat_val @ K @ V2 ) @ ( graph_nat_val @ ( fun_up810121132392276792on_val @ M @ K @ none_val ) ) ) ) ).
% graph_map_upd
thf(fact_547_graph__map__upd,axiom,
! [M: nat > option_nat,K: nat,V2: nat] :
( ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ ( some_nat @ V2 ) ) )
= ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ K @ V2 ) @ ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ none_nat ) ) ) ) ).
% graph_map_upd
thf(fact_548_notin__range__Some,axiom,
! [X3: option_option_nat] :
( ( ~ ( member6603469285749945991on_nat @ X3 @ ( image_5486090377411570711on_nat @ some_option_nat @ top_to8920198386146353926on_nat ) ) )
= ( X3 = none_option_nat ) ) ).
% notin_range_Some
thf(fact_549_notin__range__Some,axiom,
! [X3: option_option_val] :
( ( ~ ( member8465289789052943542on_val @ X3 @ ( image_3464815402931832843on_val @ some_option_val @ top_to853123400021307383on_val ) ) )
= ( X3 = none_option_val ) ) ).
% notin_range_Some
thf(fact_550_notin__range__Some,axiom,
! [X3: option2736423997234022232t_char] :
( ( ~ ( member1900516542280067841t_char @ X3 @ ( image_912670848803670091t_char @ some_P6851457240968045559t_char @ top_to1250346684925019074t_char ) ) )
= ( X3 = none_P5486107665612286067t_char ) ) ).
% notin_range_Some
thf(fact_551_notin__range__Some,axiom,
! [X3: option_val] :
( ( ~ ( member_option_val @ X3 @ ( image_val_option_val @ some_val @ top_top_set_val ) ) )
= ( X3 = none_val ) ) ).
% notin_range_Some
thf(fact_552_notin__range__Some,axiom,
! [X3: option_nat] :
( ( ~ ( member_option_nat @ X3 @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) )
= ( X3 = none_nat ) ) ).
% notin_range_Some
thf(fact_553_accp__induct__rule,axiom,
! [R: produc5122021904355189735on_val > produc5122021904355189735on_val > $o,A: produc5122021904355189735on_val,P: produc5122021904355189735on_val > $o] :
( ( accp_P4269981079083053040on_val @ R @ A )
=> ( ! [X: produc5122021904355189735on_val] :
( ( accp_P4269981079083053040on_val @ R @ X )
=> ( ! [Y5: produc5122021904355189735on_val] :
( ( R @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_554_not__accp__down,axiom,
! [R2: produc5122021904355189735on_val > produc5122021904355189735on_val > $o,X3: produc5122021904355189735on_val] :
( ~ ( accp_P4269981079083053040on_val @ R2 @ X3 )
=> ~ ! [Z2: produc5122021904355189735on_val] :
( ( R2 @ Z2 @ X3 )
=> ( accp_P4269981079083053040on_val @ R2 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_555_accp__downward,axiom,
! [R: produc5122021904355189735on_val > produc5122021904355189735on_val > $o,B: produc5122021904355189735on_val,A: produc5122021904355189735on_val] :
( ( accp_P4269981079083053040on_val @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P4269981079083053040on_val @ R @ A ) ) ) ).
% accp_downward
thf(fact_556_accp__induct,axiom,
! [R: produc5122021904355189735on_val > produc5122021904355189735on_val > $o,A: produc5122021904355189735on_val,P: produc5122021904355189735on_val > $o] :
( ( accp_P4269981079083053040on_val @ R @ A )
=> ( ! [X: produc5122021904355189735on_val] :
( ( accp_P4269981079083053040on_val @ R @ X )
=> ( ! [Y5: produc5122021904355189735on_val] :
( ( R @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_557_accpI,axiom,
! [R: produc5122021904355189735on_val > produc5122021904355189735on_val > $o,X3: produc5122021904355189735on_val] :
( ! [Y4: produc5122021904355189735on_val] :
( ( R @ Y4 @ X3 )
=> ( accp_P4269981079083053040on_val @ R @ Y4 ) )
=> ( accp_P4269981079083053040on_val @ R @ X3 ) ) ).
% accpI
thf(fact_558_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_559_surj__def,axiom,
! [F: nat > $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
= ( ! [Y: $o] :
? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_560_surj__def,axiom,
! [F: nat > val] :
( ( ( image_nat_val @ F @ top_top_set_nat )
= top_top_set_val )
= ( ! [Y: val] :
? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_561_surj__def,axiom,
! [F: val > nat] :
( ( ( image_val_nat @ F @ top_top_set_val )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X2: val] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_562_surj__def,axiom,
! [F: val > val] :
( ( ( image_val_val @ F @ top_top_set_val )
= top_top_set_val )
= ( ! [Y: val] :
? [X2: val] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_563_surj__def,axiom,
! [F: nat > set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
= ( ! [Y: set_nat] :
? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_564_surj__def,axiom,
! [F: nat > option_nat] :
( ( ( image_nat_option_nat @ F @ top_top_set_nat )
= top_to8920198386146353926on_nat )
= ( ! [Y: option_nat] :
? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_565_surj__def,axiom,
! [F: nat > option_val] :
( ( ( image_nat_option_val @ F @ top_top_set_nat )
= top_to853123400021307383on_val )
= ( ! [Y: option_val] :
? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_566_surj__def,axiom,
! [F: val > option_nat] :
( ( ( image_val_option_nat @ F @ top_top_set_val )
= top_to8920198386146353926on_nat )
= ( ! [Y: option_nat] :
? [X2: val] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_567_surj__def,axiom,
! [F: val > option_val] :
( ( ( image_val_option_val @ F @ top_top_set_val )
= top_to853123400021307383on_val )
= ( ! [Y: option_val] :
? [X2: val] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_568_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_nat2 @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_569_surjI,axiom,
! [G: nat > $o,F: $o > nat] :
( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_o @ G @ top_top_set_nat )
= top_top_set_o ) ) ).
% surjI
thf(fact_570_surjI,axiom,
! [G: nat > val,F: val > nat] :
( ! [X: val] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_val @ G @ top_top_set_nat )
= top_top_set_val ) ) ).
% surjI
thf(fact_571_surjI,axiom,
! [G: val > nat,F: nat > val] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_val_nat @ G @ top_top_set_val )
= top_top_set_nat ) ) ).
% surjI
thf(fact_572_surjI,axiom,
! [G: val > val,F: val > val] :
( ! [X: val] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_val_val @ G @ top_top_set_val )
= top_top_set_val ) ) ).
% surjI
thf(fact_573_surjI,axiom,
! [G: nat > set_nat,F: set_nat > nat] :
( ! [X: set_nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_set_nat @ G @ top_top_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_574_surjI,axiom,
! [G: nat > option_nat,F: option_nat > nat] :
( ! [X: option_nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_option_nat @ G @ top_top_set_nat )
= top_to8920198386146353926on_nat ) ) ).
% surjI
thf(fact_575_surjI,axiom,
! [G: nat > option_val,F: option_val > nat] :
( ! [X: option_val] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_option_val @ G @ top_top_set_nat )
= top_to853123400021307383on_val ) ) ).
% surjI
thf(fact_576_surjI,axiom,
! [G: val > option_nat,F: option_nat > val] :
( ! [X: option_nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_val_option_nat @ G @ top_top_set_val )
= top_to8920198386146353926on_nat ) ) ).
% surjI
thf(fact_577_surjI,axiom,
! [G: val > option_val,F: option_val > val] :
( ! [X: option_val] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_val_option_val @ G @ top_top_set_val )
= top_to853123400021307383on_val ) ) ).
% surjI
thf(fact_578_surjE,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X: nat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_579_surjE,axiom,
! [F: nat > $o,Y3: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ~ ! [X: nat] :
( Y3
= ( ~ ( F @ X ) ) ) ) ).
% surjE
thf(fact_580_surjE,axiom,
! [F: nat > val,Y3: val] :
( ( ( image_nat_val @ F @ top_top_set_nat )
= top_top_set_val )
=> ~ ! [X: nat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_581_surjE,axiom,
! [F: val > nat,Y3: nat] :
( ( ( image_val_nat @ F @ top_top_set_val )
= top_top_set_nat )
=> ~ ! [X: val] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_582_surjE,axiom,
! [F: val > val,Y3: val] :
( ( ( image_val_val @ F @ top_top_set_val )
= top_top_set_val )
=> ~ ! [X: val] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_583_surjE,axiom,
! [F: nat > set_nat,Y3: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ~ ! [X: nat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_584_surjE,axiom,
! [F: nat > option_nat,Y3: option_nat] :
( ( ( image_nat_option_nat @ F @ top_top_set_nat )
= top_to8920198386146353926on_nat )
=> ~ ! [X: nat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_585_surjE,axiom,
! [F: nat > option_val,Y3: option_val] :
( ( ( image_nat_option_val @ F @ top_top_set_nat )
= top_to853123400021307383on_val )
=> ~ ! [X: nat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_586_surjE,axiom,
! [F: val > option_nat,Y3: option_nat] :
( ( ( image_val_option_nat @ F @ top_top_set_val )
= top_to8920198386146353926on_nat )
=> ~ ! [X: val] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_587_surjE,axiom,
! [F: val > option_val,Y3: option_val] :
( ( ( image_val_option_val @ F @ top_top_set_val )
= top_to853123400021307383on_val )
=> ~ ! [X: val] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_588_surjD,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_589_surjD,axiom,
! [F: nat > $o,Y3: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_590_surjD,axiom,
! [F: nat > val,Y3: val] :
( ( ( image_nat_val @ F @ top_top_set_nat )
= top_top_set_val )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_591_surjD,axiom,
! [F: val > nat,Y3: nat] :
( ( ( image_val_nat @ F @ top_top_set_val )
= top_top_set_nat )
=> ? [X: val] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_592_surjD,axiom,
! [F: val > val,Y3: val] :
( ( ( image_val_val @ F @ top_top_set_val )
= top_top_set_val )
=> ? [X: val] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_593_surjD,axiom,
! [F: nat > set_nat,Y3: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_594_surjD,axiom,
! [F: nat > option_nat,Y3: option_nat] :
( ( ( image_nat_option_nat @ F @ top_top_set_nat )
= top_to8920198386146353926on_nat )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_595_surjD,axiom,
! [F: nat > option_val,Y3: option_val] :
( ( ( image_nat_option_val @ F @ top_top_set_nat )
= top_to853123400021307383on_val )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_596_surjD,axiom,
! [F: val > option_nat,Y3: option_nat] :
( ( ( image_val_option_nat @ F @ top_top_set_val )
= top_to8920198386146353926on_nat )
=> ? [X: val] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_597_surjD,axiom,
! [F: val > option_val,Y3: option_val] :
( ( ( image_val_option_val @ F @ top_top_set_val )
= top_to853123400021307383on_val )
=> ? [X: val] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_598_rangeI,axiom,
! [F: nat > nat,X3: nat] : ( member_nat @ ( F @ X3 ) @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_599_rangeI,axiom,
! [F: nat > val,X3: nat] : ( member_val @ ( F @ X3 ) @ ( image_nat_val @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_600_rangeI,axiom,
! [F: nat > $o,X3: nat] : ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_601_rangeI,axiom,
! [F: val > nat,X3: val] : ( member_nat @ ( F @ X3 ) @ ( image_val_nat @ F @ top_top_set_val ) ) ).
% rangeI
thf(fact_602_rangeI,axiom,
! [F: val > val,X3: val] : ( member_val @ ( F @ X3 ) @ ( image_val_val @ F @ top_top_set_val ) ) ).
% rangeI
thf(fact_603_rangeI,axiom,
! [F: val > $o,X3: val] : ( member_o @ ( F @ X3 ) @ ( image_val_o @ F @ top_top_set_val ) ) ).
% rangeI
thf(fact_604_rangeI,axiom,
! [F: nat > option_val,X3: nat] : ( member_option_val @ ( F @ X3 ) @ ( image_nat_option_val @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_605_rangeI,axiom,
! [F: nat > option_nat,X3: nat] : ( member_option_nat @ ( F @ X3 ) @ ( image_nat_option_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_606_rangeI,axiom,
! [F: nat > set_nat,X3: nat] : ( member_set_nat @ ( F @ X3 ) @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_607_rangeI,axiom,
! [F: val > option_val,X3: val] : ( member_option_val @ ( F @ X3 ) @ ( image_val_option_val @ F @ top_top_set_val ) ) ).
% rangeI
thf(fact_608_range__eqI,axiom,
! [B: nat,F: nat > nat,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_609_range__eqI,axiom,
! [B: val,F: nat > val,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member_val @ B @ ( image_nat_val @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_610_range__eqI,axiom,
! [B: $o,F: nat > $o,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_611_range__eqI,axiom,
! [B: nat,F: val > nat,X3: val] :
( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_val_nat @ F @ top_top_set_val ) ) ) ).
% range_eqI
thf(fact_612_range__eqI,axiom,
! [B: val,F: val > val,X3: val] :
( ( B
= ( F @ X3 ) )
=> ( member_val @ B @ ( image_val_val @ F @ top_top_set_val ) ) ) ).
% range_eqI
thf(fact_613_range__eqI,axiom,
! [B: $o,F: val > $o,X3: val] :
( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_val_o @ F @ top_top_set_val ) ) ) ).
% range_eqI
thf(fact_614_range__eqI,axiom,
! [B: option_val,F: nat > option_val,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member_option_val @ B @ ( image_nat_option_val @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_615_range__eqI,axiom,
! [B: option_nat,F: nat > option_nat,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member_option_nat @ B @ ( image_nat_option_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_616_range__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_617_range__eqI,axiom,
! [B: option_val,F: val > option_val,X3: val] :
( ( B
= ( F @ X3 ) )
=> ( member_option_val @ B @ ( image_val_option_val @ F @ top_top_set_val ) ) ) ).
% range_eqI
thf(fact_618_insert__UNIV,axiom,
! [X3: set_nat] :
( ( insert_set_nat @ X3 @ top_top_set_set_nat )
= top_top_set_set_nat ) ).
% insert_UNIV
thf(fact_619_insert__UNIV,axiom,
! [X3: product_prod_nat_nat] :
( ( insert8211810215607154385at_nat @ X3 @ top_to4669805908274784177at_nat )
= top_to4669805908274784177at_nat ) ).
% insert_UNIV
thf(fact_620_insert__UNIV,axiom,
! [X3: val] :
( ( insert_val @ X3 @ top_top_set_val )
= top_top_set_val ) ).
% insert_UNIV
thf(fact_621_insert__UNIV,axiom,
! [X3: option_nat] :
( ( insert_option_nat @ X3 @ top_to8920198386146353926on_nat )
= top_to8920198386146353926on_nat ) ).
% insert_UNIV
thf(fact_622_insert__UNIV,axiom,
! [X3: option_val] :
( ( insert_option_val @ X3 @ top_to853123400021307383on_val )
= top_to853123400021307383on_val ) ).
% insert_UNIV
thf(fact_623_insert__UNIV,axiom,
! [X3: produc1457211279475724562t_char] :
( ( insert3372640069787401954t_char @ X3 @ top_to1250346684925019074t_char )
= top_to1250346684925019074t_char ) ).
% insert_UNIV
thf(fact_624_insert__UNIV,axiom,
! [X3: nat] :
( ( insert_nat @ X3 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_625_UNIV__option__conv,axiom,
( top_to2947398533425636438on_nat
= ( insert504548404241883424on_nat @ none_option_nat @ ( image_5486090377411570711on_nat @ some_option_nat @ top_to8920198386146353926on_nat ) ) ) ).
% UNIV_option_conv
thf(fact_626_UNIV__option__conv,axiom,
( top_to6510155901028473021on_val
= ( insert7563226391266834525on_val @ none_option_val @ ( image_3464815402931832843on_val @ some_option_val @ top_to853123400021307383on_val ) ) ) ).
% UNIV_option_conv
thf(fact_627_UNIV__option__conv,axiom,
( top_to8912972517910612872t_char
= ( insert213255421748673832t_char @ none_P5486107665612286067t_char @ ( image_912670848803670091t_char @ some_P6851457240968045559t_char @ top_to1250346684925019074t_char ) ) ) ).
% UNIV_option_conv
thf(fact_628_UNIV__option__conv,axiom,
( top_to853123400021307383on_val
= ( insert_option_val @ none_val @ ( image_val_option_val @ some_val @ top_top_set_val ) ) ) ).
% UNIV_option_conv
thf(fact_629_UNIV__option__conv,axiom,
( top_to8920198386146353926on_nat
= ( insert_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) ) ).
% UNIV_option_conv
thf(fact_630_in__graphI,axiom,
! [M: option_val > option_option_val,K: option_val,V2: option_val] :
( ( ( M @ K )
= ( some_option_val @ V2 ) )
=> ( member7947933638605243152on_val @ ( produc904852209897547607on_val @ K @ V2 ) @ ( graph_7684138220035623358on_val @ M ) ) ) ).
% in_graphI
thf(fact_631_in__graphI,axiom,
! [M: set_nat > option_set_nat,K: set_nat,V2: set_nat] :
( ( ( M @ K )
= ( some_set_nat @ V2 ) )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ K @ V2 ) @ ( graph_8728433485795580044et_nat @ M ) ) ) ).
% in_graphI
thf(fact_632_in__graphI,axiom,
! [M: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,V2: val] :
( ( ( M @ K )
= ( some_val @ V2 ) )
=> ( member8657528774786783295ar_val @ ( produc5138639219294217606ar_val @ K @ V2 ) @ ( graph_1733029056113894637ar_val @ M ) ) ) ).
% in_graphI
thf(fact_633_in__graphI,axiom,
! [M: nat > option_nat,K: nat,V2: nat] :
( ( ( M @ K )
= ( some_nat @ V2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V2 ) @ ( graph_nat_nat @ M ) ) ) ).
% in_graphI
thf(fact_634_in__graphD,axiom,
! [K: option_val,V2: option_val,M: option_val > option_option_val] :
( ( member7947933638605243152on_val @ ( produc904852209897547607on_val @ K @ V2 ) @ ( graph_7684138220035623358on_val @ M ) )
=> ( ( M @ K )
= ( some_option_val @ V2 ) ) ) ).
% in_graphD
thf(fact_635_in__graphD,axiom,
! [K: set_nat,V2: set_nat,M: set_nat > option_set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ K @ V2 ) @ ( graph_8728433485795580044et_nat @ M ) )
=> ( ( M @ K )
= ( some_set_nat @ V2 ) ) ) ).
% in_graphD
thf(fact_636_in__graphD,axiom,
! [K: produc1457211279475724562t_char,V2: val,M: produc1457211279475724562t_char > option_val] :
( ( member8657528774786783295ar_val @ ( produc5138639219294217606ar_val @ K @ V2 ) @ ( graph_1733029056113894637ar_val @ M ) )
=> ( ( M @ K )
= ( some_val @ V2 ) ) ) ).
% in_graphD
thf(fact_637_in__graphD,axiom,
! [K: nat,V2: nat,M: nat > option_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V2 ) @ ( graph_nat_nat @ M ) )
=> ( ( M @ K )
= ( some_nat @ V2 ) ) ) ).
% in_graphD
thf(fact_638_accp_Ocases,axiom,
! [R: produc5122021904355189735on_val > produc5122021904355189735on_val > $o,A: produc5122021904355189735on_val] :
( ( accp_P4269981079083053040on_val @ R @ A )
=> ! [Y5: produc5122021904355189735on_val] :
( ( R @ Y5 @ A )
=> ( accp_P4269981079083053040on_val @ R @ Y5 ) ) ) ).
% accp.cases
thf(fact_639_accp_Osimps,axiom,
( accp_P4269981079083053040on_val
= ( ^ [R3: produc5122021904355189735on_val > produc5122021904355189735on_val > $o,A4: produc5122021904355189735on_val] :
? [X2: produc5122021904355189735on_val] :
( ( A4 = X2 )
& ! [Y: produc5122021904355189735on_val] :
( ( R3 @ Y @ X2 )
=> ( accp_P4269981079083053040on_val @ R3 @ Y ) ) ) ) ) ).
% accp.simps
thf(fact_640_finite__range__updI,axiom,
! [F: nat > option_nat,A: nat,B: nat] :
( ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ F @ top_top_set_nat ) )
=> ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ ( fun_up1493157387958331631on_nat @ F @ A @ ( some_nat @ B ) ) @ top_top_set_nat ) ) ) ).
% finite_range_updI
thf(fact_641_finite__range__updI,axiom,
! [F: val > option_val,A: val,B: val] :
( ( finite4212117559080419824on_val @ ( image_val_option_val @ F @ top_top_set_val ) )
=> ( finite4212117559080419824on_val @ ( image_val_option_val @ ( fun_up735531404784515971on_val @ F @ A @ ( some_val @ B ) ) @ top_top_set_val ) ) ) ).
% finite_range_updI
thf(fact_642_finite__range__updI,axiom,
! [F: val > option_nat,A: val,B: nat] :
( ( finite5523153139673422903on_nat @ ( image_val_option_nat @ F @ top_top_set_val ) )
=> ( finite5523153139673422903on_nat @ ( image_val_option_nat @ ( fun_up7213561917443673572on_nat @ F @ A @ ( some_nat @ B ) ) @ top_top_set_val ) ) ) ).
% finite_range_updI
thf(fact_643_finite__range__updI,axiom,
! [F: option_nat > option_val,A: option_nat,B: val] :
( ( finite4212117559080419824on_val @ ( image_2375463299254234464on_val @ F @ top_to8920198386146353926on_nat ) )
=> ( finite4212117559080419824on_val @ ( image_2375463299254234464on_val @ ( fun_up2035529056102068968on_val @ F @ A @ ( some_val @ B ) ) @ top_to8920198386146353926on_nat ) ) ) ).
% finite_range_updI
thf(fact_644_finite__range__updI,axiom,
! [F: option_nat > option_nat,A: option_nat,B: nat] :
( ( finite5523153139673422903on_nat @ ( image_2533357264035992775on_nat @ F @ top_to8920198386146353926on_nat ) )
=> ( finite5523153139673422903on_nat @ ( image_2533357264035992775on_nat @ ( fun_up5972625598298123583on_nat @ F @ A @ ( some_nat @ B ) ) @ top_to8920198386146353926on_nat ) ) ) ).
% finite_range_updI
thf(fact_645_finite__range__updI,axiom,
! [F: option_val > option_val,A: option_val,B: val] :
( ( finite4212117559080419824on_val @ ( image_5994708315261978053on_val @ F @ top_to853123400021307383on_val ) )
=> ( finite4212117559080419824on_val @ ( image_5994708315261978053on_val @ ( fun_up8761876891205509693on_val @ F @ A @ ( some_val @ B ) ) @ top_to853123400021307383on_val ) ) ) ).
% finite_range_updI
thf(fact_646_finite__range__updI,axiom,
! [F: option_val > option_nat,A: option_val,B: nat] :
( ( finite5523153139673422903on_nat @ ( image_1875112342138737442on_nat @ F @ top_to853123400021307383on_val ) )
=> ( finite5523153139673422903on_nat @ ( image_1875112342138737442on_nat @ ( fun_up1535178098986571946on_nat @ F @ A @ ( some_nat @ B ) ) @ top_to853123400021307383on_val ) ) ) ).
% finite_range_updI
thf(fact_647_finite__range__updI,axiom,
! [F: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char,B: val] :
( ( finite4212117559080419824on_val @ ( image_4269386950610408506on_val @ F @ top_to1250346684925019074t_char ) )
=> ( finite4212117559080419824on_val @ ( image_4269386950610408506on_val @ ( fun_up5735975307950539954on_val @ F @ A @ ( some_val @ B ) ) @ top_to1250346684925019074t_char ) ) ) ).
% finite_range_updI
thf(fact_648_finite__range__updI,axiom,
! [F: produc1457211279475724562t_char > option_nat,A: produc1457211279475724562t_char,B: nat] :
( ( finite5523153139673422903on_nat @ ( image_4937545321503807341on_nat @ F @ top_to1250346684925019074t_char ) )
=> ( finite5523153139673422903on_nat @ ( image_4937545321503807341on_nat @ ( fun_up1015184567166388469on_nat @ F @ A @ ( some_nat @ B ) ) @ top_to1250346684925019074t_char ) ) ) ).
% finite_range_updI
thf(fact_649_finite__range__updI,axiom,
! [F: nat > option_val,A: nat,B: val] :
( ( finite4212117559080419824on_val @ ( image_nat_option_val @ F @ top_top_set_nat ) )
=> ( finite4212117559080419824on_val @ ( image_nat_option_val @ ( fun_up810121132392276792on_val @ F @ A @ ( some_val @ B ) ) @ top_top_set_nat ) ) ) ).
% finite_range_updI
thf(fact_650_graph__restrictD_I2_J,axiom,
! [K: option_val,V2: option_val,M: option_val > option_option_val,A5: set_option_val] :
( ( member7947933638605243152on_val @ ( produc904852209897547607on_val @ K @ V2 ) @ ( graph_7684138220035623358on_val @ ( restri7220945691392438384on_val @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some_option_val @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_651_graph__restrictD_I2_J,axiom,
! [K: set_nat,V2: set_nat,M: set_nat > option_set_nat,A5: set_set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ K @ V2 ) @ ( graph_8728433485795580044et_nat @ ( restri1233848060703379006et_nat @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some_set_nat @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_652_graph__restrictD_I2_J,axiom,
! [K: produc1457211279475724562t_char,V2: val,M: produc1457211279475724562t_char > option_val,A5: set_Pr2507339680178222706t_char] :
( ( member8657528774786783295ar_val @ ( produc5138639219294217606ar_val @ K @ V2 ) @ ( graph_1733029056113894637ar_val @ ( restri4256959289591943583ar_val @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some_val @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_653_graph__restrictD_I2_J,axiom,
! [K: nat,V2: val,M: nat > option_val,A5: set_nat] :
( ( member4983101843562193725at_val @ ( product_Pair_nat_val @ K @ V2 ) @ ( graph_nat_val @ ( restrict_map_nat_val @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some_val @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_654_graph__restrictD_I2_J,axiom,
! [K: nat,V2: nat,M: nat > option_nat,A5: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V2 ) @ ( graph_nat_nat @ ( restrict_map_nat_nat @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some_nat @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_655_option_Osimps_I15_J,axiom,
! [X22: option_val] :
( ( set_op911010586042427349on_val @ ( some_option_val @ X22 ) )
= ( insert_option_val @ X22 @ bot_bo5906790151302476563on_val ) ) ).
% option.simps(15)
thf(fact_656_option_Osimps_I15_J,axiom,
! [X22: option_nat] :
( ( set_op3360498428384587026on_nat @ ( some_option_nat @ X22 ) )
= ( insert_option_nat @ X22 @ bot_bo5009843511495006442on_nat ) ) ).
% option.simps(15)
thf(fact_657_option_Osimps_I15_J,axiom,
! [X22: produc1457211279475724562t_char] :
( ( set_op7440730737645789856t_char @ ( some_P6851457240968045559t_char @ X22 ) )
= ( insert3372640069787401954t_char @ X22 @ bot_bo1414094081888755934t_char ) ) ).
% option.simps(15)
thf(fact_658_option_Osimps_I15_J,axiom,
! [X22: set_nat] :
( ( set_option_set_nat2 @ ( some_set_nat @ X22 ) )
= ( insert_set_nat @ X22 @ bot_bot_set_set_nat ) ) ).
% option.simps(15)
thf(fact_659_option_Osimps_I15_J,axiom,
! [X22: product_prod_nat_nat] :
( ( set_op5809825150163292047at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
= ( insert8211810215607154385at_nat @ X22 @ bot_bo2099793752762293965at_nat ) ) ).
% option.simps(15)
thf(fact_660_option_Osimps_I15_J,axiom,
! [X22: val] :
( ( set_option_val2 @ ( some_val @ X22 ) )
= ( insert_val @ X22 @ bot_bot_set_val ) ) ).
% option.simps(15)
thf(fact_661_option_Osimps_I15_J,axiom,
! [X22: nat] :
( ( set_option_nat2 @ ( some_nat @ X22 ) )
= ( insert_nat @ X22 @ bot_bot_set_nat ) ) ).
% option.simps(15)
thf(fact_662_finite__map__freshness,axiom,
! [F: nat > option_val] :
( ( finite_finite_nat @ ( dom_nat_val @ F ) )
=> ( ~ ( finite_finite_nat @ top_top_set_nat )
=> ? [X: nat] :
( ( F @ X )
= none_val ) ) ) ).
% finite_map_freshness
thf(fact_663_finite__map__freshness,axiom,
! [F: nat > option_nat] :
( ( finite_finite_nat @ ( dom_nat_nat @ F ) )
=> ( ~ ( finite_finite_nat @ top_top_set_nat )
=> ? [X: nat] :
( ( F @ X )
= none_nat ) ) ) ).
% finite_map_freshness
thf(fact_664_finite__map__freshness,axiom,
! [F: val > option_val] :
( ( finite_finite_val @ ( dom_val_val @ F ) )
=> ( ~ ( finite_finite_val @ top_top_set_val )
=> ? [X: val] :
( ( F @ X )
= none_val ) ) ) ).
% finite_map_freshness
thf(fact_665_finite__map__freshness,axiom,
! [F: val > option_nat] :
( ( finite_finite_val @ ( dom_val_nat @ F ) )
=> ( ~ ( finite_finite_val @ top_top_set_val )
=> ? [X: val] :
( ( F @ X )
= none_nat ) ) ) ).
% finite_map_freshness
thf(fact_666_finite__map__freshness,axiom,
! [F: option_nat > option_val] :
( ( finite5523153139673422903on_nat @ ( dom_option_nat_val @ F ) )
=> ( ~ ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ? [X: option_nat] :
( ( F @ X )
= none_val ) ) ) ).
% finite_map_freshness
thf(fact_667_finite__map__freshness,axiom,
! [F: option_nat > option_nat] :
( ( finite5523153139673422903on_nat @ ( dom_option_nat_nat @ F ) )
=> ( ~ ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
=> ? [X: option_nat] :
( ( F @ X )
= none_nat ) ) ) ).
% finite_map_freshness
thf(fact_668_finite__map__freshness,axiom,
! [F: option_val > option_val] :
( ( finite4212117559080419824on_val @ ( dom_option_val_val @ F ) )
=> ( ~ ( finite4212117559080419824on_val @ top_to853123400021307383on_val )
=> ? [X: option_val] :
( ( F @ X )
= none_val ) ) ) ).
% finite_map_freshness
thf(fact_669_finite__map__freshness,axiom,
! [F: option_val > option_nat] :
( ( finite4212117559080419824on_val @ ( dom_option_val_nat @ F ) )
=> ( ~ ( finite4212117559080419824on_val @ top_to853123400021307383on_val )
=> ? [X: option_val] :
( ( F @ X )
= none_nat ) ) ) ).
% finite_map_freshness
thf(fact_670_finite__map__freshness,axiom,
! [F: produc1457211279475724562t_char > option_val] :
( ( finite3262071724885184827t_char @ ( dom_Pr8989479155164304065ar_val @ F ) )
=> ( ~ ( finite3262071724885184827t_char @ top_to1250346684925019074t_char )
=> ? [X: produc1457211279475724562t_char] :
( ( F @ X )
= none_val ) ) ) ).
% finite_map_freshness
thf(fact_671_finite__map__freshness,axiom,
! [F: produc1457211279475724562t_char > option_nat] :
( ( finite3262071724885184827t_char @ ( dom_Pr3443001725208114576ar_nat @ F ) )
=> ( ~ ( finite3262071724885184827t_char @ top_to1250346684925019074t_char )
=> ? [X: produc1457211279475724562t_char] :
( ( F @ X )
= none_nat ) ) ) ).
% finite_map_freshness
thf(fact_672_Inf_OINF__cong,axiom,
! [A5: set_nat_o,B5: set_nat_o,C3: ( nat > $o ) > set_nat,D2: ( nat > $o ) > set_nat,Inf: set_set_nat > set_nat] :
( ( A5 = B5 )
=> ( ! [X: nat > $o] :
( ( member_nat_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_o_set_nat @ C3 @ A5 ) )
= ( Inf @ ( image_nat_o_set_nat @ D2 @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_673_Inf_OINF__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > option_val,D2: nat > option_val,Inf: set_option_val > option_val] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_option_val @ C3 @ A5 ) )
= ( Inf @ ( image_nat_option_val @ D2 @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_674_Inf_OINF__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > option_nat,D2: nat > option_nat,Inf: set_option_nat > option_nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_option_nat @ C3 @ A5 ) )
= ( Inf @ ( image_nat_option_nat @ D2 @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_675_Inf_OINF__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > $o,D2: nat > $o,Inf: set_o > $o] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_o @ C3 @ A5 ) )
= ( Inf @ ( image_nat_o @ D2 @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_676_Inf_OINF__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > set_nat,D2: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C3 @ A5 ) )
= ( Inf @ ( image_nat_set_nat @ D2 @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_677_Inf_OINF__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_nat2 @ C3 @ A5 ) )
= ( Inf @ ( image_nat_nat2 @ D2 @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_678_Sup_OSUP__cong,axiom,
! [A5: set_nat_o,B5: set_nat_o,C3: ( nat > $o ) > set_nat,D2: ( nat > $o ) > set_nat,Sup: set_set_nat > set_nat] :
( ( A5 = B5 )
=> ( ! [X: nat > $o] :
( ( member_nat_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_o_set_nat @ C3 @ A5 ) )
= ( Sup @ ( image_nat_o_set_nat @ D2 @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_679_Sup_OSUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > option_val,D2: nat > option_val,Sup: set_option_val > option_val] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_option_val @ C3 @ A5 ) )
= ( Sup @ ( image_nat_option_val @ D2 @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_680_Sup_OSUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > option_nat,D2: nat > option_nat,Sup: set_option_nat > option_nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_option_nat @ C3 @ A5 ) )
= ( Sup @ ( image_nat_option_nat @ D2 @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_681_Sup_OSUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > $o,D2: nat > $o,Sup: set_o > $o] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_o @ C3 @ A5 ) )
= ( Sup @ ( image_nat_o @ D2 @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_682_Sup_OSUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > set_nat,D2: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C3 @ A5 ) )
= ( Sup @ ( image_nat_set_nat @ D2 @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_683_Sup_OSUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_nat2 @ C3 @ A5 ) )
= ( Sup @ ( image_nat_nat2 @ D2 @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_684_range__eq__singletonD,axiom,
! [F: nat > nat,A: nat,X3: nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_685_range__eq__singletonD,axiom,
! [F: nat > $o,A: $o,X3: nat] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_686_range__eq__singletonD,axiom,
! [F: nat > val,A: val,X3: nat] :
( ( ( image_nat_val @ F @ top_top_set_nat )
= ( insert_val @ A @ bot_bot_set_val ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_687_range__eq__singletonD,axiom,
! [F: val > nat,A: nat,X3: val] :
( ( ( image_val_nat @ F @ top_top_set_val )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_688_range__eq__singletonD,axiom,
! [F: val > val,A: val,X3: val] :
( ( ( image_val_val @ F @ top_top_set_val )
= ( insert_val @ A @ bot_bot_set_val ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_689_range__eq__singletonD,axiom,
! [F: nat > option_val,A: option_val,X3: nat] :
( ( ( image_nat_option_val @ F @ top_top_set_nat )
= ( insert_option_val @ A @ bot_bo5906790151302476563on_val ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_690_range__eq__singletonD,axiom,
! [F: nat > option_nat,A: option_nat,X3: nat] :
( ( ( image_nat_option_nat @ F @ top_top_set_nat )
= ( insert_option_nat @ A @ bot_bo5009843511495006442on_nat ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_691_range__eq__singletonD,axiom,
! [F: nat > set_nat,A: set_nat,X3: nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_692_range__eq__singletonD,axiom,
! [F: val > option_val,A: option_val,X3: val] :
( ( ( image_val_option_val @ F @ top_top_set_val )
= ( insert_option_val @ A @ bot_bo5906790151302476563on_val ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_693_range__eq__singletonD,axiom,
! [F: val > option_nat,A: option_nat,X3: val] :
( ( ( image_val_option_nat @ F @ top_top_set_val )
= ( insert_option_nat @ A @ bot_bo5009843511495006442on_nat ) )
=> ( ( F @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_694_dom__minus,axiom,
! [F: val > option_val,X3: val,A5: set_val] :
( ( ( F @ X3 )
= none_val )
=> ( ( minus_minus_set_val @ ( dom_val_val @ F ) @ ( insert_val @ X3 @ A5 ) )
= ( minus_minus_set_val @ ( dom_val_val @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_695_dom__minus,axiom,
! [F: val > option_nat,X3: val,A5: set_val] :
( ( ( F @ X3 )
= none_nat )
=> ( ( minus_minus_set_val @ ( dom_val_nat @ F ) @ ( insert_val @ X3 @ A5 ) )
= ( minus_minus_set_val @ ( dom_val_nat @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_696_dom__minus,axiom,
! [F: nat > option_val,X3: nat,A5: set_nat] :
( ( ( F @ X3 )
= none_val )
=> ( ( minus_minus_set_nat @ ( dom_nat_val @ F ) @ ( insert_nat @ X3 @ A5 ) )
= ( minus_minus_set_nat @ ( dom_nat_val @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_697_dom__minus,axiom,
! [F: nat > option_nat,X3: nat,A5: set_nat] :
( ( ( F @ X3 )
= none_nat )
=> ( ( minus_minus_set_nat @ ( dom_nat_nat @ F ) @ ( insert_nat @ X3 @ A5 ) )
= ( minus_minus_set_nat @ ( dom_nat_nat @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_698_dom__minus,axiom,
! [F: option_val > option_val,X3: option_val,A5: set_option_val] :
( ( ( F @ X3 )
= none_val )
=> ( ( minus_3256168188512067470on_val @ ( dom_option_val_val @ F ) @ ( insert_option_val @ X3 @ A5 ) )
= ( minus_3256168188512067470on_val @ ( dom_option_val_val @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_699_dom__minus,axiom,
! [F: option_nat > option_val,X3: option_nat,A5: set_option_nat] :
( ( ( F @ X3 )
= none_val )
=> ( ( minus_5999362281193037231on_nat @ ( dom_option_nat_val @ F ) @ ( insert_option_nat @ X3 @ A5 ) )
= ( minus_5999362281193037231on_nat @ ( dom_option_nat_val @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_700_dom__minus,axiom,
! [F: set_nat > option_val,X3: set_nat,A5: set_set_nat] :
( ( ( F @ X3 )
= none_val )
=> ( ( minus_2163939370556025621et_nat @ ( dom_set_nat_val @ F ) @ ( insert_set_nat @ X3 @ A5 ) )
= ( minus_2163939370556025621et_nat @ ( dom_set_nat_val @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_701_dom__minus,axiom,
! [F: option_val > option_nat,X3: option_val,A5: set_option_val] :
( ( ( F @ X3 )
= none_nat )
=> ( ( minus_3256168188512067470on_val @ ( dom_option_val_nat @ F ) @ ( insert_option_val @ X3 @ A5 ) )
= ( minus_3256168188512067470on_val @ ( dom_option_val_nat @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_702_dom__minus,axiom,
! [F: option_nat > option_nat,X3: option_nat,A5: set_option_nat] :
( ( ( F @ X3 )
= none_nat )
=> ( ( minus_5999362281193037231on_nat @ ( dom_option_nat_nat @ F ) @ ( insert_option_nat @ X3 @ A5 ) )
= ( minus_5999362281193037231on_nat @ ( dom_option_nat_nat @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_703_dom__minus,axiom,
! [F: set_nat > option_nat,X3: set_nat,A5: set_set_nat] :
( ( ( F @ X3 )
= none_nat )
=> ( ( minus_2163939370556025621et_nat @ ( dom_set_nat_nat @ F ) @ ( insert_set_nat @ X3 @ A5 ) )
= ( minus_2163939370556025621et_nat @ ( dom_set_nat_nat @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_704_restrict__in,axiom,
! [X3: nat,A5: set_nat,M: nat > option_val] :
( ( member_nat @ X3 @ A5 )
=> ( ( restrict_map_nat_val @ M @ A5 @ X3 )
= ( M @ X3 ) ) ) ).
% restrict_in
thf(fact_705_image__is__empty,axiom,
! [F: nat > nat,A5: set_nat] :
( ( ( image_nat_nat2 @ F @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_706_image__is__empty,axiom,
! [F: nat > $o,A5: set_nat] :
( ( ( image_nat_o @ F @ A5 )
= bot_bot_set_o )
= ( A5 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_707_image__is__empty,axiom,
! [F: val > nat,A5: set_val] :
( ( ( image_val_nat @ F @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bot_set_val ) ) ).
% image_is_empty
thf(fact_708_image__is__empty,axiom,
! [F: nat > val,A5: set_nat] :
( ( ( image_nat_val @ F @ A5 )
= bot_bot_set_val )
= ( A5 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_709_image__is__empty,axiom,
! [F: val > val,A5: set_val] :
( ( ( image_val_val @ F @ A5 )
= bot_bot_set_val )
= ( A5 = bot_bot_set_val ) ) ).
% image_is_empty
thf(fact_710_image__is__empty,axiom,
! [F: nat > option_val,A5: set_nat] :
( ( ( image_nat_option_val @ F @ A5 )
= bot_bo5906790151302476563on_val )
= ( A5 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_711_image__is__empty,axiom,
! [F: option_nat > nat,A5: set_option_nat] :
( ( ( image_option_nat_nat @ F @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bo5009843511495006442on_nat ) ) ).
% image_is_empty
thf(fact_712_image__is__empty,axiom,
! [F: set_nat > nat,A5: set_set_nat] :
( ( ( image_set_nat_nat @ F @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_713_image__is__empty,axiom,
! [F: option_nat > val,A5: set_option_nat] :
( ( ( image_option_nat_val @ F @ A5 )
= bot_bot_set_val )
= ( A5 = bot_bo5009843511495006442on_nat ) ) ).
% image_is_empty
thf(fact_714_image__is__empty,axiom,
! [F: set_nat > val,A5: set_set_nat] :
( ( ( image_set_nat_val @ F @ A5 )
= bot_bot_set_val )
= ( A5 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_715_empty__is__image,axiom,
! [F: nat > nat,A5: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat2 @ F @ A5 ) )
= ( A5 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_716_empty__is__image,axiom,
! [F: nat > $o,A5: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o @ F @ A5 ) )
= ( A5 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_717_empty__is__image,axiom,
! [F: val > nat,A5: set_val] :
( ( bot_bot_set_nat
= ( image_val_nat @ F @ A5 ) )
= ( A5 = bot_bot_set_val ) ) ).
% empty_is_image
thf(fact_718_empty__is__image,axiom,
! [F: nat > val,A5: set_nat] :
( ( bot_bot_set_val
= ( image_nat_val @ F @ A5 ) )
= ( A5 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_719_empty__is__image,axiom,
! [F: val > val,A5: set_val] :
( ( bot_bot_set_val
= ( image_val_val @ F @ A5 ) )
= ( A5 = bot_bot_set_val ) ) ).
% empty_is_image
thf(fact_720_empty__is__image,axiom,
! [F: nat > option_val,A5: set_nat] :
( ( bot_bo5906790151302476563on_val
= ( image_nat_option_val @ F @ A5 ) )
= ( A5 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_721_empty__is__image,axiom,
! [F: option_nat > nat,A5: set_option_nat] :
( ( bot_bot_set_nat
= ( image_option_nat_nat @ F @ A5 ) )
= ( A5 = bot_bo5009843511495006442on_nat ) ) ).
% empty_is_image
thf(fact_722_empty__is__image,axiom,
! [F: set_nat > nat,A5: set_set_nat] :
( ( bot_bot_set_nat
= ( image_set_nat_nat @ F @ A5 ) )
= ( A5 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_723_empty__is__image,axiom,
! [F: option_nat > val,A5: set_option_nat] :
( ( bot_bot_set_val
= ( image_option_nat_val @ F @ A5 ) )
= ( A5 = bot_bo5009843511495006442on_nat ) ) ).
% empty_is_image
thf(fact_724_empty__is__image,axiom,
! [F: set_nat > val,A5: set_set_nat] :
( ( bot_bot_set_val
= ( image_set_nat_val @ F @ A5 ) )
= ( A5 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_725_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat2 @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_726_image__empty,axiom,
! [F: nat > $o] :
( ( image_nat_o @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_727_image__empty,axiom,
! [F: nat > val] :
( ( image_nat_val @ F @ bot_bot_set_nat )
= bot_bot_set_val ) ).
% image_empty
thf(fact_728_image__empty,axiom,
! [F: val > nat] :
( ( image_val_nat @ F @ bot_bot_set_val )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_729_image__empty,axiom,
! [F: val > val] :
( ( image_val_val @ F @ bot_bot_set_val )
= bot_bot_set_val ) ).
% image_empty
thf(fact_730_image__empty,axiom,
! [F: nat > option_val] :
( ( image_nat_option_val @ F @ bot_bot_set_nat )
= bot_bo5906790151302476563on_val ) ).
% image_empty
thf(fact_731_image__empty,axiom,
! [F: nat > option_nat] :
( ( image_nat_option_nat @ F @ bot_bot_set_nat )
= bot_bo5009843511495006442on_nat ) ).
% image_empty
thf(fact_732_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_733_image__empty,axiom,
! [F: val > option_nat] :
( ( image_val_option_nat @ F @ bot_bot_set_val )
= bot_bo5009843511495006442on_nat ) ).
% image_empty
thf(fact_734_image__empty,axiom,
! [F: val > set_nat] :
( ( image_val_set_nat @ F @ bot_bot_set_val )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_735_singletonI,axiom,
! [A: val] : ( member_val @ A @ ( insert_val @ A @ bot_bot_set_val ) ) ).
% singletonI
thf(fact_736_singletonI,axiom,
! [A: option_nat] : ( member_option_nat @ A @ ( insert_option_nat @ A @ bot_bo5009843511495006442on_nat ) ) ).
% singletonI
thf(fact_737_singletonI,axiom,
! [A: produc1457211279475724562t_char] : ( member484235747325421115t_char @ A @ ( insert3372640069787401954t_char @ A @ bot_bo1414094081888755934t_char ) ) ).
% singletonI
thf(fact_738_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_739_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_740_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_741_insert__Diff1,axiom,
! [X3: nat,B5: set_nat,A5: set_nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_742_Diff__insert0,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_743_these__empty,axiom,
( ( these_nat @ bot_bo5009843511495006442on_nat )
= bot_bot_set_nat ) ).
% these_empty
thf(fact_744_insert__Diff__single,axiom,
! [A: nat,A5: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A5 ) ) ).
% insert_Diff_single
thf(fact_745_set__empty__eq,axiom,
! [Xo: option_nat] :
( ( ( set_option_nat2 @ Xo )
= bot_bot_set_nat )
= ( Xo = none_nat ) ) ).
% set_empty_eq
thf(fact_746_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_747_Diff__insert,axiom,
! [A5: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A5 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_748_insert__Diff,axiom,
! [A: nat,A5: set_nat] :
( ( member_nat @ A @ A5 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_749_Diff__insert2,axiom,
! [A5: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_750_Diff__insert__absorb,axiom,
! [X3: nat,A5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_751_in__image__insert__iff,axiom,
! [B5: set_set_nat,X3: nat,A5: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B5 )
=> ~ ( member_nat @ X3 @ C4 ) )
=> ( ( member_set_nat @ A5 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ B5 ) )
= ( ( member_nat @ X3 @ A5 )
& ( member_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_752_insert__Diff__if,axiom,
! [X3: nat,B5: set_nat,A5: set_nat] :
( ( ( member_nat @ X3 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( minus_minus_set_nat @ A5 @ B5 ) ) )
& ( ~ ( member_nat @ X3 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( insert_nat @ X3 @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_753_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_754_insert__not__empty,axiom,
! [A: nat,A5: set_nat] :
( ( insert_nat @ A @ A5 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_755_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_756_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_757_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_758_fun__upd__image,axiom,
! [X3: nat,A5: set_nat,F: nat > nat,Y3: nat] :
( ( ( member_nat @ X3 @ A5 )
=> ( ( image_nat_nat2 @ ( fun_upd_nat_nat @ F @ X3 @ Y3 ) @ A5 )
= ( insert_nat @ Y3 @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ( image_nat_nat2 @ ( fun_upd_nat_nat @ F @ X3 @ Y3 ) @ A5 )
= ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_759_finite__range__Some,axiom,
( ( finite4212117559080419824on_val @ ( image_val_option_val @ some_val @ top_top_set_val ) )
= ( finite_finite_val @ top_top_set_val ) ) ).
% finite_range_Some
thf(fact_760_finite__range__Some,axiom,
( ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_range_Some
thf(fact_761_these__empty__eq,axiom,
! [B5: set_option_nat] :
( ( ( these_nat @ B5 )
= bot_bot_set_nat )
= ( ( B5 = bot_bo5009843511495006442on_nat )
| ( B5
= ( insert_option_nat @ none_nat @ bot_bo5009843511495006442on_nat ) ) ) ) ).
% these_empty_eq
thf(fact_762_these__not__empty__eq,axiom,
! [B5: set_option_nat] :
( ( ( these_nat @ B5 )
!= bot_bot_set_nat )
= ( ( B5 != bot_bo5009843511495006442on_nat )
& ( B5
!= ( insert_option_nat @ none_nat @ bot_bo5009843511495006442on_nat ) ) ) ) ).
% these_not_empty_eq
thf(fact_763_option_Osimps_I14_J,axiom,
( ( set_option_nat2 @ none_nat )
= bot_bot_set_nat ) ).
% option.simps(14)
thf(fact_764_finite__Diff__insert,axiom,
! [A5: set_nat,A: nat,B5: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ B5 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_765_finite__insert,axiom,
! [A: nat,A5: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A5 ) )
= ( finite_finite_nat @ A5 ) ) ).
% finite_insert
thf(fact_766_finite__imageI,axiom,
! [F3: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat2 @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_767_infinite__remove,axiom,
! [S3: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_768_infinite__coinduct,axiom,
! [X5: set_nat > $o,A5: set_nat] :
( ( X5 @ A5 )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A5 ) ) ) ).
% infinite_coinduct
thf(fact_769_finite__empty__induct,axiom,
! [A5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A5 )
=> ( ( P @ A5 )
=> ( ! [A2: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A2 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_770_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A2: nat] :
( A
= ( insert_nat @ A2 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_771_finite_OinsertI,axiom,
! [A5: set_nat,A: nat] :
( ( finite_finite_nat @ A5 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A5 ) ) ) ).
% finite.insertI
thf(fact_772_infinite__finite__induct,axiom,
! [P: set_nat > $o,A5: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X @ F4 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_773_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_774_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_775_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A7: set_nat,B4: nat] :
( ( A4
= ( insert_nat @ B4 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_776_the__elem__eq,axiom,
! [X3: nat] :
( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% the_elem_eq
thf(fact_777_is__singletonI,axiom,
! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_778_restrict__upd__same,axiom,
! [M: nat > option_val,X3: nat,Y3: val] :
( ( restrict_map_nat_val @ ( fun_up810121132392276792on_val @ M @ X3 @ ( some_val @ Y3 ) ) @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
= ( restrict_map_nat_val @ M @ ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% restrict_upd_same
thf(fact_779_remove__def,axiom,
( remove_nat
= ( ^ [X2: nat,A7: set_nat] : ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_780_finite__range__map__of__map__add,axiom,
! [F: produc1457211279475724562t_char > option_val,L: list_P375024181684815772ar_val] :
( ( finite4212117559080419824on_val @ ( image_4269386950610408506on_val @ F @ top_to1250346684925019074t_char ) )
=> ( finite4212117559080419824on_val @ ( image_4269386950610408506on_val @ ( map_ad2908576761495157028ar_val @ F @ ( map_of6844234013878170066ar_val @ L ) ) @ top_to1250346684925019074t_char ) ) ) ).
% finite_range_map_of_map_add
thf(fact_781_the__elem__image__unique,axiom,
! [A5: set_nat,F: nat > nat,X3: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ A5 )
=> ( ( F @ Y4 )
= ( F @ X3 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat2 @ F @ A5 ) )
= ( F @ X3 ) ) ) ) ).
% the_elem_image_unique
thf(fact_782_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A7: set_nat] :
( A7
= ( insert_nat @ ( the_elem_nat @ A7 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_783_Compl__insert,axiom,
! [X3: nat,A5: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ A5 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A5 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_784_is__singletonE,axiom,
! [A5: set_nat] :
( ( is_singleton_nat @ A5 )
=> ~ ! [X: nat] :
( A5
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_785_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A7: set_nat] :
? [X2: nat] :
( A7
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_786_max__ext_Omax__extI,axiom,
! [X5: set_nat,Y6: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( Y6 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ Y6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Xa2 ) @ R2 ) ) )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y6 ) @ ( max_ext_nat @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_787_max__ext_Osimps,axiom,
! [A1: set_nat,A22: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R2 ) )
= ( ( finite_finite_nat @ A1 )
& ( finite_finite_nat @ A22 )
& ( A22 != bot_bot_set_nat )
& ! [X2: nat] :
( ( member_nat @ X2 @ A1 )
=> ? [Y: nat] :
( ( member_nat @ Y @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_788_max__ext_Ocases,axiom,
! [A1: set_nat,A22: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R2 ) )
=> ~ ( ( finite_finite_nat @ A1 )
=> ( ( finite_finite_nat @ A22 )
=> ( ( A22 != bot_bot_set_nat )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ A1 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_789_inj__on__insert,axiom,
! [F: nat > nat,A: nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ ( insert_nat @ A @ A5 ) )
= ( ( inj_on_nat_nat @ F @ A5 )
& ~ ( member_nat @ ( F @ A ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_790_remove__induct,axiom,
! [P: set_nat > $o,B5: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B5 )
=> ( P @ B5 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B5 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_791_finite__remove__induct,axiom,
! [B5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B5 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_792_insert__subset,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( ( member_nat @ X3 @ B5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_793_inj__on__map__add__dom,axiom,
! [M: produc1457211279475724562t_char > option_val,M5: produc1457211279475724562t_char > option_val] :
( ( inj_on4474484569326248358on_val @ ( map_ad2908576761495157028ar_val @ M @ M5 ) @ ( dom_Pr8989479155164304065ar_val @ M5 ) )
= ( inj_on4474484569326248358on_val @ M5 @ ( dom_Pr8989479155164304065ar_val @ M5 ) ) ) ).
% inj_on_map_add_dom
thf(fact_794_singleton__insert__inj__eq_H,axiom,
! [A: nat,A5: set_nat,B: nat] :
( ( ( insert_nat @ A @ A5 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_795_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A5: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A5 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_796_subset__Compl__singleton,axiom,
! [A5: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A5 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_797_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C3: set_nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A5 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( ( image_nat_nat2 @ F @ A5 )
= ( image_nat_nat2 @ F @ B5 ) )
= ( A5 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_798_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B5: set_nat,A: nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat2 @ F @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_799_inj__image__subset__iff,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) )
= ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_800_finite__surj__inj,axiom,
! [A5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ A5 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( inj_on_nat_nat @ F @ A5 ) ) ) ).
% finite_surj_inj
thf(fact_801_inj__on__finite,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A5 ) ) ) ) ).
% inj_on_finite
thf(fact_802_endo__inj__surj,axiom,
! [A5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ A5 )
=> ( ( inj_on_nat_nat @ F @ A5 )
=> ( ( image_nat_nat2 @ F @ A5 )
= A5 ) ) ) ) ).
% endo_inj_surj
thf(fact_803_inj__on__image__set__diff,axiom,
! [F: nat > nat,C3: set_nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A5 @ B5 ) @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A5 @ B5 ) )
= ( minus_minus_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_804_inj__Some,axiom,
! [A5: set_val] : ( inj_on4074814566786472055on_val @ some_val @ A5 ) ).
% inj_Some
thf(fact_805_all__subset__image,axiom,
! [F: nat > nat,A5: set_nat,P: set_nat > $o] :
( ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A5 )
=> ( P @ ( image_nat_nat2 @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_806_subset__image__iff,axiom,
! [B5: set_nat,F: nat > nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A5 )
& ( B5
= ( image_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_807_image__subset__iff,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ ( F @ X2 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_808_subset__imageE,axiom,
! [B5: set_nat,F: nat > nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A5 )
=> ( B5
!= ( image_nat_nat2 @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_809_image__subsetI,axiom,
! [A5: set_nat,F: nat > nat,B5: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_810_image__mono,axiom,
! [A5: set_nat,B5: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ).
% image_mono
thf(fact_811_subset__emptyI,axiom,
! [A5: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A5 )
=> ( ord_less_eq_set_nat @ A5 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_812_insert__subsetI,axiom,
! [X3: nat,A5: set_nat,X5: set_nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( ord_less_eq_set_nat @ X5 @ A5 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X5 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_813_insert__mono,axiom,
! [C3: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_814_subset__insert,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_815_subset__insertI,axiom,
! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).
% subset_insertI
thf(fact_816_subset__insertI2,axiom,
! [A5: set_nat,B5: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_817_inj__image__Compl__subset,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( uminus5710092332889474511et_nat @ A5 ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_818_inj__image__mem__iff,axiom,
! [F: nat > nat,A: nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat2 @ F @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_819_inj__image__eq__iff,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat2 @ F @ A5 )
= ( image_nat_nat2 @ F @ B5 ) )
= ( A5 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_820_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
= ( ? [X2: nat] :
( ( B
= ( F @ X2 ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_821_finite__imageD,axiom,
! [F: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( inj_on_nat_nat @ F @ A5 )
=> ( finite_finite_nat @ A5 ) ) ) ).
% finite_imageD
thf(fact_822_finite__image__iff,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A5 ) )
= ( finite_finite_nat @ A5 ) ) ) ).
% finite_image_iff
thf(fact_823_inj__img__insertE,axiom,
! [F: nat > nat,A5: set_nat,X3: nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ~ ( member_nat @ X3 @ B5 )
=> ( ( ( insert_nat @ X3 @ B5 )
= ( image_nat_nat2 @ F @ A5 ) )
=> ~ ! [X7: nat,A8: set_nat] :
( ~ ( member_nat @ X7 @ A8 )
=> ( ( A5
= ( insert_nat @ X7 @ A8 ) )
=> ( ( X3
= ( F @ X7 ) )
=> ( B5
!= ( image_nat_nat2 @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_824_inj__on__fun__updI,axiom,
! [F: nat > nat,A5: set_nat,Y3: nat,X3: nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ~ ( member_nat @ Y3 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( inj_on_nat_nat @ ( fun_upd_nat_nat @ F @ X3 @ Y3 ) @ A5 ) ) ) ).
% inj_on_fun_updI
thf(fact_825_range__subsetD,axiom,
! [F: nat > nat,B5: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) @ B5 )
=> ( member_nat @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_826_finite__surj,axiom,
! [A5: set_nat,B5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( finite_finite_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_827_finite__subset__image,axiom,
! [B5: set_nat,F: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A5 )
& ( finite_finite_nat @ C4 )
& ( B5
= ( image_nat_nat2 @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_828_ex__finite__subset__image,axiom,
! [F: nat > nat,A5: set_nat,P: set_nat > $o] :
( ( ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat2 @ F @ A5 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A5 )
& ( P @ ( image_nat_nat2 @ F @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_829_all__finite__subset__image,axiom,
! [F: nat > nat,A5: set_nat,P: set_nat > $o] :
( ( ! [B7: set_nat] :
( ( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat2 @ F @ A5 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ( finite_finite_nat @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A5 ) )
=> ( P @ ( image_nat_nat2 @ F @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_830_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_831_subset__singletonD,axiom,
! [A5: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A5 = bot_bot_set_nat )
| ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_832_image__diff__subset,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_833_subset__Diff__insert,axiom,
! [A5: set_nat,B5: set_nat,X3: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X3 @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A5 @ ( minus_minus_set_nat @ B5 @ C3 ) )
& ~ ( member_nat @ X3 @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_834_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_835_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_836_image__set__diff,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A5 @ B5 ) )
= ( minus_minus_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_837_finite__subset__induct_H,axiom,
! [F3: set_nat,A5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A5 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A2 @ A5 )
=> ( ( ord_less_eq_set_nat @ F4 @ A5 )
=> ( ~ ( member_nat @ A2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A2 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_838_finite__subset__induct,axiom,
! [F3: set_nat,A5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A5 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A2 @ A5 )
=> ( ~ ( member_nat @ A2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_839_Diff__single__insert,axiom,
! [A5: set_nat,X3: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 )
=> ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_840_subset__insert__iff,axiom,
! [A5: set_nat,X3: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( ( ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_841_surj__Compl__image__subset,axiom,
! [F: nat > nat,A5: set_nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( image_nat_nat2 @ F @ ( uminus5710092332889474511et_nat @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_842_inj__on__iff__surj,axiom,
! [A5: set_nat,A9: set_nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
& ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ A5 ) @ A9 ) ) )
= ( ? [G2: nat > nat] :
( ( image_nat_nat2 @ G2 @ A9 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_843_finite__ranking__induct,axiom,
! [S3: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S4 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert_nat @ X @ S4 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_ranking_induct
thf(fact_844_subset__image__inj,axiom,
! [S3: set_nat,F: nat > nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ ( image_nat_nat2 @ F @ T2 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T2 )
& ( inj_on_nat_nat @ F @ U )
& ( S3
= ( image_nat_nat2 @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_845_infinite__iff__countable__subset,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_846_infinite__countable__subset,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F5 @ top_top_set_nat ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_847_Un__insert__right,axiom,
! [A5: set_nat,A: nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_848_Un__insert__left,axiom,
! [A: nat,B5: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B5 ) @ C3 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B5 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_849_dom__map__add,axiom,
! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val] :
( ( dom_Pr8989479155164304065ar_val @ ( map_ad2908576761495157028ar_val @ M @ N ) )
= ( sup_su7899656352754981958t_char @ ( dom_Pr8989479155164304065ar_val @ N ) @ ( dom_Pr8989479155164304065ar_val @ M ) ) ) ).
% dom_map_add
thf(fact_850_image__Un,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A5 @ B5 ) )
= ( sup_sup_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ).
% image_Un
thf(fact_851_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_852_Un__singleton__iff,axiom,
! [A5: set_nat,B5: set_nat,X3: nat] :
( ( ( sup_sup_set_nat @ A5 @ B5 )
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( ( ( A5 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_853_singleton__Un__iff,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ( ( insert_nat @ X3 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A5 @ B5 ) )
= ( ( ( A5 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_854_inj__on__Un__image__eq__iff,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A5 @ B5 ) )
=> ( ( ( image_nat_nat2 @ F @ A5 )
= ( image_nat_nat2 @ F @ B5 ) )
= ( A5 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_855_image__Fpow__mono,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( finite_Fpow_nat @ A5 ) ) @ ( finite_Fpow_nat @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_856_Sup__fin_Oremove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A5 )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A5 )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_857_Sup__fin_Oinsert__remove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A5 ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_858_Sup__fin_Oinsert,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A5 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_859_ran__map__add,axiom,
! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val] :
( ( ( inf_in3544766959038609504t_char @ ( dom_Pr8989479155164304065ar_val @ M1 ) @ ( dom_Pr8989479155164304065ar_val @ M2 ) )
= bot_bo1414094081888755934t_char )
=> ( ( ran_Pr786515303459910110ar_val @ ( map_ad2908576761495157028ar_val @ M1 @ M2 ) )
= ( sup_sup_set_val @ ( ran_Pr786515303459910110ar_val @ M1 ) @ ( ran_Pr786515303459910110ar_val @ M2 ) ) ) ) ).
% ran_map_add
thf(fact_860_Int__insert__left__if0,axiom,
! [A: nat,C3: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C3 )
= ( inf_inf_set_nat @ B5 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_861_Int__insert__left__if1,axiom,
! [A: nat,C3: set_nat,B5: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C3 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_862_insert__inter__insert,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A5 ) @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A5 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_863_Int__insert__right__if0,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ A5 )
=> ( ( inf_inf_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_864_Int__insert__right__if1,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ A @ A5 )
=> ( ( inf_inf_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_865_insert__disjoint_I1_J,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A5 ) @ B5 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ A5 @ B5 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_866_insert__disjoint_I2_J,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A5 ) @ B5 ) )
= ( ~ ( member_nat @ A @ B5 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_867_disjoint__insert_I1_J,axiom,
! [B5: set_nat,A: nat,A5: set_nat] :
( ( ( inf_inf_set_nat @ B5 @ ( insert_nat @ A @ A5 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ B5 @ A5 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_868_disjoint__insert_I2_J,axiom,
! [A5: set_nat,B: nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A5 @ ( insert_nat @ B @ B5 ) ) )
= ( ~ ( member_nat @ B @ A5 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_869_Sup__fin_Osingleton,axiom,
! [X3: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Sup_fin.singleton
thf(fact_870_Int__insert__right,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ A5 )
=> ( ( inf_inf_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A5 @ B5 ) ) ) )
& ( ~ ( member_nat @ A @ A5 )
=> ( ( inf_inf_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_871_Int__insert__left,axiom,
! [A: nat,C3: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C3 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C3 ) ) ) )
& ( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C3 )
= ( inf_inf_set_nat @ B5 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_872_image__Int__subset,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( inf_inf_set_nat @ A5 @ B5 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_873_image__Int,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ ( inf_inf_set_nat @ A5 @ B5 ) )
= ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_874_inj__on__image__Int,axiom,
! [F: nat > nat,C3: set_nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A5 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_nat2 @ F @ ( inf_inf_set_nat @ A5 @ B5 ) )
= ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_875_map__add__comm,axiom,
! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val] :
( ( ( inf_in3544766959038609504t_char @ ( dom_Pr8989479155164304065ar_val @ M1 ) @ ( dom_Pr8989479155164304065ar_val @ M2 ) )
= bot_bo1414094081888755934t_char )
=> ( ( map_ad2908576761495157028ar_val @ M1 @ M2 )
= ( map_ad2908576761495157028ar_val @ M2 @ M1 ) ) ) ).
% map_add_comm
thf(fact_876_inj__on__image__Fpow,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat2 @ F ) @ ( finite_Fpow_nat @ A5 ) ) ) ).
% inj_on_image_Fpow
thf(fact_877_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y4: nat] :
( ( H @ ( sup_sup_nat @ X @ Y4 ) )
= ( sup_sup_nat @ ( H @ X ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N2 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat2 @ H @ N2 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_878_graph__map__add,axiom,
! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val] :
( ( ( inf_in3544766959038609504t_char @ ( dom_Pr8989479155164304065ar_val @ M1 ) @ ( dom_Pr8989479155164304065ar_val @ M2 ) )
= bot_bo1414094081888755934t_char )
=> ( ( graph_1733029056113894637ar_val @ ( map_ad2908576761495157028ar_val @ M1 @ M2 ) )
= ( sup_su5754804325903363146ar_val @ ( graph_1733029056113894637ar_val @ M1 ) @ ( graph_1733029056113894637ar_val @ M2 ) ) ) ) ).
% graph_map_add
thf(fact_879_Sup__fin_Oinsert__not__elem,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ~ ( member_nat @ X3 @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A5 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_880_Sup__fin_Oclosed,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat,Y4: nat] : ( member_nat @ ( sup_sup_nat @ X @ Y4 ) @ ( insert_nat @ X @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A5 ) @ A5 ) ) ) ) ).
% Sup_fin.closed
thf(fact_881_inj__on__Un,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A5 @ B5 ) )
= ( ( inj_on_nat_nat @ F @ A5 )
& ( inj_on_nat_nat @ F @ B5 )
& ( ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A5 @ B5 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ B5 @ A5 ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_882_Inf__fin_Oinsert__remove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A5 ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_883_Inf__fin_Oremove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A5 )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A5 )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_884_Inf__fin_Oinsert,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A5 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_885_UNION__fun__upd,axiom,
! [A5: nat > set_nat,I: nat,B5: set_nat,J: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( fun_upd_nat_set_nat @ A5 @ I @ B5 ) @ J ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A5 @ ( minus_minus_set_nat @ J @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) @ ( if_set_nat @ ( member_nat @ I @ J ) @ B5 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_886_image__Pow__mono,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A5 ) ) @ ( pow_nat @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_887_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_888_Pow__singleton__iff,axiom,
! [X5: set_nat,Y6: set_nat] :
( ( ( pow_nat @ X5 )
= ( insert_set_nat @ Y6 @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
& ( Y6 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_889_Inf__fin_Osingleton,axiom,
! [X3: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Inf_fin.singleton
thf(fact_890_SUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > nat,D2: nat > nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ C3 @ A5 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ D2 @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_891_insert__partition,axiom,
! [X3: set_nat,F3: set_set_nat] :
( ~ ( member_set_nat @ X3 @ F3 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( insert_set_nat @ X3 @ F3 ) )
=> ! [Xa3: set_nat] :
( ( member_set_nat @ Xa3 @ ( insert_set_nat @ X3 @ F3 ) )
=> ( ( X != Xa3 )
=> ( ( inf_inf_set_nat @ X @ Xa3 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( comple7399068483239264473et_nat @ F3 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_892_inj__on__image,axiom,
! [F: nat > nat,A5: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A5 ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat2 @ F ) @ A5 ) ) ).
% inj_on_image
thf(fact_893_inj__on__image__Pow,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A5 ) ) ) ).
% inj_on_image_Pow
thf(fact_894_image__Pow__surj,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( ( image_nat_nat2 @ F @ A5 )
= B5 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A5 ) )
= ( pow_nat @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_895_Pow__insert,axiom,
! [A: nat,A5: set_nat] :
( ( pow_nat @ ( insert_nat @ A @ A5 ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A5 ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A ) @ ( pow_nat @ A5 ) ) ) ) ).
% Pow_insert
thf(fact_896_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y4: nat] :
( ( H @ ( inf_inf_nat @ X @ Y4 ) )
= ( inf_inf_nat @ ( H @ X ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N2 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat2 @ H @ N2 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_897_Inf__fin_Oclosed,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat,Y4: nat] : ( member_nat @ ( inf_inf_nat @ X @ Y4 ) @ ( insert_nat @ X @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A5 ) @ A5 ) ) ) ) ).
% Inf_fin.closed
thf(fact_898_Inf__fin_Oinsert__not__elem,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ~ ( member_nat @ X3 @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A5 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_899_cSup__singleton,axiom,
! [X3: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% cSup_singleton
thf(fact_900_cSUP__least,axiom,
! [A5: set_nat,F: nat > nat,M4: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M4 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ M4 ) ) ) ).
% cSUP_least
thf(fact_901_cSUP__union,axiom,
! [A5: set_nat,F: nat > nat,B5: set_nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ B5 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A5 @ B5 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_902_cSUP__insert,axiom,
! [A5: set_nat,F: nat > nat,A: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A @ A5 ) ) )
= ( sup_sup_nat @ ( F @ A ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_903_bdd__above__insert,axiom,
! [A: nat,A5: set_nat] :
( ( condit2214826472909112428ve_nat @ ( insert_nat @ A @ A5 ) )
= ( condit2214826472909112428ve_nat @ A5 ) ) ).
% bdd_above_insert
thf(fact_904_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A5: set_nat] :
( ( Sup @ ( image_nat_nat2 @ G @ ( image_nat_nat2 @ F @ A5 ) ) )
= ( Sup @ ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F ) @ A5 ) ) ) ).
% Sup.SUP_image
thf(fact_905_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A5: set_nat] :
( ( Inf @ ( image_nat_nat2 @ G @ ( image_nat_nat2 @ F @ A5 ) ) )
= ( Inf @ ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F ) @ A5 ) ) ) ).
% Inf.INF_image
thf(fact_906_image__comp,axiom,
! [F: nat > nat,G: nat > nat,R: set_nat] :
( ( image_nat_nat2 @ F @ ( image_nat_nat2 @ G @ R ) )
= ( image_nat_nat2 @ ( comp_nat_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_907_image__eq__imp__comp,axiom,
! [F: nat > nat,A5: set_nat,G: nat > nat,B5: set_nat,H: nat > nat] :
( ( ( image_nat_nat2 @ F @ A5 )
= ( image_nat_nat2 @ G @ B5 ) )
=> ( ( image_nat_nat2 @ ( comp_nat_nat_nat @ H @ F ) @ A5 )
= ( image_nat_nat2 @ ( comp_nat_nat_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_908_comp__surj,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_nat2 @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_909_SUP__image,axiom,
! [G: nat > nat,F: nat > nat,A5: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ ( image_nat_nat2 @ F @ A5 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F ) @ A5 ) ) ) ).
% SUP_image
thf(fact_910_bdd__above_OI2,axiom,
! [A5: set_nat,F: nat > nat,M4: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M4 ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ).
% bdd_above.I2
thf(fact_911_cSUP__upper2,axiom,
! [F: nat > nat,A5: set_nat,X3: nat,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( ord_less_eq_nat @ U2 @ ( F @ X3 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_912_cSUP__upper,axiom,
! [X3: nat,A5: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ).
% cSUP_upper
thf(fact_913_cSUP__le__iff,axiom,
! [A5: set_nat,F: nat > nat,U2: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ U2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_914_cSUP__mono,axiom,
! [A5: set_nat,G: nat > nat,B5: set_nat,F: nat > nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B5 ) )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ A5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_915_cSup__insert,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X5 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ).
% cSup_insert
thf(fact_916_cSup__insert__If,axiom,
! [X5: set_nat,A: nat] :
( ( condit2214826472909112428ve_nat @ X5 )
=> ( ( ( X5 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X5 ) )
= A ) )
& ( ( X5 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X5 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_917_cSUP__subset__mono,axiom,
! [A5: set_nat,G: nat > nat,B5: set_nat,F: nat > nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_918_fun_Oset__map,axiom,
! [F: nat > nat,V2: nat > nat] :
( ( image_nat_nat2 @ ( comp_nat_nat_nat @ F @ V2 ) @ top_top_set_nat )
= ( image_nat_nat2 @ F @ ( image_nat_nat2 @ V2 @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_919_fun_Omap__ident__strong,axiom,
! [T: nat > nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( image_nat_nat2 @ T @ top_top_set_nat ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_nat_nat_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_920_the__inv__into__comp,axiom,
! [F: nat > nat,G: nat > nat,A5: set_nat,X3: nat] :
( ( inj_on_nat_nat @ F @ ( image_nat_nat2 @ G @ A5 ) )
=> ( ( inj_on_nat_nat @ G @ A5 )
=> ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ ( image_nat_nat2 @ G @ A5 ) ) )
=> ( ( the_inv_into_nat_nat @ A5 @ ( comp_nat_nat_nat @ F @ G ) @ X3 )
= ( comp_nat_nat_nat @ ( the_inv_into_nat_nat @ A5 @ G ) @ ( the_inv_into_nat_nat @ ( image_nat_nat2 @ G @ A5 ) @ F ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_921_the__inv__into__onto,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( image_nat_nat2 @ ( the_inv_into_nat_nat @ A5 @ F ) @ ( image_nat_nat2 @ F @ A5 ) )
= A5 ) ) ).
% the_inv_into_onto
thf(fact_922_union__fold__insert,axiom,
! [A5: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( sup_sup_set_nat @ A5 @ B5 )
= ( finite5529483035118572448et_nat @ insert_nat @ B5 @ A5 ) ) ) ).
% union_fold_insert
thf(fact_923_inj__on__the__inv__into,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A5 @ F ) @ ( image_nat_nat2 @ F @ A5 ) ) ) ).
% inj_on_the_inv_into
thf(fact_924_f__the__inv__into__f,axiom,
! [F: nat > nat,A5: set_nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( member_nat @ Y3 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A5 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_925_Inf__fin_Oeq__fold,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( finite_fold_nat_nat @ inf_inf_nat @ X3 @ A5 ) ) ) ).
% Inf_fin.eq_fold
thf(fact_926_Sup__fin_Oeq__fold,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( finite_fold_nat_nat @ sup_sup_nat @ X3 @ A5 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_927_the__inv__into__into,axiom,
! [F: nat > nat,A5: set_nat,X3: nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A5 @ F @ X3 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_928_type__definition_OAbs__image,axiom,
! [Rep: nat > nat,Abs: nat > nat,A5: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ A5 )
=> ( ( image_nat_nat2 @ Abs @ A5 )
= top_top_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_929_type__definition_ORep__range,axiom,
! [Rep: nat > nat,Abs: nat > nat,A5: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ A5 )
=> ( ( image_nat_nat2 @ Rep @ top_top_set_nat )
= A5 ) ) ).
% type_definition.Rep_range
thf(fact_930_inf__img__fin__domE_H,axiom,
! [F: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ~ ! [Y4: nat] :
( ( member_nat @ Y4 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_931_cInf__singleton,axiom,
! [X3: nat] :
( ( complete_Inf_Inf_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% cInf_singleton
thf(fact_932_image__vimage__eq,axiom,
! [F: nat > nat,A5: set_nat] :
( ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A5 ) )
= ( inf_inf_set_nat @ A5 @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_933_INF__cong,axiom,
! [A5: set_nat,B5: set_nat,C3: nat > nat,D2: nat > nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ C3 @ A5 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ D2 @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_934_surj__image__vimage__eq,axiom,
! [F: nat > nat,A5: set_nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A5 ) )
= A5 ) ) ).
% surj_image_vimage_eq
thf(fact_935_image__vimage__subset,axiom,
! [F: nat > nat,A5: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A5 ) ) @ A5 ) ).
% image_vimage_subset
thf(fact_936_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
= ( ord_less_eq_set_nat @ A5 @ ( vimage_nat_nat @ F @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_937_vimage__singleton__eq,axiom,
! [A: nat,F: nat > nat,B: nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_938_INF__image,axiom,
! [G: nat > nat,F: nat > nat,A5: set_nat] :
( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ ( image_nat_nat2 @ F @ A5 ) ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F ) @ A5 ) ) ) ).
% INF_image
thf(fact_939_surj__vimage__empty,axiom,
! [F: nat > nat,A5: set_nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( vimage_nat_nat @ F @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_940_finite__vimageD,axiom,
! [H: nat > nat,F3: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ H @ F3 ) )
=> ( ( ( image_nat_nat2 @ H @ top_top_set_nat )
= top_top_set_nat )
=> ( finite_finite_nat @ F3 ) ) ) ).
% finite_vimageD
thf(fact_941_vimage__subsetD,axiom,
! [F: nat > nat,B5: set_nat,A5: set_nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B5 ) @ A5 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% vimage_subsetD
thf(fact_942_inj__vimage__image__eq,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( vimage_nat_nat @ F @ ( image_nat_nat2 @ F @ A5 ) )
= A5 ) ) ).
% inj_vimage_image_eq
thf(fact_943_cINF__greatest,axiom,
! [A5: set_nat,M: nat,F: nat > nat] :
( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ord_less_eq_nat @ M @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ M @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ).
% cINF_greatest
thf(fact_944_finite__vimageD_H,axiom,
! [F: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ F @ A5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
=> ( finite_finite_nat @ A5 ) ) ) ).
% finite_vimageD'
thf(fact_945_inf__img__fin__dom,axiom,
! [F: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat2 @ F @ A5 ) )
& ~ ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_946_inf__img__fin__domE,axiom,
! [F: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ~ ! [Y4: nat] :
( ( member_nat @ Y4 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_947_vimage__subsetI,axiom,
! [F: nat > nat,B5: set_nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B5 ) @ A5 ) ) ) ).
% vimage_subsetI
thf(fact_948_inf__img__fin__dom_H,axiom,
! [F: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat2 @ F @ A5 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_949_cINF__union,axiom,
! [A5: set_nat,F: nat > nat,B5: set_nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ B5 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A5 @ B5 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_950_cINF__insert,axiom,
! [A5: set_nat,F: nat > nat,A: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A @ A5 ) ) )
= ( inf_inf_nat @ ( F @ A ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_951_card__vimage__inj,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A5 @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ) ).
% card_vimage_inj
thf(fact_952_cINF__superset__mono,axiom,
! [A5: set_nat,G: nat > nat,B5: set_nat,F: nat > nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ G @ B5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B5 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_953_bdd__below__insert,axiom,
! [A: nat,A5: set_nat] :
( ( condit1738341127787009408ow_nat @ ( insert_nat @ A @ A5 ) )
= ( condit1738341127787009408ow_nat @ A5 ) ) ).
% bdd_below_insert
thf(fact_954_card__insert__le,axiom,
! [A5: set_nat,X3: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_nat @ ( insert_nat @ X3 @ A5 ) ) ) ).
% card_insert_le
thf(fact_955_card__image__le,axiom,
! [A5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( finite_card_nat @ A5 ) ) ) ).
% card_image_le
thf(fact_956_bdd__below_OI2,axiom,
! [A5: set_nat,M4: nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ord_less_eq_nat @ M4 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ).
% bdd_below.I2
thf(fact_957_bdd__belowI2,axiom,
! [A5: set_nat,M: nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ord_less_eq_nat @ M @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ).
% bdd_belowI2
thf(fact_958_card__image,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( finite_card_nat @ ( image_nat_nat2 @ F @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ).
% card_image
thf(fact_959_surj__card__le,axiom,
! [A5: set_nat,B5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_960_inj__on__iff__eq__card,axiom,
! [A5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( inj_on_nat_nat @ F @ A5 )
= ( ( finite_card_nat @ ( image_nat_nat2 @ F @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_961_eq__card__imp__inj__on,axiom,
! [A5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( finite_card_nat @ ( image_nat_nat2 @ F @ A5 ) )
= ( finite_card_nat @ A5 ) )
=> ( inj_on_nat_nat @ F @ A5 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_962_card__Diff1__le,axiom,
! [A5: set_nat,X3: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A5 ) ) ).
% card_Diff1_le
thf(fact_963_cINF__lower,axiom,
! [F: nat > nat,A5: set_nat,X3: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( F @ X3 ) ) ) ) ).
% cINF_lower
thf(fact_964_cINF__lower2,axiom,
! [F: nat > nat,A5: set_nat,X3: nat,U2: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( ord_less_eq_nat @ ( F @ X3 ) @ U2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_965_card__bij__eq,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
=> ( ( inj_on_nat_nat @ G @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ G @ B5 ) @ A5 )
=> ( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite_card_nat @ A5 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_966_surjective__iff__injective__gen,axiom,
! [S3: set_nat,T2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T2 )
=> ( ( ( finite_card_nat @ S3 )
= ( finite_card_nat @ T2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ S3 ) @ T2 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ? [Y: nat] :
( ( member_nat @ Y @ S3 )
& ( ( F @ Y )
= X2 ) ) ) )
= ( inj_on_nat_nat @ F @ S3 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_967_card__le__inj,axiom,
! [A5: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_nat @ B5 ) )
=> ? [F5: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F5 @ A5 ) @ B5 )
& ( inj_on_nat_nat @ F5 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_968_card__inj__on__le,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_969_inj__on__iff__card__le,axiom,
! [A5: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
& ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ A5 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_970_le__cINF__iff,axiom,
! [A5: set_nat,F: nat > nat,U2: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( ord_less_eq_nat @ U2 @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( ord_less_eq_nat @ U2 @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_971_cINF__mono,axiom,
! [B5: set_nat,F: nat > nat,A5: set_nat,G: nat > nat] :
( ( B5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ! [M6: nat] :
( ( member_nat @ M6 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A5 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_972_cInf__insert,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X5 )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A @ X5 ) )
= ( inf_inf_nat @ A @ ( complete_Inf_Inf_nat @ X5 ) ) ) ) ) ).
% cInf_insert
thf(fact_973_cInf__insert__If,axiom,
! [X5: set_nat,A: nat] :
( ( condit1738341127787009408ow_nat @ X5 )
=> ( ( ( X5 = bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A @ X5 ) )
= A ) )
& ( ( X5 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A @ X5 ) )
= ( inf_inf_nat @ A @ ( complete_Inf_Inf_nat @ X5 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_974_card__Diff1__less__iff,axiom,
! [A5: set_nat,X3: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A5 ) )
= ( ( finite_finite_nat @ A5 )
& ( member_nat @ X3 @ A5 ) ) ) ).
% card_Diff1_less_iff
thf(fact_975_card__Diff2__less,axiom,
! [A5: set_nat,X3: nat,Y3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A5 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_976_card__Diff1__less,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A5 ) ) ) ) ).
% card_Diff1_less
thf(fact_977_card__Suc__Diff1,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A5 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_978_card__insert__disjoint,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ~ ( member_nat @ X3 @ A5 )
=> ( ( finite_card_nat @ ( insert_nat @ X3 @ A5 ) )
= ( suc @ ( finite_card_nat @ A5 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_979_finite__linorder__max__induct,axiom,
! [A5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A5 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B2: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( ord_less_nat @ X6 @ B2 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B2 @ A6 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_980_finite__linorder__min__induct,axiom,
! [A5: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A5 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B2: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( ord_less_nat @ B2 @ X6 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B2 @ A6 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_981_card__Suc__eq__finite,axiom,
! [A5: set_nat,K: nat] :
( ( ( finite_card_nat @ A5 )
= ( suc @ K ) )
= ( ? [B4: nat,B7: set_nat] :
( ( A5
= ( insert_nat @ B4 @ B7 ) )
& ~ ( member_nat @ B4 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( finite_finite_nat @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_982_card__insert__if,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( member_nat @ X3 @ A5 )
=> ( ( finite_card_nat @ ( insert_nat @ X3 @ A5 ) )
= ( finite_card_nat @ A5 ) ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ( finite_card_nat @ ( insert_nat @ X3 @ A5 ) )
= ( suc @ ( finite_card_nat @ A5 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_983_card__le__Suc__iff,axiom,
! [N: nat,A5: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A5 ) )
= ( ? [A4: nat,B7: set_nat] :
( ( A5
= ( insert_nat @ A4 @ B7 ) )
& ~ ( member_nat @ A4 @ B7 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B7 ) )
& ( finite_finite_nat @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_984_pigeonhole,axiom,
! [F: nat > nat,A5: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( finite_card_nat @ A5 ) )
=> ~ ( inj_on_nat_nat @ F @ A5 ) ) ).
% pigeonhole
thf(fact_985_card_Oremove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( finite_card_nat @ A5 )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_986_card_Oinsert__remove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_card_nat @ ( insert_nat @ X3 @ A5 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_987_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ! [N3: nat] :
( ( ( F @ N3 )
= ( F @ ( suc @ N3 ) ) )
=> ( ( F @ ( suc @ N3 ) )
= ( F @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N4: nat] :
( ! [N5: nat] :
( ( ord_less_eq_nat @ N5 @ N4 )
=> ! [M7: nat] :
( ( ord_less_eq_nat @ M7 @ N4 )
=> ( ( ord_less_nat @ M7 @ N5 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N5 ) ) ) ) )
& ! [N5: nat] :
( ( ord_less_eq_nat @ N4 @ N5 )
=> ( ( F @ N4 )
= ( F @ N5 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_988_card__range__greater__zero,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_989_ord_Omono__onD,axiom,
! [A5: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A5 )
=> ( ( member_nat @ S @ A5 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_990_ord_Omono__onI,axiom,
! [A5: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R4: nat,S5: nat] :
( ( member_nat @ R4 @ A5 )
=> ( ( member_nat @ S5 @ A5 )
=> ( ( Less_eq @ R4 @ S5 )
=> ( ord_less_eq_nat @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_991_ord_Omono__on__def,axiom,
! [A5: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S6: nat] :
( ( ( member_nat @ R3 @ A5 )
& ( member_nat @ S6 @ A5 )
& ( Less_eq @ R3 @ S6 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_992_mono__onD,axiom,
! [A5: set_nat,F: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A5 )
=> ( ( member_nat @ S @ A5 )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_993_mono__onI,axiom,
! [A5: set_nat,F: nat > nat] :
( ! [R4: nat,S5: nat] :
( ( member_nat @ R4 @ A5 )
=> ( ( member_nat @ S5 @ A5 )
=> ( ( ord_less_eq_nat @ R4 @ S5 )
=> ( ord_less_eq_nat @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_994_mono__on__greaterD,axiom,
! [A5: set_nat,G: nat > nat,X3: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( ord_less_nat @ ( G @ Y3 ) @ ( G @ X3 ) )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_995_strict__mono__on__leD,axiom,
! [A5: set_nat,F: nat > nat,X3: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_996_strict__mono__on__imp__mono__on,axiom,
! [A5: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_997_mono__imp__mono__on,axiom,
! [F: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_998_monoI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_999_monoE,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) ).
% monoE
thf(fact_1000_monoD,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ) ).
% monoD
thf(fact_1001_mono__on__subset,axiom,
! [A5: set_nat,F: nat > nat,B5: set_nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B5 @ A5 )
=> ( monotone_on_nat_nat @ B5 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_1002_image__strict__mono,axiom,
! [F: nat > nat,B5: set_nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ B5 )
=> ( ( ord_less_set_nat @ A5 @ B5 )
=> ( ord_less_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ).
% image_strict_mono
thf(fact_1003_strict__mono__less__eq,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
= ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_1004_mono__strict__invE,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% mono_strict_invE
thf(fact_1005_strict__mono__mono,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_1006_mono__invE,axiom,
! [F: nat > nat,X3: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% mono_invE
thf(fact_1007_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1008_mono__inf,axiom,
! [F: nat > nat,A5: nat,B5: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_nat @ A5 @ B5 ) ) @ ( inf_inf_nat @ ( F @ A5 ) @ ( F @ B5 ) ) ) ) ).
% mono_inf
thf(fact_1009_mono__sup,axiom,
! [F: nat > nat,A5: nat,B5: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F @ A5 ) @ ( F @ B5 ) ) @ ( F @ ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% mono_sup
thf(fact_1010_bdd__below__image__mono,axiom,
! [F: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ A5 )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_1011_bdd__above__image__mono,axiom,
! [F: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A5 )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1012_finite__induct__select,axiom,
! [S3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S3 )
=> ( ( P @ T3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S3 @ T3 ) )
& ( P @ ( insert_nat @ X6 @ T3 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_1013_psubset__insert__iff,axiom,
! [A5: set_nat,X3: nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( ( ( member_nat @ X3 @ B5 )
=> ( ord_less_set_nat @ A5 @ B5 ) )
& ( ~ ( member_nat @ X3 @ B5 )
=> ( ( ( member_nat @ X3 @ A5 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1014_card__Suc__eq,axiom,
! [A5: set_nat,K: nat] :
( ( ( finite_card_nat @ A5 )
= ( suc @ K ) )
= ( ? [B4: nat,B7: set_nat] :
( ( A5
= ( insert_nat @ B4 @ B7 ) )
& ~ ( member_nat @ B4 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1015_card__eq__SucD,axiom,
! [A5: set_nat,K: nat] :
( ( ( finite_card_nat @ A5 )
= ( suc @ K ) )
=> ? [B2: nat,B6: set_nat] :
( ( A5
= ( insert_nat @ B2 @ B6 ) )
& ~ ( member_nat @ B2 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1016_card__1__singleton__iff,axiom,
! [A5: set_nat] :
( ( ( finite_card_nat @ A5 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( A5
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1017_option_Osize_I4_J,axiom,
! [X22: val] :
( ( size_size_option_val @ ( some_val @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1018_zero__notin__Suc__image,axiom,
! [A5: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat2 @ suc @ A5 ) ) ).
% zero_notin_Suc_image
thf(fact_1019_mono__Max__commute,axiom,
! [F: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( F @ ( lattic8265883725875713057ax_nat @ A5 ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_1020_mono__Min__commute,axiom,
! [F: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( F @ ( lattic8721135487736765967in_nat @ A5 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_1021_Max__singleton,axiom,
! [X3: nat] :
( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Max_singleton
thf(fact_1022_Min__singleton,axiom,
! [X3: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Min_singleton
thf(fact_1023_Min__insert2,axiom,
! [A5: set_nat,A: nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ A5 )
=> ( ord_less_eq_nat @ A @ B2 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A @ A5 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_1024_Max__insert2,axiom,
! [A5: set_nat,A: nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ A5 )
=> ( ord_less_eq_nat @ B2 @ A ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A @ A5 ) )
= A ) ) ) ).
% Max_insert2
thf(fact_1025_card__insert__le__m1,axiom,
! [N: nat,Y3: set_nat,X3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X3 @ Y3 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1026_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat2 @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_1027_option_Osize__gen_I2_J,axiom,
! [X3: val > nat,X22: val] :
( ( size_option_val @ X3 @ ( some_val @ X22 ) )
= ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1028_image__add__0,axiom,
! [S3: set_nat] :
( ( image_nat_nat2 @ ( plus_plus_nat @ zero_zero_nat ) @ S3 )
= S3 ) ).
% image_add_0
thf(fact_1029_card__Diff__insert,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ A @ A5 )
=> ( ~ ( member_nat @ A @ B5 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ B5 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ B5 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1030_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_1031_mono__add,axiom,
! [A: nat] : ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( plus_plus_nat @ A ) ) ).
% mono_add
thf(fact_1032_card__1__singletonE,axiom,
! [A5: set_nat] :
( ( ( finite_card_nat @ A5 )
= one_one_nat )
=> ~ ! [X: nat] :
( A5
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1033_card__Diff__singleton__if,axiom,
! [X3: nat,A5: set_nat] :
( ( ( member_nat @ X3 @ A5 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A5 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A5 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1034_card__Diff__singleton,axiom,
! [X3: nat,A5: set_nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A5 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1035_Min_Oinsert__remove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X3 @ A5 ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( ord_min_nat @ X3 @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_1036_Min__insert,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( ord_min_nat @ X3 @ ( lattic8721135487736765967in_nat @ A5 ) ) ) ) ) ).
% Min_insert
thf(fact_1037_min__of__mono,axiom,
! [F: nat > nat,M: nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_min_nat @ ( F @ M ) @ ( F @ N ) )
= ( F @ ( ord_min_nat @ M @ N ) ) ) ) ).
% min_of_mono
thf(fact_1038_pred__on_Ochain__extend,axiom,
! [A5: set_nat,P: nat > nat > $o,C3: set_nat,Z: nat] :
( ( pred_chain_nat @ A5 @ P @ C3 )
=> ( ( member_nat @ Z @ A5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ C3 )
=> ( sup_sup_nat_nat_o @ P
@ ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 )
@ X
@ Z ) )
=> ( pred_chain_nat @ A5 @ P @ ( sup_sup_set_nat @ ( insert_nat @ Z @ bot_bot_set_nat ) @ C3 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1039_Inf__insert__finite,axiom,
! [S3: set_nat,X3: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ( S3 = bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ X3 @ S3 ) )
= X3 ) )
& ( ( S3 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ X3 @ S3 ) )
= ( ord_min_nat @ X3 @ ( complete_Inf_Inf_nat @ S3 ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_1040_hom__Min__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y4: nat] :
( ( H @ ( ord_min_nat @ X @ Y4 ) )
= ( ord_min_nat @ ( H @ X ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic8721135487736765967in_nat @ N2 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat2 @ H @ N2 ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_1041_Min_Oclosed,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat,Y4: nat] : ( member_nat @ ( ord_min_nat @ X @ Y4 ) @ ( insert_nat @ X @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A5 ) @ A5 ) ) ) ) ).
% Min.closed
thf(fact_1042_Min_Oinsert__not__elem,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ~ ( member_nat @ X3 @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( ord_min_nat @ X3 @ ( lattic8721135487736765967in_nat @ A5 ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_1043_Min_Oeq__fold,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X3 @ A5 ) )
= ( finite_fold_nat_nat @ ord_min_nat @ X3 @ A5 ) ) ) ).
% Min.eq_fold
thf(fact_1044_Min_Oremove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A5 )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A5 )
= ( ord_min_nat @ X3 @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_1045_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_1046_image__add__atLeast,axiom,
! [K: nat,I: nat] :
( ( image_nat_nat2 @ ( plus_plus_nat @ K ) @ ( set_ord_atLeast_nat @ I ) )
= ( set_ord_atLeast_nat @ ( plus_plus_nat @ K @ I ) ) ) ).
% image_add_atLeast
thf(fact_1047_ivl__disj__un__singleton_I1_J,axiom,
! [L: nat] :
( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or1210151606488870762an_nat @ L ) )
= ( set_ord_atLeast_nat @ L ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_1048_Max_Oinsert__remove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X3 @ A5 ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X3 @ A5 ) )
= ( ord_max_nat @ X3 @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_1049_Max_Oremove,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A5 )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A5 )
= ( ord_max_nat @ X3 @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_1050_mono__image__least,axiom,
! [F: nat > nat,M: nat,N: nat,M5: nat,N6: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ( image_nat_nat2 @ F @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( set_or4665077453230672383an_nat @ M5 @ N6 ) )
=> ( ( ord_less_nat @ M @ N )
=> ( ( F @ M )
= M5 ) ) ) ) ).
% mono_image_least
thf(fact_1051_image__Suc__atLeastLessThan,axiom,
! [I: nat,J2: nat] :
( ( image_nat_nat2 @ suc @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1052_image__add__atLeastLessThan,axiom,
! [K: nat,I: nat,J2: nat] :
( ( image_nat_nat2 @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_1053_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_1054_Max__insert,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X3 @ A5 ) )
= ( ord_max_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A5 ) ) ) ) ) ).
% Max_insert
thf(fact_1055_max__of__mono,axiom,
! [F: nat > nat,M: nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_max_nat @ ( F @ M ) @ ( F @ N ) )
= ( F @ ( ord_max_nat @ M @ N ) ) ) ) ).
% max_of_mono
thf(fact_1056_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_1057_Sup__insert__finite,axiom,
! [S3: set_nat,X3: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ( S3 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ X3 @ S3 ) )
= X3 ) )
& ( ( S3 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ X3 @ S3 ) )
= ( ord_max_nat @ X3 @ ( complete_Sup_Sup_nat @ S3 ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_1058_hom__Max__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y4: nat] :
( ( H @ ( ord_max_nat @ X @ Y4 ) )
= ( ord_max_nat @ ( H @ X ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic8265883725875713057ax_nat @ N2 ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ H @ N2 ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_1059_Max_Oclosed,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat,Y4: nat] : ( member_nat @ ( ord_max_nat @ X @ Y4 ) @ ( insert_nat @ X @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A5 ) @ A5 ) ) ) ) ).
% Max.closed
thf(fact_1060_Max_Oinsert__not__elem,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ~ ( member_nat @ X3 @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X3 @ A5 ) )
= ( ord_max_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A5 ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_1061_Max_Oeq__fold,axiom,
! [A5: set_nat,X3: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X3 @ A5 ) )
= ( finite_fold_nat_nat @ ord_max_nat @ X3 @ A5 ) ) ) ).
% Max.eq_fold
thf(fact_1062_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_1063_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat2 @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_1064_ivl__disj__un__singleton_I3_J,axiom,
! [L: nat,U2: nat] :
( ( ord_less_nat @ L @ U2 )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L @ U2 ) )
= ( set_or4665077453230672383an_nat @ L @ U2 ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_1065_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_1066_ivl__disj__un__singleton_I6_J,axiom,
! [L: nat,U2: nat] :
( ( ord_less_eq_nat @ L @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U2 ) @ ( insert_nat @ U2 @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L @ U2 ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_1067_image__Suc__atLeastAtMost,axiom,
! [I: nat,J2: nat] :
( ( image_nat_nat2 @ suc @ ( set_or1269000886237332187st_nat @ I @ J2 ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1068_image__add__atLeastAtMost,axiom,
! [K: nat,I: nat,J2: nat] :
( ( image_nat_nat2 @ ( plus_plus_nat @ K ) @ ( set_or1269000886237332187st_nat @ I @ J2 ) )
= ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% image_add_atLeastAtMost
thf(fact_1069_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1070_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_1071_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_1072_atLeastAtMost__diff__ends,axiom,
! [A: nat,B: nat] :
( ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( set_or5834768355832116004an_nat @ A @ B ) ) ).
% atLeastAtMost_diff_ends
thf(fact_1073_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_1074_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image_nat_nat2 @ suc @ ( set_ord_lessThan_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% image_Suc_lessThan
thf(fact_1075_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_1076_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_1077_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1078_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1079_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or1269000886237332187st_nat @ M @ N )
= ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1080_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A4: nat,B4: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B4 ) @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1081_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat2 @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_1082_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat2 @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_1083_Iio__Int__singleton,axiom,
! [X3: nat,K: nat] :
( ( ( ord_less_nat @ X3 @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X3 @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ) ).
% Iio_Int_singleton
thf(fact_1084_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_1085_ivl__disj__un__singleton_I4_J,axiom,
! [L: nat,U2: nat] :
( ( ord_less_nat @ L @ U2 )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U2 ) @ ( insert_nat @ U2 @ bot_bot_set_nat ) )
= ( set_or6659071591806873216st_nat @ L @ U2 ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_1086_image__add__greaterThanAtMost,axiom,
! [C: nat,A: nat,B: nat] :
( ( image_nat_nat2 @ ( plus_plus_nat @ C ) @ ( set_or6659071591806873216st_nat @ A @ B ) )
= ( set_or6659071591806873216st_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_1087_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1088_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_1089_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat2 @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_1090_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat2 @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_1091_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or6659071591806873216st_nat
= ( ^ [A4: nat,B4: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B4 ) @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_1092_ivl__disj__un__singleton_I2_J,axiom,
! [U2: nat] :
( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ U2 ) @ ( insert_nat @ U2 @ bot_bot_set_nat ) )
= ( set_ord_atMost_nat @ U2 ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_1093_atMost__Int__atLeast,axiom,
! [N: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ N ) @ ( set_ord_atLeast_nat @ N ) )
= ( insert_nat @ N @ bot_bot_set_nat ) ) ).
% atMost_Int_atLeast
thf(fact_1094_ivl__disj__un__singleton_I5_J,axiom,
! [L: nat,U2: nat] :
( ( ord_less_eq_nat @ L @ U2 )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or6659071591806873216st_nat @ L @ U2 ) )
= ( set_or1269000886237332187st_nat @ L @ U2 ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_1095_Field__insert,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).
% Field_insert
thf(fact_1096_bij__betw__add,axiom,
! [A: nat,A5: set_nat,B5: set_nat] :
( ( bij_betw_nat_nat @ ( plus_plus_nat @ A ) @ A5 @ B5 )
= ( ( image_nat_nat2 @ ( plus_plus_nat @ A ) @ A5 )
= B5 ) ) ).
% bij_betw_add
thf(fact_1097_bij__betw__the__inv__into,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( bij_betw_nat_nat @ ( the_inv_into_nat_nat @ A5 @ F ) @ B5 @ A5 ) ) ).
% bij_betw_the_inv_into
thf(fact_1098_f__the__inv__into__f__bij__betw,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat,X3: nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( ( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A5 @ F @ X3 ) )
= X3 ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1099_bij__betw__imp__inj__on,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( inj_on_nat_nat @ F @ A5 ) ) ).
% bij_betw_imp_inj_on
thf(fact_1100_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A7: set_nat,B7: set_nat] :
? [G2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A7 )
=> ( ( member_nat @ ( F2 @ X2 ) @ B7 )
& ( ( G2 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B7 )
=> ( ( member_nat @ ( G2 @ X2 ) @ A7 )
& ( ( F2 @ ( G2 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1101_bij__betw__apply,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat,A: nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( ( member_nat @ A @ A5 )
=> ( member_nat @ ( F @ A ) @ B5 ) ) ) ).
% bij_betw_apply
thf(fact_1102_bij__betw__cong,axiom,
! [A5: set_nat,F: nat > nat,G: nat > nat,A9: set_nat] :
( ! [A2: nat] :
( ( member_nat @ A2 @ A5 )
=> ( ( F @ A2 )
= ( G @ A2 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A5 @ A9 )
= ( bij_betw_nat_nat @ G @ A5 @ A9 ) ) ) ).
% bij_betw_cong
thf(fact_1103_bij__betw__ball,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat,Phi: nat > $o] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ B5 )
=> ( Phi @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( Phi @ ( F @ X2 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_1104_bij__betw__inv,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ? [G3: nat > nat] : ( bij_betw_nat_nat @ G3 @ B5 @ A5 ) ) ).
% bij_betw_inv
thf(fact_1105_bij__betwE,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A5 )
=> ( member_nat @ ( F @ X6 ) @ B5 ) ) ) ).
% bij_betwE
thf(fact_1106_bij__betwI_H,axiom,
! [X5: set_nat,F: nat > nat,Y6: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ X5 )
=> ( ( ( F @ X )
= ( F @ Y4 ) )
= ( X = Y4 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( member_nat @ ( F @ X ) @ Y6 ) )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ Y6 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ X5 )
& ( Y4
= ( F @ X6 ) ) ) )
=> ( bij_betw_nat_nat @ F @ X5 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_1107_bij__betw__comp__iff,axiom,
! [F: nat > nat,A5: set_nat,A9: set_nat,F6: nat > nat,A10: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ A9 )
=> ( ( bij_betw_nat_nat @ F6 @ A9 @ A10 )
= ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ F6 @ F ) @ A5 @ A10 ) ) ) ).
% bij_betw_comp_iff
thf(fact_1108_bij__betw__trans,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat,G: nat > nat,C3: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( ( bij_betw_nat_nat @ G @ B5 @ C3 )
=> ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A5 @ C3 ) ) ) ).
% bij_betw_trans
thf(fact_1109_bij__betw__Pow,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( bij_be3438014552859920132et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A5 ) @ ( pow_nat @ B5 ) ) ) ).
% bij_betw_Pow
thf(fact_1110_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( ( image_nat_nat2 @ F @ A5 )
= B5 ) ) ).
% bij_betw_imp_surj_on
thf(fact_1111_bij__iff,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ! [X2: nat] :
? [Y: nat] :
( ( ( F @ Y )
= X2 )
& ! [Z3: nat] :
( ( ( F @ Z3 )
= X2 )
=> ( Z3 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1112_bij__pointE,axiom,
! [F: nat > nat,Y3: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ~ ! [X: nat] :
( ( Y3
= ( F @ X ) )
=> ~ ! [X8: nat] :
( ( Y3
= ( F @ X8 ) )
=> ( X8 = X ) ) ) ) ).
% bij_pointE
thf(fact_1113_involuntory__imp__bij,axiom,
! [F: nat > nat] :
( ! [X: nat] :
( ( F @ ( F @ X ) )
= X )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ).
% involuntory_imp_bij
thf(fact_1114_bij__is__inj,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% bij_is_inj
thf(fact_1115_bij__betw__def,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A7: set_nat,B7: set_nat] :
( ( inj_on_nat_nat @ F2 @ A7 )
& ( ( image_nat_nat2 @ F2 @ A7 )
= B7 ) ) ) ) ).
% bij_betw_def
thf(fact_1116_bij__betw__imageI,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( ( image_nat_nat2 @ F @ A5 )
= B5 )
=> ( bij_betw_nat_nat @ F @ A5 @ B5 ) ) ) ).
% bij_betw_imageI
thf(fact_1117_inj__on__imp__bij__betw,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( bij_betw_nat_nat @ F @ A5 @ ( image_nat_nat2 @ F @ A5 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_1118_bij__comp,axiom,
! [F: nat > nat,G: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( bij_betw_nat_nat @ G @ top_top_set_nat @ top_top_set_nat )
=> ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bij_comp
thf(fact_1119_bij__betw__subset,axiom,
! [F: nat > nat,A5: set_nat,A9: set_nat,B5: set_nat,B8: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ A9 )
=> ( ( ord_less_eq_set_nat @ B5 @ A5 )
=> ( ( ( image_nat_nat2 @ F @ B5 )
= B8 )
=> ( bij_betw_nat_nat @ F @ B5 @ B8 ) ) ) ) ).
% bij_betw_subset
thf(fact_1120_bij__betw__byWitness,axiom,
! [A5: set_nat,F6: nat > nat,F: nat > nat,A9: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ( F6 @ ( F @ X ) )
= X ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A9 )
=> ( ( F @ ( F6 @ X ) )
= X ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ A9 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F6 @ A9 ) @ A5 )
=> ( bij_betw_nat_nat @ F @ A5 @ A9 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_1121_bij__betw__imp__surj,axiom,
! [F: nat > nat,A5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_betw_imp_surj
thf(fact_1122_bij__is__surj,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_is_surj
thf(fact_1123_FieldI2,axiom,
! [I: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J2 ) @ R2 )
=> ( member_nat @ J2 @ ( field_nat @ R2 ) ) ) ).
% FieldI2
thf(fact_1124_FieldI1,axiom,
! [I: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J2 ) @ R2 )
=> ( member_nat @ I @ ( field_nat @ R2 ) ) ) ).
% FieldI1
thf(fact_1125_bij__betw__empty2,axiom,
! [F: nat > nat,A5: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ bot_bot_set_nat )
=> ( A5 = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_1126_bij__betw__empty1,axiom,
! [F: nat > nat,A5: set_nat] :
( ( bij_betw_nat_nat @ F @ bot_bot_set_nat @ A5 )
=> ( A5 = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_1127_bij__betw__comp__iff2,axiom,
! [F6: nat > nat,A9: set_nat,A10: set_nat,F: nat > nat,A5: set_nat] :
( ( bij_betw_nat_nat @ F6 @ A9 @ A10 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ A9 )
=> ( ( bij_betw_nat_nat @ F @ A5 @ A9 )
= ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ F6 @ F ) @ A5 @ A10 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_1128_bijI,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijI
thf(fact_1129_bij__def,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ( inj_on_nat_nat @ F @ top_top_set_nat )
& ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% bij_def
thf(fact_1130_notIn__Un__bij__betw3,axiom,
! [B: nat,A5: set_nat,F: nat > nat,A9: set_nat] :
( ~ ( member_nat @ B @ A5 )
=> ( ~ ( member_nat @ ( F @ B ) @ A9 )
=> ( ( bij_betw_nat_nat @ F @ A5 @ A9 )
= ( bij_betw_nat_nat @ F @ ( sup_sup_set_nat @ A5 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( sup_sup_set_nat @ A9 @ ( insert_nat @ ( F @ B ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_1131_notIn__Un__bij__betw,axiom,
! [B: nat,A5: set_nat,F: nat > nat,A9: set_nat] :
( ~ ( member_nat @ B @ A5 )
=> ( ~ ( member_nat @ ( F @ B ) @ A9 )
=> ( ( bij_betw_nat_nat @ F @ A5 @ A9 )
=> ( bij_betw_nat_nat @ F @ ( sup_sup_set_nat @ A5 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( sup_sup_set_nat @ A9 @ ( insert_nat @ ( F @ B ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_1132_bij__betw__combine,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat,C3: set_nat,D2: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ B5 )
=> ( ( bij_betw_nat_nat @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_nat @ B5 @ D2 )
= bot_bot_set_nat )
=> ( bij_betw_nat_nat @ F @ ( sup_sup_set_nat @ A5 @ C3 ) @ ( sup_sup_set_nat @ B5 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_1133_bij__betw__partition,axiom,
! [F: nat > nat,A5: set_nat,C3: set_nat,B5: set_nat,D2: set_nat] :
( ( bij_betw_nat_nat @ F @ ( sup_sup_set_nat @ A5 @ C3 ) @ ( sup_sup_set_nat @ B5 @ D2 ) )
=> ( ( bij_betw_nat_nat @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_nat @ A5 @ C3 )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat @ B5 @ D2 )
= bot_bot_set_nat )
=> ( bij_betw_nat_nat @ F @ A5 @ B5 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1134_bij__image__Compl__eq,axiom,
! [F: nat > nat,A5: set_nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ ( uminus5710092332889474511et_nat @ A5 ) )
= ( uminus5710092332889474511et_nat @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1135_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S7: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S7 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1136_infinite__imp__bij__betw2,axiom,
! [A5: set_nat,A: nat] :
( ~ ( finite_finite_nat @ A5 )
=> ? [H5: nat > nat] : ( bij_betw_nat_nat @ H5 @ A5 @ ( sup_sup_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_1137_infinite__imp__bij__betw,axiom,
! [A5: set_nat,A: nat] :
( ~ ( finite_finite_nat @ A5 )
=> ? [H5: nat > nat] : ( bij_betw_nat_nat @ H5 @ A5 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_1138_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S7: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S7 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1139_vimage__subset__eq,axiom,
! [F: nat > nat,B5: set_nat,A5: set_nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B5 ) @ A5 )
= ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ).
% vimage_subset_eq
thf(fact_1140_bij__betw__Suc,axiom,
! [M4: set_nat,N2: set_nat] :
( ( bij_betw_nat_nat @ suc @ M4 @ N2 )
= ( ( image_nat_nat2 @ suc @ M4 )
= N2 ) ) ).
% bij_betw_Suc
thf(fact_1141_Schroeder__Bernstein,axiom,
! [F: nat > nat,A5: set_nat,B5: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A5 ) @ B5 )
=> ( ( inj_on_nat_nat @ G @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ G @ B5 ) @ A5 )
=> ? [H5: nat > nat] : ( bij_betw_nat_nat @ H5 @ A5 @ B5 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_1142_refl__on__singleton,axiom,
! [X3: nat] : ( refl_on_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ bot_bo2099793752762293965at_nat ) ) ).
% refl_on_singleton
thf(fact_1143_reflI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R )
=> ( refl_on_nat @ top_top_set_nat @ R ) ) ).
% reflI
thf(fact_1144_reflD,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat] :
( ( refl_on_nat @ top_top_set_nat @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ).
% reflD
thf(fact_1145_refl__onD2,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( refl_on_nat @ A5 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( member_nat @ Y3 @ A5 ) ) ) ).
% refl_onD2
thf(fact_1146_refl__onD1,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( refl_on_nat @ A5 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( member_nat @ X3 @ A5 ) ) ) ).
% refl_onD1
thf(fact_1147_refl__onD,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( refl_on_nat @ A5 @ R )
=> ( ( member_nat @ A @ A5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_1148_refl__on__domain,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( refl_on_nat @ A5 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A5 )
& ( member_nat @ B @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_1149_linear__order__on__singleton,axiom,
! [X3: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_1150_Linear__order__Well__order__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
= ( ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R ) )
=> ( ( A7 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A7 )
& ! [Y: nat] :
( ( member_nat @ Y @ A7 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1151_underS__incl__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( member_nat @ B @ ( field_nat @ R ) )
=> ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A ) @ ( order_underS_nat @ R @ B ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1152_well__order__on__domain,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_2888998067076097458on_nat @ A5 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A5 )
& ( member_nat @ B @ A5 ) ) ) ) ).
% well_order_on_domain
thf(fact_1153_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I: nat,R2: set_Pr1261947904930325089at_nat,J2: nat] :
( ( member_nat @ I @ ( order_underS_nat @ R2 @ J2 ) )
=> ( member_nat @ I @ ( field_nat @ R2 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1154_underS__I,axiom,
! [I: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( I != J2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J2 ) @ R2 )
=> ( member_nat @ I @ ( order_underS_nat @ R2 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1155_underS__E,axiom,
! [I: nat,R2: set_Pr1261947904930325089at_nat,J2: nat] :
( ( member_nat @ I @ ( order_underS_nat @ R2 @ J2 ) )
=> ( ( I != J2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J2 ) @ R2 ) ) ) ).
% underS_E
thf(fact_1156_Refl__under__underS,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat] :
( ( refl_on_nat @ ( field_nat @ R ) @ R )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( order_under_nat @ R @ A )
= ( sup_sup_set_nat @ ( order_underS_nat @ R @ A ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1157_trans__singleton,axiom,
! [A: nat] : ( trans_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ bot_bo2099793752762293965at_nat ) ) ).
% trans_singleton
thf(fact_1158_trans__onD,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat,Z: nat] :
( ( trans_on_nat @ A5 @ R )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( member_nat @ Z @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_1159_trans__onI,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y4: nat,Z2: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y4 @ A5 )
=> ( ( member_nat @ Z2 @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z2 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R ) ) ) ) ) )
=> ( trans_on_nat @ A5 @ R ) ) ).
% trans_onI
thf(fact_1160_transD,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat,Z: nat] :
( ( trans_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_1161_transE,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat,Z: nat] :
( ( trans_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_1162_transI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y4: nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z2 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R ) ) )
=> ( trans_on_nat @ top_top_set_nat @ R ) ) ).
% transI
thf(fact_1163_under__incr,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( trans_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ord_less_eq_set_nat @ ( order_under_nat @ R @ A ) @ ( order_under_nat @ R @ B ) ) ) ) ).
% under_incr
thf(fact_1164_wellorders__totally__ordered__aux,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,A: nat,F: nat > nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( order_underS_nat @ R @ A ) )
=> ( bij_betw_nat_nat @ F @ ( order_under_nat @ R @ X ) @ ( order_under_nat @ R5 @ ( F @ X ) ) ) )
=> ( ( ( image_nat_nat2 @ F @ ( order_underS_nat @ R @ A ) )
!= ( field_nat @ R5 ) )
=> ( ( ( F @ A )
= ( bNF_We6243639482952820783uc_nat @ R5 @ ( image_nat_nat2 @ F @ ( order_underS_nat @ R @ A ) ) ) )
=> ( bij_betw_nat_nat @ F @ ( order_under_nat @ R @ A ) @ ( order_under_nat @ R5 @ ( F @ A ) ) ) ) ) ) ) ) ) ).
% wellorders_totally_ordered_aux
thf(fact_1165_underS__incr,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( trans_on_nat @ top_top_set_nat @ R )
=> ( ( antisym_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A ) @ ( order_underS_nat @ R @ B ) ) ) ) ) ).
% underS_incr
thf(fact_1166_antisym__onI,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y4: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y4 @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X ) @ R )
=> ( X = Y4 ) ) ) ) )
=> ( antisym_on_nat @ A5 @ R ) ) ).
% antisym_onI
thf(fact_1167_antisym__onD,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( antisym_on_nat @ A5 @ R )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1168_antisymD,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( antisym_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R )
=> ( X3 = Y3 ) ) ) ) ).
% antisymD
thf(fact_1169_antisymI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X ) @ R )
=> ( X = Y4 ) ) )
=> ( antisym_on_nat @ top_top_set_nat @ R ) ) ).
% antisymI
thf(fact_1170_antisym__singleton,axiom,
! [X3: product_prod_nat_nat] : ( antisym_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ).
% antisym_singleton
thf(fact_1171_wellorders__totally__ordered__aux2,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,G: nat > $o,F: nat > nat,A: nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
=> ( ! [A2: nat] :
( ( ( ~ ( member_o @ $false @ ( image_nat_o @ G @ ( order_underS_nat @ R @ A2 ) ) )
& ( ( image_nat_nat2 @ F @ ( order_underS_nat @ R @ A2 ) )
!= ( field_nat @ R5 ) ) )
=> ( ( ( F @ A2 )
= ( bNF_We6243639482952820783uc_nat @ R5 @ ( image_nat_nat2 @ F @ ( order_underS_nat @ R @ A2 ) ) ) )
& ( G @ A2 ) ) )
& ( ~ ( ~ ( member_o @ $false @ ( image_nat_o @ G @ ( order_underS_nat @ R @ A2 ) ) )
& ( ( image_nat_nat2 @ F @ ( order_underS_nat @ R @ A2 ) )
!= ( field_nat @ R5 ) ) )
=> ~ ( G @ A2 ) ) )
=> ( ! [A2: nat] :
( ( ( member_nat @ A2 @ ( field_nat @ R ) )
& ~ ( member_o @ $false @ ( image_nat_o @ G @ ( order_under_nat @ R @ A2 ) ) ) )
=> ( bij_betw_nat_nat @ F @ ( order_under_nat @ R @ A2 ) @ ( order_under_nat @ R5 @ ( F @ A2 ) ) ) )
=> ( ( ( member_nat @ A @ ( field_nat @ R ) )
& ( member_o @ $false @ ( image_nat_o @ G @ ( order_under_nat @ R @ A ) ) ) )
=> ? [X_1: nat > nat] : ( bNF_We650101353357441270at_nat @ R5 @ R @ X_1 ) ) ) ) ) ) ).
% wellorders_totally_ordered_aux2
thf(fact_1172_embedS__Field,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( bNF_We2139224379931913761at_nat @ R @ R5 @ F )
=> ( ord_less_set_nat @ ( image_nat_nat2 @ F @ ( field_nat @ R ) ) @ ( field_nat @ R5 ) ) ) ) ).
% embedS_Field
thf(fact_1173_embedS__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
=> ( ( bNF_We2139224379931913761at_nat @ R @ R5 @ F )
= ( ord_less_set_nat @ ( image_nat_nat2 @ F @ ( field_nat @ R ) ) @ ( field_nat @ R5 ) ) ) ) ) ).
% embedS_iff
thf(fact_1174_embed__Field,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
=> ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( field_nat @ R ) ) @ ( field_nat @ R5 ) ) ) ).
% embed_Field
thf(fact_1175_embed__determined,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat,A: nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
=> ( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( F @ A )
= ( bNF_We6243639482952820783uc_nat @ R5 @ ( image_nat_nat2 @ F @ ( order_underS_nat @ R @ A ) ) ) ) ) ) ) ) ).
% embed_determined
thf(fact_1176_inv__into__underS__embed,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,F: nat > nat,R5: set_Pr1261947904930325089at_nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( order_underS_nat @ R @ A ) )
=> ( bij_betw_nat_nat @ F @ ( order_under_nat @ R @ X ) @ ( order_under_nat @ R5 @ ( F @ X ) ) ) )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( ( image_nat_nat2 @ F @ ( order_underS_nat @ R @ A ) )
= ( field_nat @ R5 ) )
=> ( bNF_We650101353357441270at_nat @ R5 @ R @ ( hilber3633877196798814958at_nat @ ( order_underS_nat @ R @ A ) @ F ) ) ) ) ) ) ).
% inv_into_underS_embed
thf(fact_1177_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( refl_on_nat @ ( field_nat @ R ) @ R )
=> ( ( antisym_on_nat @ top_top_set_nat @ R )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( member_nat @ B @ ( field_nat @ R ) )
=> ( ( ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( A = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1178_ImageI,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,A5: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A5 )
=> ( member_nat @ B @ ( image_nat_nat @ R @ A5 ) ) ) ) ).
% ImageI
thf(fact_1179_Image__singleton__iff,axiom,
! [B: nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1180_inv__into__image__cancel,axiom,
! [F: nat > nat,A5: set_nat,S3: set_nat] :
( ( inj_on_nat_nat @ F @ A5 )
=> ( ( ord_less_eq_set_nat @ S3 @ A5 )
=> ( ( image_nat_nat2 @ ( hilber3633877196798814958at_nat @ A5 @ F ) @ ( image_nat_nat2 @ F @ S3 ) )
= S3 ) ) ) ).
% inv_into_image_cancel
thf(fact_1181_image__inv__into__cancel,axiom,
! [F: nat > nat,A5: set_nat,A9: set_nat,B8: set_nat] :
( ( ( image_nat_nat2 @ F @ A5 )
= A9 )
=> ( ( ord_less_eq_set_nat @ B8 @ A9 )
=> ( ( image_nat_nat2 @ F @ ( image_nat_nat2 @ ( hilber3633877196798814958at_nat @ A5 @ F ) @ B8 ) )
= B8 ) ) ) ).
% image_inv_into_cancel
thf(fact_1182_f__inv__into__f,axiom,
! [Y3: nat,F: nat > nat,A5: set_nat] :
( ( member_nat @ Y3 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( F @ ( hilber3633877196798814958at_nat @ A5 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_1183_inv__into__into,axiom,
! [X3: nat,F: nat > nat,A5: set_nat] :
( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( member_nat @ ( hilber3633877196798814958at_nat @ A5 @ F @ X3 ) @ A5 ) ) ).
% inv_into_into
thf(fact_1184_inv__into__injective,axiom,
! [A5: set_nat,F: nat > nat,X3: nat,Y3: nat] :
( ( ( hilber3633877196798814958at_nat @ A5 @ F @ X3 )
= ( hilber3633877196798814958at_nat @ A5 @ F @ Y3 ) )
=> ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( ( member_nat @ Y3 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( X3 = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_1185_surj__f__inv__f,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_1186_surj__iff__all,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [X2: nat] :
( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_1187_image__f__inv__f,axiom,
! [F: nat > nat,A5: set_nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat2 @ F @ ( image_nat_nat2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ A5 ) )
= A5 ) ) ).
% image_f_inv_f
thf(fact_1188_surj__imp__inv__eq,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_1189_ImageE,axiom,
! [B: nat,R: set_Pr1261947904930325089at_nat,A5: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R @ A5 ) )
=> ~ ! [X: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ B ) @ R )
=> ~ ( member_nat @ X @ A5 ) ) ) ).
% ImageE
thf(fact_1190_rev__ImageI,axiom,
! [A: nat,A5: set_nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member_nat @ B @ ( image_nat_nat @ R @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1191_inj__transfer,axiom,
! [F: nat > nat,P: nat > $o,X3: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
=> ( P @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y4 ) ) )
=> ( P @ X3 ) ) ) ).
% inj_transfer
thf(fact_1192_image__inv__f__f,axiom,
! [F: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ ( image_nat_nat2 @ F @ A5 ) )
= A5 ) ) ).
% image_inv_f_f
thf(fact_1193_inj__imp__surj__inv,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_1194_surj__imp__inj__inv,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_1195_inj__on__inv__into,axiom,
! [B5: set_nat,F: nat > nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A5 ) )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ A5 @ F ) @ B5 ) ) ).
% inj_on_inv_into
thf(fact_1196_inv__into__comp,axiom,
! [F: nat > nat,G: nat > nat,A5: set_nat,X3: nat] :
( ( inj_on_nat_nat @ F @ ( image_nat_nat2 @ G @ A5 ) )
=> ( ( inj_on_nat_nat @ G @ A5 )
=> ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ ( image_nat_nat2 @ G @ A5 ) ) )
=> ( ( hilber3633877196798814958at_nat @ A5 @ ( comp_nat_nat_nat @ F @ G ) @ X3 )
= ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ A5 @ G ) @ ( hilber3633877196798814958at_nat @ ( image_nat_nat2 @ G @ A5 ) @ F ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_1197_bij__betw__inv__into__subset,axiom,
! [F: nat > nat,A5: set_nat,A9: set_nat,B5: set_nat,B8: set_nat] :
( ( bij_betw_nat_nat @ F @ A5 @ A9 )
=> ( ( ord_less_eq_set_nat @ B5 @ A5 )
=> ( ( ( image_nat_nat2 @ F @ B5 )
= B8 )
=> ( bij_betw_nat_nat @ ( hilber3633877196798814958at_nat @ A5 @ F ) @ B8 @ B5 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_1198_strict__mono__inv__on__range,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) @ ord_less_nat @ ord_less_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_1199_inj__imp__bij__betw__inv,axiom,
! [F: nat > nat,M4: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( bij_betw_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ ( image_nat_nat2 @ F @ M4 ) @ M4 ) ) ).
% inj_imp_bij_betw_inv
thf(fact_1200_bij__vimage__eq__inv__image,axiom,
! [F: nat > nat,A5: set_nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( vimage_nat_nat @ F @ A5 )
= ( image_nat_nat2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ A5 ) ) ) ).
% bij_vimage_eq_inv_image
thf(fact_1201_inv__into__Field__embed,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
=> ( ( ord_less_eq_set_nat @ ( field_nat @ R5 ) @ ( image_nat_nat2 @ F @ ( field_nat @ R ) ) )
=> ( bNF_We650101353357441270at_nat @ R5 @ R @ ( hilber3633877196798814958at_nat @ ( field_nat @ R ) @ F ) ) ) ) ) ).
% inv_into_Field_embed
thf(fact_1202_subset__Image1__Image1__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_4861654808422542329on_nat @ ( field_nat @ R ) @ R )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( member_nat @ B @ ( field_nat @ R ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1203_inv__into__ofilter__embed,axiom,
! [R: set_Pr1261947904930325089at_nat,A5: set_nat,F: nat > nat,R5: set_Pr1261947904930325089at_nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_ofilter_nat @ R @ A5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( bij_betw_nat_nat @ F @ ( order_under_nat @ R @ X ) @ ( order_under_nat @ R5 @ ( F @ X ) ) ) )
=> ( ( ( image_nat_nat2 @ F @ A5 )
= ( field_nat @ R5 ) )
=> ( bNF_We650101353357441270at_nat @ R5 @ R @ ( hilber3633877196798814958at_nat @ A5 @ F ) ) ) ) ) ) ).
% inv_into_ofilter_embed
thf(fact_1204_embed__preserves__ofilter,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat,A5: set_nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
=> ( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
=> ( ( order_ofilter_nat @ R @ A5 )
=> ( order_ofilter_nat @ R5 @ ( image_nat_nat2 @ F @ A5 ) ) ) ) ) ) ).
% embed_preserves_ofilter
thf(fact_1205_embed__Field__ofilter,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
=> ( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
=> ( order_ofilter_nat @ R5 @ ( image_nat_nat2 @ F @ ( field_nat @ R ) ) ) ) ) ) ).
% embed_Field_ofilter
thf(fact_1206_embed__iff__compat__inj__on__ofilter,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R5 ) @ R5 )
=> ( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
= ( ( bNF_We5309077942755642313at_nat @ R @ R5 @ F )
& ( inj_on_nat_nat @ F @ ( field_nat @ R ) )
& ( order_ofilter_nat @ R5 @ ( image_nat_nat2 @ F @ ( field_nat @ R ) ) ) ) ) ) ) ).
% embed_iff_compat_inj_on_ofilter
thf(fact_1207_Partial__order__eq__Image1__Image1__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_5251275573222108571on_nat @ ( field_nat @ R ) @ R )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( member_nat @ B @ ( field_nat @ R ) )
=> ( ( ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1208_iso__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
=> ( ( bNF_We2525705117968701890at_nat @ R @ R5 @ F )
= ( ( bNF_We650101353357441270at_nat @ R @ R5 @ F )
& ( ( image_nat_nat2 @ F @ ( field_nat @ R ) )
= ( field_nat @ R5 ) ) ) ) ) ).
% iso_iff
thf(fact_1209_iso__Field,axiom,
! [R: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( bNF_We2525705117968701890at_nat @ R @ R5 @ F )
=> ( ( image_nat_nat2 @ F @ ( field_nat @ R ) )
= ( field_nat @ R5 ) ) ) ).
% iso_Field
thf(fact_1210_iso__iff2,axiom,
( bNF_We2525705117968701890at_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat,F2: nat > nat] :
( ( bij_betw_nat_nat @ F2 @ ( field_nat @ R3 ) @ ( field_nat @ R6 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( field_nat @ R3 ) )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( field_nat @ R3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R3 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) @ R6 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_1211_bijection_Osurj__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj_inv
thf(fact_1212_singleton__quotient,axiom,
! [X3: nat,R: set_Pr1261947904930325089at_nat] :
( ( equiv_quotient_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ R )
= ( insert_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ bot_bot_set_set_nat ) ) ).
% singleton_quotient
thf(fact_1213_bijection_Osurj,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat2 @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj
thf(fact_1214_quotientI,axiom,
! [X3: nat,A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ ( equiv_quotient_nat @ A5 @ R ) ) ) ).
% quotientI
thf(fact_1215_quotientE,axiom,
! [X5: set_nat,A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A5 @ R ) )
=> ~ ! [X: nat] :
( ( X5
= ( image_nat_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
=> ~ ( member_nat @ X @ A5 ) ) ) ).
% quotientE
thf(fact_1216_in__quotient__imp__in__rel,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A5 @ R ) )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ X5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1217_quotient__eqI,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,Y6: set_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A5 @ R ) )
=> ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A5 @ R ) )
=> ( ( member_nat @ X3 @ X5 )
=> ( ( member_nat @ Y3 @ Y6 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( X5 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_1218_quotient__eq__iff,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,Y6: set_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A5 @ R ) )
=> ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A5 @ R ) )
=> ( ( member_nat @ X3 @ X5 )
=> ( ( member_nat @ Y3 @ Y6 )
=> ( ( X5 = Y6 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_1219_in__quotient__imp__closed,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X5: set_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_set_nat @ X5 @ ( equiv_quotient_nat @ A5 @ R ) )
=> ( ( member_nat @ X3 @ X5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( member_nat @ Y3 @ X5 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_1220_proj__iff,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ A5 )
=> ( ( ( equiv_proj_nat_nat @ R @ X3 )
= ( equiv_proj_nat_nat @ R @ Y3 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ).
% proj_iff
thf(fact_1221_equiv__class__self,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_nat @ A @ A5 )
=> ( member_nat @ A @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_self
thf(fact_1222_equiv__class__eq__iff,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
= ( ( ( image_nat_nat @ R @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
& ( member_nat @ X3 @ A5 )
& ( member_nat @ Y3 @ A5 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1223_eq__equiv__class__iff,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( ( image_nat_nat @ R @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1224_equiv__class__eq,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1225_eq__equiv__class,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,A5: set_nat] :
( ( ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
=> ( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_nat @ B @ A5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_1226_eq__equiv__class__iff2,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( ( equiv_quotient_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ R )
= ( equiv_quotient_nat @ ( insert_nat @ Y3 @ bot_bot_set_nat ) @ R ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_1227_refines__equiv__class__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A5: set_nat,A: nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S3 )
=> ( ( equiv_equiv_nat @ A5 @ R2 )
=> ( ( equiv_equiv_nat @ A5 @ S3 )
=> ( ( image_nat_nat @ R2 @ ( image_nat_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( image_nat_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1228_refines__equiv__class__eq2,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A5: set_nat,A: nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S3 )
=> ( ( equiv_equiv_nat @ A5 @ R2 )
=> ( ( equiv_equiv_nat @ A5 @ S3 )
=> ( ( image_nat_nat @ S3 @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( image_nat_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1229_equiv__class__subset,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1230_subset__equiv__class,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,B: nat,A: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ B @ A5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_1231_equiv__class__nondisjoint,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,A: nat,B: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( member_nat @ X3 @ ( inf_inf_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1232_disjnt__equiv__class,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( equiv_equiv_nat @ A5 @ R )
=> ( ( disjnt_nat @ ( image_nat_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1233_disjnt__insert2,axiom,
! [Y6: set_nat,A: nat,X5: set_nat] :
( ( disjnt_nat @ Y6 @ ( insert_nat @ A @ X5 ) )
= ( ~ ( member_nat @ A @ Y6 )
& ( disjnt_nat @ Y6 @ X5 ) ) ) ).
% disjnt_insert2
thf(fact_1234_disjnt__insert1,axiom,
! [A: nat,X5: set_nat,Y6: set_nat] :
( ( disjnt_nat @ ( insert_nat @ A @ X5 ) @ Y6 )
= ( ~ ( member_nat @ A @ Y6 )
& ( disjnt_nat @ X5 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_1235_disjnt__insert,axiom,
! [X3: nat,N2: set_nat,M4: set_nat] :
( ~ ( member_nat @ X3 @ N2 )
=> ( ( disjnt_nat @ M4 @ N2 )
=> ( disjnt_nat @ ( insert_nat @ X3 @ M4 ) @ N2 ) ) ) ).
% disjnt_insert
thf(fact_1236_disjnt__inj__on__iff,axiom,
! [F: nat > nat,A11: set_set_nat,X5: set_nat,Y6: set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A11 ) )
=> ( ( member_set_nat @ X5 @ A11 )
=> ( ( member_set_nat @ Y6 @ A11 )
=> ( ( disjnt_nat @ ( image_nat_nat2 @ F @ X5 ) @ ( image_nat_nat2 @ F @ Y6 ) )
= ( disjnt_nat @ X5 @ Y6 ) ) ) ) ) ).
% disjnt_inj_on_iff
thf(fact_1237_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R7: nat > nat > $o,S7: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ S7 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( minus_minus_set_nat @ S7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( R7 @ X2 @ Y ) ) ) ) ) ).
% pairwise_alt
thf(fact_1238_pairwise__insert,axiom,
! [R: nat > nat > $o,X3: nat,S: set_nat] :
( ( pairwise_nat @ R @ ( insert_nat @ X3 @ S ) )
= ( ! [Y: nat] :
( ( ( member_nat @ Y @ S )
& ( Y != X3 ) )
=> ( ( R @ X3 @ Y )
& ( R @ Y @ X3 ) ) )
& ( pairwise_nat @ R @ S ) ) ) ).
% pairwise_insert
thf(fact_1239_pairwise__imageI,axiom,
! [A5: set_nat,F: nat > nat,P: nat > nat > $o] :
( ! [X: nat,Y4: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y4 @ A5 )
=> ( ( X != Y4 )
=> ( ( ( F @ X )
!= ( F @ Y4 ) )
=> ( P @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) )
=> ( pairwise_nat @ P @ ( image_nat_nat2 @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1240_pairwise__singleton,axiom,
! [P: nat > nat > $o,A5: nat] : ( pairwise_nat @ P @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_1241_Pow__fold,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( pow_nat @ A5 )
= ( finite4178521680790401110et_nat
@ ^ [X2: nat,A7: set_set_nat] : ( sup_sup_set_set_nat @ A7 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ A7 ) )
@ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
@ A5 ) ) ) ).
% Pow_fold
thf(fact_1242_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat2
@ ^ [X2: nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_1243_SUP__identity__eq,axiom,
! [A5: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : X2
@ A5 ) )
= ( complete_Sup_Sup_nat @ A5 ) ) ).
% SUP_identity_eq
thf(fact_1244_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1245_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1246_INF__identity__eq,axiom,
! [A5: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : X2
@ A5 ) )
= ( complete_Inf_Inf_nat @ A5 ) ) ).
% INF_identity_eq
thf(fact_1247_UN__I,axiom,
! [A: nat,A5: set_nat,B: nat,B5: nat > set_nat] :
( ( member_nat @ A @ A5 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A5 ) ) ) ) ) ).
% UN_I
thf(fact_1248_INT__I,axiom,
! [A5: set_nat,B: nat,B5: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ B @ ( B5 @ X ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A5 ) ) ) ) ).
% INT_I
thf(fact_1249_empty__map__add,axiom,
! [M: produc1457211279475724562t_char > option_val] :
( ( map_ad2908576761495157028ar_val
@ ^ [X2: produc1457211279475724562t_char] : none_val
@ M )
= M ) ).
% empty_map_add
thf(fact_1250_map__add__empty,axiom,
! [M: produc1457211279475724562t_char > option_val] :
( ( map_ad2908576761495157028ar_val @ M
@ ^ [X2: produc1457211279475724562t_char] : none_val )
= M ) ).
% map_add_empty
thf(fact_1251_empty__eq__map__add__iff,axiom,
! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
( ( ( ^ [X2: produc1457211279475724562t_char] : none_val )
= ( map_ad2908576761495157028ar_val @ F @ G ) )
= ( ( F
= ( ^ [X2: produc1457211279475724562t_char] : none_val ) )
& ( G
= ( ^ [X2: produc1457211279475724562t_char] : none_val ) ) ) ) ).
% empty_eq_map_add_iff
thf(fact_1252_cSUP__const,axiom,
! [A5: set_nat,C: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : C
@ A5 ) )
= C ) ) ).
% cSUP_const
thf(fact_1253_image__add__atLeastAtMost_H,axiom,
! [K: nat,I: nat,J2: nat] :
( ( image_nat_nat2
@ ^ [N7: nat] : ( plus_plus_nat @ N7 @ K )
@ ( set_or1269000886237332187st_nat @ I @ J2 ) )
= ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_1254_cINF__const,axiom,
! [A5: set_nat,C: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : C
@ A5 ) )
= C ) ) ).
% cINF_const
thf(fact_1255_image__add__atLeastLessThan_H,axiom,
! [K: nat,I: nat,J2: nat] :
( ( image_nat_nat2
@ ^ [N7: nat] : ( plus_plus_nat @ N7 @ K )
@ ( set_or4665077453230672383an_nat @ I @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_1256_if__image__distrib,axiom,
! [P: nat > $o,F: nat > nat,G: nat > nat,S3: set_nat] :
( ( image_nat_nat2
@ ^ [X2: nat] : ( if_nat @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ S3 )
= ( sup_sup_set_nat @ ( image_nat_nat2 @ F @ ( inf_inf_set_nat @ S3 @ ( collect_nat @ P ) ) )
@ ( image_nat_nat2 @ G
@ ( inf_inf_set_nat @ S3
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1257_UN__constant,axiom,
! [A5: set_nat,C: set_nat] :
( ( ( A5 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A5 ) )
= bot_bot_set_nat ) )
& ( ( A5 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A5 ) )
= C ) ) ) ).
% UN_constant
thf(fact_1258_bdd__below__image__inf,axiom,
! [F: nat > nat,G: nat > nat,A5: set_nat] :
( ( condit1738341127787009408ow_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : ( inf_inf_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 ) )
= ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A5 ) )
& ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ G @ A5 ) ) ) ) ).
% bdd_below_image_inf
thf(fact_1259_bdd__above__image__sup,axiom,
! [F: nat > nat,G: nat > nat,A5: set_nat] :
( ( condit2214826472909112428ve_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : ( sup_sup_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 ) )
= ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A5 ) )
& ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ A5 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1260_dom__const,axiom,
! [F: nat > val] :
( ( dom_nat_val
@ ^ [X2: nat] : ( some_val @ ( F @ X2 ) ) )
= top_top_set_nat ) ).
% dom_const
thf(fact_1261_range__constant,axiom,
! [X3: nat] :
( ( image_nat_nat2
@ ^ [Uu3: nat] : X3
@ top_top_set_nat )
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_1262_UN__singleton,axiom,
! [A5: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( insert_nat @ X2 @ bot_bot_set_nat )
@ A5 ) )
= A5 ) ).
% UN_singleton
thf(fact_1263_UN__simps_I1_J,axiom,
! [C3: set_nat,A: nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( insert_nat @ A @ ( B5 @ X2 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( insert_nat @ A @ ( B5 @ X2 ) )
@ C3 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1264_UN__simps_I3_J,axiom,
! [C3: set_nat,A5: set_nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ A5 @ ( B5 @ X2 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ A5 @ ( B5 @ X2 ) )
@ C3 ) )
= ( sup_sup_set_nat @ A5 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1265_UN__simps_I2_J,axiom,
! [C3: set_nat,A5: nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ ( A5 @ X2 ) @ B5 )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ ( A5 @ X2 ) @ B5 )
@ C3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A5 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_1266_INT__constant,axiom,
! [A5: set_nat,C: set_nat] :
( ( ( A5 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A5 ) )
= top_top_set_nat ) )
& ( ( A5 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A5 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1267_Max__const,axiom,
! [A5: set_nat,C: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( image_nat_nat2
@ ^ [Uu3: nat] : C
@ A5 ) )
= C ) ) ) ).
% Max_const
thf(fact_1268_Min__const,axiom,
! [A5: set_nat,C: nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat
@ ( image_nat_nat2
@ ^ [Uu3: nat] : C
@ A5 ) )
= C ) ) ) ).
% Min_const
thf(fact_1269_INT__simps_I1_J,axiom,
! [C3: set_nat,A5: nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ ( A5 @ X2 ) @ B5 )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ ( A5 @ X2 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A5 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_1270_INT__simps_I2_J,axiom,
! [C3: set_nat,A5: set_nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ A5 @ ( B5 @ X2 ) )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ A5 @ ( B5 @ X2 ) )
@ C3 ) )
= ( inf_inf_set_nat @ A5 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1271_INT__simps_I3_J,axiom,
! [C3: set_nat,A5: nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ ( A5 @ X2 ) @ B5 )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ ( A5 @ X2 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A5 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_1272_INT__simps_I4_J,axiom,
! [C3: set_nat,A5: set_nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ A5 @ ( B5 @ X2 ) )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ A5 @ ( B5 @ X2 ) )
@ C3 ) )
= ( minus_minus_set_nat @ A5 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_1273_pairwise__image,axiom,
! [R: nat > nat > $o,F: nat > nat,S: set_nat] :
( ( pairwise_nat @ R @ ( image_nat_nat2 @ F @ S ) )
= ( pairwise_nat
@ ^ [X2: nat,Y: nat] :
( ( ( F @ X2 )
!= ( F @ Y ) )
=> ( R @ ( F @ X2 ) @ ( F @ Y ) ) )
@ S ) ) ).
% pairwise_image
thf(fact_1274_mono__cSUP,axiom,
! [F: nat > nat,A5: nat > nat,I2: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ A5 @ I2 ) )
=> ( ( I2 != bot_bot_set_nat )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : ( F @ ( A5 @ X2 ) )
@ I2 ) )
@ ( F @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ A5 @ I2 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_1275_mono__cSup,axiom,
! [F: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A5 ) ) @ ( F @ ( complete_Sup_Sup_nat @ A5 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_1276_mono__cINF,axiom,
! [F: nat > nat,A5: nat > nat,I2: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ A5 @ I2 ) )
=> ( ( I2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ A5 @ I2 ) ) )
@ ( complete_Inf_Inf_nat
@ ( image_nat_nat2
@ ^ [X2: nat] : ( F @ ( A5 @ X2 ) )
@ I2 ) ) ) ) ) ) ).
% mono_cINF
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( if_set_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( if_set_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__Mask__Oval_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Mask__Oval_J_T,axiom,
! [X3: option_val,Y3: option_val] :
( ( if_option_val @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Mask__Oval_J_T,axiom,
! [X3: option_val,Y3: option_val] :
( ( if_option_val @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
aa = ba ).
%------------------------------------------------------------------------------