TPTP Problem File: SLH0186^1.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/0003_CombinableWands/prob_00163_005228__7577656_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1519 ( 581 unt; 236 typ; 0 def)
% Number of atoms : 3795 (1674 equ; 0 cnn)
% Maximal formula atoms : 21 ( 2 avg)
% Number of connectives : 10599 ( 357 ~; 34 |; 267 &;8331 @)
% ( 0 <=>;1610 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 1295 (1295 >; 0 *; 0 +; 0 <<)
% Number of symbols : 218 ( 215 usr; 33 con; 0-5 aty)
% Number of variables : 3680 ( 177 ^;3382 !; 121 ?;3680 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:08:53.866
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J_J,type,
set_op381248089174005275_state: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
option6833441738159790651_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
set_Pr1688445902015331925_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
produc3142500478612311029_state: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
set_op4912175446517189883_state: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
option2250103068101548571_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
set_Pr1785066336555260981_state: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
set_op9003753404445127824_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
produc8023240190789890773_state: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
set_option_set_state: $tType ).
thf(ty_n_t__Option__Ooption_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
option_option_state: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
set_option_state: $tType ).
thf(ty_n_t__Option__Ooption_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
option_set_state: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
set_set_state: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__PartialHeapSA__Ostate_M_Eo_J_J,type,
set_state_o: $tType ).
thf(ty_n_t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
option_state: $tType ).
thf(ty_n_t__Filter__Ofilter_It__PartialHeapSA__Ostate_J,type,
filter_state: $tType ).
thf(ty_n_t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
set_state: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__PartialHeapSA__Ostate,type,
state: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (215)
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
boolea5298875108296682874_state: ( set_state > set_state > set_state ) > ( set_state > set_state > set_state ) > ( set_state > set_state ) > set_state > set_state > $o ).
thf(sy_c_CombinableWands_OR,type,
r: state > state > state ).
thf(sy_c_CombinableWands_Ocwand,type,
cwand: set_state > set_state > set_state ).
thf(sy_c_CombinableWands_Ointuitionistic,type,
intuitionistic: set_state > $o ).
thf(sy_c_CombinableWands_Oscaled,type,
scaled: state > set_state ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
comple1664268212576242562tate_o: set_state_o > state > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
comple4352483261711748803_state: set_set_state > set_state ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
condit2486170202803819276_state: set_set_state > $o ).
thf(sy_c_Filter_Ocofinite_001t__PartialHeapSA__Ostate,type,
cofinite_state: filter_state ).
thf(sy_c_Finite__Set_OFpow_001t__PartialHeapSA__Ostate,type,
finite_Fpow_state: set_state > set_set_state ).
thf(sy_c_Finite__Set_Ocard_001t__PartialHeapSA__Ostate,type,
finite_card_state: set_state > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
finite3180955649987104801_state: set_option_state > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__PartialHeapSA__Ostate,type,
finite_finite_state: set_state > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
finite4951987536711252743_state: set_set_state > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
fun_up8433615654909047651_state: ( option_state > option_option_state ) > option_state > option_option_state > option_state > option_option_state ).
thf(sy_c_Fun_Ofun__upd_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
fun_up8843634000204221123_state: ( state > option_state ) > state > option_state > state > option_state ).
thf(sy_c_Fun_Ofun__upd_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
fun_upd_state_state: ( state > state ) > state > state > state > state ).
thf(sy_c_Fun_Ofun__upd_001t__PartialHeapSA__Ostate_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
fun_up837824997938104489_state: ( state > set_state ) > state > set_state > state > set_state ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__PartialHeapSA__Ostate,type,
inj_on_nat_state: ( nat > state ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
inj_on3577428053172332983_state: ( state > option_state ) > set_state > $o ).
thf(sy_c_Fun_Oinj__on_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
inj_on_state_state: ( state > state ) > set_state > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
the_in3035302284364921129_state: set_state > ( state > state ) > state > state ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
minus_3933957440811877961_state: set_state > set_state > set_state ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
uminus472742206872269241_state: set_state > set_state ).
thf(sy_c_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
if_option_state: $o > option_state > option_state > option_state ).
thf(sy_c_If_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
if_set_state: $o > set_state > set_state > set_state ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
inf_inf_set_state: set_state > set_state > set_state ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
inf_in8939913482472430008_state: set_set_state > set_set_state > set_set_state ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
sup_su99422728725954156_state: set_option_state > set_option_state > set_option_state ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
sup_sup_set_state: set_state > set_state > set_state ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
sup_su4188871578264421970_state: set_set_state > set_set_state > set_set_state ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
lattic4879230916095660051_state: set_set_state > set_state ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
lattic1454283544731368441_state: set_set_state > set_state ).
thf(sy_c_Map_Odom_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
dom_state_state: ( state > option_state ) > set_state ).
thf(sy_c_Map_Ograph_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
graph_628825510059920660_state: ( option_state > option_option_state ) > set_Pr1688445902015331925_state ).
thf(sy_c_Map_Omap__add_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
map_add_state_state: ( state > option_state ) > ( state > option_state ) > state > option_state ).
thf(sy_c_Map_Orestrict__map_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
restri6741197069250724038_state: ( option_state > option_option_state ) > set_option_state > option_state > option_option_state ).
thf(sy_c_Map_Orestrict__map_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
restri2287918369865870758_state: ( state > option_state ) > set_state > state > option_state ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
none_option_state: option_option_state ).
thf(sy_c_Option_Ooption_ONone_001t__PartialHeapSA__Ostate,type,
none_state: option_state ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
none_P2148290358184556502_state: option6833441738159790651_state ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
none_P348328851727313334_state: option2250103068101548571_state ).
thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
none_set_state: option_set_state ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
some_option_state: option_state > option_option_state ).
thf(sy_c_Option_Ooption_OSome_001t__PartialHeapSA__Ostate,type,
some_state: state > option_state ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
some_P3186401494017672538_state: produc3142500478612311029_state > option6833441738159790651_state ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
some_P8120450764674687802_state: produc8023240190789890773_state > option2250103068101548571_state ).
thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
some_set_state: set_state > option_set_state ).
thf(sy_c_Option_Ooption_Oset__option_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
set_op5162993263733338108_state: option_option_state > set_option_state ).
thf(sy_c_Option_Ooption_Oset__option_001t__PartialHeapSA__Ostate,type,
set_option_state2: option_state > set_state ).
thf(sy_c_Option_Ooption_Oset__option_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
set_op4197959913983417475_state: option6833441738159790651_state > set_Pr1688445902015331925_state ).
thf(sy_c_Option_Ooption_Oset__option_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
set_op4970154267292507491_state: option2250103068101548571_state > set_Pr1785066336555260981_state ).
thf(sy_c_Option_Ooption_Oset__option_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
set_option_set_state2: option_set_state > set_set_state ).
thf(sy_c_Option_Ooption_Othe_001t__PartialHeapSA__Ostate,type,
the_state: option_state > state ).
thf(sy_c_Option_Othese_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
these_option_state: set_op9003753404445127824_state > set_option_state ).
thf(sy_c_Option_Othese_001t__PartialHeapSA__Ostate,type,
these_state: set_option_state > set_state ).
thf(sy_c_Option_Othese_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
these_5013389664457929650_state: set_op381248089174005275_state > set_Pr1688445902015331925_state ).
thf(sy_c_Option_Othese_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
these_9162046556185909650_state: set_op4912175446517189883_state > set_Pr1785066336555260981_state ).
thf(sy_c_Option_Othese_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
these_set_state: set_option_set_state > set_set_state ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
order_7502633587729487893_state: set_option_state > set_Pr1688445902015331925_state > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__PartialHeapSA__Ostate,type,
order_7174064491916216133_state: set_state > set_Pr1785066336555260981_state > $o ).
thf(sy_c_Order__Relation_Owell__order__on_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
order_4461428600198971116_state: set_option_state > set_Pr1688445902015331925_state > $o ).
thf(sy_c_Order__Relation_Owell__order__on_001t__PartialHeapSA__Ostate,type,
order_3972865417222356124_state: set_state > set_Pr1785066336555260981_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Option__Ooption_It__PartialHeapSA__Ostate_J_M_Eo_J,type,
bot_bo4453335400789057457tate_o: option_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
bot_bot_state_o: state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_M_Eo_J,type,
bot_bo4049596492272799580tate_o: produc3142500478612311029_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_M_Eo_J,type,
bot_bo2162275076026452348tate_o: produc8023240190789890773_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__PartialHeapSA__Ostate_J_M_Eo_J,type,
bot_bot_set_state_o: set_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__PartialHeapSA__Ostate_J,type,
bot_bot_filter_state: filter_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__PartialHeapSA__Ostate_M_Eo_J_J,type,
bot_bot_set_state_o2: set_state_o ).
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__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
bot_bo489212050006660900_state: set_op9003753404445127824_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
bot_bo710180891245420500_state: set_option_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J_J,type,
bot_bo2763976174278612103_state: set_op381248089174005275_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
bot_bo3922232019306180455_state: set_op4912175446517189883_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
bot_bo7912781682106412938_state: set_option_set_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
bot_bot_set_state: set_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
bot_bo1080640394036989633_state: set_Pr1688445902015331925_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
bot_bo9041262728264437921_state: set_Pr1785066336555260981_state ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
bot_bo2271482359692755898_state: set_set_state ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
ord_less_set_state: set_state > set_state > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
ord_less_eq_state_o: ( state > $o ) > ( state > $o ) > $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__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
ord_le7116032884704190368_state: set_option_state > set_option_state > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
ord_le2494988322063910608_state: set_state > set_state > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
ord_le6423325748750870005_state: set_Pr1688445902015331925_state > set_Pr1688445902015331925_state > $o ).
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is_sin7571815095334105321_state: set_Pr1688445902015331925_state > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
is_sin3624314468123612105_state: set_Pr1785066336555260981_state > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
is_sin3747817797122030204_state: set_set_state > $o ).
thf(sy_c_Set_Opairwise_001t__PartialHeapSA__Ostate,type,
pairwise_state: ( state > state > $o ) > set_state > $o ).
thf(sy_c_Set_Oremove_001t__PartialHeapSA__Ostate,type,
remove_state: state > set_state > set_state ).
thf(sy_c_Set_Othe__elem_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
the_el1618976816499768149_state: set_option_state > option_state ).
thf(sy_c_Set_Othe__elem_001t__PartialHeapSA__Ostate,type,
the_elem_state: set_state > state ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
the_el7118915247011979882_state: set_Pr1688445902015331925_state > produc3142500478612311029_state ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
the_el1727186808309674058_state: set_Pr1785066336555260981_state > produc8023240190789890773_state ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
the_elem_set_state: set_set_state > set_state ).
thf(sy_c_Set_Ovimage_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
vimage_state_state: ( state > state ) > set_state > set_state ).
thf(sy_c_member_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
member_state_o: ( state > $o ) > set_state_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
member1079230918592710257_state: option_option_state > set_op9003753404445127824_state > $o ).
thf(sy_c_member_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
member_option_state: option_state > set_option_state > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
member3754053363737695844_state: option6833441738159790651_state > set_op381248089174005275_state > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
member5076014802351601988_state: option2250103068101548571_state > set_op4912175446517189883_state > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
member1412897518142203351_state: option_set_state > set_option_set_state > $o ).
thf(sy_c_member_001t__PartialHeapSA__Ostate,type,
member_state: state > set_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
member3029510603097127326_state: produc3142500478612311029_state > set_Pr1688445902015331925_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
member753036827967488894_state: produc8023240190789890773_state > set_Pr1785066336555260981_state > $o ).
thf(sy_c_member_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
member_set_state: set_state > set_set_state > $o ).
thf(sy_v_A,type,
a: set_state ).
thf(sy_v_B,type,
b: set_state ).
thf(sy_v_a____,type,
a2: state ).
thf(sy_v_w,type,
w: state ).
thf(sy_v_x____,type,
x: state ).
% Relevant facts (1277)
thf(fact_0__092_060open_062Some_Ax_A_061_Aa_A_092_060oplus_062_Aw_092_060close_062,axiom,
( ( some_state @ x )
= ( plus @ a2 @ w ) ) ).
% \<open>Some x = a \<oplus> w\<close>
thf(fact_1_assms,axiom,
! [A: state,X: state] :
( ( ( member_state @ A @ a )
& ( ( some_state @ X )
= ( plus @ A @ w ) ) )
=> ( member_state @ X @ b ) ) ).
% assms
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_O_A_092_060lbrakk_062Some_Ax_A_061_Aa_A_092_060oplus_062_Aw_059_Aa_A_092_060in_062_AA_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [A2: state] :
( ( ( some_state @ x )
= ( plus @ A2 @ w ) )
=> ~ ( member_state @ A2 @ a ) ) ).
% \<open>\<And>thesis. (\<And>a. \<lbrakk>Some x = a \<oplus> w; a \<in> A\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3__092_060open_062x_A_092_060in_062_AA_A_092_060otimes_062_A_123w_125_092_060close_062,axiom,
member_state @ x @ ( add_set @ a @ ( insert_state @ w @ bot_bot_set_state ) ) ).
% \<open>x \<in> A \<otimes> {w}\<close>
thf(fact_4_w__in__scaled,axiom,
! [W: state] : ( member_state @ W @ ( scaled @ W ) ) ).
% w_in_scaled
thf(fact_5__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
member_state @ a2 @ a ).
% \<open>a \<in> A\<close>
thf(fact_6_PartialSA_Osum__then__singleton,axiom,
! [A: state,B: state,C: state] :
( ( ( some_state @ A )
= ( plus @ B @ C ) )
= ( ( insert_state @ A @ bot_bot_set_state )
= ( add_set @ ( insert_state @ B @ bot_bot_set_state ) @ ( insert_state @ C @ bot_bot_set_state ) ) ) ) ).
% PartialSA.sum_then_singleton
thf(fact_7_singletonI,axiom,
! [A: produc8023240190789890773_state] : ( member753036827967488894_state @ A @ ( insert7525286303735658661_state @ A @ bot_bo9041262728264437921_state ) ) ).
% singletonI
thf(fact_8_singletonI,axiom,
! [A: produc3142500478612311029_state] : ( member3029510603097127326_state @ A @ ( insert4171857611248116165_state @ A @ bot_bo1080640394036989633_state ) ) ).
% singletonI
thf(fact_9_singletonI,axiom,
! [A: option_state] : ( member_option_state @ A @ ( insert_option_state @ A @ bot_bo710180891245420500_state ) ) ).
% singletonI
thf(fact_10_singletonI,axiom,
! [A: set_state] : ( member_set_state @ A @ ( insert_set_state @ A @ bot_bo2271482359692755898_state ) ) ).
% singletonI
thf(fact_11_singletonI,axiom,
! [A: state] : ( member_state @ A @ ( insert_state @ A @ bot_bot_set_state ) ) ).
% singletonI
thf(fact_12_PartialSA_Oadd__set__elem,axiom,
! [Phi: state,A3: set_state,B2: set_state] :
( ( member_state @ Phi @ ( add_set @ A3 @ B2 ) )
= ( ? [A4: state,B3: state] :
( ( ( some_state @ Phi )
= ( plus @ A4 @ B3 ) )
& ( member_state @ A4 @ A3 )
& ( member_state @ B3 @ B2 ) ) ) ) ).
% PartialSA.add_set_elem
thf(fact_13_PartialSA_Ox__elem__set__product,axiom,
! [X: state,A3: set_state,B2: set_state] :
( ( member_state @ X @ ( add_set @ A3 @ B2 ) )
= ( ? [A4: state,B3: state] :
( ( member_state @ A4 @ A3 )
& ( member_state @ B3 @ B2 )
& ( ( some_state @ X )
= ( plus @ A4 @ B3 ) ) ) ) ) ).
% PartialSA.x_elem_set_product
thf(fact_14_PartialSA_Oempty__set__sum,axiom,
! [A3: set_state] :
( ( add_set @ bot_bot_set_state @ A3 )
= bot_bot_set_state ) ).
% PartialSA.empty_set_sum
thf(fact_15_option_Oinject,axiom,
! [X2: option_state,Y2: option_state] :
( ( ( some_option_state @ X2 )
= ( some_option_state @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_16_option_Oinject,axiom,
! [X2: state,Y2: state] :
( ( ( some_state @ X2 )
= ( some_state @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_17_insertCI,axiom,
! [A: option_state,B2: set_option_state,B: option_state] :
( ( ~ ( member_option_state @ A @ B2 )
=> ( A = B ) )
=> ( member_option_state @ A @ ( insert_option_state @ B @ B2 ) ) ) ).
% insertCI
thf(fact_18_insertCI,axiom,
! [A: set_state,B2: set_set_state,B: set_state] :
( ( ~ ( member_set_state @ A @ B2 )
=> ( A = B ) )
=> ( member_set_state @ A @ ( insert_set_state @ B @ B2 ) ) ) ).
% insertCI
thf(fact_19_insertCI,axiom,
! [A: produc3142500478612311029_state,B2: set_Pr1688445902015331925_state,B: produc3142500478612311029_state] :
( ( ~ ( member3029510603097127326_state @ A @ B2 )
=> ( A = B ) )
=> ( member3029510603097127326_state @ A @ ( insert4171857611248116165_state @ B @ B2 ) ) ) ).
% insertCI
thf(fact_20_insertCI,axiom,
! [A: produc8023240190789890773_state,B2: set_Pr1785066336555260981_state,B: produc8023240190789890773_state] :
( ( ~ ( member753036827967488894_state @ A @ B2 )
=> ( A = B ) )
=> ( member753036827967488894_state @ A @ ( insert7525286303735658661_state @ B @ B2 ) ) ) ).
% insertCI
thf(fact_21_insertCI,axiom,
! [A: state,B2: set_state,B: state] :
( ( ~ ( member_state @ A @ B2 )
=> ( A = B ) )
=> ( member_state @ A @ ( insert_state @ B @ B2 ) ) ) ).
% insertCI
thf(fact_22_insert__iff,axiom,
! [A: option_state,B: option_state,A3: set_option_state] :
( ( member_option_state @ A @ ( insert_option_state @ B @ A3 ) )
= ( ( A = B )
| ( member_option_state @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_23_insert__iff,axiom,
! [A: set_state,B: set_state,A3: set_set_state] :
( ( member_set_state @ A @ ( insert_set_state @ B @ A3 ) )
= ( ( A = B )
| ( member_set_state @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_24_insert__iff,axiom,
! [A: produc3142500478612311029_state,B: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ A @ ( insert4171857611248116165_state @ B @ A3 ) )
= ( ( A = B )
| ( member3029510603097127326_state @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_25_insert__iff,axiom,
! [A: produc8023240190789890773_state,B: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( member753036827967488894_state @ A @ ( insert7525286303735658661_state @ B @ A3 ) )
= ( ( A = B )
| ( member753036827967488894_state @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_26_insert__iff,axiom,
! [A: state,B: state,A3: set_state] :
( ( member_state @ A @ ( insert_state @ B @ A3 ) )
= ( ( A = B )
| ( member_state @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_27_insert__absorb2,axiom,
! [X: option_state,A3: set_option_state] :
( ( insert_option_state @ X @ ( insert_option_state @ X @ A3 ) )
= ( insert_option_state @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_28_insert__absorb2,axiom,
! [X: set_state,A3: set_set_state] :
( ( insert_set_state @ X @ ( insert_set_state @ X @ A3 ) )
= ( insert_set_state @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_29_insert__absorb2,axiom,
! [X: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( insert4171857611248116165_state @ X @ ( insert4171857611248116165_state @ X @ A3 ) )
= ( insert4171857611248116165_state @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_30_insert__absorb2,axiom,
! [X: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( insert7525286303735658661_state @ X @ ( insert7525286303735658661_state @ X @ A3 ) )
= ( insert7525286303735658661_state @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_31_insert__absorb2,axiom,
! [X: state,A3: set_state] :
( ( insert_state @ X @ ( insert_state @ X @ A3 ) )
= ( insert_state @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_32_empty__iff,axiom,
! [C: produc8023240190789890773_state] :
~ ( member753036827967488894_state @ C @ bot_bo9041262728264437921_state ) ).
% empty_iff
thf(fact_33_empty__iff,axiom,
! [C: produc3142500478612311029_state] :
~ ( member3029510603097127326_state @ C @ bot_bo1080640394036989633_state ) ).
% empty_iff
thf(fact_34_empty__iff,axiom,
! [C: option_state] :
~ ( member_option_state @ C @ bot_bo710180891245420500_state ) ).
% empty_iff
thf(fact_35_empty__iff,axiom,
! [C: set_state] :
~ ( member_set_state @ C @ bot_bo2271482359692755898_state ) ).
% empty_iff
thf(fact_36_empty__iff,axiom,
! [C: state] :
~ ( member_state @ C @ bot_bot_set_state ) ).
% empty_iff
thf(fact_37_all__not__in__conv,axiom,
! [A3: set_Pr1785066336555260981_state] :
( ( ! [X3: produc8023240190789890773_state] :
~ ( member753036827967488894_state @ X3 @ A3 ) )
= ( A3 = bot_bo9041262728264437921_state ) ) ).
% all_not_in_conv
thf(fact_38_all__not__in__conv,axiom,
! [A3: set_Pr1688445902015331925_state] :
( ( ! [X3: produc3142500478612311029_state] :
~ ( member3029510603097127326_state @ X3 @ A3 ) )
= ( A3 = bot_bo1080640394036989633_state ) ) ).
% all_not_in_conv
thf(fact_39_all__not__in__conv,axiom,
! [A3: set_option_state] :
( ( ! [X3: option_state] :
~ ( member_option_state @ X3 @ A3 ) )
= ( A3 = bot_bo710180891245420500_state ) ) ).
% all_not_in_conv
thf(fact_40_all__not__in__conv,axiom,
! [A3: set_set_state] :
( ( ! [X3: set_state] :
~ ( member_set_state @ X3 @ A3 ) )
= ( A3 = bot_bo2271482359692755898_state ) ) ).
% all_not_in_conv
thf(fact_41_all__not__in__conv,axiom,
! [A3: set_state] :
( ( ! [X3: state] :
~ ( member_state @ X3 @ A3 ) )
= ( A3 = bot_bot_set_state ) ) ).
% all_not_in_conv
thf(fact_42_Collect__empty__eq,axiom,
! [P: produc8023240190789890773_state > $o] :
( ( ( collec7320380983431419584_state @ P )
= bot_bo9041262728264437921_state )
= ( ! [X3: produc8023240190789890773_state] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_43_Collect__empty__eq,axiom,
! [P: produc3142500478612311029_state > $o] :
( ( ( collec8144523193623705312_state @ P )
= bot_bo1080640394036989633_state )
= ( ! [X3: produc3142500478612311029_state] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_44_Collect__empty__eq,axiom,
! [P: option_state > $o] :
( ( ( collect_option_state @ P )
= bot_bo710180891245420500_state )
= ( ! [X3: option_state] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_45_Collect__empty__eq,axiom,
! [P: set_state > $o] :
( ( ( collect_set_state @ P )
= bot_bo2271482359692755898_state )
= ( ! [X3: set_state] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_46_Collect__empty__eq,axiom,
! [P: state > $o] :
( ( ( collect_state @ P )
= bot_bot_set_state )
= ( ! [X3: state] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_47_empty__Collect__eq,axiom,
! [P: produc8023240190789890773_state > $o] :
( ( bot_bo9041262728264437921_state
= ( collec7320380983431419584_state @ P ) )
= ( ! [X3: produc8023240190789890773_state] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_48_empty__Collect__eq,axiom,
! [P: produc3142500478612311029_state > $o] :
( ( bot_bo1080640394036989633_state
= ( collec8144523193623705312_state @ P ) )
= ( ! [X3: produc3142500478612311029_state] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_49_empty__Collect__eq,axiom,
! [P: option_state > $o] :
( ( bot_bo710180891245420500_state
= ( collect_option_state @ P ) )
= ( ! [X3: option_state] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_50_empty__Collect__eq,axiom,
! [P: set_state > $o] :
( ( bot_bo2271482359692755898_state
= ( collect_set_state @ P ) )
= ( ! [X3: set_state] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_51_empty__Collect__eq,axiom,
! [P: state > $o] :
( ( bot_bot_set_state
= ( collect_state @ P ) )
= ( ! [X3: state] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_52_ex__in__conv,axiom,
! [A3: set_Pr1785066336555260981_state] :
( ( ? [X3: produc8023240190789890773_state] : ( member753036827967488894_state @ X3 @ A3 ) )
= ( A3 != bot_bo9041262728264437921_state ) ) ).
% ex_in_conv
thf(fact_53_ex__in__conv,axiom,
! [A3: set_Pr1688445902015331925_state] :
( ( ? [X3: produc3142500478612311029_state] : ( member3029510603097127326_state @ X3 @ A3 ) )
= ( A3 != bot_bo1080640394036989633_state ) ) ).
% ex_in_conv
thf(fact_54_ex__in__conv,axiom,
! [A3: set_option_state] :
( ( ? [X3: option_state] : ( member_option_state @ X3 @ A3 ) )
= ( A3 != bot_bo710180891245420500_state ) ) ).
% ex_in_conv
thf(fact_55_ex__in__conv,axiom,
! [A3: set_set_state] :
( ( ? [X3: set_state] : ( member_set_state @ X3 @ A3 ) )
= ( A3 != bot_bo2271482359692755898_state ) ) ).
% ex_in_conv
thf(fact_56_ex__in__conv,axiom,
! [A3: set_state] :
( ( ? [X3: state] : ( member_state @ X3 @ A3 ) )
= ( A3 != bot_bot_set_state ) ) ).
% ex_in_conv
thf(fact_57_equals0I,axiom,
! [A3: set_Pr1785066336555260981_state] :
( ! [Y: produc8023240190789890773_state] :
~ ( member753036827967488894_state @ Y @ A3 )
=> ( A3 = bot_bo9041262728264437921_state ) ) ).
% equals0I
thf(fact_58_equals0I,axiom,
! [A3: set_Pr1688445902015331925_state] :
( ! [Y: produc3142500478612311029_state] :
~ ( member3029510603097127326_state @ Y @ A3 )
=> ( A3 = bot_bo1080640394036989633_state ) ) ).
% equals0I
thf(fact_59_equals0I,axiom,
! [A3: set_option_state] :
( ! [Y: option_state] :
~ ( member_option_state @ Y @ A3 )
=> ( A3 = bot_bo710180891245420500_state ) ) ).
% equals0I
thf(fact_60_equals0I,axiom,
! [A3: set_set_state] :
( ! [Y: set_state] :
~ ( member_set_state @ Y @ A3 )
=> ( A3 = bot_bo2271482359692755898_state ) ) ).
% equals0I
thf(fact_61_equals0I,axiom,
! [A3: set_state] :
( ! [Y: state] :
~ ( member_state @ Y @ A3 )
=> ( A3 = bot_bot_set_state ) ) ).
% equals0I
thf(fact_62_equals0D,axiom,
! [A3: set_Pr1785066336555260981_state,A: produc8023240190789890773_state] :
( ( A3 = bot_bo9041262728264437921_state )
=> ~ ( member753036827967488894_state @ A @ A3 ) ) ).
% equals0D
thf(fact_63_equals0D,axiom,
! [A3: set_Pr1688445902015331925_state,A: produc3142500478612311029_state] :
( ( A3 = bot_bo1080640394036989633_state )
=> ~ ( member3029510603097127326_state @ A @ A3 ) ) ).
% equals0D
thf(fact_64_equals0D,axiom,
! [A3: set_option_state,A: option_state] :
( ( A3 = bot_bo710180891245420500_state )
=> ~ ( member_option_state @ A @ A3 ) ) ).
% equals0D
thf(fact_65_equals0D,axiom,
! [A3: set_set_state,A: set_state] :
( ( A3 = bot_bo2271482359692755898_state )
=> ~ ( member_set_state @ A @ A3 ) ) ).
% equals0D
thf(fact_66_equals0D,axiom,
! [A3: set_state,A: state] :
( ( A3 = bot_bot_set_state )
=> ~ ( member_state @ A @ A3 ) ) ).
% equals0D
thf(fact_67_emptyE,axiom,
! [A: produc8023240190789890773_state] :
~ ( member753036827967488894_state @ A @ bot_bo9041262728264437921_state ) ).
% emptyE
thf(fact_68_emptyE,axiom,
! [A: produc3142500478612311029_state] :
~ ( member3029510603097127326_state @ A @ bot_bo1080640394036989633_state ) ).
% emptyE
thf(fact_69_emptyE,axiom,
! [A: option_state] :
~ ( member_option_state @ A @ bot_bo710180891245420500_state ) ).
% emptyE
thf(fact_70_emptyE,axiom,
! [A: set_state] :
~ ( member_set_state @ A @ bot_bo2271482359692755898_state ) ).
% emptyE
thf(fact_71_emptyE,axiom,
! [A: state] :
~ ( member_state @ A @ bot_bot_set_state ) ).
% emptyE
thf(fact_72_mk__disjoint__insert,axiom,
! [A: option_state,A3: set_option_state] :
( ( member_option_state @ A @ A3 )
=> ? [B4: set_option_state] :
( ( A3
= ( insert_option_state @ A @ B4 ) )
& ~ ( member_option_state @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_73_mk__disjoint__insert,axiom,
! [A: set_state,A3: set_set_state] :
( ( member_set_state @ A @ A3 )
=> ? [B4: set_set_state] :
( ( A3
= ( insert_set_state @ A @ B4 ) )
& ~ ( member_set_state @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_74_mk__disjoint__insert,axiom,
! [A: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ A @ A3 )
=> ? [B4: set_Pr1688445902015331925_state] :
( ( A3
= ( insert4171857611248116165_state @ A @ B4 ) )
& ~ ( member3029510603097127326_state @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_75_mk__disjoint__insert,axiom,
! [A: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( member753036827967488894_state @ A @ A3 )
=> ? [B4: set_Pr1785066336555260981_state] :
( ( A3
= ( insert7525286303735658661_state @ A @ B4 ) )
& ~ ( member753036827967488894_state @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_76_mk__disjoint__insert,axiom,
! [A: state,A3: set_state] :
( ( member_state @ A @ A3 )
=> ? [B4: set_state] :
( ( A3
= ( insert_state @ A @ B4 ) )
& ~ ( member_state @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_77_insert__commute,axiom,
! [X: option_state,Y3: option_state,A3: set_option_state] :
( ( insert_option_state @ X @ ( insert_option_state @ Y3 @ A3 ) )
= ( insert_option_state @ Y3 @ ( insert_option_state @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_78_insert__commute,axiom,
! [X: set_state,Y3: set_state,A3: set_set_state] :
( ( insert_set_state @ X @ ( insert_set_state @ Y3 @ A3 ) )
= ( insert_set_state @ Y3 @ ( insert_set_state @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_79_insert__commute,axiom,
! [X: produc3142500478612311029_state,Y3: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( insert4171857611248116165_state @ X @ ( insert4171857611248116165_state @ Y3 @ A3 ) )
= ( insert4171857611248116165_state @ Y3 @ ( insert4171857611248116165_state @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_80_insert__commute,axiom,
! [X: produc8023240190789890773_state,Y3: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( insert7525286303735658661_state @ X @ ( insert7525286303735658661_state @ Y3 @ A3 ) )
= ( insert7525286303735658661_state @ Y3 @ ( insert7525286303735658661_state @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_81_insert__commute,axiom,
! [X: state,Y3: state,A3: set_state] :
( ( insert_state @ X @ ( insert_state @ Y3 @ A3 ) )
= ( insert_state @ Y3 @ ( insert_state @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_82_insert__eq__iff,axiom,
! [A: option_state,A3: set_option_state,B: option_state,B2: set_option_state] :
( ~ ( member_option_state @ A @ A3 )
=> ( ~ ( member_option_state @ B @ B2 )
=> ( ( ( insert_option_state @ A @ A3 )
= ( insert_option_state @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C2: set_option_state] :
( ( A3
= ( insert_option_state @ B @ C2 ) )
& ~ ( member_option_state @ B @ C2 )
& ( B2
= ( insert_option_state @ A @ C2 ) )
& ~ ( member_option_state @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_83_insert__eq__iff,axiom,
! [A: set_state,A3: set_set_state,B: set_state,B2: set_set_state] :
( ~ ( member_set_state @ A @ A3 )
=> ( ~ ( member_set_state @ B @ B2 )
=> ( ( ( insert_set_state @ A @ A3 )
= ( insert_set_state @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C2: set_set_state] :
( ( A3
= ( insert_set_state @ B @ C2 ) )
& ~ ( member_set_state @ B @ C2 )
& ( B2
= ( insert_set_state @ A @ C2 ) )
& ~ ( member_set_state @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_84_insert__eq__iff,axiom,
! [A: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state,B: produc3142500478612311029_state,B2: set_Pr1688445902015331925_state] :
( ~ ( member3029510603097127326_state @ A @ A3 )
=> ( ~ ( member3029510603097127326_state @ B @ B2 )
=> ( ( ( insert4171857611248116165_state @ A @ A3 )
= ( insert4171857611248116165_state @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C2: set_Pr1688445902015331925_state] :
( ( A3
= ( insert4171857611248116165_state @ B @ C2 ) )
& ~ ( member3029510603097127326_state @ B @ C2 )
& ( B2
= ( insert4171857611248116165_state @ A @ C2 ) )
& ~ ( member3029510603097127326_state @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_85_insert__eq__iff,axiom,
! [A: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state,B: produc8023240190789890773_state,B2: set_Pr1785066336555260981_state] :
( ~ ( member753036827967488894_state @ A @ A3 )
=> ( ~ ( member753036827967488894_state @ B @ B2 )
=> ( ( ( insert7525286303735658661_state @ A @ A3 )
= ( insert7525286303735658661_state @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C2: set_Pr1785066336555260981_state] :
( ( A3
= ( insert7525286303735658661_state @ B @ C2 ) )
& ~ ( member753036827967488894_state @ B @ C2 )
& ( B2
= ( insert7525286303735658661_state @ A @ C2 ) )
& ~ ( member753036827967488894_state @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_86_insert__eq__iff,axiom,
! [A: state,A3: set_state,B: state,B2: set_state] :
( ~ ( member_state @ A @ A3 )
=> ( ~ ( member_state @ B @ B2 )
=> ( ( ( insert_state @ A @ A3 )
= ( insert_state @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C2: set_state] :
( ( A3
= ( insert_state @ B @ C2 ) )
& ~ ( member_state @ B @ C2 )
& ( B2
= ( insert_state @ A @ C2 ) )
& ~ ( member_state @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_87_insert__absorb,axiom,
! [A: option_state,A3: set_option_state] :
( ( member_option_state @ A @ A3 )
=> ( ( insert_option_state @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_88_insert__absorb,axiom,
! [A: set_state,A3: set_set_state] :
( ( member_set_state @ A @ A3 )
=> ( ( insert_set_state @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_89_insert__absorb,axiom,
! [A: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ A @ A3 )
=> ( ( insert4171857611248116165_state @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_90_insert__absorb,axiom,
! [A: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( member753036827967488894_state @ A @ A3 )
=> ( ( insert7525286303735658661_state @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_91_insert__absorb,axiom,
! [A: state,A3: set_state] :
( ( member_state @ A @ A3 )
=> ( ( insert_state @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_92_insert__ident,axiom,
! [X: option_state,A3: set_option_state,B2: set_option_state] :
( ~ ( member_option_state @ X @ A3 )
=> ( ~ ( member_option_state @ X @ B2 )
=> ( ( ( insert_option_state @ X @ A3 )
= ( insert_option_state @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_93_insert__ident,axiom,
! [X: set_state,A3: set_set_state,B2: set_set_state] :
( ~ ( member_set_state @ X @ A3 )
=> ( ~ ( member_set_state @ X @ B2 )
=> ( ( ( insert_set_state @ X @ A3 )
= ( insert_set_state @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_94_insert__ident,axiom,
! [X: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state] :
( ~ ( member3029510603097127326_state @ X @ A3 )
=> ( ~ ( member3029510603097127326_state @ X @ B2 )
=> ( ( ( insert4171857611248116165_state @ X @ A3 )
= ( insert4171857611248116165_state @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_95_insert__ident,axiom,
! [X: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state,B2: set_Pr1785066336555260981_state] :
( ~ ( member753036827967488894_state @ X @ A3 )
=> ( ~ ( member753036827967488894_state @ X @ B2 )
=> ( ( ( insert7525286303735658661_state @ X @ A3 )
= ( insert7525286303735658661_state @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_96_insert__ident,axiom,
! [X: state,A3: set_state,B2: set_state] :
( ~ ( member_state @ X @ A3 )
=> ( ~ ( member_state @ X @ B2 )
=> ( ( ( insert_state @ X @ A3 )
= ( insert_state @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_97_Set_Oset__insert,axiom,
! [X: option_state,A3: set_option_state] :
( ( member_option_state @ X @ A3 )
=> ~ ! [B4: set_option_state] :
( ( A3
= ( insert_option_state @ X @ B4 ) )
=> ( member_option_state @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_98_Set_Oset__insert,axiom,
! [X: set_state,A3: set_set_state] :
( ( member_set_state @ X @ A3 )
=> ~ ! [B4: set_set_state] :
( ( A3
= ( insert_set_state @ X @ B4 ) )
=> ( member_set_state @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_99_Set_Oset__insert,axiom,
! [X: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ X @ A3 )
=> ~ ! [B4: set_Pr1688445902015331925_state] :
( ( A3
= ( insert4171857611248116165_state @ X @ B4 ) )
=> ( member3029510603097127326_state @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_100_Set_Oset__insert,axiom,
! [X: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( member753036827967488894_state @ X @ A3 )
=> ~ ! [B4: set_Pr1785066336555260981_state] :
( ( A3
= ( insert7525286303735658661_state @ X @ B4 ) )
=> ( member753036827967488894_state @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_101_Set_Oset__insert,axiom,
! [X: state,A3: set_state] :
( ( member_state @ X @ A3 )
=> ~ ! [B4: set_state] :
( ( A3
= ( insert_state @ X @ B4 ) )
=> ( member_state @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_102_insertI2,axiom,
! [A: option_state,B2: set_option_state,B: option_state] :
( ( member_option_state @ A @ B2 )
=> ( member_option_state @ A @ ( insert_option_state @ B @ B2 ) ) ) ).
% insertI2
thf(fact_103_insertI2,axiom,
! [A: set_state,B2: set_set_state,B: set_state] :
( ( member_set_state @ A @ B2 )
=> ( member_set_state @ A @ ( insert_set_state @ B @ B2 ) ) ) ).
% insertI2
thf(fact_104_insertI2,axiom,
! [A: produc3142500478612311029_state,B2: set_Pr1688445902015331925_state,B: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ A @ B2 )
=> ( member3029510603097127326_state @ A @ ( insert4171857611248116165_state @ B @ B2 ) ) ) ).
% insertI2
thf(fact_105_insertI2,axiom,
! [A: produc8023240190789890773_state,B2: set_Pr1785066336555260981_state,B: produc8023240190789890773_state] :
( ( member753036827967488894_state @ A @ B2 )
=> ( member753036827967488894_state @ A @ ( insert7525286303735658661_state @ B @ B2 ) ) ) ).
% insertI2
thf(fact_106_insertI2,axiom,
! [A: state,B2: set_state,B: state] :
( ( member_state @ A @ B2 )
=> ( member_state @ A @ ( insert_state @ B @ B2 ) ) ) ).
% insertI2
thf(fact_107_insertI1,axiom,
! [A: option_state,B2: set_option_state] : ( member_option_state @ A @ ( insert_option_state @ A @ B2 ) ) ).
% insertI1
thf(fact_108_insertI1,axiom,
! [A: set_state,B2: set_set_state] : ( member_set_state @ A @ ( insert_set_state @ A @ B2 ) ) ).
% insertI1
thf(fact_109_insertI1,axiom,
! [A: produc3142500478612311029_state,B2: set_Pr1688445902015331925_state] : ( member3029510603097127326_state @ A @ ( insert4171857611248116165_state @ A @ B2 ) ) ).
% insertI1
thf(fact_110_insertI1,axiom,
! [A: produc8023240190789890773_state,B2: set_Pr1785066336555260981_state] : ( member753036827967488894_state @ A @ ( insert7525286303735658661_state @ A @ B2 ) ) ).
% insertI1
thf(fact_111_insertI1,axiom,
! [A: state,B2: set_state] : ( member_state @ A @ ( insert_state @ A @ B2 ) ) ).
% insertI1
thf(fact_112_insertE,axiom,
! [A: option_state,B: option_state,A3: set_option_state] :
( ( member_option_state @ A @ ( insert_option_state @ B @ A3 ) )
=> ( ( A != B )
=> ( member_option_state @ A @ A3 ) ) ) ).
% insertE
thf(fact_113_insertE,axiom,
! [A: set_state,B: set_state,A3: set_set_state] :
( ( member_set_state @ A @ ( insert_set_state @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_state @ A @ A3 ) ) ) ).
% insertE
thf(fact_114_insertE,axiom,
! [A: produc3142500478612311029_state,B: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ A @ ( insert4171857611248116165_state @ B @ A3 ) )
=> ( ( A != B )
=> ( member3029510603097127326_state @ A @ A3 ) ) ) ).
% insertE
thf(fact_115_insertE,axiom,
! [A: produc8023240190789890773_state,B: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( member753036827967488894_state @ A @ ( insert7525286303735658661_state @ B @ A3 ) )
=> ( ( A != B )
=> ( member753036827967488894_state @ A @ A3 ) ) ) ).
% insertE
thf(fact_116_insertE,axiom,
! [A: state,B: state,A3: set_state] :
( ( member_state @ A @ ( insert_state @ B @ A3 ) )
=> ( ( A != B )
=> ( member_state @ A @ A3 ) ) ) ).
% insertE
thf(fact_117_commutative,axiom,
( plus
= ( ^ [A4: state,B3: state] : ( plus @ B3 @ A4 ) ) ) ).
% commutative
thf(fact_118_PartialSA_Oadd__set__commm,axiom,
( add_set
= ( ^ [A5: set_state,B5: set_state] : ( add_set @ B5 @ A5 ) ) ) ).
% PartialSA.add_set_commm
thf(fact_119_PartialSA_Oadd__set__asso,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( add_set @ ( add_set @ A3 @ B2 ) @ C3 )
= ( add_set @ A3 @ ( add_set @ B2 @ C3 ) ) ) ).
% PartialSA.add_set_asso
thf(fact_120_singleton__inject,axiom,
! [A: produc8023240190789890773_state,B: produc8023240190789890773_state] :
( ( ( insert7525286303735658661_state @ A @ bot_bo9041262728264437921_state )
= ( insert7525286303735658661_state @ B @ bot_bo9041262728264437921_state ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_121_singleton__inject,axiom,
! [A: produc3142500478612311029_state,B: produc3142500478612311029_state] :
( ( ( insert4171857611248116165_state @ A @ bot_bo1080640394036989633_state )
= ( insert4171857611248116165_state @ B @ bot_bo1080640394036989633_state ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_122_singleton__inject,axiom,
! [A: option_state,B: option_state] :
( ( ( insert_option_state @ A @ bot_bo710180891245420500_state )
= ( insert_option_state @ B @ bot_bo710180891245420500_state ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_123_singleton__inject,axiom,
! [A: set_state,B: set_state] :
( ( ( insert_set_state @ A @ bot_bo2271482359692755898_state )
= ( insert_set_state @ B @ bot_bo2271482359692755898_state ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_124_singleton__inject,axiom,
! [A: state,B: state] :
( ( ( insert_state @ A @ bot_bot_set_state )
= ( insert_state @ B @ bot_bot_set_state ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_125_insert__not__empty,axiom,
! [A: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( insert7525286303735658661_state @ A @ A3 )
!= bot_bo9041262728264437921_state ) ).
% insert_not_empty
thf(fact_126_insert__not__empty,axiom,
! [A: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( insert4171857611248116165_state @ A @ A3 )
!= bot_bo1080640394036989633_state ) ).
% insert_not_empty
thf(fact_127_insert__not__empty,axiom,
! [A: option_state,A3: set_option_state] :
( ( insert_option_state @ A @ A3 )
!= bot_bo710180891245420500_state ) ).
% insert_not_empty
thf(fact_128_insert__not__empty,axiom,
! [A: set_state,A3: set_set_state] :
( ( insert_set_state @ A @ A3 )
!= bot_bo2271482359692755898_state ) ).
% insert_not_empty
thf(fact_129_insert__not__empty,axiom,
! [A: state,A3: set_state] :
( ( insert_state @ A @ A3 )
!= bot_bot_set_state ) ).
% insert_not_empty
thf(fact_130_doubleton__eq__iff,axiom,
! [A: produc8023240190789890773_state,B: produc8023240190789890773_state,C: produc8023240190789890773_state,D: produc8023240190789890773_state] :
( ( ( insert7525286303735658661_state @ A @ ( insert7525286303735658661_state @ B @ bot_bo9041262728264437921_state ) )
= ( insert7525286303735658661_state @ C @ ( insert7525286303735658661_state @ D @ bot_bo9041262728264437921_state ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_131_doubleton__eq__iff,axiom,
! [A: produc3142500478612311029_state,B: produc3142500478612311029_state,C: produc3142500478612311029_state,D: produc3142500478612311029_state] :
( ( ( insert4171857611248116165_state @ A @ ( insert4171857611248116165_state @ B @ bot_bo1080640394036989633_state ) )
= ( insert4171857611248116165_state @ C @ ( insert4171857611248116165_state @ D @ bot_bo1080640394036989633_state ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_132_doubleton__eq__iff,axiom,
! [A: option_state,B: option_state,C: option_state,D: option_state] :
( ( ( insert_option_state @ A @ ( insert_option_state @ B @ bot_bo710180891245420500_state ) )
= ( insert_option_state @ C @ ( insert_option_state @ D @ bot_bo710180891245420500_state ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_133_doubleton__eq__iff,axiom,
! [A: set_state,B: set_state,C: set_state,D: set_state] :
( ( ( insert_set_state @ A @ ( insert_set_state @ B @ bot_bo2271482359692755898_state ) )
= ( insert_set_state @ C @ ( insert_set_state @ D @ bot_bo2271482359692755898_state ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_134_doubleton__eq__iff,axiom,
! [A: state,B: state,C: state,D: state] :
( ( ( insert_state @ A @ ( insert_state @ B @ bot_bot_set_state ) )
= ( insert_state @ C @ ( insert_state @ D @ bot_bot_set_state ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_135_singleton__iff,axiom,
! [B: produc8023240190789890773_state,A: produc8023240190789890773_state] :
( ( member753036827967488894_state @ B @ ( insert7525286303735658661_state @ A @ bot_bo9041262728264437921_state ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_136_singleton__iff,axiom,
! [B: produc3142500478612311029_state,A: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ B @ ( insert4171857611248116165_state @ A @ bot_bo1080640394036989633_state ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_137_singleton__iff,axiom,
! [B: option_state,A: option_state] :
( ( member_option_state @ B @ ( insert_option_state @ A @ bot_bo710180891245420500_state ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_138_singleton__iff,axiom,
! [B: set_state,A: set_state] :
( ( member_set_state @ B @ ( insert_set_state @ A @ bot_bo2271482359692755898_state ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_139_singleton__iff,axiom,
! [B: state,A: state] :
( ( member_state @ B @ ( insert_state @ A @ bot_bot_set_state ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_140_singletonD,axiom,
! [B: produc8023240190789890773_state,A: produc8023240190789890773_state] :
( ( member753036827967488894_state @ B @ ( insert7525286303735658661_state @ A @ bot_bo9041262728264437921_state ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_141_singletonD,axiom,
! [B: produc3142500478612311029_state,A: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ B @ ( insert4171857611248116165_state @ A @ bot_bo1080640394036989633_state ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_142_singletonD,axiom,
! [B: option_state,A: option_state] :
( ( member_option_state @ B @ ( insert_option_state @ A @ bot_bo710180891245420500_state ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_143_singletonD,axiom,
! [B: set_state,A: set_state] :
( ( member_set_state @ B @ ( insert_set_state @ A @ bot_bo2271482359692755898_state ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_144_singletonD,axiom,
! [B: state,A: state] :
( ( member_state @ B @ ( insert_state @ A @ bot_bot_set_state ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_145_positivity,axiom,
! [A: state,B: state,C: state] :
( ( ( plus @ A @ B )
= ( some_state @ C ) )
=> ( ( ( some_state @ C )
= ( plus @ C @ C ) )
=> ( ( some_state @ A )
= ( plus @ A @ A ) ) ) ) ).
% positivity
thf(fact_146_asso1,axiom,
! [A: state,B: state,Ab: state,C: state,Bc: state] :
( ( ( ( plus @ A @ B )
= ( some_state @ Ab ) )
& ( ( plus @ B @ C )
= ( some_state @ Bc ) ) )
=> ( ( plus @ Ab @ C )
= ( plus @ A @ Bc ) ) ) ).
% asso1
thf(fact_147_the__elem__eq,axiom,
! [X: produc8023240190789890773_state] :
( ( the_el1727186808309674058_state @ ( insert7525286303735658661_state @ X @ bot_bo9041262728264437921_state ) )
= X ) ).
% the_elem_eq
thf(fact_148_the__elem__eq,axiom,
! [X: produc3142500478612311029_state] :
( ( the_el7118915247011979882_state @ ( insert4171857611248116165_state @ X @ bot_bo1080640394036989633_state ) )
= X ) ).
% the_elem_eq
thf(fact_149_the__elem__eq,axiom,
! [X: option_state] :
( ( the_el1618976816499768149_state @ ( insert_option_state @ X @ bot_bo710180891245420500_state ) )
= X ) ).
% the_elem_eq
thf(fact_150_the__elem__eq,axiom,
! [X: set_state] :
( ( the_elem_set_state @ ( insert_set_state @ X @ bot_bo2271482359692755898_state ) )
= X ) ).
% the_elem_eq
thf(fact_151_the__elem__eq,axiom,
! [X: state] :
( ( the_elem_state @ ( insert_state @ X @ bot_bot_set_state ) )
= X ) ).
% the_elem_eq
thf(fact_152_bot__apply,axiom,
( bot_bot_state_o
= ( ^ [X3: state] : bot_bot_o ) ) ).
% bot_apply
thf(fact_153_is__singletonI,axiom,
! [X: produc8023240190789890773_state] : ( is_sin3624314468123612105_state @ ( insert7525286303735658661_state @ X @ bot_bo9041262728264437921_state ) ) ).
% is_singletonI
thf(fact_154_is__singletonI,axiom,
! [X: produc3142500478612311029_state] : ( is_sin7571815095334105321_state @ ( insert4171857611248116165_state @ X @ bot_bo1080640394036989633_state ) ) ).
% is_singletonI
thf(fact_155_is__singletonI,axiom,
! [X: option_state] : ( is_sin8559911096322084886_state @ ( insert_option_state @ X @ bot_bo710180891245420500_state ) ) ).
% is_singletonI
thf(fact_156_is__singletonI,axiom,
! [X: set_state] : ( is_sin3747817797122030204_state @ ( insert_set_state @ X @ bot_bo2271482359692755898_state ) ) ).
% is_singletonI
thf(fact_157_is__singletonI,axiom,
! [X: state] : ( is_singleton_state @ ( insert_state @ X @ bot_bot_set_state ) ) ).
% is_singletonI
thf(fact_158_mem__Collect__eq,axiom,
! [A: option_state,P: option_state > $o] :
( ( member_option_state @ A @ ( collect_option_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A: set_state,P: set_state > $o] :
( ( member_set_state @ A @ ( collect_set_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A: produc3142500478612311029_state,P: produc3142500478612311029_state > $o] :
( ( member3029510603097127326_state @ A @ ( collec8144523193623705312_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A: produc8023240190789890773_state,P: produc8023240190789890773_state > $o] :
( ( member753036827967488894_state @ A @ ( collec7320380983431419584_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
! [A: state,P: state > $o] :
( ( member_state @ A @ ( collect_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A3: set_option_state] :
( ( collect_option_state
@ ^ [X3: option_state] : ( member_option_state @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A3: set_set_state] :
( ( collect_set_state
@ ^ [X3: set_state] : ( member_set_state @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A3: set_Pr1688445902015331925_state] :
( ( collec8144523193623705312_state
@ ^ [X3: produc3142500478612311029_state] : ( member3029510603097127326_state @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_166_Collect__mem__eq,axiom,
! [A3: set_Pr1785066336555260981_state] :
( ( collec7320380983431419584_state
@ ^ [X3: produc8023240190789890773_state] : ( member753036827967488894_state @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A3: set_state] :
( ( collect_state
@ ^ [X3: state] : ( member_state @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_168_Collect__cong,axiom,
! [P: state > $o,Q: state > $o] :
( ! [X4: state] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_state @ P )
= ( collect_state @ Q ) ) ) ).
% Collect_cong
thf(fact_169_PartialSA_Osetify__sum,axiom,
! [P: state > $o,A3: set_state,B2: set_state] :
( ( sep_setify_state @ P @ ( add_set @ A3 @ B2 ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ( sep_setify_state @ P @ ( add_set @ ( insert_state @ X3 @ bot_bot_set_state ) @ B2 ) ) ) ) ) ).
% PartialSA.setify_sum
thf(fact_170_these__insert__Some,axiom,
! [X: set_state,A3: set_option_set_state] :
( ( these_set_state @ ( insert1104837854041676528_state @ ( some_set_state @ X ) @ A3 ) )
= ( insert_set_state @ X @ ( these_set_state @ A3 ) ) ) ).
% these_insert_Some
thf(fact_171_these__insert__Some,axiom,
! [X: produc3142500478612311029_state,A3: set_op381248089174005275_state] :
( ( these_5013389664457929650_state @ ( insert4808919046788495371_state @ ( some_P3186401494017672538_state @ X ) @ A3 ) )
= ( insert4171857611248116165_state @ X @ ( these_5013389664457929650_state @ A3 ) ) ) ).
% these_insert_Some
thf(fact_172_these__insert__Some,axiom,
! [X: produc8023240190789890773_state,A3: set_op4912175446517189883_state] :
( ( these_9162046556185909650_state @ ( insert4154903780514342891_state @ ( some_P8120450764674687802_state @ X ) @ A3 ) )
= ( insert7525286303735658661_state @ X @ ( these_9162046556185909650_state @ A3 ) ) ) ).
% these_insert_Some
thf(fact_173_these__insert__Some,axiom,
! [X: option_state,A3: set_op9003753404445127824_state] :
( ( these_option_state @ ( insert7535573567550883338_state @ ( some_option_state @ X ) @ A3 ) )
= ( insert_option_state @ X @ ( these_option_state @ A3 ) ) ) ).
% these_insert_Some
thf(fact_174_these__insert__Some,axiom,
! [X: state,A3: set_option_state] :
( ( these_state @ ( insert_option_state @ ( some_state @ X ) @ A3 ) )
= ( insert_state @ X @ ( these_state @ A3 ) ) ) ).
% these_insert_Some
thf(fact_175_Set_Ois__empty__def,axiom,
( is_emp7935198893844832935_state
= ( ^ [A5: set_Pr1785066336555260981_state] : ( A5 = bot_bo9041262728264437921_state ) ) ) ).
% Set.is_empty_def
thf(fact_176_Set_Ois__empty__def,axiom,
( is_emp6065741494043489735_state
= ( ^ [A5: set_Pr1688445902015331925_state] : ( A5 = bot_bo1080640394036989633_state ) ) ) ).
% Set.is_empty_def
thf(fact_177_Set_Ois__empty__def,axiom,
( is_emp7138872393943660984_state
= ( ^ [A5: set_option_state] : ( A5 = bot_bo710180891245420500_state ) ) ) ).
% Set.is_empty_def
thf(fact_178_Set_Ois__empty__def,axiom,
( is_empty_set_state
= ( ^ [A5: set_set_state] : ( A5 = bot_bo2271482359692755898_state ) ) ) ).
% Set.is_empty_def
thf(fact_179_Set_Ois__empty__def,axiom,
( is_empty_state
= ( ^ [A5: set_state] : ( A5 = bot_bot_set_state ) ) ) ).
% Set.is_empty_def
thf(fact_180_is__singleton__def,axiom,
( is_sin3624314468123612105_state
= ( ^ [A5: set_Pr1785066336555260981_state] :
? [X3: produc8023240190789890773_state] :
( A5
= ( insert7525286303735658661_state @ X3 @ bot_bo9041262728264437921_state ) ) ) ) ).
% is_singleton_def
thf(fact_181_is__singleton__def,axiom,
( is_sin7571815095334105321_state
= ( ^ [A5: set_Pr1688445902015331925_state] :
? [X3: produc3142500478612311029_state] :
( A5
= ( insert4171857611248116165_state @ X3 @ bot_bo1080640394036989633_state ) ) ) ) ).
% is_singleton_def
thf(fact_182_is__singleton__def,axiom,
( is_sin8559911096322084886_state
= ( ^ [A5: set_option_state] :
? [X3: option_state] :
( A5
= ( insert_option_state @ X3 @ bot_bo710180891245420500_state ) ) ) ) ).
% is_singleton_def
thf(fact_183_is__singleton__def,axiom,
( is_sin3747817797122030204_state
= ( ^ [A5: set_set_state] :
? [X3: set_state] :
( A5
= ( insert_set_state @ X3 @ bot_bo2271482359692755898_state ) ) ) ) ).
% is_singleton_def
thf(fact_184_is__singleton__def,axiom,
( is_singleton_state
= ( ^ [A5: set_state] :
? [X3: state] :
( A5
= ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) ).
% is_singleton_def
thf(fact_185_is__singletonE,axiom,
! [A3: set_Pr1785066336555260981_state] :
( ( is_sin3624314468123612105_state @ A3 )
=> ~ ! [X4: produc8023240190789890773_state] :
( A3
!= ( insert7525286303735658661_state @ X4 @ bot_bo9041262728264437921_state ) ) ) ).
% is_singletonE
thf(fact_186_is__singletonE,axiom,
! [A3: set_Pr1688445902015331925_state] :
( ( is_sin7571815095334105321_state @ A3 )
=> ~ ! [X4: produc3142500478612311029_state] :
( A3
!= ( insert4171857611248116165_state @ X4 @ bot_bo1080640394036989633_state ) ) ) ).
% is_singletonE
thf(fact_187_is__singletonE,axiom,
! [A3: set_option_state] :
( ( is_sin8559911096322084886_state @ A3 )
=> ~ ! [X4: option_state] :
( A3
!= ( insert_option_state @ X4 @ bot_bo710180891245420500_state ) ) ) ).
% is_singletonE
thf(fact_188_is__singletonE,axiom,
! [A3: set_set_state] :
( ( is_sin3747817797122030204_state @ A3 )
=> ~ ! [X4: set_state] :
( A3
!= ( insert_set_state @ X4 @ bot_bo2271482359692755898_state ) ) ) ).
% is_singletonE
thf(fact_189_is__singletonE,axiom,
! [A3: set_state] :
( ( is_singleton_state @ A3 )
=> ~ ! [X4: state] :
( A3
!= ( insert_state @ X4 @ bot_bot_set_state ) ) ) ).
% is_singletonE
thf(fact_190_PartialSA_Ox__elem__set__product__splus,axiom,
! [X: state,A3: set_state,B2: set_state] :
( ( member_state @ X @ ( add_set @ A3 @ B2 ) )
= ( ? [A4: state,B3: state] :
( ( member_state @ A4 @ A3 )
& ( member_state @ B3 @ B2 )
& ( ( some_state @ X )
= ( sep_splus_state @ plus @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) ) ) ) ) ).
% PartialSA.x_elem_set_product_splus
thf(fact_191_option_Osimps_I15_J,axiom,
! [X2: produc8023240190789890773_state] :
( ( set_op4970154267292507491_state @ ( some_P8120450764674687802_state @ X2 ) )
= ( insert7525286303735658661_state @ X2 @ bot_bo9041262728264437921_state ) ) ).
% option.simps(15)
thf(fact_192_option_Osimps_I15_J,axiom,
! [X2: produc3142500478612311029_state] :
( ( set_op4197959913983417475_state @ ( some_P3186401494017672538_state @ X2 ) )
= ( insert4171857611248116165_state @ X2 @ bot_bo1080640394036989633_state ) ) ).
% option.simps(15)
thf(fact_193_option_Osimps_I15_J,axiom,
! [X2: option_state] :
( ( set_op5162993263733338108_state @ ( some_option_state @ X2 ) )
= ( insert_option_state @ X2 @ bot_bo710180891245420500_state ) ) ).
% option.simps(15)
thf(fact_194_option_Osimps_I15_J,axiom,
! [X2: set_state] :
( ( set_option_set_state2 @ ( some_set_state @ X2 ) )
= ( insert_set_state @ X2 @ bot_bo2271482359692755898_state ) ) ).
% option.simps(15)
thf(fact_195_option_Osimps_I15_J,axiom,
! [X2: state] :
( ( set_option_state2 @ ( some_state @ X2 ) )
= ( insert_state @ X2 @ bot_bot_set_state ) ) ).
% option.simps(15)
thf(fact_196_PartialSA_Oequiv__stable__sum,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( sep_equiv_state @ plus @ A3 @ B2 )
=> ( sep_equiv_state @ plus @ ( add_set @ A3 @ C3 ) @ ( add_set @ B2 @ C3 ) ) ) ).
% PartialSA.equiv_stable_sum
thf(fact_197_in__cwand,axiom,
! [A3: set_state,W: state,B2: set_state] :
( ! [A2: state,X4: state] :
( ( ( member_state @ A2 @ A3 )
& ( ( some_state @ X4 )
= ( plus @ ( r @ A2 @ W ) @ A2 ) ) )
=> ( member_state @ X4 @ B2 ) )
=> ( member_state @ W @ ( cwand @ A3 @ B2 ) ) ) ).
% in_cwand
thf(fact_198_elem__set,axiom,
! [X: set_state,Xo: option_set_state] :
( ( member_set_state @ X @ ( set_option_set_state2 @ Xo ) )
= ( Xo
= ( some_set_state @ X ) ) ) ).
% elem_set
thf(fact_199_elem__set,axiom,
! [X: produc3142500478612311029_state,Xo: option6833441738159790651_state] :
( ( member3029510603097127326_state @ X @ ( set_op4197959913983417475_state @ Xo ) )
= ( Xo
= ( some_P3186401494017672538_state @ X ) ) ) ).
% elem_set
thf(fact_200_elem__set,axiom,
! [X: produc8023240190789890773_state,Xo: option2250103068101548571_state] :
( ( member753036827967488894_state @ X @ ( set_op4970154267292507491_state @ Xo ) )
= ( Xo
= ( some_P8120450764674687802_state @ X ) ) ) ).
% elem_set
thf(fact_201_elem__set,axiom,
! [X: option_state,Xo: option_option_state] :
( ( member_option_state @ X @ ( set_op5162993263733338108_state @ Xo ) )
= ( Xo
= ( some_option_state @ X ) ) ) ).
% elem_set
thf(fact_202_elem__set,axiom,
! [X: state,Xo: option_state] :
( ( member_state @ X @ ( set_option_state2 @ Xo ) )
= ( Xo
= ( some_state @ X ) ) ) ).
% elem_set
thf(fact_203_these__empty,axiom,
( ( these_9162046556185909650_state @ bot_bo3922232019306180455_state )
= bot_bo9041262728264437921_state ) ).
% these_empty
thf(fact_204_these__empty,axiom,
( ( these_5013389664457929650_state @ bot_bo2763976174278612103_state )
= bot_bo1080640394036989633_state ) ).
% these_empty
thf(fact_205_these__empty,axiom,
( ( these_option_state @ bot_bo489212050006660900_state )
= bot_bo710180891245420500_state ) ).
% these_empty
thf(fact_206_these__empty,axiom,
( ( these_set_state @ bot_bo7912781682106412938_state )
= bot_bo2271482359692755898_state ) ).
% these_empty
thf(fact_207_these__empty,axiom,
( ( these_state @ bot_bo710180891245420500_state )
= bot_bot_set_state ) ).
% these_empty
thf(fact_208_PartialSA_Osetify__def,axiom,
( sep_setify_state
= ( ^ [P2: state > $o,A5: set_state] :
! [X3: state] :
( ( member_state @ X3 @ A5 )
=> ( P2 @ X3 ) ) ) ) ).
% PartialSA.setify_def
thf(fact_209_bot__set__def,axiom,
( bot_bo9041262728264437921_state
= ( collec7320380983431419584_state @ bot_bo2162275076026452348tate_o ) ) ).
% bot_set_def
thf(fact_210_bot__set__def,axiom,
( bot_bo1080640394036989633_state
= ( collec8144523193623705312_state @ bot_bo4049596492272799580tate_o ) ) ).
% bot_set_def
thf(fact_211_bot__set__def,axiom,
( bot_bo710180891245420500_state
= ( collect_option_state @ bot_bo4453335400789057457tate_o ) ) ).
% bot_set_def
thf(fact_212_bot__set__def,axiom,
( bot_bo2271482359692755898_state
= ( collect_set_state @ bot_bot_set_state_o ) ) ).
% bot_set_def
thf(fact_213_bot__set__def,axiom,
( bot_bot_set_state
= ( collect_state @ bot_bot_state_o ) ) ).
% bot_set_def
thf(fact_214_PartialSA_Osplus__asso,axiom,
! [A: option_state,B: option_state,C: option_state] :
( ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ A @ B ) @ C )
= ( sep_splus_state @ plus @ A @ ( sep_splus_state @ plus @ B @ C ) ) ) ).
% PartialSA.splus_asso
thf(fact_215_PartialSA_Osplus__comm,axiom,
! [A: option_state,B: option_state] :
( ( sep_splus_state @ plus @ A @ B )
= ( sep_splus_state @ plus @ B @ A ) ) ).
% PartialSA.splus_comm
thf(fact_216_option_Oset__cases,axiom,
! [E: set_state,A: option_set_state] :
( ( member_set_state @ E @ ( set_option_set_state2 @ A ) )
=> ( A
= ( some_set_state @ E ) ) ) ).
% option.set_cases
thf(fact_217_option_Oset__cases,axiom,
! [E: produc3142500478612311029_state,A: option6833441738159790651_state] :
( ( member3029510603097127326_state @ E @ ( set_op4197959913983417475_state @ A ) )
=> ( A
= ( some_P3186401494017672538_state @ E ) ) ) ).
% option.set_cases
thf(fact_218_option_Oset__cases,axiom,
! [E: produc8023240190789890773_state,A: option2250103068101548571_state] :
( ( member753036827967488894_state @ E @ ( set_op4970154267292507491_state @ A ) )
=> ( A
= ( some_P8120450764674687802_state @ E ) ) ) ).
% option.set_cases
thf(fact_219_option_Oset__cases,axiom,
! [E: option_state,A: option_option_state] :
( ( member_option_state @ E @ ( set_op5162993263733338108_state @ A ) )
=> ( A
= ( some_option_state @ E ) ) ) ).
% option.set_cases
thf(fact_220_option_Oset__cases,axiom,
! [E: state,A: option_state] :
( ( member_state @ E @ ( set_option_state2 @ A ) )
=> ( A
= ( some_state @ E ) ) ) ).
% option.set_cases
thf(fact_221_option_Oset__intros,axiom,
! [X2: set_state] : ( member_set_state @ X2 @ ( set_option_set_state2 @ ( some_set_state @ X2 ) ) ) ).
% option.set_intros
thf(fact_222_option_Oset__intros,axiom,
! [X2: produc3142500478612311029_state] : ( member3029510603097127326_state @ X2 @ ( set_op4197959913983417475_state @ ( some_P3186401494017672538_state @ X2 ) ) ) ).
% option.set_intros
thf(fact_223_option_Oset__intros,axiom,
! [X2: produc8023240190789890773_state] : ( member753036827967488894_state @ X2 @ ( set_op4970154267292507491_state @ ( some_P8120450764674687802_state @ X2 ) ) ) ).
% option.set_intros
thf(fact_224_option_Oset__intros,axiom,
! [X2: option_state] : ( member_option_state @ X2 @ ( set_op5162993263733338108_state @ ( some_option_state @ X2 ) ) ) ).
% option.set_intros
thf(fact_225_option_Oset__intros,axiom,
! [X2: state] : ( member_state @ X2 @ ( set_option_state2 @ ( some_state @ X2 ) ) ) ).
% option.set_intros
thf(fact_226_ospec,axiom,
! [A3: option_option_state,P: option_state > $o,X: option_state] :
( ! [X4: option_state] :
( ( member_option_state @ X4 @ ( set_op5162993263733338108_state @ A3 ) )
=> ( P @ X4 ) )
=> ( ( A3
= ( some_option_state @ X ) )
=> ( P @ X ) ) ) ).
% ospec
thf(fact_227_ospec,axiom,
! [A3: option_state,P: state > $o,X: state] :
( ! [X4: state] :
( ( member_state @ X4 @ ( set_option_state2 @ A3 ) )
=> ( P @ X4 ) )
=> ( ( A3
= ( some_state @ X ) )
=> ( P @ X ) ) ) ).
% ospec
thf(fact_228_in__these__eq,axiom,
! [X: set_state,A3: set_option_set_state] :
( ( member_set_state @ X @ ( these_set_state @ A3 ) )
= ( member1412897518142203351_state @ ( some_set_state @ X ) @ A3 ) ) ).
% in_these_eq
thf(fact_229_in__these__eq,axiom,
! [X: produc3142500478612311029_state,A3: set_op381248089174005275_state] :
( ( member3029510603097127326_state @ X @ ( these_5013389664457929650_state @ A3 ) )
= ( member3754053363737695844_state @ ( some_P3186401494017672538_state @ X ) @ A3 ) ) ).
% in_these_eq
thf(fact_230_in__these__eq,axiom,
! [X: produc8023240190789890773_state,A3: set_op4912175446517189883_state] :
( ( member753036827967488894_state @ X @ ( these_9162046556185909650_state @ A3 ) )
= ( member5076014802351601988_state @ ( some_P8120450764674687802_state @ X ) @ A3 ) ) ).
% in_these_eq
thf(fact_231_in__these__eq,axiom,
! [X: option_state,A3: set_op9003753404445127824_state] :
( ( member_option_state @ X @ ( these_option_state @ A3 ) )
= ( member1079230918592710257_state @ ( some_option_state @ X ) @ A3 ) ) ).
% in_these_eq
thf(fact_232_in__these__eq,axiom,
! [X: state,A3: set_option_state] :
( ( member_state @ X @ ( these_state @ A3 ) )
= ( member_option_state @ ( some_state @ X ) @ A3 ) ) ).
% in_these_eq
thf(fact_233_is__singleton__the__elem,axiom,
( is_sin3624314468123612105_state
= ( ^ [A5: set_Pr1785066336555260981_state] :
( A5
= ( insert7525286303735658661_state @ ( the_el1727186808309674058_state @ A5 ) @ bot_bo9041262728264437921_state ) ) ) ) ).
% is_singleton_the_elem
thf(fact_234_is__singleton__the__elem,axiom,
( is_sin7571815095334105321_state
= ( ^ [A5: set_Pr1688445902015331925_state] :
( A5
= ( insert4171857611248116165_state @ ( the_el7118915247011979882_state @ A5 ) @ bot_bo1080640394036989633_state ) ) ) ) ).
% is_singleton_the_elem
thf(fact_235_is__singleton__the__elem,axiom,
( is_sin8559911096322084886_state
= ( ^ [A5: set_option_state] :
( A5
= ( insert_option_state @ ( the_el1618976816499768149_state @ A5 ) @ bot_bo710180891245420500_state ) ) ) ) ).
% is_singleton_the_elem
thf(fact_236_is__singleton__the__elem,axiom,
( is_sin3747817797122030204_state
= ( ^ [A5: set_set_state] :
( A5
= ( insert_set_state @ ( the_elem_set_state @ A5 ) @ bot_bo2271482359692755898_state ) ) ) ) ).
% is_singleton_the_elem
thf(fact_237_is__singleton__the__elem,axiom,
( is_singleton_state
= ( ^ [A5: set_state] :
( A5
= ( insert_state @ ( the_elem_state @ A5 ) @ bot_bot_set_state ) ) ) ) ).
% is_singleton_the_elem
thf(fact_238_PartialSA_Osplus_Osimps_I3_J,axiom,
! [A: state,B: state] :
( ( sep_splus_state @ plus @ ( some_state @ A ) @ ( some_state @ B ) )
= ( plus @ A @ B ) ) ).
% PartialSA.splus.simps(3)
thf(fact_239_PartialSA_Osplus__develop,axiom,
! [A: state,B: state,C: state,D: state] :
( ( ( some_state @ A )
= ( plus @ B @ C ) )
=> ( ( plus @ A @ D )
= ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ ( some_state @ B ) @ ( some_state @ C ) ) @ ( some_state @ D ) ) ) ) ).
% PartialSA.splus_develop
thf(fact_240_is__singletonI_H,axiom,
! [A3: set_Pr1785066336555260981_state] :
( ( A3 != bot_bo9041262728264437921_state )
=> ( ! [X4: produc8023240190789890773_state,Y: produc8023240190789890773_state] :
( ( member753036827967488894_state @ X4 @ A3 )
=> ( ( member753036827967488894_state @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_sin3624314468123612105_state @ A3 ) ) ) ).
% is_singletonI'
thf(fact_241_is__singletonI_H,axiom,
! [A3: set_Pr1688445902015331925_state] :
( ( A3 != bot_bo1080640394036989633_state )
=> ( ! [X4: produc3142500478612311029_state,Y: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ X4 @ A3 )
=> ( ( member3029510603097127326_state @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_sin7571815095334105321_state @ A3 ) ) ) ).
% is_singletonI'
thf(fact_242_is__singletonI_H,axiom,
! [A3: set_option_state] :
( ( A3 != bot_bo710180891245420500_state )
=> ( ! [X4: option_state,Y: option_state] :
( ( member_option_state @ X4 @ A3 )
=> ( ( member_option_state @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_sin8559911096322084886_state @ A3 ) ) ) ).
% is_singletonI'
thf(fact_243_is__singletonI_H,axiom,
! [A3: set_set_state] :
( ( A3 != bot_bo2271482359692755898_state )
=> ( ! [X4: set_state,Y: set_state] :
( ( member_set_state @ X4 @ A3 )
=> ( ( member_set_state @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_sin3747817797122030204_state @ A3 ) ) ) ).
% is_singletonI'
thf(fact_244_is__singletonI_H,axiom,
! [A3: set_state] :
( ( A3 != bot_bot_set_state )
=> ( ! [X4: state,Y: state] :
( ( member_state @ X4 @ A3 )
=> ( ( member_state @ Y @ A3 )
=> ( X4 = Y ) ) )
=> ( is_singleton_state @ A3 ) ) ) ).
% is_singletonI'
thf(fact_245_bot__fun__def,axiom,
( bot_bot_state_o
= ( ^ [X3: state] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_246_PartialSA_Omono__prop__set__equiv,axiom,
! [P: state > $o,A3: set_state,B2: set_state] :
( ( sep_mono_prop_state @ plus @ P )
=> ( ( sep_equiv_state @ plus @ A3 @ B2 )
=> ( ( sep_setify_state @ P @ A3 )
= ( sep_setify_state @ P @ B2 ) ) ) ) ).
% PartialSA.mono_prop_set_equiv
thf(fact_247_PartialSA_Osetify__sum__image,axiom,
! [P: state > $o,F: nat > state,A3: set_nat,B2: set_state] :
( ( sep_setify_state @ P @ ( add_set @ ( image_nat_state @ F @ A3 ) @ B2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( sep_setify_state @ P @ ( add_set @ ( insert_state @ ( F @ X3 ) @ bot_bot_set_state ) @ B2 ) ) ) ) ) ).
% PartialSA.setify_sum_image
thf(fact_248_sep__algebra_Oequiv_Ocong,axiom,
sep_equiv_state = sep_equiv_state ).
% sep_algebra.equiv.cong
thf(fact_249_sep__algebra_Osplus_Ocong,axiom,
sep_splus_state = sep_splus_state ).
% sep_algebra.splus.cong
thf(fact_250_these__not__empty__eq,axiom,
! [B2: set_op4912175446517189883_state] :
( ( ( these_9162046556185909650_state @ B2 )
!= bot_bo9041262728264437921_state )
= ( ( B2 != bot_bo3922232019306180455_state )
& ( B2
!= ( insert4154903780514342891_state @ none_P348328851727313334_state @ bot_bo3922232019306180455_state ) ) ) ) ).
% these_not_empty_eq
thf(fact_251_these__not__empty__eq,axiom,
! [B2: set_op381248089174005275_state] :
( ( ( these_5013389664457929650_state @ B2 )
!= bot_bo1080640394036989633_state )
= ( ( B2 != bot_bo2763976174278612103_state )
& ( B2
!= ( insert4808919046788495371_state @ none_P2148290358184556502_state @ bot_bo2763976174278612103_state ) ) ) ) ).
% these_not_empty_eq
thf(fact_252_these__not__empty__eq,axiom,
! [B2: set_op9003753404445127824_state] :
( ( ( these_option_state @ B2 )
!= bot_bo710180891245420500_state )
= ( ( B2 != bot_bo489212050006660900_state )
& ( B2
!= ( insert7535573567550883338_state @ none_option_state @ bot_bo489212050006660900_state ) ) ) ) ).
% these_not_empty_eq
thf(fact_253_these__not__empty__eq,axiom,
! [B2: set_option_set_state] :
( ( ( these_set_state @ B2 )
!= bot_bo2271482359692755898_state )
= ( ( B2 != bot_bo7912781682106412938_state )
& ( B2
!= ( insert1104837854041676528_state @ none_set_state @ bot_bo7912781682106412938_state ) ) ) ) ).
% these_not_empty_eq
thf(fact_254_these__not__empty__eq,axiom,
! [B2: set_option_state] :
( ( ( these_state @ B2 )
!= bot_bot_set_state )
= ( ( B2 != bot_bo710180891245420500_state )
& ( B2
!= ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).
% these_not_empty_eq
thf(fact_255_these__empty__eq,axiom,
! [B2: set_op4912175446517189883_state] :
( ( ( these_9162046556185909650_state @ B2 )
= bot_bo9041262728264437921_state )
= ( ( B2 = bot_bo3922232019306180455_state )
| ( B2
= ( insert4154903780514342891_state @ none_P348328851727313334_state @ bot_bo3922232019306180455_state ) ) ) ) ).
% these_empty_eq
thf(fact_256_these__empty__eq,axiom,
! [B2: set_op381248089174005275_state] :
( ( ( these_5013389664457929650_state @ B2 )
= bot_bo1080640394036989633_state )
= ( ( B2 = bot_bo2763976174278612103_state )
| ( B2
= ( insert4808919046788495371_state @ none_P2148290358184556502_state @ bot_bo2763976174278612103_state ) ) ) ) ).
% these_empty_eq
thf(fact_257_these__empty__eq,axiom,
! [B2: set_op9003753404445127824_state] :
( ( ( these_option_state @ B2 )
= bot_bo710180891245420500_state )
= ( ( B2 = bot_bo489212050006660900_state )
| ( B2
= ( insert7535573567550883338_state @ none_option_state @ bot_bo489212050006660900_state ) ) ) ) ).
% these_empty_eq
thf(fact_258_these__empty__eq,axiom,
! [B2: set_option_set_state] :
( ( ( these_set_state @ B2 )
= bot_bo2271482359692755898_state )
= ( ( B2 = bot_bo7912781682106412938_state )
| ( B2
= ( insert1104837854041676528_state @ none_set_state @ bot_bo7912781682106412938_state ) ) ) ) ).
% these_empty_eq
thf(fact_259_these__empty__eq,axiom,
! [B2: set_option_state] :
( ( ( these_state @ B2 )
= bot_bot_set_state )
= ( ( B2 = bot_bo710180891245420500_state )
| ( B2
= ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).
% these_empty_eq
thf(fact_260_PartialSA_OequivI,axiom,
! [A3: set_state,B2: set_state] :
( ( greater_set @ A3 @ B2 )
=> ( ( greater_set @ B2 @ A3 )
=> ( sep_equiv_state @ plus @ A3 @ B2 ) ) ) ).
% PartialSA.equivI
thf(fact_261_PartialSA_Oequiv__def,axiom,
! [A3: set_state,B2: set_state] :
( ( sep_equiv_state @ plus @ A3 @ B2 )
= ( ( greater_set @ A3 @ B2 )
& ( greater_set @ B2 @ A3 ) ) ) ).
% PartialSA.equiv_def
thf(fact_262_these__insert__None,axiom,
! [A3: set_op9003753404445127824_state] :
( ( these_option_state @ ( insert7535573567550883338_state @ none_option_state @ A3 ) )
= ( these_option_state @ A3 ) ) ).
% these_insert_None
thf(fact_263_these__insert__None,axiom,
! [A3: set_option_state] :
( ( these_state @ ( insert_option_state @ none_state @ A3 ) )
= ( these_state @ A3 ) ) ).
% these_insert_None
thf(fact_264_PartialSA_Osplus_Osimps_I2_J,axiom,
! [V: state] :
( ( sep_splus_state @ plus @ ( some_state @ V ) @ none_state )
= none_state ) ).
% PartialSA.splus.simps(2)
thf(fact_265_PartialSA_Osplus_Oelims,axiom,
! [X: option_state,Xa: option_state,Y3: option_state] :
( ( ( sep_splus_state @ plus @ X @ Xa )
= Y3 )
=> ( ( ( X = none_state )
=> ( Y3 != none_state ) )
=> ( ( ? [V2: state] :
( X
= ( some_state @ V2 ) )
=> ( ( Xa = none_state )
=> ( Y3 != none_state ) ) )
=> ~ ! [A2: state] :
( ( X
= ( some_state @ A2 ) )
=> ! [B6: state] :
( ( Xa
= ( some_state @ B6 ) )
=> ( Y3
!= ( plus @ A2 @ B6 ) ) ) ) ) ) ) ).
% PartialSA.splus.elims
thf(fact_266_image__eqI,axiom,
! [B: state,F: state > state,X: state,A3: set_state] :
( ( B
= ( F @ X ) )
=> ( ( member_state @ X @ A3 )
=> ( member_state @ B @ ( image_state_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_267_image__eqI,axiom,
! [B: state,F: nat > state,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_state @ B @ ( image_nat_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_268_image__eqI,axiom,
! [B: option_state,F: state > option_state,X: state,A3: set_state] :
( ( B
= ( F @ X ) )
=> ( ( member_state @ X @ A3 )
=> ( member_option_state @ B @ ( image_6076465424260689483_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_269_image__eqI,axiom,
! [B: set_state,F: state > set_state,X: state,A3: set_state] :
( ( B
= ( F @ X ) )
=> ( ( member_state @ X @ A3 )
=> ( member_set_state @ B @ ( image_4774290769506072625_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_270_image__eqI,axiom,
! [B: state,F: option_state > state,X: option_state,A3: set_option_state] :
( ( B
= ( F @ X ) )
=> ( ( member_option_state @ X @ A3 )
=> ( member_state @ B @ ( image_3532137647693456075_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_271_image__eqI,axiom,
! [B: state,F: set_state > state,X: set_state,A3: set_set_state] :
( ( B
= ( F @ X ) )
=> ( ( member_set_state @ X @ A3 )
=> ( member_state @ B @ ( image_4575879259649255985_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_272_image__eqI,axiom,
! [B: produc8023240190789890773_state,F: state > produc8023240190789890773_state,X: state,A3: set_state] :
( ( B
= ( F @ X ) )
=> ( ( member_state @ X @ A3 )
=> ( member753036827967488894_state @ B @ ( image_2153010417677216468_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_273_image__eqI,axiom,
! [B: option_state,F: option_state > option_state,X: option_state,A3: set_option_state] :
( ( B
= ( F @ X ) )
=> ( ( member_option_state @ X @ A3 )
=> ( member_option_state @ B @ ( image_8299601973402907291_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_274_image__eqI,axiom,
! [B: set_state,F: option_state > set_state,X: option_state,A3: set_option_state] :
( ( B
= ( F @ X ) )
=> ( ( member_option_state @ X @ A3 )
=> ( member_set_state @ B @ ( image_3284240608559150209_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_275_image__eqI,axiom,
! [B: option_state,F: set_state > option_state,X: set_state,A3: set_set_state] :
( ( B
= ( F @ X ) )
=> ( ( member_set_state @ X @ A3 )
=> ( member_option_state @ B @ ( image_8427833934475094273_state @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_276_image__is__empty,axiom,
! [F: state > state,A3: set_state] :
( ( ( image_state_state @ F @ A3 )
= bot_bot_set_state )
= ( A3 = bot_bot_set_state ) ) ).
% image_is_empty
thf(fact_277_image__is__empty,axiom,
! [F: nat > state,A3: set_nat] :
( ( ( image_nat_state @ F @ A3 )
= bot_bot_set_state )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_278_image__is__empty,axiom,
! [F: option_state > state,A3: set_option_state] :
( ( ( image_3532137647693456075_state @ F @ A3 )
= bot_bot_set_state )
= ( A3 = bot_bo710180891245420500_state ) ) ).
% image_is_empty
thf(fact_279_image__is__empty,axiom,
! [F: set_state > state,A3: set_set_state] :
( ( ( image_4575879259649255985_state @ F @ A3 )
= bot_bot_set_state )
= ( A3 = bot_bo2271482359692755898_state ) ) ).
% image_is_empty
thf(fact_280_image__is__empty,axiom,
! [F: state > option_state,A3: set_state] :
( ( ( image_6076465424260689483_state @ F @ A3 )
= bot_bo710180891245420500_state )
= ( A3 = bot_bot_set_state ) ) ).
% image_is_empty
thf(fact_281_image__is__empty,axiom,
! [F: state > set_state,A3: set_state] :
( ( ( image_4774290769506072625_state @ F @ A3 )
= bot_bo2271482359692755898_state )
= ( A3 = bot_bot_set_state ) ) ).
% image_is_empty
thf(fact_282_image__is__empty,axiom,
! [F: produc8023240190789890773_state > state,A3: set_Pr1785066336555260981_state] :
( ( ( image_5145414166468158282_state @ F @ A3 )
= bot_bot_set_state )
= ( A3 = bot_bo9041262728264437921_state ) ) ).
% image_is_empty
thf(fact_283_image__is__empty,axiom,
! [F: state > produc8023240190789890773_state,A3: set_state] :
( ( ( image_2153010417677216468_state @ F @ A3 )
= bot_bo9041262728264437921_state )
= ( A3 = bot_bot_set_state ) ) ).
% image_is_empty
thf(fact_284_image__is__empty,axiom,
! [F: option_state > option_state,A3: set_option_state] :
( ( ( image_8299601973402907291_state @ F @ A3 )
= bot_bo710180891245420500_state )
= ( A3 = bot_bo710180891245420500_state ) ) ).
% image_is_empty
thf(fact_285_image__is__empty,axiom,
! [F: set_state > option_state,A3: set_set_state] :
( ( ( image_8427833934475094273_state @ F @ A3 )
= bot_bo710180891245420500_state )
= ( A3 = bot_bo2271482359692755898_state ) ) ).
% image_is_empty
thf(fact_286_empty__is__image,axiom,
! [F: state > state,A3: set_state] :
( ( bot_bot_set_state
= ( image_state_state @ F @ A3 ) )
= ( A3 = bot_bot_set_state ) ) ).
% empty_is_image
thf(fact_287_empty__is__image,axiom,
! [F: nat > state,A3: set_nat] :
( ( bot_bot_set_state
= ( image_nat_state @ F @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_288_empty__is__image,axiom,
! [F: option_state > state,A3: set_option_state] :
( ( bot_bot_set_state
= ( image_3532137647693456075_state @ F @ A3 ) )
= ( A3 = bot_bo710180891245420500_state ) ) ).
% empty_is_image
thf(fact_289_empty__is__image,axiom,
! [F: set_state > state,A3: set_set_state] :
( ( bot_bot_set_state
= ( image_4575879259649255985_state @ F @ A3 ) )
= ( A3 = bot_bo2271482359692755898_state ) ) ).
% empty_is_image
thf(fact_290_empty__is__image,axiom,
! [F: state > option_state,A3: set_state] :
( ( bot_bo710180891245420500_state
= ( image_6076465424260689483_state @ F @ A3 ) )
= ( A3 = bot_bot_set_state ) ) ).
% empty_is_image
thf(fact_291_empty__is__image,axiom,
! [F: state > set_state,A3: set_state] :
( ( bot_bo2271482359692755898_state
= ( image_4774290769506072625_state @ F @ A3 ) )
= ( A3 = bot_bot_set_state ) ) ).
% empty_is_image
thf(fact_292_empty__is__image,axiom,
! [F: produc8023240190789890773_state > state,A3: set_Pr1785066336555260981_state] :
( ( bot_bot_set_state
= ( image_5145414166468158282_state @ F @ A3 ) )
= ( A3 = bot_bo9041262728264437921_state ) ) ).
% empty_is_image
thf(fact_293_empty__is__image,axiom,
! [F: state > produc8023240190789890773_state,A3: set_state] :
( ( bot_bo9041262728264437921_state
= ( image_2153010417677216468_state @ F @ A3 ) )
= ( A3 = bot_bot_set_state ) ) ).
% empty_is_image
thf(fact_294_empty__is__image,axiom,
! [F: option_state > option_state,A3: set_option_state] :
( ( bot_bo710180891245420500_state
= ( image_8299601973402907291_state @ F @ A3 ) )
= ( A3 = bot_bo710180891245420500_state ) ) ).
% empty_is_image
thf(fact_295_empty__is__image,axiom,
! [F: set_state > option_state,A3: set_set_state] :
( ( bot_bo710180891245420500_state
= ( image_8427833934475094273_state @ F @ A3 ) )
= ( A3 = bot_bo2271482359692755898_state ) ) ).
% empty_is_image
thf(fact_296_image__empty,axiom,
! [F: state > state] :
( ( image_state_state @ F @ bot_bot_set_state )
= bot_bot_set_state ) ).
% image_empty
thf(fact_297_image__empty,axiom,
! [F: nat > state] :
( ( image_nat_state @ F @ bot_bot_set_nat )
= bot_bot_set_state ) ).
% image_empty
thf(fact_298_image__empty,axiom,
! [F: state > option_state] :
( ( image_6076465424260689483_state @ F @ bot_bot_set_state )
= bot_bo710180891245420500_state ) ).
% image_empty
thf(fact_299_image__empty,axiom,
! [F: state > set_state] :
( ( image_4774290769506072625_state @ F @ bot_bot_set_state )
= bot_bo2271482359692755898_state ) ).
% image_empty
thf(fact_300_image__empty,axiom,
! [F: option_state > state] :
( ( image_3532137647693456075_state @ F @ bot_bo710180891245420500_state )
= bot_bot_set_state ) ).
% image_empty
thf(fact_301_image__empty,axiom,
! [F: set_state > state] :
( ( image_4575879259649255985_state @ F @ bot_bo2271482359692755898_state )
= bot_bot_set_state ) ).
% image_empty
thf(fact_302_image__empty,axiom,
! [F: state > produc8023240190789890773_state] :
( ( image_2153010417677216468_state @ F @ bot_bot_set_state )
= bot_bo9041262728264437921_state ) ).
% image_empty
thf(fact_303_image__empty,axiom,
! [F: produc8023240190789890773_state > state] :
( ( image_5145414166468158282_state @ F @ bot_bo9041262728264437921_state )
= bot_bot_set_state ) ).
% image_empty
thf(fact_304_image__empty,axiom,
! [F: option_state > option_state] :
( ( image_8299601973402907291_state @ F @ bot_bo710180891245420500_state )
= bot_bo710180891245420500_state ) ).
% image_empty
thf(fact_305_image__empty,axiom,
! [F: option_state > set_state] :
( ( image_3284240608559150209_state @ F @ bot_bo710180891245420500_state )
= bot_bo2271482359692755898_state ) ).
% image_empty
thf(fact_306_insert__image,axiom,
! [X: state,A3: set_state,F: state > state] :
( ( member_state @ X @ A3 )
=> ( ( insert_state @ ( F @ X ) @ ( image_state_state @ F @ A3 ) )
= ( image_state_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_307_insert__image,axiom,
! [X: nat,A3: set_nat,F: nat > state] :
( ( member_nat @ X @ A3 )
=> ( ( insert_state @ ( F @ X ) @ ( image_nat_state @ F @ A3 ) )
= ( image_nat_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_308_insert__image,axiom,
! [X: state,A3: set_state,F: state > option_state] :
( ( member_state @ X @ A3 )
=> ( ( insert_option_state @ ( F @ X ) @ ( image_6076465424260689483_state @ F @ A3 ) )
= ( image_6076465424260689483_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_309_insert__image,axiom,
! [X: state,A3: set_state,F: state > set_state] :
( ( member_state @ X @ A3 )
=> ( ( insert_set_state @ ( F @ X ) @ ( image_4774290769506072625_state @ F @ A3 ) )
= ( image_4774290769506072625_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_310_insert__image,axiom,
! [X: option_state,A3: set_option_state,F: option_state > state] :
( ( member_option_state @ X @ A3 )
=> ( ( insert_state @ ( F @ X ) @ ( image_3532137647693456075_state @ F @ A3 ) )
= ( image_3532137647693456075_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_311_insert__image,axiom,
! [X: set_state,A3: set_set_state,F: set_state > state] :
( ( member_set_state @ X @ A3 )
=> ( ( insert_state @ ( F @ X ) @ ( image_4575879259649255985_state @ F @ A3 ) )
= ( image_4575879259649255985_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_312_insert__image,axiom,
! [X: state,A3: set_state,F: state > produc8023240190789890773_state] :
( ( member_state @ X @ A3 )
=> ( ( insert7525286303735658661_state @ ( F @ X ) @ ( image_2153010417677216468_state @ F @ A3 ) )
= ( image_2153010417677216468_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_313_insert__image,axiom,
! [X: option_state,A3: set_option_state,F: option_state > option_state] :
( ( member_option_state @ X @ A3 )
=> ( ( insert_option_state @ ( F @ X ) @ ( image_8299601973402907291_state @ F @ A3 ) )
= ( image_8299601973402907291_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_314_insert__image,axiom,
! [X: option_state,A3: set_option_state,F: option_state > set_state] :
( ( member_option_state @ X @ A3 )
=> ( ( insert_set_state @ ( F @ X ) @ ( image_3284240608559150209_state @ F @ A3 ) )
= ( image_3284240608559150209_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_315_insert__image,axiom,
! [X: set_state,A3: set_set_state,F: set_state > option_state] :
( ( member_set_state @ X @ A3 )
=> ( ( insert_option_state @ ( F @ X ) @ ( image_8427833934475094273_state @ F @ A3 ) )
= ( image_8427833934475094273_state @ F @ A3 ) ) ) ).
% insert_image
thf(fact_316_image__insert,axiom,
! [F: state > state,A: state,B2: set_state] :
( ( image_state_state @ F @ ( insert_state @ A @ B2 ) )
= ( insert_state @ ( F @ A ) @ ( image_state_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_317_image__insert,axiom,
! [F: nat > state,A: nat,B2: set_nat] :
( ( image_nat_state @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_state @ ( F @ A ) @ ( image_nat_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_318_image__insert,axiom,
! [F: state > option_state,A: state,B2: set_state] :
( ( image_6076465424260689483_state @ F @ ( insert_state @ A @ B2 ) )
= ( insert_option_state @ ( F @ A ) @ ( image_6076465424260689483_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_319_image__insert,axiom,
! [F: state > set_state,A: state,B2: set_state] :
( ( image_4774290769506072625_state @ F @ ( insert_state @ A @ B2 ) )
= ( insert_set_state @ ( F @ A ) @ ( image_4774290769506072625_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_320_image__insert,axiom,
! [F: option_state > state,A: option_state,B2: set_option_state] :
( ( image_3532137647693456075_state @ F @ ( insert_option_state @ A @ B2 ) )
= ( insert_state @ ( F @ A ) @ ( image_3532137647693456075_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_321_image__insert,axiom,
! [F: set_state > state,A: set_state,B2: set_set_state] :
( ( image_4575879259649255985_state @ F @ ( insert_set_state @ A @ B2 ) )
= ( insert_state @ ( F @ A ) @ ( image_4575879259649255985_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_322_image__insert,axiom,
! [F: state > produc8023240190789890773_state,A: state,B2: set_state] :
( ( image_2153010417677216468_state @ F @ ( insert_state @ A @ B2 ) )
= ( insert7525286303735658661_state @ ( F @ A ) @ ( image_2153010417677216468_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_323_image__insert,axiom,
! [F: option_state > option_state,A: option_state,B2: set_option_state] :
( ( image_8299601973402907291_state @ F @ ( insert_option_state @ A @ B2 ) )
= ( insert_option_state @ ( F @ A ) @ ( image_8299601973402907291_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_324_image__insert,axiom,
! [F: option_state > set_state,A: option_state,B2: set_option_state] :
( ( image_3284240608559150209_state @ F @ ( insert_option_state @ A @ B2 ) )
= ( insert_set_state @ ( F @ A ) @ ( image_3284240608559150209_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_325_image__insert,axiom,
! [F: set_state > option_state,A: set_state,B2: set_set_state] :
( ( image_8427833934475094273_state @ F @ ( insert_set_state @ A @ B2 ) )
= ( insert_option_state @ ( F @ A ) @ ( image_8427833934475094273_state @ F @ B2 ) ) ) ).
% image_insert
thf(fact_326_not__Some__eq,axiom,
! [X: option_option_state] :
( ( ! [Y4: option_state] :
( X
!= ( some_option_state @ Y4 ) ) )
= ( X = none_option_state ) ) ).
% not_Some_eq
thf(fact_327_not__Some__eq,axiom,
! [X: option_state] :
( ( ! [Y4: state] :
( X
!= ( some_state @ Y4 ) ) )
= ( X = none_state ) ) ).
% not_Some_eq
thf(fact_328_not__None__eq,axiom,
! [X: option_option_state] :
( ( X != none_option_state )
= ( ? [Y4: option_state] :
( X
= ( some_option_state @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_329_not__None__eq,axiom,
! [X: option_state] :
( ( X != none_state )
= ( ? [Y4: state] :
( X
= ( some_state @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_330_set__empty__eq,axiom,
! [Xo: option2250103068101548571_state] :
( ( ( set_op4970154267292507491_state @ Xo )
= bot_bo9041262728264437921_state )
= ( Xo = none_P348328851727313334_state ) ) ).
% set_empty_eq
thf(fact_331_set__empty__eq,axiom,
! [Xo: option6833441738159790651_state] :
( ( ( set_op4197959913983417475_state @ Xo )
= bot_bo1080640394036989633_state )
= ( Xo = none_P2148290358184556502_state ) ) ).
% set_empty_eq
thf(fact_332_set__empty__eq,axiom,
! [Xo: option_option_state] :
( ( ( set_op5162993263733338108_state @ Xo )
= bot_bo710180891245420500_state )
= ( Xo = none_option_state ) ) ).
% set_empty_eq
thf(fact_333_set__empty__eq,axiom,
! [Xo: option_set_state] :
( ( ( set_option_set_state2 @ Xo )
= bot_bo2271482359692755898_state )
= ( Xo = none_set_state ) ) ).
% set_empty_eq
thf(fact_334_set__empty__eq,axiom,
! [Xo: option_state] :
( ( ( set_option_state2 @ Xo )
= bot_bot_set_state )
= ( Xo = none_state ) ) ).
% set_empty_eq
thf(fact_335_sep__algebra_Omono__prop_Ocong,axiom,
sep_mono_prop_state = sep_mono_prop_state ).
% sep_algebra.mono_prop.cong
thf(fact_336_PartialSA_Osucc__set__trans,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( greater_set @ A3 @ B2 )
=> ( ( greater_set @ B2 @ C3 )
=> ( greater_set @ A3 @ C3 ) ) ) ).
% PartialSA.succ_set_trans
thf(fact_337_PartialSA_Olarger__set__refl,axiom,
! [A3: set_state] : ( greater_set @ A3 @ A3 ) ).
% PartialSA.larger_set_refl
thf(fact_338_imageI,axiom,
! [X: state,A3: set_state,F: state > state] :
( ( member_state @ X @ A3 )
=> ( member_state @ ( F @ X ) @ ( image_state_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_339_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > state] :
( ( member_nat @ X @ A3 )
=> ( member_state @ ( F @ X ) @ ( image_nat_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_340_imageI,axiom,
! [X: state,A3: set_state,F: state > option_state] :
( ( member_state @ X @ A3 )
=> ( member_option_state @ ( F @ X ) @ ( image_6076465424260689483_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_341_imageI,axiom,
! [X: state,A3: set_state,F: state > set_state] :
( ( member_state @ X @ A3 )
=> ( member_set_state @ ( F @ X ) @ ( image_4774290769506072625_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_342_imageI,axiom,
! [X: option_state,A3: set_option_state,F: option_state > state] :
( ( member_option_state @ X @ A3 )
=> ( member_state @ ( F @ X ) @ ( image_3532137647693456075_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_343_imageI,axiom,
! [X: set_state,A3: set_set_state,F: set_state > state] :
( ( member_set_state @ X @ A3 )
=> ( member_state @ ( F @ X ) @ ( image_4575879259649255985_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_344_imageI,axiom,
! [X: state,A3: set_state,F: state > produc8023240190789890773_state] :
( ( member_state @ X @ A3 )
=> ( member753036827967488894_state @ ( F @ X ) @ ( image_2153010417677216468_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_345_imageI,axiom,
! [X: option_state,A3: set_option_state,F: option_state > option_state] :
( ( member_option_state @ X @ A3 )
=> ( member_option_state @ ( F @ X ) @ ( image_8299601973402907291_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_346_imageI,axiom,
! [X: option_state,A3: set_option_state,F: option_state > set_state] :
( ( member_option_state @ X @ A3 )
=> ( member_set_state @ ( F @ X ) @ ( image_3284240608559150209_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_347_imageI,axiom,
! [X: set_state,A3: set_set_state,F: set_state > option_state] :
( ( member_set_state @ X @ A3 )
=> ( member_option_state @ ( F @ X ) @ ( image_8427833934475094273_state @ F @ A3 ) ) ) ).
% imageI
thf(fact_348_image__iff,axiom,
! [Z: state,F: nat > state,A3: set_nat] :
( ( member_state @ Z @ ( image_nat_state @ F @ A3 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_349_image__iff,axiom,
! [Z: option_state,F: state > option_state,A3: set_state] :
( ( member_option_state @ Z @ ( image_6076465424260689483_state @ F @ A3 ) )
= ( ? [X3: state] :
( ( member_state @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_350_image__iff,axiom,
! [Z: set_state,F: state > set_state,A3: set_state] :
( ( member_set_state @ Z @ ( image_4774290769506072625_state @ F @ A3 ) )
= ( ? [X3: state] :
( ( member_state @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_351_image__iff,axiom,
! [Z: set_state,F: set_state > set_state,A3: set_set_state] :
( ( member_set_state @ Z @ ( image_2476256681063834599_state @ F @ A3 ) )
= ( ? [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_352_image__iff,axiom,
! [Z: set_state,F: ( state > $o ) > set_state,A3: set_state_o] :
( ( member_set_state @ Z @ ( image_7376656169852520768_state @ F @ A3 ) )
= ( ? [X3: state > $o] :
( ( member_state_o @ X3 @ A3 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_353_bex__imageD,axiom,
! [F: state > option_state,A3: set_state,P: option_state > $o] :
( ? [X5: option_state] :
( ( member_option_state @ X5 @ ( image_6076465424260689483_state @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X4: state] :
( ( member_state @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_354_bex__imageD,axiom,
! [F: state > set_state,A3: set_state,P: set_state > $o] :
( ? [X5: set_state] :
( ( member_set_state @ X5 @ ( image_4774290769506072625_state @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X4: state] :
( ( member_state @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_355_bex__imageD,axiom,
! [F: set_state > set_state,A3: set_set_state,P: set_state > $o] :
( ? [X5: set_state] :
( ( member_set_state @ X5 @ ( image_2476256681063834599_state @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X4: set_state] :
( ( member_set_state @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_356_bex__imageD,axiom,
! [F: nat > state,A3: set_nat,P: state > $o] :
( ? [X5: state] :
( ( member_state @ X5 @ ( image_nat_state @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_357_bex__imageD,axiom,
! [F: ( state > $o ) > set_state,A3: set_state_o,P: set_state > $o] :
( ? [X5: set_state] :
( ( member_set_state @ X5 @ ( image_7376656169852520768_state @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X4: state > $o] :
( ( member_state_o @ X4 @ A3 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_358_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > state,G: nat > state] :
( ( M = N )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_state @ F @ M )
= ( image_nat_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_359_image__cong,axiom,
! [M: set_state_o,N: set_state_o,F: ( state > $o ) > set_state,G: ( state > $o ) > set_state] :
( ( M = N )
=> ( ! [X4: state > $o] :
( ( member_state_o @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7376656169852520768_state @ F @ M )
= ( image_7376656169852520768_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_360_image__cong,axiom,
! [M: set_state,N: set_state,F: state > option_state,G: state > option_state] :
( ( M = N )
=> ( ! [X4: state] :
( ( member_state @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_6076465424260689483_state @ F @ M )
= ( image_6076465424260689483_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_361_image__cong,axiom,
! [M: set_state,N: set_state,F: state > set_state,G: state > set_state] :
( ( M = N )
=> ( ! [X4: state] :
( ( member_state @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_4774290769506072625_state @ F @ M )
= ( image_4774290769506072625_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_362_image__cong,axiom,
! [M: set_set_state,N: set_set_state,F: set_state > set_state,G: set_state > set_state] :
( ( M = N )
=> ( ! [X4: set_state] :
( ( member_set_state @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_2476256681063834599_state @ F @ M )
= ( image_2476256681063834599_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_363_ball__imageD,axiom,
! [F: state > option_state,A3: set_state,P: option_state > $o] :
( ! [X4: option_state] :
( ( member_option_state @ X4 @ ( image_6076465424260689483_state @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X5: state] :
( ( member_state @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_364_ball__imageD,axiom,
! [F: state > set_state,A3: set_state,P: set_state > $o] :
( ! [X4: set_state] :
( ( member_set_state @ X4 @ ( image_4774290769506072625_state @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X5: state] :
( ( member_state @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_365_ball__imageD,axiom,
! [F: set_state > set_state,A3: set_set_state,P: set_state > $o] :
( ! [X4: set_state] :
( ( member_set_state @ X4 @ ( image_2476256681063834599_state @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X5: set_state] :
( ( member_set_state @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_366_ball__imageD,axiom,
! [F: nat > state,A3: set_nat,P: state > $o] :
( ! [X4: state] :
( ( member_state @ X4 @ ( image_nat_state @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_367_ball__imageD,axiom,
! [F: ( state > $o ) > set_state,A3: set_state_o,P: set_state > $o] :
( ! [X4: set_state] :
( ( member_set_state @ X4 @ ( image_7376656169852520768_state @ F @ A3 ) )
=> ( P @ X4 ) )
=> ! [X5: state > $o] :
( ( member_state_o @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_368_rev__image__eqI,axiom,
! [X: state,A3: set_state,B: state,F: state > state] :
( ( member_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_state @ B @ ( image_state_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_369_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: state,F: nat > state] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_state @ B @ ( image_nat_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_370_rev__image__eqI,axiom,
! [X: state,A3: set_state,B: option_state,F: state > option_state] :
( ( member_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_option_state @ B @ ( image_6076465424260689483_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_371_rev__image__eqI,axiom,
! [X: state,A3: set_state,B: set_state,F: state > set_state] :
( ( member_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_state @ B @ ( image_4774290769506072625_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_372_rev__image__eqI,axiom,
! [X: option_state,A3: set_option_state,B: state,F: option_state > state] :
( ( member_option_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_state @ B @ ( image_3532137647693456075_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_373_rev__image__eqI,axiom,
! [X: set_state,A3: set_set_state,B: state,F: set_state > state] :
( ( member_set_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_state @ B @ ( image_4575879259649255985_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_374_rev__image__eqI,axiom,
! [X: state,A3: set_state,B: produc8023240190789890773_state,F: state > produc8023240190789890773_state] :
( ( member_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member753036827967488894_state @ B @ ( image_2153010417677216468_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_375_rev__image__eqI,axiom,
! [X: option_state,A3: set_option_state,B: option_state,F: option_state > option_state] :
( ( member_option_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_option_state @ B @ ( image_8299601973402907291_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_376_rev__image__eqI,axiom,
! [X: option_state,A3: set_option_state,B: set_state,F: option_state > set_state] :
( ( member_option_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_state @ B @ ( image_3284240608559150209_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_377_rev__image__eqI,axiom,
! [X: set_state,A3: set_set_state,B: option_state,F: set_state > option_state] :
( ( member_set_state @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_option_state @ B @ ( image_8427833934475094273_state @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_378_combine__options__cases,axiom,
! [X: option_state,P: option_state > option_option_state > $o,Y3: option_option_state] :
( ( ( X = none_state )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_option_state )
=> ( P @ X @ Y3 ) )
=> ( ! [A2: state,B6: option_state] :
( ( X
= ( some_state @ A2 ) )
=> ( ( Y3
= ( some_option_state @ B6 ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_379_combine__options__cases,axiom,
! [X: option_option_state,P: option_option_state > option_state > $o,Y3: option_state] :
( ( ( X = none_option_state )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_state )
=> ( P @ X @ Y3 ) )
=> ( ! [A2: option_state,B6: state] :
( ( X
= ( some_option_state @ A2 ) )
=> ( ( Y3
= ( some_state @ B6 ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_380_combine__options__cases,axiom,
! [X: option_option_state,P: option_option_state > option_option_state > $o,Y3: option_option_state] :
( ( ( X = none_option_state )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_option_state )
=> ( P @ X @ Y3 ) )
=> ( ! [A2: option_state,B6: option_state] :
( ( X
= ( some_option_state @ A2 ) )
=> ( ( Y3
= ( some_option_state @ B6 ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_381_combine__options__cases,axiom,
! [X: option_state,P: option_state > option_state > $o,Y3: option_state] :
( ( ( X = none_state )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_state )
=> ( P @ X @ Y3 ) )
=> ( ! [A2: state,B6: state] :
( ( X
= ( some_state @ A2 ) )
=> ( ( Y3
= ( some_state @ B6 ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_382_split__option__all,axiom,
( ( ^ [P3: option_option_state > $o] :
! [X6: option_option_state] : ( P3 @ X6 ) )
= ( ^ [P2: option_option_state > $o] :
( ( P2 @ none_option_state )
& ! [X3: option_state] : ( P2 @ ( some_option_state @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_383_split__option__all,axiom,
( ( ^ [P3: option_state > $o] :
! [X6: option_state] : ( P3 @ X6 ) )
= ( ^ [P2: option_state > $o] :
( ( P2 @ none_state )
& ! [X3: state] : ( P2 @ ( some_state @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_384_split__option__ex,axiom,
( ( ^ [P3: option_option_state > $o] :
? [X6: option_option_state] : ( P3 @ X6 ) )
= ( ^ [P2: option_option_state > $o] :
( ( P2 @ none_option_state )
| ? [X3: option_state] : ( P2 @ ( some_option_state @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_385_split__option__ex,axiom,
( ( ^ [P3: option_state > $o] :
? [X6: option_state] : ( P3 @ X6 ) )
= ( ^ [P2: option_state > $o] :
( ( P2 @ none_state )
| ? [X3: state] : ( P2 @ ( some_state @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_386_option_Oexhaust,axiom,
! [Y3: option_option_state] :
( ( Y3 != none_option_state )
=> ~ ! [X22: option_state] :
( Y3
!= ( some_option_state @ X22 ) ) ) ).
% option.exhaust
thf(fact_387_option_Oexhaust,axiom,
! [Y3: option_state] :
( ( Y3 != none_state )
=> ~ ! [X22: state] :
( Y3
!= ( some_state @ X22 ) ) ) ).
% option.exhaust
thf(fact_388_option_OdiscI,axiom,
! [Option: option_option_state,X2: option_state] :
( ( Option
= ( some_option_state @ X2 ) )
=> ( Option != none_option_state ) ) ).
% option.discI
thf(fact_389_option_OdiscI,axiom,
! [Option: option_state,X2: state] :
( ( Option
= ( some_state @ X2 ) )
=> ( Option != none_state ) ) ).
% option.discI
thf(fact_390_option_Odistinct_I1_J,axiom,
! [X2: option_state] :
( none_option_state
!= ( some_option_state @ X2 ) ) ).
% option.distinct(1)
thf(fact_391_option_Odistinct_I1_J,axiom,
! [X2: state] :
( none_state
!= ( some_state @ X2 ) ) ).
% option.distinct(1)
thf(fact_392_PartialSA_Omono__prop__set,axiom,
! [A3: set_state,B2: set_state,P: state > $o] :
( ( greater_set @ A3 @ B2 )
=> ( ( sep_setify_state @ P @ B2 )
=> ( ( sep_mono_prop_state @ plus @ P )
=> ( sep_setify_state @ P @ A3 ) ) ) ) ).
% PartialSA.mono_prop_set
thf(fact_393_PartialSA_Obigger__set,axiom,
! [A6: set_state,A3: set_state,B2: set_state] :
( ( greater_set @ A6 @ A3 )
=> ( greater_set @ ( add_set @ A6 @ B2 ) @ ( add_set @ A3 @ B2 ) ) ) ).
% PartialSA.bigger_set
thf(fact_394_asso2,axiom,
! [A: state,B: state,Ab: state,C: state] :
( ( ( ( plus @ A @ B )
= ( some_state @ Ab ) )
& ( ( plus @ B @ C )
= none_state ) )
=> ( ( plus @ Ab @ C )
= none_state ) ) ).
% asso2
thf(fact_395_PartialSA_Oasso3,axiom,
! [A: state,B: state,C: state,Bc: state] :
( ( ( plus @ A @ B )
= none_state )
=> ( ( ( plus @ B @ C )
= ( some_state @ Bc ) )
=> ( ( plus @ A @ Bc )
= none_state ) ) ) ).
% PartialSA.asso3
thf(fact_396_option_Osimps_I14_J,axiom,
( ( set_op4970154267292507491_state @ none_P348328851727313334_state )
= bot_bo9041262728264437921_state ) ).
% option.simps(14)
thf(fact_397_option_Osimps_I14_J,axiom,
( ( set_op4197959913983417475_state @ none_P2148290358184556502_state )
= bot_bo1080640394036989633_state ) ).
% option.simps(14)
thf(fact_398_option_Osimps_I14_J,axiom,
( ( set_op5162993263733338108_state @ none_option_state )
= bot_bo710180891245420500_state ) ).
% option.simps(14)
thf(fact_399_option_Osimps_I14_J,axiom,
( ( set_option_set_state2 @ none_set_state )
= bot_bo2271482359692755898_state ) ).
% option.simps(14)
thf(fact_400_option_Osimps_I14_J,axiom,
( ( set_option_state2 @ none_state )
= bot_bot_set_state ) ).
% option.simps(14)
thf(fact_401_PartialSA_Osplus_Osimps_I1_J,axiom,
! [Uu: option_state] :
( ( sep_splus_state @ plus @ none_state @ Uu )
= none_state ) ).
% PartialSA.splus.simps(1)
thf(fact_402_the__elem__image__unique,axiom,
! [A3: set_state_o,F: ( state > $o ) > set_state,X: state > $o] :
( ( A3 != bot_bot_set_state_o2 )
=> ( ! [Y: state > $o] :
( ( member_state_o @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_set_state @ ( image_7376656169852520768_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_403_the__elem__image__unique,axiom,
! [A3: set_nat,F: nat > state,X: nat] :
( ( A3 != bot_bot_set_nat )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_state @ ( image_nat_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_404_the__elem__image__unique,axiom,
! [A3: set_state,F: state > option_state,X: state] :
( ( A3 != bot_bot_set_state )
=> ( ! [Y: state] :
( ( member_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_el1618976816499768149_state @ ( image_6076465424260689483_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_405_the__elem__image__unique,axiom,
! [A3: set_state,F: state > set_state,X: state] :
( ( A3 != bot_bot_set_state )
=> ( ! [Y: state] :
( ( member_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_set_state @ ( image_4774290769506072625_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_406_the__elem__image__unique,axiom,
! [A3: set_state,F: state > state,X: state] :
( ( A3 != bot_bot_set_state )
=> ( ! [Y: state] :
( ( member_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_state @ ( image_state_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_407_the__elem__image__unique,axiom,
! [A3: set_Pr1785066336555260981_state,F: produc8023240190789890773_state > state,X: produc8023240190789890773_state] :
( ( A3 != bot_bo9041262728264437921_state )
=> ( ! [Y: produc8023240190789890773_state] :
( ( member753036827967488894_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_state @ ( image_5145414166468158282_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_408_the__elem__image__unique,axiom,
! [A3: set_Pr1688445902015331925_state,F: produc3142500478612311029_state > state,X: produc3142500478612311029_state] :
( ( A3 != bot_bo1080640394036989633_state )
=> ( ! [Y: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_state @ ( image_393491784373993066_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_409_the__elem__image__unique,axiom,
! [A3: set_option_state,F: option_state > state,X: option_state] :
( ( A3 != bot_bo710180891245420500_state )
=> ( ! [Y: option_state] :
( ( member_option_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_state @ ( image_3532137647693456075_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_410_the__elem__image__unique,axiom,
! [A3: set_set_state,F: set_state > set_state,X: set_state] :
( ( A3 != bot_bo2271482359692755898_state )
=> ( ! [Y: set_state] :
( ( member_set_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_set_state @ ( image_2476256681063834599_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_411_the__elem__image__unique,axiom,
! [A3: set_set_state,F: set_state > state,X: set_state] :
( ( A3 != bot_bo2271482359692755898_state )
=> ( ! [Y: set_state] :
( ( member_set_state @ Y @ A3 )
=> ( ( F @ Y )
= ( F @ X ) ) )
=> ( ( the_elem_state @ ( image_4575879259649255985_state @ F @ A3 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_412_Collect__empty__eq__bot,axiom,
! [P: produc8023240190789890773_state > $o] :
( ( ( collec7320380983431419584_state @ P )
= bot_bo9041262728264437921_state )
= ( P = bot_bo2162275076026452348tate_o ) ) ).
% Collect_empty_eq_bot
thf(fact_413_Collect__empty__eq__bot,axiom,
! [P: produc3142500478612311029_state > $o] :
( ( ( collec8144523193623705312_state @ P )
= bot_bo1080640394036989633_state )
= ( P = bot_bo4049596492272799580tate_o ) ) ).
% Collect_empty_eq_bot
thf(fact_414_Collect__empty__eq__bot,axiom,
! [P: option_state > $o] :
( ( ( collect_option_state @ P )
= bot_bo710180891245420500_state )
= ( P = bot_bo4453335400789057457tate_o ) ) ).
% Collect_empty_eq_bot
thf(fact_415_Collect__empty__eq__bot,axiom,
! [P: set_state > $o] :
( ( ( collect_set_state @ P )
= bot_bo2271482359692755898_state )
= ( P = bot_bot_set_state_o ) ) ).
% Collect_empty_eq_bot
thf(fact_416_Collect__empty__eq__bot,axiom,
! [P: state > $o] :
( ( ( collect_state @ P )
= bot_bot_set_state )
= ( P = bot_bot_state_o ) ) ).
% Collect_empty_eq_bot
thf(fact_417_bot__empty__eq,axiom,
( bot_bo2162275076026452348tate_o
= ( ^ [X3: produc8023240190789890773_state] : ( member753036827967488894_state @ X3 @ bot_bo9041262728264437921_state ) ) ) ).
% bot_empty_eq
thf(fact_418_bot__empty__eq,axiom,
( bot_bo4049596492272799580tate_o
= ( ^ [X3: produc3142500478612311029_state] : ( member3029510603097127326_state @ X3 @ bot_bo1080640394036989633_state ) ) ) ).
% bot_empty_eq
thf(fact_419_bot__empty__eq,axiom,
( bot_bo4453335400789057457tate_o
= ( ^ [X3: option_state] : ( member_option_state @ X3 @ bot_bo710180891245420500_state ) ) ) ).
% bot_empty_eq
thf(fact_420_bot__empty__eq,axiom,
( bot_bot_set_state_o
= ( ^ [X3: set_state] : ( member_set_state @ X3 @ bot_bo2271482359692755898_state ) ) ) ).
% bot_empty_eq
thf(fact_421_bot__empty__eq,axiom,
( bot_bot_state_o
= ( ^ [X3: state] : ( member_state @ X3 @ bot_bot_set_state ) ) ) ).
% bot_empty_eq
thf(fact_422_compatible__options_Oelims_I2_J,axiom,
! [X: option_option_state,Xa: option_option_state] :
( ( compat839418488507666022_state @ X @ Xa )
=> ( ! [A2: option_state] :
( ( X
= ( some_option_state @ A2 ) )
=> ! [B6: option_state] :
( ( Xa
= ( some_option_state @ B6 ) )
=> ( A2 != B6 ) ) )
=> ( ( X != none_option_state )
=> ( Xa = none_option_state ) ) ) ) ).
% compatible_options.elims(2)
thf(fact_423_compatible__options_Oelims_I2_J,axiom,
! [X: option_state,Xa: option_state] :
( ( compat2278460363914054422_state @ X @ Xa )
=> ( ! [A2: state] :
( ( X
= ( some_state @ A2 ) )
=> ! [B6: state] :
( ( Xa
= ( some_state @ B6 ) )
=> ( A2 != B6 ) ) )
=> ( ( X != none_state )
=> ( Xa = none_state ) ) ) ) ).
% compatible_options.elims(2)
thf(fact_424_compatible__options_Oelims_I1_J,axiom,
! [X: option_option_state,Xa: option_option_state,Y3: $o] :
( ( ( compat839418488507666022_state @ X @ Xa )
= Y3 )
=> ( ! [A2: option_state] :
( ( X
= ( some_option_state @ A2 ) )
=> ! [B6: option_state] :
( ( Xa
= ( some_option_state @ B6 ) )
=> ( Y3
= ( A2 != B6 ) ) ) )
=> ( ( ( X = none_option_state )
=> ~ Y3 )
=> ~ ( ( Xa = none_option_state )
=> ~ Y3 ) ) ) ) ).
% compatible_options.elims(1)
thf(fact_425_compatible__options_Oelims_I1_J,axiom,
! [X: option_state,Xa: option_state,Y3: $o] :
( ( ( compat2278460363914054422_state @ X @ Xa )
= Y3 )
=> ( ! [A2: state] :
( ( X
= ( some_state @ A2 ) )
=> ! [B6: state] :
( ( Xa
= ( some_state @ B6 ) )
=> ( Y3
= ( A2 != B6 ) ) ) )
=> ( ( ( X = none_state )
=> ~ Y3 )
=> ~ ( ( Xa = none_state )
=> ~ Y3 ) ) ) ) ).
% compatible_options.elims(1)
thf(fact_426_PartialSA_Obigger__singleton,axiom,
! [Phi2: state,Phi: state] :
( ( greater @ Phi2 @ Phi )
=> ( greater_set @ ( insert_state @ Phi2 @ bot_bot_set_state ) @ ( insert_state @ Phi @ bot_bot_set_state ) ) ) ).
% PartialSA.bigger_singleton
thf(fact_427_sep__algebra_Omono__prop__set__equiv,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_mono_prop_state @ Plus @ P )
=> ( ( sep_equiv_state @ Plus @ A3 @ B2 )
=> ( ( sep_setify_state @ P @ A3 )
= ( sep_setify_state @ P @ B2 ) ) ) ) ) ).
% sep_algebra.mono_prop_set_equiv
thf(fact_428_PartialSA_Oup__close__equiv,axiom,
! [A3: set_state,B2: set_state] :
( ( sep_up_closed_state @ plus @ A3 )
=> ( ( sep_up_closed_state @ plus @ B2 )
=> ( ( sep_equiv_state @ plus @ A3 @ B2 )
= ( A3 = B2 ) ) ) ) ).
% PartialSA.up_close_equiv
thf(fact_429_greater__set__def,axiom,
( greater_set
= ( sep_gr7105985528888466643_state @ plus ) ) ).
% greater_set_def
thf(fact_430_Sup_OSUP__cong,axiom,
! [A3: set_nat,B2: set_nat,C3: nat > state,D2: nat > state,Sup: set_state > state] :
( ( A3 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_nat_state @ C3 @ A3 ) )
= ( Sup @ ( image_nat_state @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_431_Sup_OSUP__cong,axiom,
! [A3: set_state_o,B2: set_state_o,C3: ( state > $o ) > set_state,D2: ( state > $o ) > set_state,Sup: set_set_state > set_state] :
( ( A3 = B2 )
=> ( ! [X4: state > $o] :
( ( member_state_o @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_7376656169852520768_state @ C3 @ A3 ) )
= ( Sup @ ( image_7376656169852520768_state @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_432_Sup_OSUP__cong,axiom,
! [A3: set_state,B2: set_state,C3: state > option_state,D2: state > option_state,Sup: set_option_state > option_state] :
( ( A3 = B2 )
=> ( ! [X4: state] :
( ( member_state @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_6076465424260689483_state @ C3 @ A3 ) )
= ( Sup @ ( image_6076465424260689483_state @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_433_Sup_OSUP__cong,axiom,
! [A3: set_state,B2: set_state,C3: state > set_state,D2: state > set_state,Sup: set_set_state > set_state] :
( ( A3 = B2 )
=> ( ! [X4: state] :
( ( member_state @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_4774290769506072625_state @ C3 @ A3 ) )
= ( Sup @ ( image_4774290769506072625_state @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_434_Sup_OSUP__cong,axiom,
! [A3: set_set_state,B2: set_set_state,C3: set_state > set_state,D2: set_state > set_state,Sup: set_set_state > set_state] :
( ( A3 = B2 )
=> ( ! [X4: set_state] :
( ( member_set_state @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_2476256681063834599_state @ C3 @ A3 ) )
= ( Sup @ ( image_2476256681063834599_state @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_435_these__image__Some__eq,axiom,
! [A3: set_option_state] :
( ( these_option_state @ ( image_4760606469461393131_state @ some_option_state @ A3 ) )
= A3 ) ).
% these_image_Some_eq
thf(fact_436_these__image__Some__eq,axiom,
! [A3: set_state] :
( ( these_state @ ( image_6076465424260689483_state @ some_state @ A3 ) )
= A3 ) ).
% these_image_Some_eq
thf(fact_437_sep__algebra_Ogreater__set_Ocong,axiom,
sep_gr7105985528888466643_state = sep_gr7105985528888466643_state ).
% sep_algebra.greater_set.cong
thf(fact_438_sep__algebra_Olarger__set__refl,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( sep_gr7105985528888466643_state @ Plus @ A3 @ A3 ) ) ).
% sep_algebra.larger_set_refl
thf(fact_439_sep__algebra_Oup__closed_Ocong,axiom,
sep_up_closed_state = sep_up_closed_state ).
% sep_algebra.up_closed.cong
thf(fact_440_sep__algebra_Osucc__set__trans,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state,C3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 )
=> ( ( sep_gr7105985528888466643_state @ Plus @ B2 @ C3 )
=> ( sep_gr7105985528888466643_state @ Plus @ A3 @ C3 ) ) ) ) ).
% sep_algebra.succ_set_trans
thf(fact_441_sep__algebra_Ocommutative,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( Plus @ A @ B )
= ( Plus @ B @ A ) ) ) ).
% sep_algebra.commutative
thf(fact_442_PartialSA_Osucc__antisym,axiom,
! [A: state,B: state] :
( ( greater @ A @ B )
=> ( ( greater @ B @ A )
=> ( A = B ) ) ) ).
% PartialSA.succ_antisym
thf(fact_443_PartialSA_Osucc__trans,axiom,
! [A: state,B: state,C: state] :
( ( greater @ A @ B )
=> ( ( greater @ B @ C )
=> ( greater @ A @ C ) ) ) ).
% PartialSA.succ_trans
thf(fact_444_PartialSA_Osucc__refl,axiom,
! [A: state] : ( greater @ A @ A ) ).
% PartialSA.succ_refl
thf(fact_445_PartialSA_Oup__closed__def,axiom,
! [A3: set_state] :
( ( sep_up_closed_state @ plus @ A3 )
= ( ! [Phi3: state] :
( ? [X3: state] :
( ( member_state @ X3 @ A3 )
& ( greater @ Phi3 @ X3 ) )
=> ( member_state @ Phi3 @ A3 ) ) ) ) ).
% PartialSA.up_closed_def
thf(fact_446_PartialSA_Oup__closedI,axiom,
! [A3: set_state] :
( ! [Phi4: state,Phi5: state] :
( ( ( greater @ Phi4 @ Phi5 )
& ( member_state @ Phi5 @ A3 ) )
=> ( member_state @ Phi4 @ A3 ) )
=> ( sep_up_closed_state @ plus @ A3 ) ) ).
% PartialSA.up_closedI
thf(fact_447_sep__algebra_Oequiv__def,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_equiv_state @ Plus @ A3 @ B2 )
= ( ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 )
& ( sep_gr7105985528888466643_state @ Plus @ B2 @ A3 ) ) ) ) ).
% sep_algebra.equiv_def
thf(fact_448_sep__algebra_OequivI,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 )
=> ( ( sep_gr7105985528888466643_state @ Plus @ B2 @ A3 )
=> ( sep_equiv_state @ Plus @ A3 @ B2 ) ) ) ) ).
% sep_algebra.equivI
thf(fact_449_sep__algebra_Oup__close__equiv,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_up_closed_state @ Plus @ A3 )
=> ( ( sep_up_closed_state @ Plus @ B2 )
=> ( ( sep_equiv_state @ Plus @ A3 @ B2 )
= ( A3 = B2 ) ) ) ) ) ).
% sep_algebra.up_close_equiv
thf(fact_450_sep__algebra_Oasso1,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state,Ab: option_state,C: option_state,Bc: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( ( Plus @ A @ B )
= ( some_option_state @ Ab ) )
& ( ( Plus @ B @ C )
= ( some_option_state @ Bc ) ) )
=> ( ( Plus @ Ab @ C )
= ( Plus @ A @ Bc ) ) ) ) ).
% sep_algebra.asso1
thf(fact_451_sep__algebra_Oasso1,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,Ab: state,C: state,Bc: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( ( Plus @ A @ B )
= ( some_state @ Ab ) )
& ( ( Plus @ B @ C )
= ( some_state @ Bc ) ) )
=> ( ( Plus @ Ab @ C )
= ( Plus @ A @ Bc ) ) ) ) ).
% sep_algebra.asso1
thf(fact_452_sep__algebra_Ocore__max,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,X: option_state,C: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( some_option_state @ X )
= ( Plus @ X @ C ) )
=> ? [R: option_state] :
( ( some_option_state @ ( Core @ X ) )
= ( Plus @ C @ R ) ) ) ) ).
% sep_algebra.core_max
thf(fact_453_sep__algebra_Ocore__max,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state,C: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ X )
= ( Plus @ X @ C ) )
=> ? [R: state] :
( ( some_state @ ( Core @ X ) )
= ( Plus @ C @ R ) ) ) ) ).
% sep_algebra.core_max
thf(fact_454_sep__algebra_Ocore__sum,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,C: option_state,A: option_state,B: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( some_option_state @ C )
= ( Plus @ A @ B ) )
=> ( ( some_option_state @ ( Core @ C ) )
= ( Plus @ ( Core @ A ) @ ( Core @ B ) ) ) ) ) ).
% sep_algebra.core_sum
thf(fact_455_sep__algebra_Ocore__sum,axiom,
! [Plus: state > state > option_state,Core: state > state,C: state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ C )
= ( Plus @ A @ B ) )
=> ( ( some_state @ ( Core @ C ) )
= ( Plus @ ( Core @ A ) @ ( Core @ B ) ) ) ) ) ).
% sep_algebra.core_sum
thf(fact_456_sep__algebra_Opositivity,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state,C: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( Plus @ A @ B )
= ( some_option_state @ C ) )
=> ( ( ( some_option_state @ C )
= ( Plus @ C @ C ) )
=> ( ( some_option_state @ A )
= ( Plus @ A @ A ) ) ) ) ) ).
% sep_algebra.positivity
thf(fact_457_sep__algebra_Opositivity,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( Plus @ A @ B )
= ( some_state @ C ) )
=> ( ( ( some_state @ C )
= ( Plus @ C @ C ) )
=> ( ( some_state @ A )
= ( Plus @ A @ A ) ) ) ) ) ).
% sep_algebra.positivity
thf(fact_458_sep__algebra_Ocancellative,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state,X: option_state,Y3: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( some_option_state @ A )
= ( Plus @ B @ X ) )
=> ( ( ( some_option_state @ A )
= ( Plus @ B @ Y3 ) )
=> ( ( ( Core @ X )
= ( Core @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ).
% sep_algebra.cancellative
thf(fact_459_sep__algebra_Ocancellative,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,X: state,Y3: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ A )
= ( Plus @ B @ X ) )
=> ( ( ( some_state @ A )
= ( Plus @ B @ Y3 ) )
=> ( ( ( Core @ X )
= ( Core @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ).
% sep_algebra.cancellative
thf(fact_460_sep__algebra_Ocore__is__pure,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,X: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( some_option_state @ ( Core @ X ) )
= ( Plus @ ( Core @ X ) @ ( Core @ X ) ) ) ) ).
% sep_algebra.core_is_pure
thf(fact_461_sep__algebra_Ocore__is__pure,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( some_state @ ( Core @ X ) )
= ( Plus @ ( Core @ X ) @ ( Core @ X ) ) ) ) ).
% sep_algebra.core_is_pure
thf(fact_462_sep__algebra_Ocore__is__smaller,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,X: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( some_option_state @ X )
= ( Plus @ X @ ( Core @ X ) ) ) ) ).
% sep_algebra.core_is_smaller
thf(fact_463_sep__algebra_Ocore__is__smaller,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( some_state @ X )
= ( Plus @ X @ ( Core @ X ) ) ) ) ).
% sep_algebra.core_is_smaller
thf(fact_464_sep__algebra_Osplus__asso,axiom,
! [Plus: state > state > option_state,Core: state > state,A: option_state,B: option_state,C: option_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ A @ B ) @ C )
= ( sep_splus_state @ Plus @ A @ ( sep_splus_state @ Plus @ B @ C ) ) ) ) ).
% sep_algebra.splus_asso
thf(fact_465_sep__algebra_Osplus__comm,axiom,
! [Plus: state > state > option_state,Core: state > state,A: option_state,B: option_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_splus_state @ Plus @ A @ B )
= ( sep_splus_state @ Plus @ B @ A ) ) ) ).
% sep_algebra.splus_comm
thf(fact_466_sep__algebra_Osetify__def,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_setify_state @ P @ A3 )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ( P @ X3 ) ) ) ) ) ).
% sep_algebra.setify_def
thf(fact_467_PartialSA_Ogreater__set__def,axiom,
( greater_set
= ( ^ [A5: set_state,B5: set_state] :
! [X3: state] :
( ( member_state @ X3 @ A5 )
=> ? [Y4: state] :
( ( member_state @ Y4 @ B5 )
& ( greater @ X3 @ Y4 ) ) ) ) ) ).
% PartialSA.greater_set_def
thf(fact_468_PartialSA_Ogreater__setI,axiom,
! [A3: set_state,B2: set_state] :
( ! [A2: state] :
( ( member_state @ A2 @ A3 )
=> ? [X5: state] :
( ( member_state @ X5 @ B2 )
& ( greater @ A2 @ X5 ) ) )
=> ( greater_set @ A3 @ B2 ) ) ).
% PartialSA.greater_setI
thf(fact_469_sep__algebra_Omono__prop__set,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state,P: state > $o] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 )
=> ( ( sep_setify_state @ P @ B2 )
=> ( ( sep_mono_prop_state @ Plus @ P )
=> ( sep_setify_state @ P @ A3 ) ) ) ) ) ).
% sep_algebra.mono_prop_set
thf(fact_470_R__smaller,axiom,
! [W: state,A: state] : ( greater @ W @ ( r @ A @ W ) ) ).
% R_smaller
thf(fact_471_sep__algebra_Oasso2,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state,Ab: option_state,C: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( ( Plus @ A @ B )
= ( some_option_state @ Ab ) )
& ( ( Plus @ B @ C )
= none_option_state ) )
=> ( ( Plus @ Ab @ C )
= none_option_state ) ) ) ).
% sep_algebra.asso2
thf(fact_472_sep__algebra_Oasso2,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,Ab: state,C: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( ( Plus @ A @ B )
= ( some_state @ Ab ) )
& ( ( Plus @ B @ C )
= none_state ) )
=> ( ( Plus @ Ab @ C )
= none_state ) ) ) ).
% sep_algebra.asso2
thf(fact_473_sep__algebra_Oasso3,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state,C: option_state,Bc: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( Plus @ A @ B )
= none_option_state )
=> ( ( ( Plus @ B @ C )
= ( some_option_state @ Bc ) )
=> ( ( Plus @ A @ Bc )
= none_option_state ) ) ) ) ).
% sep_algebra.asso3
thf(fact_474_sep__algebra_Oasso3,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state,Bc: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( Plus @ A @ B )
= none_state )
=> ( ( ( Plus @ B @ C )
= ( some_state @ Bc ) )
=> ( ( Plus @ A @ Bc )
= none_state ) ) ) ) ).
% sep_algebra.asso3
thf(fact_475_sep__algebra_Ointro,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state] :
( ! [A2: option_state,B6: option_state] :
( ( Plus @ A2 @ B6 )
= ( Plus @ B6 @ A2 ) )
=> ( ! [A2: option_state,B6: option_state,Ab2: option_state,C4: option_state,Bc2: option_state] :
( ( ( ( Plus @ A2 @ B6 )
= ( some_option_state @ Ab2 ) )
& ( ( Plus @ B6 @ C4 )
= ( some_option_state @ Bc2 ) ) )
=> ( ( Plus @ Ab2 @ C4 )
= ( Plus @ A2 @ Bc2 ) ) )
=> ( ! [A2: option_state,B6: option_state,Ab2: option_state,C4: option_state] :
( ( ( ( Plus @ A2 @ B6 )
= ( some_option_state @ Ab2 ) )
& ( ( Plus @ B6 @ C4 )
= none_option_state ) )
=> ( ( Plus @ Ab2 @ C4 )
= none_option_state ) )
=> ( ! [X4: option_state] :
( ( some_option_state @ X4 )
= ( Plus @ X4 @ ( Core @ X4 ) ) )
=> ( ! [X4: option_state] :
( ( some_option_state @ ( Core @ X4 ) )
= ( Plus @ ( Core @ X4 ) @ ( Core @ X4 ) ) )
=> ( ! [X4: option_state,C4: option_state] :
( ( ( some_option_state @ X4 )
= ( Plus @ X4 @ C4 ) )
=> ? [R2: option_state] :
( ( some_option_state @ ( Core @ X4 ) )
= ( Plus @ C4 @ R2 ) ) )
=> ( ! [C4: option_state,A2: option_state,B6: option_state] :
( ( ( some_option_state @ C4 )
= ( Plus @ A2 @ B6 ) )
=> ( ( some_option_state @ ( Core @ C4 ) )
= ( Plus @ ( Core @ A2 ) @ ( Core @ B6 ) ) ) )
=> ( ! [A2: option_state,B6: option_state,C4: option_state] :
( ( ( Plus @ A2 @ B6 )
= ( some_option_state @ C4 ) )
=> ( ( ( some_option_state @ C4 )
= ( Plus @ C4 @ C4 ) )
=> ( ( some_option_state @ A2 )
= ( Plus @ A2 @ A2 ) ) ) )
=> ( ! [A2: option_state,B6: option_state,X4: option_state,Y: option_state] :
( ( ( some_option_state @ A2 )
= ( Plus @ B6 @ X4 ) )
=> ( ( ( some_option_state @ A2 )
= ( Plus @ B6 @ Y ) )
=> ( ( ( Core @ X4 )
= ( Core @ Y ) )
=> ( X4 = Y ) ) ) )
=> ( sep_al511465913568422763_state @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).
% sep_algebra.intro
thf(fact_476_sep__algebra_Ointro,axiom,
! [Plus: state > state > option_state,Core: state > state] :
( ! [A2: state,B6: state] :
( ( Plus @ A2 @ B6 )
= ( Plus @ B6 @ A2 ) )
=> ( ! [A2: state,B6: state,Ab2: state,C4: state,Bc2: state] :
( ( ( ( Plus @ A2 @ B6 )
= ( some_state @ Ab2 ) )
& ( ( Plus @ B6 @ C4 )
= ( some_state @ Bc2 ) ) )
=> ( ( Plus @ Ab2 @ C4 )
= ( Plus @ A2 @ Bc2 ) ) )
=> ( ! [A2: state,B6: state,Ab2: state,C4: state] :
( ( ( ( Plus @ A2 @ B6 )
= ( some_state @ Ab2 ) )
& ( ( Plus @ B6 @ C4 )
= none_state ) )
=> ( ( Plus @ Ab2 @ C4 )
= none_state ) )
=> ( ! [X4: state] :
( ( some_state @ X4 )
= ( Plus @ X4 @ ( Core @ X4 ) ) )
=> ( ! [X4: state] :
( ( some_state @ ( Core @ X4 ) )
= ( Plus @ ( Core @ X4 ) @ ( Core @ X4 ) ) )
=> ( ! [X4: state,C4: state] :
( ( ( some_state @ X4 )
= ( Plus @ X4 @ C4 ) )
=> ? [R2: state] :
( ( some_state @ ( Core @ X4 ) )
= ( Plus @ C4 @ R2 ) ) )
=> ( ! [C4: state,A2: state,B6: state] :
( ( ( some_state @ C4 )
= ( Plus @ A2 @ B6 ) )
=> ( ( some_state @ ( Core @ C4 ) )
= ( Plus @ ( Core @ A2 ) @ ( Core @ B6 ) ) ) )
=> ( ! [A2: state,B6: state,C4: state] :
( ( ( Plus @ A2 @ B6 )
= ( some_state @ C4 ) )
=> ( ( ( some_state @ C4 )
= ( Plus @ C4 @ C4 ) )
=> ( ( some_state @ A2 )
= ( Plus @ A2 @ A2 ) ) ) )
=> ( ! [A2: state,B6: state,X4: state,Y: state] :
( ( ( some_state @ A2 )
= ( Plus @ B6 @ X4 ) )
=> ( ( ( some_state @ A2 )
= ( Plus @ B6 @ Y ) )
=> ( ( ( Core @ X4 )
= ( Core @ Y ) )
=> ( X4 = Y ) ) ) )
=> ( sep_algebra_state @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).
% sep_algebra.intro
thf(fact_477_sep__algebra__def,axiom,
( sep_al511465913568422763_state
= ( ^ [Plus2: option_state > option_state > option_option_state,Core2: option_state > option_state] :
( ! [A4: option_state,B3: option_state] :
( ( Plus2 @ A4 @ B3 )
= ( Plus2 @ B3 @ A4 ) )
& ! [A4: option_state,B3: option_state,Ab3: option_state,C5: option_state,Bc3: option_state] :
( ( ( ( Plus2 @ A4 @ B3 )
= ( some_option_state @ Ab3 ) )
& ( ( Plus2 @ B3 @ C5 )
= ( some_option_state @ Bc3 ) ) )
=> ( ( Plus2 @ Ab3 @ C5 )
= ( Plus2 @ A4 @ Bc3 ) ) )
& ! [A4: option_state,B3: option_state,Ab3: option_state,C5: option_state] :
( ( ( ( Plus2 @ A4 @ B3 )
= ( some_option_state @ Ab3 ) )
& ( ( Plus2 @ B3 @ C5 )
= none_option_state ) )
=> ( ( Plus2 @ Ab3 @ C5 )
= none_option_state ) )
& ! [X3: option_state] :
( ( some_option_state @ X3 )
= ( Plus2 @ X3 @ ( Core2 @ X3 ) ) )
& ! [X3: option_state] :
( ( some_option_state @ ( Core2 @ X3 ) )
= ( Plus2 @ ( Core2 @ X3 ) @ ( Core2 @ X3 ) ) )
& ! [X3: option_state,C5: option_state] :
( ( ( some_option_state @ X3 )
= ( Plus2 @ X3 @ C5 ) )
=> ? [R3: option_state] :
( ( some_option_state @ ( Core2 @ X3 ) )
= ( Plus2 @ C5 @ R3 ) ) )
& ! [C5: option_state,A4: option_state,B3: option_state] :
( ( ( some_option_state @ C5 )
= ( Plus2 @ A4 @ B3 ) )
=> ( ( some_option_state @ ( Core2 @ C5 ) )
= ( Plus2 @ ( Core2 @ A4 ) @ ( Core2 @ B3 ) ) ) )
& ! [A4: option_state,B3: option_state,C5: option_state] :
( ( ( Plus2 @ A4 @ B3 )
= ( some_option_state @ C5 ) )
=> ( ( ( some_option_state @ C5 )
= ( Plus2 @ C5 @ C5 ) )
=> ( ( some_option_state @ A4 )
= ( Plus2 @ A4 @ A4 ) ) ) )
& ! [A4: option_state,B3: option_state,X3: option_state,Y4: option_state] :
( ( ( some_option_state @ A4 )
= ( Plus2 @ B3 @ X3 ) )
=> ( ( ( some_option_state @ A4 )
= ( Plus2 @ B3 @ Y4 ) )
=> ( ( ( Core2 @ X3 )
= ( Core2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ) ) ) ) ).
% sep_algebra_def
thf(fact_478_sep__algebra__def,axiom,
( sep_algebra_state
= ( ^ [Plus2: state > state > option_state,Core2: state > state] :
( ! [A4: state,B3: state] :
( ( Plus2 @ A4 @ B3 )
= ( Plus2 @ B3 @ A4 ) )
& ! [A4: state,B3: state,Ab3: state,C5: state,Bc3: state] :
( ( ( ( Plus2 @ A4 @ B3 )
= ( some_state @ Ab3 ) )
& ( ( Plus2 @ B3 @ C5 )
= ( some_state @ Bc3 ) ) )
=> ( ( Plus2 @ Ab3 @ C5 )
= ( Plus2 @ A4 @ Bc3 ) ) )
& ! [A4: state,B3: state,Ab3: state,C5: state] :
( ( ( ( Plus2 @ A4 @ B3 )
= ( some_state @ Ab3 ) )
& ( ( Plus2 @ B3 @ C5 )
= none_state ) )
=> ( ( Plus2 @ Ab3 @ C5 )
= none_state ) )
& ! [X3: state] :
( ( some_state @ X3 )
= ( Plus2 @ X3 @ ( Core2 @ X3 ) ) )
& ! [X3: state] :
( ( some_state @ ( Core2 @ X3 ) )
= ( Plus2 @ ( Core2 @ X3 ) @ ( Core2 @ X3 ) ) )
& ! [X3: state,C5: state] :
( ( ( some_state @ X3 )
= ( Plus2 @ X3 @ C5 ) )
=> ? [R3: state] :
( ( some_state @ ( Core2 @ X3 ) )
= ( Plus2 @ C5 @ R3 ) ) )
& ! [C5: state,A4: state,B3: state] :
( ( ( some_state @ C5 )
= ( Plus2 @ A4 @ B3 ) )
=> ( ( some_state @ ( Core2 @ C5 ) )
= ( Plus2 @ ( Core2 @ A4 ) @ ( Core2 @ B3 ) ) ) )
& ! [A4: state,B3: state,C5: state] :
( ( ( Plus2 @ A4 @ B3 )
= ( some_state @ C5 ) )
=> ( ( ( some_state @ C5 )
= ( Plus2 @ C5 @ C5 ) )
=> ( ( some_state @ A4 )
= ( Plus2 @ A4 @ A4 ) ) ) )
& ! [A4: state,B3: state,X3: state,Y4: state] :
( ( ( some_state @ A4 )
= ( Plus2 @ B3 @ X3 ) )
=> ( ( ( some_state @ A4 )
= ( Plus2 @ B3 @ Y4 ) )
=> ( ( ( Core2 @ X3 )
= ( Core2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ) ) ) ) ).
% sep_algebra_def
thf(fact_479_PartialSA_Obigger__sum,axiom,
! [Phi: state,A: state,B: state,Phi2: state] :
( ( ( some_state @ Phi )
= ( plus @ A @ B ) )
=> ( ( greater @ Phi2 @ Phi )
=> ? [B7: state] :
( ( greater @ B7 @ B )
& ( ( some_state @ Phi2 )
= ( plus @ A @ B7 ) ) ) ) ) ).
% PartialSA.bigger_sum
thf(fact_480_PartialSA_Ogreater__def,axiom,
( greater
= ( ^ [A4: state,B3: state] :
? [C5: state] :
( ( some_state @ A4 )
= ( plus @ B3 @ C5 ) ) ) ) ).
% PartialSA.greater_def
thf(fact_481_PartialSA_Ogreater__equiv,axiom,
( greater
= ( ^ [A4: state,B3: state] :
? [C5: state] :
( ( some_state @ A4 )
= ( plus @ C5 @ B3 ) ) ) ) ).
% PartialSA.greater_equiv
thf(fact_482_PartialSA_Oaddition__bigger,axiom,
! [A7: state,A: state,X7: state,B: state,X: state] :
( ( greater @ A7 @ A )
=> ( ( ( some_state @ X7 )
= ( plus @ A7 @ B ) )
=> ( ( ( some_state @ X )
= ( plus @ A @ B ) )
=> ( greater @ X7 @ X ) ) ) ) ).
% PartialSA.addition_bigger
thf(fact_483_PartialSA_Obigger__sum__smaller,axiom,
! [C: state,A: state,B: state,A7: state] :
( ( ( some_state @ C )
= ( plus @ A @ B ) )
=> ( ( greater @ A @ A7 )
=> ? [B7: state] :
( ( greater @ B7 @ B )
& ( ( some_state @ C )
= ( plus @ A7 @ B7 ) ) ) ) ) ).
% PartialSA.bigger_sum_smaller
thf(fact_484_sep__algebra_Osplus_Osimps_I3_J,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( sep_sp2496543662054378344_state @ Plus @ ( some_option_state @ A ) @ ( some_option_state @ B ) )
= ( Plus @ A @ B ) ) ) ).
% sep_algebra.splus.simps(3)
thf(fact_485_sep__algebra_Osplus_Osimps_I3_J,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_splus_state @ Plus @ ( some_state @ A ) @ ( some_state @ B ) )
= ( Plus @ A @ B ) ) ) ).
% sep_algebra.splus.simps(3)
thf(fact_486_sep__algebra_Osplus__develop,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state,C: option_state,D: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( some_option_state @ A )
= ( Plus @ B @ C ) )
=> ( ( Plus @ A @ D )
= ( sep_sp2496543662054378344_state @ Plus @ ( sep_sp2496543662054378344_state @ Plus @ ( some_option_state @ B ) @ ( some_option_state @ C ) ) @ ( some_option_state @ D ) ) ) ) ) ).
% sep_algebra.splus_develop
thf(fact_487_sep__algebra_Osplus__develop,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state,D: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ A )
= ( Plus @ B @ C ) )
=> ( ( Plus @ A @ D )
= ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ ( some_state @ B ) @ ( some_state @ C ) ) @ ( some_state @ D ) ) ) ) ) ).
% sep_algebra.splus_develop
thf(fact_488_sep__algebra_Osplus_Osimps_I1_J,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,Uu: option_option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( sep_sp2496543662054378344_state @ Plus @ none_option_state @ Uu )
= none_option_state ) ) ).
% sep_algebra.splus.simps(1)
thf(fact_489_sep__algebra_Osplus_Osimps_I1_J,axiom,
! [Plus: state > state > option_state,Core: state > state,Uu: option_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_splus_state @ Plus @ none_state @ Uu )
= none_state ) ) ).
% sep_algebra.splus.simps(1)
thf(fact_490_None__notin__image__Some,axiom,
! [A3: set_option_state] :
~ ( member1079230918592710257_state @ none_option_state @ ( image_4760606469461393131_state @ some_option_state @ A3 ) ) ).
% None_notin_image_Some
thf(fact_491_None__notin__image__Some,axiom,
! [A3: set_state] :
~ ( member_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ A3 ) ) ).
% None_notin_image_Some
thf(fact_492_compatible__options_Oelims_I3_J,axiom,
! [X: option_option_state,Xa: option_option_state] :
( ~ ( compat839418488507666022_state @ X @ Xa )
=> ~ ! [A2: option_state] :
( ( X
= ( some_option_state @ A2 ) )
=> ! [B6: option_state] :
( ( Xa
= ( some_option_state @ B6 ) )
=> ( A2 = B6 ) ) ) ) ).
% compatible_options.elims(3)
thf(fact_493_compatible__options_Oelims_I3_J,axiom,
! [X: option_state,Xa: option_state] :
( ~ ( compat2278460363914054422_state @ X @ Xa )
=> ~ ! [A2: state] :
( ( X
= ( some_state @ A2 ) )
=> ! [B6: state] :
( ( Xa
= ( some_state @ B6 ) )
=> ( A2 = B6 ) ) ) ) ).
% compatible_options.elims(3)
thf(fact_494_compatible__options_Osimps_I1_J,axiom,
! [A: option_state,B: option_state] :
( ( compat839418488507666022_state @ ( some_option_state @ A ) @ ( some_option_state @ B ) )
= ( A = B ) ) ).
% compatible_options.simps(1)
thf(fact_495_compatible__options_Osimps_I1_J,axiom,
! [A: state,B: state] :
( ( compat2278460363914054422_state @ ( some_state @ A ) @ ( some_state @ B ) )
= ( A = B ) ) ).
% compatible_options.simps(1)
thf(fact_496_compatible__options_Osimps_I2_J,axiom,
! [Uv: option_option_state] : ( compat839418488507666022_state @ none_option_state @ Uv ) ).
% compatible_options.simps(2)
thf(fact_497_compatible__options_Osimps_I2_J,axiom,
! [Uv: option_state] : ( compat2278460363914054422_state @ none_state @ Uv ) ).
% compatible_options.simps(2)
thf(fact_498_compatible__options_Osimps_I3_J,axiom,
! [Uu: option_option_state] : ( compat839418488507666022_state @ Uu @ none_option_state ) ).
% compatible_options.simps(3)
thf(fact_499_compatible__options_Osimps_I3_J,axiom,
! [Uu: option_state] : ( compat2278460363914054422_state @ Uu @ none_state ) ).
% compatible_options.simps(3)
thf(fact_500_PartialSA_Omono__prop__def,axiom,
! [P: state > $o] :
( ( sep_mono_prop_state @ plus @ P )
= ( ! [X3: state,Y4: state] :
( ( ( greater @ Y4 @ X3 )
& ( P @ X3 ) )
=> ( P @ Y4 ) ) ) ) ).
% PartialSA.mono_prop_def
thf(fact_501_PartialSA_Omono__propI,axiom,
! [P: state > $o] :
( ! [X4: state,Y: state] :
( ( ( greater @ Y @ X4 )
& ( P @ X4 ) )
=> ( P @ Y ) )
=> ( sep_mono_prop_state @ plus @ P ) ) ).
% PartialSA.mono_propI
thf(fact_502_sep__algebra_Osplus_Osimps_I2_J,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,V: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( sep_sp2496543662054378344_state @ Plus @ ( some_option_state @ V ) @ none_option_state )
= none_option_state ) ) ).
% sep_algebra.splus.simps(2)
thf(fact_503_sep__algebra_Osplus_Osimps_I2_J,axiom,
! [Plus: state > state > option_state,Core: state > state,V: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_splus_state @ Plus @ ( some_state @ V ) @ none_state )
= none_state ) ) ).
% sep_algebra.splus.simps(2)
thf(fact_504_sep__algebra_Osplus_Oelims,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,X: option_option_state,Xa: option_option_state,Y3: option_option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( sep_sp2496543662054378344_state @ Plus @ X @ Xa )
= Y3 )
=> ( ( ( X = none_option_state )
=> ( Y3 != none_option_state ) )
=> ( ( ? [V2: option_state] :
( X
= ( some_option_state @ V2 ) )
=> ( ( Xa = none_option_state )
=> ( Y3 != none_option_state ) ) )
=> ~ ! [A2: option_state] :
( ( X
= ( some_option_state @ A2 ) )
=> ! [B6: option_state] :
( ( Xa
= ( some_option_state @ B6 ) )
=> ( Y3
!= ( Plus @ A2 @ B6 ) ) ) ) ) ) ) ) ).
% sep_algebra.splus.elims
thf(fact_505_sep__algebra_Osplus_Oelims,axiom,
! [Plus: state > state > option_state,Core: state > state,X: option_state,Xa: option_state,Y3: option_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( sep_splus_state @ Plus @ X @ Xa )
= Y3 )
=> ( ( ( X = none_state )
=> ( Y3 != none_state ) )
=> ( ( ? [V2: state] :
( X
= ( some_state @ V2 ) )
=> ( ( Xa = none_state )
=> ( Y3 != none_state ) ) )
=> ~ ! [A2: state] :
( ( X
= ( some_state @ A2 ) )
=> ! [B6: state] :
( ( Xa
= ( some_state @ B6 ) )
=> ( Y3
!= ( Plus @ A2 @ B6 ) ) ) ) ) ) ) ) ).
% sep_algebra.splus.elims
thf(fact_506_PartialSA_Oup__closed__sum,axiom,
! [A3: set_state,B2: set_state] :
( ( sep_up_closed_state @ plus @ A3 )
=> ( sep_up_closed_state @ plus @ ( add_set @ A3 @ B2 ) ) ) ).
% PartialSA.up_closed_sum
thf(fact_507_Inf_OINF__cong,axiom,
! [A3: set_nat,B2: set_nat,C3: nat > state,D2: nat > state,Inf: set_state > state] :
( ( A3 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_nat_state @ C3 @ A3 ) )
= ( Inf @ ( image_nat_state @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_508_Inf_OINF__cong,axiom,
! [A3: set_state_o,B2: set_state_o,C3: ( state > $o ) > set_state,D2: ( state > $o ) > set_state,Inf: set_set_state > set_state] :
( ( A3 = B2 )
=> ( ! [X4: state > $o] :
( ( member_state_o @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_7376656169852520768_state @ C3 @ A3 ) )
= ( Inf @ ( image_7376656169852520768_state @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_509_Inf_OINF__cong,axiom,
! [A3: set_state,B2: set_state,C3: state > option_state,D2: state > option_state,Inf: set_option_state > option_state] :
( ( A3 = B2 )
=> ( ! [X4: state] :
( ( member_state @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_6076465424260689483_state @ C3 @ A3 ) )
= ( Inf @ ( image_6076465424260689483_state @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_510_Inf_OINF__cong,axiom,
! [A3: set_state,B2: set_state,C3: state > set_state,D2: state > set_state,Inf: set_set_state > set_state] :
( ( A3 = B2 )
=> ( ! [X4: state] :
( ( member_state @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_4774290769506072625_state @ C3 @ A3 ) )
= ( Inf @ ( image_4774290769506072625_state @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_511_Inf_OINF__cong,axiom,
! [A3: set_set_state,B2: set_set_state,C3: set_state > set_state,D2: set_state > set_state,Inf: set_set_state > set_state] :
( ( A3 = B2 )
=> ( ! [X4: set_state] :
( ( member_set_state @ X4 @ B2 )
=> ( ( C3 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_2476256681063834599_state @ C3 @ A3 ) )
= ( Inf @ ( image_2476256681063834599_state @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_512_intuitionistic__def,axiom,
( intuitionistic
= ( ^ [A5: set_state] :
! [A4: state,B3: state] :
( ( ( greater @ A4 @ B3 )
& ( member_state @ B3 @ A5 ) )
=> ( member_state @ A4 @ A5 ) ) ) ) ).
% intuitionistic_def
thf(fact_513_sep__algebra_Osetify__sum__image,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,P: option_state > $o,F: state > option_state,A3: set_state,B2: set_option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( sep_se2722818254322476245_state @ P @ ( sep_ad494385427648137084_state @ Plus @ ( image_6076465424260689483_state @ F @ A3 ) @ B2 ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ( sep_se2722818254322476245_state @ P @ ( sep_ad494385427648137084_state @ Plus @ ( insert_option_state @ ( F @ X3 ) @ bot_bo710180891245420500_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum_image
thf(fact_514_sep__algebra_Osetify__sum__image,axiom,
! [Plus: set_state > set_state > option_set_state,Core: set_state > set_state,P: set_state > $o,F: state > set_state,A3: set_state,B2: set_set_state] :
( ( sep_al3379035725709354321_state @ Plus @ Core )
=> ( ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ ( image_4774290769506072625_state @ F @ A3 ) @ B2 ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ ( insert_set_state @ ( F @ X3 ) @ bot_bo2271482359692755898_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum_image
thf(fact_515_sep__algebra_Osetify__sum__image,axiom,
! [Plus: set_state > set_state > option_set_state,Core: set_state > set_state,P: set_state > $o,F: set_state > set_state,A3: set_set_state,B2: set_set_state] :
( ( sep_al3379035725709354321_state @ Plus @ Core )
=> ( ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ ( image_2476256681063834599_state @ F @ A3 ) @ B2 ) )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ ( insert_set_state @ ( F @ X3 ) @ bot_bo2271482359692755898_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum_image
thf(fact_516_sep__algebra_Osetify__sum__image,axiom,
! [Plus: set_state > set_state > option_set_state,Core: set_state > set_state,P: set_state > $o,F: ( state > $o ) > set_state,A3: set_state_o,B2: set_set_state] :
( ( sep_al3379035725709354321_state @ Plus @ Core )
=> ( ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ ( image_7376656169852520768_state @ F @ A3 ) @ B2 ) )
= ( ! [X3: state > $o] :
( ( member_state_o @ X3 @ A3 )
=> ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ ( insert_set_state @ ( F @ X3 ) @ bot_bo2271482359692755898_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum_image
thf(fact_517_sep__algebra_Osetify__sum__image,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: nat > state,A3: set_nat,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( image_nat_state @ F @ A3 ) @ B2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( insert_state @ ( F @ X3 ) @ bot_bot_set_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum_image
thf(fact_518_sep__algebra_Oupper__closure__up__closed,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( sep_up_closed_state @ Plus @ ( sep_up1246176804924251236_state @ Plus @ A3 ) ) ) ).
% sep_algebra.upper_closure_up_closed
thf(fact_519_sep__algebra_Osetify__sum,axiom,
! [Plus: produc8023240190789890773_state > produc8023240190789890773_state > option2250103068101548571_state,Core: produc8023240190789890773_state > produc8023240190789890773_state,P: produc8023240190789890773_state > $o,A3: set_Pr1785066336555260981_state,B2: set_Pr1785066336555260981_state] :
( ( sep_al7026638520560055732_state @ Plus @ Core )
=> ( ( sep_se8263300668621121226_state @ P @ ( sep_ad3780630032383550947_state @ Plus @ A3 @ B2 ) )
= ( ! [X3: produc8023240190789890773_state] :
( ( member753036827967488894_state @ X3 @ A3 )
=> ( sep_se8263300668621121226_state @ P @ ( sep_ad3780630032383550947_state @ Plus @ ( insert7525286303735658661_state @ X3 @ bot_bo9041262728264437921_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum
thf(fact_520_sep__algebra_Osetify__sum,axiom,
! [Plus: produc3142500478612311029_state > produc3142500478612311029_state > option6833441738159790651_state,Core: produc3142500478612311029_state > produc3142500478612311029_state,P: produc3142500478612311029_state > $o,A3: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state] :
( ( sep_al3542390230161005012_state @ Plus @ Core )
=> ( ( sep_se7172211557308979434_state @ P @ ( sep_ad4751565379959694595_state @ Plus @ A3 @ B2 ) )
= ( ! [X3: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ X3 @ A3 )
=> ( sep_se7172211557308979434_state @ P @ ( sep_ad4751565379959694595_state @ Plus @ ( insert4171857611248116165_state @ X3 @ bot_bo1080640394036989633_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum
thf(fact_521_sep__algebra_Osetify__sum,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,P: option_state > $o,A3: set_option_state,B2: set_option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( sep_se2722818254322476245_state @ P @ ( sep_ad494385427648137084_state @ Plus @ A3 @ B2 ) )
= ( ! [X3: option_state] :
( ( member_option_state @ X3 @ A3 )
=> ( sep_se2722818254322476245_state @ P @ ( sep_ad494385427648137084_state @ Plus @ ( insert_option_state @ X3 @ bot_bo710180891245420500_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum
thf(fact_522_sep__algebra_Osetify__sum,axiom,
! [Plus: set_state > set_state > option_set_state,Core: set_state > set_state,P: set_state > $o,A3: set_set_state,B2: set_set_state] :
( ( sep_al3379035725709354321_state @ Plus @ Core )
=> ( ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ A3 @ B2 ) )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( sep_setify_set_state @ P @ ( sep_ad7561664654817782498_state @ Plus @ ( insert_set_state @ X3 @ bot_bo2271482359692755898_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum
thf(fact_523_sep__algebra_Osetify__sum,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ A3 @ B2 ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( insert_state @ X3 @ bot_bot_set_state ) @ B2 ) ) ) ) ) ) ).
% sep_algebra.setify_sum
thf(fact_524_sep__algebra_Osum__then__singleton,axiom,
! [Plus: produc8023240190789890773_state > produc8023240190789890773_state > option2250103068101548571_state,Core: produc8023240190789890773_state > produc8023240190789890773_state,A: produc8023240190789890773_state,B: produc8023240190789890773_state,C: produc8023240190789890773_state] :
( ( sep_al7026638520560055732_state @ Plus @ Core )
=> ( ( ( some_P8120450764674687802_state @ A )
= ( Plus @ B @ C ) )
= ( ( insert7525286303735658661_state @ A @ bot_bo9041262728264437921_state )
= ( sep_ad3780630032383550947_state @ Plus @ ( insert7525286303735658661_state @ B @ bot_bo9041262728264437921_state ) @ ( insert7525286303735658661_state @ C @ bot_bo9041262728264437921_state ) ) ) ) ) ).
% sep_algebra.sum_then_singleton
thf(fact_525_sep__algebra_Osum__then__singleton,axiom,
! [Plus: produc3142500478612311029_state > produc3142500478612311029_state > option6833441738159790651_state,Core: produc3142500478612311029_state > produc3142500478612311029_state,A: produc3142500478612311029_state,B: produc3142500478612311029_state,C: produc3142500478612311029_state] :
( ( sep_al3542390230161005012_state @ Plus @ Core )
=> ( ( ( some_P3186401494017672538_state @ A )
= ( Plus @ B @ C ) )
= ( ( insert4171857611248116165_state @ A @ bot_bo1080640394036989633_state )
= ( sep_ad4751565379959694595_state @ Plus @ ( insert4171857611248116165_state @ B @ bot_bo1080640394036989633_state ) @ ( insert4171857611248116165_state @ C @ bot_bo1080640394036989633_state ) ) ) ) ) ).
% sep_algebra.sum_then_singleton
thf(fact_526_sep__algebra_Osum__then__singleton,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,A: option_state,B: option_state,C: option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( ( some_option_state @ A )
= ( Plus @ B @ C ) )
= ( ( insert_option_state @ A @ bot_bo710180891245420500_state )
= ( sep_ad494385427648137084_state @ Plus @ ( insert_option_state @ B @ bot_bo710180891245420500_state ) @ ( insert_option_state @ C @ bot_bo710180891245420500_state ) ) ) ) ) ).
% sep_algebra.sum_then_singleton
thf(fact_527_sep__algebra_Osum__then__singleton,axiom,
! [Plus: set_state > set_state > option_set_state,Core: set_state > set_state,A: set_state,B: set_state,C: set_state] :
( ( sep_al3379035725709354321_state @ Plus @ Core )
=> ( ( ( some_set_state @ A )
= ( Plus @ B @ C ) )
= ( ( insert_set_state @ A @ bot_bo2271482359692755898_state )
= ( sep_ad7561664654817782498_state @ Plus @ ( insert_set_state @ B @ bot_bo2271482359692755898_state ) @ ( insert_set_state @ C @ bot_bo2271482359692755898_state ) ) ) ) ) ).
% sep_algebra.sum_then_singleton
thf(fact_528_sep__algebra_Osum__then__singleton,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ A )
= ( Plus @ B @ C ) )
= ( ( insert_state @ A @ bot_bot_set_state )
= ( sep_add_set_state @ Plus @ ( insert_state @ B @ bot_bot_set_state ) @ ( insert_state @ C @ bot_bot_set_state ) ) ) ) ) ).
% sep_algebra.sum_then_singleton
thf(fact_529_PartialSA_Oupper__closure__up__closed,axiom,
! [A3: set_state] : ( sep_up_closed_state @ plus @ ( sep_up1246176804924251236_state @ plus @ A3 ) ) ).
% PartialSA.upper_closure_up_closed
thf(fact_530_sep__algebra_Ox__elem__set__product__splus,axiom,
! [Plus: set_state > set_state > option_set_state,Core: set_state > set_state,X: set_state,A3: set_set_state,B2: set_set_state] :
( ( sep_al3379035725709354321_state @ Plus @ Core )
=> ( ( member_set_state @ X @ ( sep_ad7561664654817782498_state @ Plus @ A3 @ B2 ) )
= ( ? [A4: set_state,B3: set_state] :
( ( member_set_state @ A4 @ A3 )
& ( member_set_state @ B3 @ B2 )
& ( ( some_set_state @ X )
= ( sep_splus_set_state @ Plus @ ( some_set_state @ A4 ) @ ( some_set_state @ B3 ) ) ) ) ) ) ) ).
% sep_algebra.x_elem_set_product_splus
thf(fact_531_sep__algebra_Ox__elem__set__product__splus,axiom,
! [Plus: produc3142500478612311029_state > produc3142500478612311029_state > option6833441738159790651_state,Core: produc3142500478612311029_state > produc3142500478612311029_state,X: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state] :
( ( sep_al3542390230161005012_state @ Plus @ Core )
=> ( ( member3029510603097127326_state @ X @ ( sep_ad4751565379959694595_state @ Plus @ A3 @ B2 ) )
= ( ? [A4: produc3142500478612311029_state,B3: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ A4 @ A3 )
& ( member3029510603097127326_state @ B3 @ B2 )
& ( ( some_P3186401494017672538_state @ X )
= ( sep_sp2357042210444369943_state @ Plus @ ( some_P3186401494017672538_state @ A4 ) @ ( some_P3186401494017672538_state @ B3 ) ) ) ) ) ) ) ).
% sep_algebra.x_elem_set_product_splus
thf(fact_532_sep__algebra_Ox__elem__set__product__splus,axiom,
! [Plus: produc8023240190789890773_state > produc8023240190789890773_state > option2250103068101548571_state,Core: produc8023240190789890773_state > produc8023240190789890773_state,X: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state,B2: set_Pr1785066336555260981_state] :
( ( sep_al7026638520560055732_state @ Plus @ Core )
=> ( ( member753036827967488894_state @ X @ ( sep_ad3780630032383550947_state @ Plus @ A3 @ B2 ) )
= ( ? [A4: produc8023240190789890773_state,B3: produc8023240190789890773_state] :
( ( member753036827967488894_state @ A4 @ A3 )
& ( member753036827967488894_state @ B3 @ B2 )
& ( ( some_P8120450764674687802_state @ X )
= ( sep_sp5678321299423575287_state @ Plus @ ( some_P8120450764674687802_state @ A4 ) @ ( some_P8120450764674687802_state @ B3 ) ) ) ) ) ) ) ).
% sep_algebra.x_elem_set_product_splus
thf(fact_533_sep__algebra_Ox__elem__set__product__splus,axiom,
! [Plus: option_state > option_state > option_option_state,Core: option_state > option_state,X: option_state,A3: set_option_state,B2: set_option_state] :
( ( sep_al511465913568422763_state @ Plus @ Core )
=> ( ( member_option_state @ X @ ( sep_ad494385427648137084_state @ Plus @ A3 @ B2 ) )
= ( ? [A4: option_state,B3: option_state] :
( ( member_option_state @ A4 @ A3 )
& ( member_option_state @ B3 @ B2 )
& ( ( some_option_state @ X )
= ( sep_sp2496543662054378344_state @ Plus @ ( some_option_state @ A4 ) @ ( some_option_state @ B3 ) ) ) ) ) ) ) ).
% sep_algebra.x_elem_set_product_splus
thf(fact_534_sep__algebra_Ox__elem__set__product__splus,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( member_state @ X @ ( sep_add_set_state @ Plus @ A3 @ B2 ) )
= ( ? [A4: state,B3: state] :
( ( member_state @ A4 @ A3 )
& ( member_state @ B3 @ B2 )
& ( ( some_state @ X )
= ( sep_splus_state @ Plus @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) ) ) ) ) ) ).
% sep_algebra.x_elem_set_product_splus
thf(fact_535_PartialSA_Oequiv__up__closed__subset,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( sep_up_closed_state @ plus @ A3 )
=> ( ( sep_equiv_state @ plus @ B2 @ C3 )
=> ( ( ord_le2494988322063910608_state @ B2 @ A3 )
= ( ord_le2494988322063910608_state @ C3 @ A3 ) ) ) ) ).
% PartialSA.equiv_up_closed_subset
thf(fact_536_PartialSA_Oup__closed__plus__UNIV,axiom,
! [A3: set_state] : ( sep_up_closed_state @ plus @ ( add_set @ A3 @ top_top_set_state ) ) ).
% PartialSA.up_closed_plus_UNIV
thf(fact_537_order__refl,axiom,
! [X: set_set_state] : ( ord_le5175021213330142598_state @ X @ X ) ).
% order_refl
thf(fact_538_order__refl,axiom,
! [X: set_Pr1688445902015331925_state] : ( ord_le6423325748750870005_state @ X @ X ) ).
% order_refl
thf(fact_539_order__refl,axiom,
! [X: set_Pr1785066336555260981_state] : ( ord_le2777189432094499797_state @ X @ X ) ).
% order_refl
thf(fact_540_order__refl,axiom,
! [X: set_option_state] : ( ord_le7116032884704190368_state @ X @ X ) ).
% order_refl
thf(fact_541_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_542_order__refl,axiom,
! [X: set_state] : ( ord_le2494988322063910608_state @ X @ X ) ).
% order_refl
thf(fact_543_dual__order_Orefl,axiom,
! [A: set_set_state] : ( ord_le5175021213330142598_state @ A @ A ) ).
% dual_order.refl
thf(fact_544_dual__order_Orefl,axiom,
! [A: set_Pr1688445902015331925_state] : ( ord_le6423325748750870005_state @ A @ A ) ).
% dual_order.refl
thf(fact_545_dual__order_Orefl,axiom,
! [A: set_Pr1785066336555260981_state] : ( ord_le2777189432094499797_state @ A @ A ) ).
% dual_order.refl
thf(fact_546_dual__order_Orefl,axiom,
! [A: set_option_state] : ( ord_le7116032884704190368_state @ A @ A ) ).
% dual_order.refl
thf(fact_547_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_548_dual__order_Orefl,axiom,
! [A: set_state] : ( ord_le2494988322063910608_state @ A @ A ) ).
% dual_order.refl
thf(fact_549_top__apply,axiom,
( top_top_state_o
= ( ^ [X3: state] : top_top_o ) ) ).
% top_apply
thf(fact_550_UNIV__I,axiom,
! [X: produc3142500478612311029_state] : ( member3029510603097127326_state @ X @ top_to5248747511432852389_state ) ).
% UNIV_I
thf(fact_551_UNIV__I,axiom,
! [X: produc8023240190789890773_state] : ( member753036827967488894_state @ X @ top_to4183510833390158213_state ) ).
% UNIV_I
thf(fact_552_UNIV__I,axiom,
! [X: option_state] : ( member_option_state @ X @ top_to7666338855062656496_state ) ).
% UNIV_I
thf(fact_553_UNIV__I,axiom,
! [X: set_state] : ( member_set_state @ X @ top_to5262587396890829782_state ) ).
% UNIV_I
thf(fact_554_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_555_UNIV__I,axiom,
! [X: state] : ( member_state @ X @ top_top_set_state ) ).
% UNIV_I
thf(fact_556_subsetI,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ! [X4: set_state] :
( ( member_set_state @ X4 @ A3 )
=> ( member_set_state @ X4 @ B2 ) )
=> ( ord_le5175021213330142598_state @ A3 @ B2 ) ) ).
% subsetI
thf(fact_557_subsetI,axiom,
! [A3: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state] :
( ! [X4: produc3142500478612311029_state] :
( ( member3029510603097127326_state @ X4 @ A3 )
=> ( member3029510603097127326_state @ X4 @ B2 ) )
=> ( ord_le6423325748750870005_state @ A3 @ B2 ) ) ).
% subsetI
thf(fact_558_subsetI,axiom,
! [A3: set_Pr1785066336555260981_state,B2: set_Pr1785066336555260981_state] :
( ! [X4: produc8023240190789890773_state] :
( ( member753036827967488894_state @ X4 @ A3 )
=> ( member753036827967488894_state @ X4 @ B2 ) )
=> ( ord_le2777189432094499797_state @ A3 @ B2 ) ) ).
% subsetI
thf(fact_559_subsetI,axiom,
! [A3: set_option_state,B2: set_option_state] :
( ! [X4: option_state] :
( ( member_option_state @ X4 @ A3 )
=> ( member_option_state @ X4 @ B2 ) )
=> ( ord_le7116032884704190368_state @ A3 @ B2 ) ) ).
% subsetI
thf(fact_560_subsetI,axiom,
! [A3: set_state,B2: set_state] :
( ! [X4: state] :
( ( member_state @ X4 @ A3 )
=> ( member_state @ X4 @ B2 ) )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).
% subsetI
thf(fact_561_empty__subsetI,axiom,
! [A3: set_set_state] : ( ord_le5175021213330142598_state @ bot_bo2271482359692755898_state @ A3 ) ).
% empty_subsetI
thf(fact_562_empty__subsetI,axiom,
! [A3: set_Pr1688445902015331925_state] : ( ord_le6423325748750870005_state @ bot_bo1080640394036989633_state @ A3 ) ).
% empty_subsetI
thf(fact_563_empty__subsetI,axiom,
! [A3: set_Pr1785066336555260981_state] : ( ord_le2777189432094499797_state @ bot_bo9041262728264437921_state @ A3 ) ).
% empty_subsetI
thf(fact_564_empty__subsetI,axiom,
! [A3: set_option_state] : ( ord_le7116032884704190368_state @ bot_bo710180891245420500_state @ A3 ) ).
% empty_subsetI
thf(fact_565_empty__subsetI,axiom,
! [A3: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A3 ) ).
% empty_subsetI
thf(fact_566_subset__empty,axiom,
! [A3: set_set_state] :
( ( ord_le5175021213330142598_state @ A3 @ bot_bo2271482359692755898_state )
= ( A3 = bot_bo2271482359692755898_state ) ) ).
% subset_empty
thf(fact_567_subset__empty,axiom,
! [A3: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ A3 @ bot_bo1080640394036989633_state )
= ( A3 = bot_bo1080640394036989633_state ) ) ).
% subset_empty
thf(fact_568_subset__empty,axiom,
! [A3: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ A3 @ bot_bo9041262728264437921_state )
= ( A3 = bot_bo9041262728264437921_state ) ) ).
% subset_empty
thf(fact_569_subset__empty,axiom,
! [A3: set_option_state] :
( ( ord_le7116032884704190368_state @ A3 @ bot_bo710180891245420500_state )
= ( A3 = bot_bo710180891245420500_state ) ) ).
% subset_empty
thf(fact_570_subset__empty,axiom,
! [A3: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ bot_bot_set_state )
= ( A3 = bot_bot_set_state ) ) ).
% subset_empty
thf(fact_571_insert__subset,axiom,
! [X: set_state,A3: set_set_state,B2: set_set_state] :
( ( ord_le5175021213330142598_state @ ( insert_set_state @ X @ A3 ) @ B2 )
= ( ( member_set_state @ X @ B2 )
& ( ord_le5175021213330142598_state @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_572_insert__subset,axiom,
! [X: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ ( insert4171857611248116165_state @ X @ A3 ) @ B2 )
= ( ( member3029510603097127326_state @ X @ B2 )
& ( ord_le6423325748750870005_state @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_573_insert__subset,axiom,
! [X: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state,B2: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ ( insert7525286303735658661_state @ X @ A3 ) @ B2 )
= ( ( member753036827967488894_state @ X @ B2 )
& ( ord_le2777189432094499797_state @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_574_insert__subset,axiom,
! [X: option_state,A3: set_option_state,B2: set_option_state] :
( ( ord_le7116032884704190368_state @ ( insert_option_state @ X @ A3 ) @ B2 )
= ( ( member_option_state @ X @ B2 )
& ( ord_le7116032884704190368_state @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_575_insert__subset,axiom,
! [X: state,A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ ( insert_state @ X @ A3 ) @ B2 )
= ( ( member_state @ X @ B2 )
& ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_576_singleton__insert__inj__eq_H,axiom,
! [A: set_state,A3: set_set_state,B: set_state] :
( ( ( insert_set_state @ A @ A3 )
= ( insert_set_state @ B @ bot_bo2271482359692755898_state ) )
= ( ( A = B )
& ( ord_le5175021213330142598_state @ A3 @ ( insert_set_state @ B @ bot_bo2271482359692755898_state ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_577_singleton__insert__inj__eq_H,axiom,
! [A: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state,B: produc3142500478612311029_state] :
( ( ( insert4171857611248116165_state @ A @ A3 )
= ( insert4171857611248116165_state @ B @ bot_bo1080640394036989633_state ) )
= ( ( A = B )
& ( ord_le6423325748750870005_state @ A3 @ ( insert4171857611248116165_state @ B @ bot_bo1080640394036989633_state ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_578_singleton__insert__inj__eq_H,axiom,
! [A: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state,B: produc8023240190789890773_state] :
( ( ( insert7525286303735658661_state @ A @ A3 )
= ( insert7525286303735658661_state @ B @ bot_bo9041262728264437921_state ) )
= ( ( A = B )
& ( ord_le2777189432094499797_state @ A3 @ ( insert7525286303735658661_state @ B @ bot_bo9041262728264437921_state ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_579_singleton__insert__inj__eq_H,axiom,
! [A: option_state,A3: set_option_state,B: option_state] :
( ( ( insert_option_state @ A @ A3 )
= ( insert_option_state @ B @ bot_bo710180891245420500_state ) )
= ( ( A = B )
& ( ord_le7116032884704190368_state @ A3 @ ( insert_option_state @ B @ bot_bo710180891245420500_state ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_580_singleton__insert__inj__eq_H,axiom,
! [A: state,A3: set_state,B: state] :
( ( ( insert_state @ A @ A3 )
= ( insert_state @ B @ bot_bot_set_state ) )
= ( ( A = B )
& ( ord_le2494988322063910608_state @ A3 @ ( insert_state @ B @ bot_bot_set_state ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_581_singleton__insert__inj__eq,axiom,
! [B: set_state,A: set_state,A3: set_set_state] :
( ( ( insert_set_state @ B @ bot_bo2271482359692755898_state )
= ( insert_set_state @ A @ A3 ) )
= ( ( A = B )
& ( ord_le5175021213330142598_state @ A3 @ ( insert_set_state @ B @ bot_bo2271482359692755898_state ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_582_singleton__insert__inj__eq,axiom,
! [B: produc3142500478612311029_state,A: produc3142500478612311029_state,A3: set_Pr1688445902015331925_state] :
( ( ( insert4171857611248116165_state @ B @ bot_bo1080640394036989633_state )
= ( insert4171857611248116165_state @ A @ A3 ) )
= ( ( A = B )
& ( ord_le6423325748750870005_state @ A3 @ ( insert4171857611248116165_state @ B @ bot_bo1080640394036989633_state ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_583_singleton__insert__inj__eq,axiom,
! [B: produc8023240190789890773_state,A: produc8023240190789890773_state,A3: set_Pr1785066336555260981_state] :
( ( ( insert7525286303735658661_state @ B @ bot_bo9041262728264437921_state )
= ( insert7525286303735658661_state @ A @ A3 ) )
= ( ( A = B )
& ( ord_le2777189432094499797_state @ A3 @ ( insert7525286303735658661_state @ B @ bot_bo9041262728264437921_state ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_584_singleton__insert__inj__eq,axiom,
! [B: option_state,A: option_state,A3: set_option_state] :
( ( ( insert_option_state @ B @ bot_bo710180891245420500_state )
= ( insert_option_state @ A @ A3 ) )
= ( ( A = B )
& ( ord_le7116032884704190368_state @ A3 @ ( insert_option_state @ B @ bot_bo710180891245420500_state ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_585_singleton__insert__inj__eq,axiom,
! [B: state,A: state,A3: set_state] :
( ( ( insert_state @ B @ bot_bot_set_state )
= ( insert_state @ A @ A3 ) )
= ( ( A = B )
& ( ord_le2494988322063910608_state @ A3 @ ( insert_state @ B @ bot_bot_set_state ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_586_range__subsetD,axiom,
! [F: state > state,B2: set_state,I: state] :
( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ top_top_set_state ) @ B2 )
=> ( member_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_587_range__subsetD,axiom,
! [F: nat > state,B2: set_state,I: nat] :
( ( ord_le2494988322063910608_state @ ( image_nat_state @ F @ top_top_set_nat ) @ B2 )
=> ( member_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_588_range__subsetD,axiom,
! [F: option_state > state,B2: set_state,I: option_state] :
( ( ord_le2494988322063910608_state @ ( image_3532137647693456075_state @ F @ top_to7666338855062656496_state ) @ B2 )
=> ( member_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_589_range__subsetD,axiom,
! [F: set_state > state,B2: set_state,I: set_state] :
( ( ord_le2494988322063910608_state @ ( image_4575879259649255985_state @ F @ top_to5262587396890829782_state ) @ B2 )
=> ( member_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_590_range__subsetD,axiom,
! [F: state > set_state,B2: set_set_state,I: state] :
( ( ord_le5175021213330142598_state @ ( image_4774290769506072625_state @ F @ top_top_set_state ) @ B2 )
=> ( member_set_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_591_range__subsetD,axiom,
! [F: nat > set_state,B2: set_set_state,I: nat] :
( ( ord_le5175021213330142598_state @ ( image_nat_set_state @ F @ top_top_set_nat ) @ B2 )
=> ( member_set_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_592_range__subsetD,axiom,
! [F: state > option_state,B2: set_option_state,I: state] :
( ( ord_le7116032884704190368_state @ ( image_6076465424260689483_state @ F @ top_top_set_state ) @ B2 )
=> ( member_option_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_593_range__subsetD,axiom,
! [F: nat > option_state,B2: set_option_state,I: nat] :
( ( ord_le7116032884704190368_state @ ( image_8240249968597053537_state @ F @ top_top_set_nat ) @ B2 )
=> ( member_option_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_594_range__subsetD,axiom,
! [F: option_state > set_state,B2: set_set_state,I: option_state] :
( ( ord_le5175021213330142598_state @ ( image_3284240608559150209_state @ F @ top_to7666338855062656496_state ) @ B2 )
=> ( member_set_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_595_range__subsetD,axiom,
! [F: set_state > set_state,B2: set_set_state,I: set_state] :
( ( ord_le5175021213330142598_state @ ( image_2476256681063834599_state @ F @ top_to5262587396890829782_state ) @ B2 )
=> ( member_set_state @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_596_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_597_le__cases3,axiom,
! [X: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_598_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_state,Z2: set_set_state] : ( Y5 = Z2 ) )
= ( ^ [X3: set_set_state,Y4: set_set_state] :
( ( ord_le5175021213330142598_state @ X3 @ Y4 )
& ( ord_le5175021213330142598_state @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_599_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr1688445902015331925_state,Z2: set_Pr1688445902015331925_state] : ( Y5 = Z2 ) )
= ( ^ [X3: set_Pr1688445902015331925_state,Y4: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ X3 @ Y4 )
& ( ord_le6423325748750870005_state @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_600_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr1785066336555260981_state,Z2: set_Pr1785066336555260981_state] : ( Y5 = Z2 ) )
= ( ^ [X3: set_Pr1785066336555260981_state,Y4: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ X3 @ Y4 )
& ( ord_le2777189432094499797_state @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_601_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_option_state,Z2: set_option_state] : ( Y5 = Z2 ) )
= ( ^ [X3: set_option_state,Y4: set_option_state] :
( ( ord_le7116032884704190368_state @ X3 @ Y4 )
& ( ord_le7116032884704190368_state @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_602_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_603_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
= ( ^ [X3: set_state,Y4: set_state] :
( ( ord_le2494988322063910608_state @ X3 @ Y4 )
& ( ord_le2494988322063910608_state @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_604_ord__eq__le__trans,axiom,
! [A: set_set_state,B: set_set_state,C: set_set_state] :
( ( A = B )
=> ( ( ord_le5175021213330142598_state @ B @ C )
=> ( ord_le5175021213330142598_state @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_605_ord__eq__le__trans,axiom,
! [A: set_Pr1688445902015331925_state,B: set_Pr1688445902015331925_state,C: set_Pr1688445902015331925_state] :
( ( A = B )
=> ( ( ord_le6423325748750870005_state @ B @ C )
=> ( ord_le6423325748750870005_state @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_606_ord__eq__le__trans,axiom,
! [A: set_Pr1785066336555260981_state,B: set_Pr1785066336555260981_state,C: set_Pr1785066336555260981_state] :
( ( A = B )
=> ( ( ord_le2777189432094499797_state @ B @ C )
=> ( ord_le2777189432094499797_state @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_607_ord__eq__le__trans,axiom,
! [A: set_option_state,B: set_option_state,C: set_option_state] :
( ( A = B )
=> ( ( ord_le7116032884704190368_state @ B @ C )
=> ( ord_le7116032884704190368_state @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_608_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_609_ord__eq__le__trans,axiom,
! [A: set_state,B: set_state,C: set_state] :
( ( A = B )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ord_le2494988322063910608_state @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_610_ord__le__eq__trans,axiom,
! [A: set_set_state,B: set_set_state,C: set_set_state] :
( ( ord_le5175021213330142598_state @ A @ B )
=> ( ( B = C )
=> ( ord_le5175021213330142598_state @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_611_ord__le__eq__trans,axiom,
! [A: set_Pr1688445902015331925_state,B: set_Pr1688445902015331925_state,C: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ A @ B )
=> ( ( B = C )
=> ( ord_le6423325748750870005_state @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_612_ord__le__eq__trans,axiom,
! [A: set_Pr1785066336555260981_state,B: set_Pr1785066336555260981_state,C: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ A @ B )
=> ( ( B = C )
=> ( ord_le2777189432094499797_state @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_613_ord__le__eq__trans,axiom,
! [A: set_option_state,B: set_option_state,C: set_option_state] :
( ( ord_le7116032884704190368_state @ A @ B )
=> ( ( B = C )
=> ( ord_le7116032884704190368_state @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_614_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_615_ord__le__eq__trans,axiom,
! [A: set_state,B: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( B = C )
=> ( ord_le2494988322063910608_state @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_616_order__antisym,axiom,
! [X: set_set_state,Y3: set_set_state] :
( ( ord_le5175021213330142598_state @ X @ Y3 )
=> ( ( ord_le5175021213330142598_state @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_617_order__antisym,axiom,
! [X: set_Pr1688445902015331925_state,Y3: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ X @ Y3 )
=> ( ( ord_le6423325748750870005_state @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_618_order__antisym,axiom,
! [X: set_Pr1785066336555260981_state,Y3: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ X @ Y3 )
=> ( ( ord_le2777189432094499797_state @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_619_order__antisym,axiom,
! [X: set_option_state,Y3: set_option_state] :
( ( ord_le7116032884704190368_state @ X @ Y3 )
=> ( ( ord_le7116032884704190368_state @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_620_order__antisym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_621_order__antisym,axiom,
! [X: set_state,Y3: set_state] :
( ( ord_le2494988322063910608_state @ X @ Y3 )
=> ( ( ord_le2494988322063910608_state @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_622_order_Otrans,axiom,
! [A: set_set_state,B: set_set_state,C: set_set_state] :
( ( ord_le5175021213330142598_state @ A @ B )
=> ( ( ord_le5175021213330142598_state @ B @ C )
=> ( ord_le5175021213330142598_state @ A @ C ) ) ) ).
% order.trans
thf(fact_623_order_Otrans,axiom,
! [A: set_Pr1688445902015331925_state,B: set_Pr1688445902015331925_state,C: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ A @ B )
=> ( ( ord_le6423325748750870005_state @ B @ C )
=> ( ord_le6423325748750870005_state @ A @ C ) ) ) ).
% order.trans
thf(fact_624_order_Otrans,axiom,
! [A: set_Pr1785066336555260981_state,B: set_Pr1785066336555260981_state,C: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ A @ B )
=> ( ( ord_le2777189432094499797_state @ B @ C )
=> ( ord_le2777189432094499797_state @ A @ C ) ) ) ).
% order.trans
thf(fact_625_order_Otrans,axiom,
! [A: set_option_state,B: set_option_state,C: set_option_state] :
( ( ord_le7116032884704190368_state @ A @ B )
=> ( ( ord_le7116032884704190368_state @ B @ C )
=> ( ord_le7116032884704190368_state @ A @ C ) ) ) ).
% order.trans
thf(fact_626_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_627_order_Otrans,axiom,
! [A: set_state,B: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ord_le2494988322063910608_state @ A @ C ) ) ) ).
% order.trans
thf(fact_628_order__trans,axiom,
! [X: set_set_state,Y3: set_set_state,Z: set_set_state] :
( ( ord_le5175021213330142598_state @ X @ Y3 )
=> ( ( ord_le5175021213330142598_state @ Y3 @ Z )
=> ( ord_le5175021213330142598_state @ X @ Z ) ) ) ).
% order_trans
thf(fact_629_order__trans,axiom,
! [X: set_Pr1688445902015331925_state,Y3: set_Pr1688445902015331925_state,Z: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ X @ Y3 )
=> ( ( ord_le6423325748750870005_state @ Y3 @ Z )
=> ( ord_le6423325748750870005_state @ X @ Z ) ) ) ).
% order_trans
thf(fact_630_order__trans,axiom,
! [X: set_Pr1785066336555260981_state,Y3: set_Pr1785066336555260981_state,Z: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ X @ Y3 )
=> ( ( ord_le2777189432094499797_state @ Y3 @ Z )
=> ( ord_le2777189432094499797_state @ X @ Z ) ) ) ).
% order_trans
thf(fact_631_order__trans,axiom,
! [X: set_option_state,Y3: set_option_state,Z: set_option_state] :
( ( ord_le7116032884704190368_state @ X @ Y3 )
=> ( ( ord_le7116032884704190368_state @ Y3 @ Z )
=> ( ord_le7116032884704190368_state @ X @ Z ) ) ) ).
% order_trans
thf(fact_632_order__trans,axiom,
! [X: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_633_order__trans,axiom,
! [X: set_state,Y3: set_state,Z: set_state] :
( ( ord_le2494988322063910608_state @ X @ Y3 )
=> ( ( ord_le2494988322063910608_state @ Y3 @ Z )
=> ( ord_le2494988322063910608_state @ X @ Z ) ) ) ).
% order_trans
thf(fact_634_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B6: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
=> ( P @ A2 @ B6 ) )
=> ( ! [A2: nat,B6: nat] :
( ( P @ B6 @ A2 )
=> ( P @ A2 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_635_top__greatest,axiom,
! [A: state > $o] : ( ord_less_eq_state_o @ A @ top_top_state_o ) ).
% top_greatest
thf(fact_636_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_637_top__greatest,axiom,
! [A: set_set_state] : ( ord_le5175021213330142598_state @ A @ top_to5262587396890829782_state ) ).
% top_greatest
thf(fact_638_top__greatest,axiom,
! [A: set_Pr1688445902015331925_state] : ( ord_le6423325748750870005_state @ A @ top_to5248747511432852389_state ) ).
% top_greatest
thf(fact_639_top__greatest,axiom,
! [A: set_Pr1785066336555260981_state] : ( ord_le2777189432094499797_state @ A @ top_to4183510833390158213_state ) ).
% top_greatest
thf(fact_640_top__greatest,axiom,
! [A: set_option_state] : ( ord_le7116032884704190368_state @ A @ top_to7666338855062656496_state ) ).
% top_greatest
thf(fact_641_top__greatest,axiom,
! [A: set_state] : ( ord_le2494988322063910608_state @ A @ top_top_set_state ) ).
% top_greatest
thf(fact_642_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_state,Z2: set_set_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_state,B3: set_set_state] :
( ( ord_le5175021213330142598_state @ B3 @ A4 )
& ( ord_le5175021213330142598_state @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_643_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Pr1688445902015331925_state,Z2: set_Pr1688445902015331925_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_Pr1688445902015331925_state,B3: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ B3 @ A4 )
& ( ord_le6423325748750870005_state @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_644_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Pr1785066336555260981_state,Z2: set_Pr1785066336555260981_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_Pr1785066336555260981_state,B3: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ B3 @ A4 )
& ( ord_le2777189432094499797_state @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_645_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_option_state,Z2: set_option_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_option_state,B3: set_option_state] :
( ( ord_le7116032884704190368_state @ B3 @ A4 )
& ( ord_le7116032884704190368_state @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_646_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_647_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_state,B3: set_state] :
( ( ord_le2494988322063910608_state @ B3 @ A4 )
& ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_648_dual__order_Oantisym,axiom,
! [B: set_set_state,A: set_set_state] :
( ( ord_le5175021213330142598_state @ B @ A )
=> ( ( ord_le5175021213330142598_state @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_649_dual__order_Oantisym,axiom,
! [B: set_Pr1688445902015331925_state,A: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ B @ A )
=> ( ( ord_le6423325748750870005_state @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_650_dual__order_Oantisym,axiom,
! [B: set_Pr1785066336555260981_state,A: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ B @ A )
=> ( ( ord_le2777189432094499797_state @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_651_dual__order_Oantisym,axiom,
! [B: set_option_state,A: set_option_state] :
( ( ord_le7116032884704190368_state @ B @ A )
=> ( ( ord_le7116032884704190368_state @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_652_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_653_dual__order_Oantisym,axiom,
! [B: set_state,A: set_state] :
( ( ord_le2494988322063910608_state @ B @ A )
=> ( ( ord_le2494988322063910608_state @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_654_dual__order_Otrans,axiom,
! [B: set_set_state,A: set_set_state,C: set_set_state] :
( ( ord_le5175021213330142598_state @ B @ A )
=> ( ( ord_le5175021213330142598_state @ C @ B )
=> ( ord_le5175021213330142598_state @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_655_dual__order_Otrans,axiom,
! [B: set_Pr1688445902015331925_state,A: set_Pr1688445902015331925_state,C: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ B @ A )
=> ( ( ord_le6423325748750870005_state @ C @ B )
=> ( ord_le6423325748750870005_state @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_656_dual__order_Otrans,axiom,
! [B: set_Pr1785066336555260981_state,A: set_Pr1785066336555260981_state,C: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ B @ A )
=> ( ( ord_le2777189432094499797_state @ C @ B )
=> ( ord_le2777189432094499797_state @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_657_dual__order_Otrans,axiom,
! [B: set_option_state,A: set_option_state,C: set_option_state] :
( ( ord_le7116032884704190368_state @ B @ A )
=> ( ( ord_le7116032884704190368_state @ C @ B )
=> ( ord_le7116032884704190368_state @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_658_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_659_dual__order_Otrans,axiom,
! [B: set_state,A: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ B @ A )
=> ( ( ord_le2494988322063910608_state @ C @ B )
=> ( ord_le2494988322063910608_state @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_660_top_Oextremum__unique,axiom,
! [A: state > $o] :
( ( ord_less_eq_state_o @ top_top_state_o @ A )
= ( A = top_top_state_o ) ) ).
% top.extremum_unique
thf(fact_661_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_662_top_Oextremum__unique,axiom,
! [A: set_set_state] :
( ( ord_le5175021213330142598_state @ top_to5262587396890829782_state @ A )
= ( A = top_to5262587396890829782_state ) ) ).
% top.extremum_unique
thf(fact_663_top_Oextremum__unique,axiom,
! [A: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ top_to5248747511432852389_state @ A )
= ( A = top_to5248747511432852389_state ) ) ).
% top.extremum_unique
thf(fact_664_top_Oextremum__unique,axiom,
! [A: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ top_to4183510833390158213_state @ A )
= ( A = top_to4183510833390158213_state ) ) ).
% top.extremum_unique
thf(fact_665_top_Oextremum__unique,axiom,
! [A: set_option_state] :
( ( ord_le7116032884704190368_state @ top_to7666338855062656496_state @ A )
= ( A = top_to7666338855062656496_state ) ) ).
% top.extremum_unique
thf(fact_666_top_Oextremum__unique,axiom,
! [A: set_state] :
( ( ord_le2494988322063910608_state @ top_top_set_state @ A )
= ( A = top_top_set_state ) ) ).
% top.extremum_unique
thf(fact_667_top_Oextremum__uniqueI,axiom,
! [A: state > $o] :
( ( ord_less_eq_state_o @ top_top_state_o @ A )
=> ( A = top_top_state_o ) ) ).
% top.extremum_uniqueI
thf(fact_668_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_669_top_Oextremum__uniqueI,axiom,
! [A: set_set_state] :
( ( ord_le5175021213330142598_state @ top_to5262587396890829782_state @ A )
=> ( A = top_to5262587396890829782_state ) ) ).
% top.extremum_uniqueI
thf(fact_670_top_Oextremum__uniqueI,axiom,
! [A: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ top_to5248747511432852389_state @ A )
=> ( A = top_to5248747511432852389_state ) ) ).
% top.extremum_uniqueI
thf(fact_671_top_Oextremum__uniqueI,axiom,
! [A: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ top_to4183510833390158213_state @ A )
=> ( A = top_to4183510833390158213_state ) ) ).
% top.extremum_uniqueI
thf(fact_672_top_Oextremum__uniqueI,axiom,
! [A: set_option_state] :
( ( ord_le7116032884704190368_state @ top_to7666338855062656496_state @ A )
=> ( A = top_to7666338855062656496_state ) ) ).
% top.extremum_uniqueI
thf(fact_673_top_Oextremum__uniqueI,axiom,
! [A: set_state] :
( ( ord_le2494988322063910608_state @ top_top_set_state @ A )
=> ( A = top_top_set_state ) ) ).
% top.extremum_uniqueI
thf(fact_674_antisym,axiom,
! [A: set_set_state,B: set_set_state] :
( ( ord_le5175021213330142598_state @ A @ B )
=> ( ( ord_le5175021213330142598_state @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_675_antisym,axiom,
! [A: set_Pr1688445902015331925_state,B: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ A @ B )
=> ( ( ord_le6423325748750870005_state @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_676_antisym,axiom,
! [A: set_Pr1785066336555260981_state,B: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ A @ B )
=> ( ( ord_le2777189432094499797_state @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_677_antisym,axiom,
! [A: set_option_state,B: set_option_state] :
( ( ord_le7116032884704190368_state @ A @ B )
=> ( ( ord_le7116032884704190368_state @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_678_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_679_antisym,axiom,
! [A: set_state,B: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_le2494988322063910608_state @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_680_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_state,Z2: set_set_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_state,B3: set_set_state] :
( ( ord_le5175021213330142598_state @ A4 @ B3 )
& ( ord_le5175021213330142598_state @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_681_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr1688445902015331925_state,Z2: set_Pr1688445902015331925_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_Pr1688445902015331925_state,B3: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ A4 @ B3 )
& ( ord_le6423325748750870005_state @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_682_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr1785066336555260981_state,Z2: set_Pr1785066336555260981_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_Pr1785066336555260981_state,B3: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ A4 @ B3 )
& ( ord_le2777189432094499797_state @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_683_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_option_state,Z2: set_option_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_option_state,B3: set_option_state] :
( ( ord_le7116032884704190368_state @ A4 @ B3 )
& ( ord_le7116032884704190368_state @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_684_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_685_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
= ( ^ [A4: set_state,B3: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B3 )
& ( ord_le2494988322063910608_state @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_686_order__subst1,axiom,
! [A: set_state,F: set_state > set_state,B: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A @ ( F @ B ) )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_687_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_688_order__subst1,axiom,
! [A: set_state,F: nat > set_state,B: nat,C: nat] :
( ( ord_le2494988322063910608_state @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_689_order__subst1,axiom,
! [A: nat,F: set_state > nat,B: set_state,C: set_state] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_690_order__subst1,axiom,
! [A: set_set_state,F: nat > set_set_state,B: nat,C: nat] :
( ( ord_le5175021213330142598_state @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le5175021213330142598_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le5175021213330142598_state @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_691_order__subst1,axiom,
! [A: set_option_state,F: nat > set_option_state,B: nat,C: nat] :
( ( ord_le7116032884704190368_state @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le7116032884704190368_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7116032884704190368_state @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_692_order__subst1,axiom,
! [A: nat,F: set_set_state > nat,B: set_set_state,C: set_set_state] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le5175021213330142598_state @ B @ C )
=> ( ! [X4: set_set_state,Y: set_set_state] :
( ( ord_le5175021213330142598_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_693_order__subst1,axiom,
! [A: nat,F: set_option_state > nat,B: set_option_state,C: set_option_state] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le7116032884704190368_state @ B @ C )
=> ( ! [X4: set_option_state,Y: set_option_state] :
( ( ord_le7116032884704190368_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_694_order__subst1,axiom,
! [A: set_state,F: set_set_state > set_state,B: set_set_state,C: set_set_state] :
( ( ord_le2494988322063910608_state @ A @ ( F @ B ) )
=> ( ( ord_le5175021213330142598_state @ B @ C )
=> ( ! [X4: set_set_state,Y: set_set_state] :
( ( ord_le5175021213330142598_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_695_order__subst1,axiom,
! [A: set_state,F: set_option_state > set_state,B: set_option_state,C: set_option_state] :
( ( ord_le2494988322063910608_state @ A @ ( F @ B ) )
=> ( ( ord_le7116032884704190368_state @ B @ C )
=> ( ! [X4: set_option_state,Y: set_option_state] :
( ( ord_le7116032884704190368_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_696_order__subst2,axiom,
! [A: set_state,B: set_state,F: set_state > set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_le2494988322063910608_state @ ( F @ B ) @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_697_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_698_order__subst2,axiom,
! [A: set_state,B: set_state,F: set_state > nat,C: nat] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_699_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_state,C: set_state] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2494988322063910608_state @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_700_order__subst2,axiom,
! [A: set_set_state,B: set_set_state,F: set_set_state > nat,C: nat] :
( ( ord_le5175021213330142598_state @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_set_state,Y: set_set_state] :
( ( ord_le5175021213330142598_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_701_order__subst2,axiom,
! [A: set_option_state,B: set_option_state,F: set_option_state > nat,C: nat] :
( ( ord_le7116032884704190368_state @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: set_option_state,Y: set_option_state] :
( ( ord_le7116032884704190368_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_702_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_state,C: set_set_state] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le5175021213330142598_state @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le5175021213330142598_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le5175021213330142598_state @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_703_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_option_state,C: set_option_state] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le7116032884704190368_state @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le7116032884704190368_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7116032884704190368_state @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_704_order__subst2,axiom,
! [A: set_state,B: set_state,F: set_state > set_set_state,C: set_set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_le5175021213330142598_state @ ( F @ B ) @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le5175021213330142598_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le5175021213330142598_state @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_705_order__subst2,axiom,
! [A: set_state,B: set_state,F: set_state > set_option_state,C: set_option_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_le7116032884704190368_state @ ( F @ B ) @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le7116032884704190368_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7116032884704190368_state @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_706_order__eq__refl,axiom,
! [X: set_set_state,Y3: set_set_state] :
( ( X = Y3 )
=> ( ord_le5175021213330142598_state @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_707_order__eq__refl,axiom,
! [X: set_Pr1688445902015331925_state,Y3: set_Pr1688445902015331925_state] :
( ( X = Y3 )
=> ( ord_le6423325748750870005_state @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_708_order__eq__refl,axiom,
! [X: set_Pr1785066336555260981_state,Y3: set_Pr1785066336555260981_state] :
( ( X = Y3 )
=> ( ord_le2777189432094499797_state @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_709_order__eq__refl,axiom,
! [X: set_option_state,Y3: set_option_state] :
( ( X = Y3 )
=> ( ord_le7116032884704190368_state @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_710_order__eq__refl,axiom,
! [X: nat,Y3: nat] :
( ( X = Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_711_order__eq__refl,axiom,
! [X: set_state,Y3: set_state] :
( ( X = Y3 )
=> ( ord_le2494988322063910608_state @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_712_linorder__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_713_ord__eq__le__subst,axiom,
! [A: set_state,F: set_state > set_state,B: set_state,C: set_state] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_714_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_715_ord__eq__le__subst,axiom,
! [A: nat,F: set_state > nat,B: set_state,C: set_state] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_716_ord__eq__le__subst,axiom,
! [A: set_state,F: nat > set_state,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_717_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_state > nat,B: set_set_state,C: set_set_state] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5175021213330142598_state @ B @ C )
=> ( ! [X4: set_set_state,Y: set_set_state] :
( ( ord_le5175021213330142598_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_718_ord__eq__le__subst,axiom,
! [A: nat,F: set_option_state > nat,B: set_option_state,C: set_option_state] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7116032884704190368_state @ B @ C )
=> ( ! [X4: set_option_state,Y: set_option_state] :
( ( ord_le7116032884704190368_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_719_ord__eq__le__subst,axiom,
! [A: set_set_state,F: nat > set_set_state,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le5175021213330142598_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le5175021213330142598_state @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_720_ord__eq__le__subst,axiom,
! [A: set_option_state,F: nat > set_option_state,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le7116032884704190368_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7116032884704190368_state @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_721_ord__eq__le__subst,axiom,
! [A: set_set_state,F: set_state > set_set_state,B: set_state,C: set_state] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le5175021213330142598_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le5175021213330142598_state @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_722_ord__eq__le__subst,axiom,
! [A: set_option_state,F: set_state > set_option_state,B: set_state,C: set_state] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le7116032884704190368_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7116032884704190368_state @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_723_ord__le__eq__subst,axiom,
! [A: set_state,B: set_state,F: set_state > set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_724_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_725_ord__le__eq__subst,axiom,
! [A: set_state,B: set_state,F: set_state > nat,C: nat] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_726_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_state,C: set_state] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le2494988322063910608_state @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_727_ord__le__eq__subst,axiom,
! [A: set_set_state,B: set_set_state,F: set_set_state > nat,C: nat] :
( ( ord_le5175021213330142598_state @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_set_state,Y: set_set_state] :
( ( ord_le5175021213330142598_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_728_ord__le__eq__subst,axiom,
! [A: set_option_state,B: set_option_state,F: set_option_state > nat,C: nat] :
( ( ord_le7116032884704190368_state @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_option_state,Y: set_option_state] :
( ( ord_le7116032884704190368_state @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_729_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_state,C: set_set_state] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le5175021213330142598_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le5175021213330142598_state @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_730_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_option_state,C: set_option_state] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le7116032884704190368_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7116032884704190368_state @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_731_ord__le__eq__subst,axiom,
! [A: set_state,B: set_state,F: set_state > set_set_state,C: set_set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le5175021213330142598_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le5175021213330142598_state @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_732_ord__le__eq__subst,axiom,
! [A: set_state,B: set_state,F: set_state > set_option_state,C: set_option_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le7116032884704190368_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le7116032884704190368_state @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_733_linorder__le__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_734_order__antisym__conv,axiom,
! [Y3: set_set_state,X: set_set_state] :
( ( ord_le5175021213330142598_state @ Y3 @ X )
=> ( ( ord_le5175021213330142598_state @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_735_order__antisym__conv,axiom,
! [Y3: set_Pr1688445902015331925_state,X: set_Pr1688445902015331925_state] :
( ( ord_le6423325748750870005_state @ Y3 @ X )
=> ( ( ord_le6423325748750870005_state @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_736_order__antisym__conv,axiom,
! [Y3: set_Pr1785066336555260981_state,X: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ Y3 @ X )
=> ( ( ord_le2777189432094499797_state @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_737_order__antisym__conv,axiom,
! [Y3: set_option_state,X: set_option_state] :
( ( ord_le7116032884704190368_state @ Y3 @ X )
=> ( ( ord_le7116032884704190368_state @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_738_order__antisym__conv,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_739_order__antisym__conv,axiom,
! [Y3: set_state,X: set_state] :
( ( ord_le2494988322063910608_state @ Y3 @ X )
=> ( ( ord_le2494988322063910608_state @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_740_in__mono,axiom,
! [A3: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state,X: produc3142500478612311029_state] :
( ( ord_le6423325748750870005_state @ A3 @ B2 )
=> ( ( member3029510603097127326_state @ X @ A3 )
=> ( member3029510603097127326_state @ X @ B2 ) ) ) ).
% in_mono
thf(fact_741_in__mono,axiom,
! [A3: set_Pr1785066336555260981_state,B2: set_Pr1785066336555260981_state,X: produc8023240190789890773_state] :
( ( ord_le2777189432094499797_state @ A3 @ B2 )
=> ( ( member753036827967488894_state @ X @ A3 )
=> ( member753036827967488894_state @ X @ B2 ) ) ) ).
% in_mono
thf(fact_742_in__mono,axiom,
! [A3: set_option_state,B2: set_option_state,X: option_state] :
( ( ord_le7116032884704190368_state @ A3 @ B2 )
=> ( ( member_option_state @ X @ A3 )
=> ( member_option_state @ X @ B2 ) ) ) ).
% in_mono
thf(fact_743_in__mono,axiom,
! [A3: set_state,B2: set_state,X: state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( member_state @ X @ A3 )
=> ( member_state @ X @ B2 ) ) ) ).
% in_mono
thf(fact_744_subsetD,axiom,
! [A3: set_state,B2: set_state,C: state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( member_state @ C @ A3 )
=> ( member_state @ C @ B2 ) ) ) ).
% subsetD
thf(fact_745_UNIV__eq__I,axiom,
! [A3: set_state] :
( ! [X4: state] : ( member_state @ X4 @ A3 )
=> ( top_top_set_state = A3 ) ) ).
% UNIV_eq_I
thf(fact_746_equalityE,axiom,
! [A3: set_state,B2: set_state] :
( ( A3 = B2 )
=> ~ ( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ~ ( ord_le2494988322063910608_state @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_747_subset__eq,axiom,
( ord_le2494988322063910608_state
= ( ^ [A5: set_state,B5: set_state] :
! [X3: state] :
( ( member_state @ X3 @ A5 )
=> ( member_state @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_748_equalityD1,axiom,
! [A3: set_state,B2: set_state] :
( ( A3 = B2 )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_749_equalityD2,axiom,
! [A3: set_state,B2: set_state] :
( ( A3 = B2 )
=> ( ord_le2494988322063910608_state @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_750_subset__iff,axiom,
( ord_le2494988322063910608_state
= ( ^ [A5: set_state,B5: set_state] :
! [T: state] :
( ( member_state @ T @ A5 )
=> ( member_state @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_751_subset__UNIV,axiom,
! [A3: set_state] : ( ord_le2494988322063910608_state @ A3 @ top_top_set_state ) ).
% subset_UNIV
thf(fact_752_subset__refl,axiom,
! [A3: set_state] : ( ord_le2494988322063910608_state @ A3 @ A3 ) ).
% subset_refl
thf(fact_753_Collect__mono,axiom,
! [P: state > $o,Q: state > $o] :
( ! [X4: state] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le2494988322063910608_state @ ( collect_state @ P ) @ ( collect_state @ Q ) ) ) ).
% Collect_mono
thf(fact_754_UNIV__witness,axiom,
? [X4: state] : ( member_state @ X4 @ top_top_set_state ) ).
% UNIV_witness
thf(fact_755_subset__trans,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( ord_le2494988322063910608_state @ B2 @ C3 )
=> ( ord_le2494988322063910608_state @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_756_set__eq__subset,axiom,
( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
= ( ^ [A5: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A5 @ B5 )
& ( ord_le2494988322063910608_state @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_757_Collect__mono__iff,axiom,
! [P: state > $o,Q: state > $o] :
( ( ord_le2494988322063910608_state @ ( collect_state @ P ) @ ( collect_state @ Q ) )
= ( ! [X3: state] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_758_sep__algebra_Oadd__set_Ocong,axiom,
sep_add_set_state = sep_add_set_state ).
% sep_algebra.add_set.cong
thf(fact_759_sep__algebra_Oupper__closure_Ocong,axiom,
sep_up1246176804924251236_state = sep_up1246176804924251236_state ).
% sep_algebra.upper_closure.cong
thf(fact_760_bot_Oextremum__uniqueI,axiom,
! [A: set_state] :
( ( ord_le2494988322063910608_state @ A @ bot_bot_set_state )
=> ( A = bot_bot_set_state ) ) ).
% bot.extremum_uniqueI
thf(fact_761_bot_Oextremum__unique,axiom,
! [A: set_state] :
( ( ord_le2494988322063910608_state @ A @ bot_bot_set_state )
= ( A = bot_bot_set_state ) ) ).
% bot.extremum_unique
thf(fact_762_bot_Oextremum,axiom,
! [A: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A ) ).
% bot.extremum
thf(fact_763_image__mono,axiom,
! [A3: set_state,B2: set_state,F: state > state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A3 ) @ ( image_state_state @ F @ B2 ) ) ) ).
% image_mono
thf(fact_764_image__subsetI,axiom,
! [A3: set_state,F: state > state,B2: set_state] :
( ! [X4: state] :
( ( member_state @ X4 @ A3 )
=> ( member_state @ ( F @ X4 ) @ B2 ) )
=> ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_765_subset__imageE,axiom,
! [B2: set_state,F: state > state,A3: set_state] :
( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A3 ) )
=> ~ ! [C6: set_state] :
( ( ord_le2494988322063910608_state @ C6 @ A3 )
=> ( B2
!= ( image_state_state @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_766_subset__image__iff,axiom,
! [B2: set_state,F: state > state,A3: set_state] :
( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A3 ) )
= ( ? [AA: set_state] :
( ( ord_le2494988322063910608_state @ AA @ A3 )
& ( B2
= ( image_state_state @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_767_subset__insertI2,axiom,
! [A3: set_state,B2: set_state,B: state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ A3 @ ( insert_state @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_768_subset__insertI,axiom,
! [B2: set_state,A: state] : ( ord_le2494988322063910608_state @ B2 @ ( insert_state @ A @ B2 ) ) ).
% subset_insertI
thf(fact_769_subset__insert,axiom,
! [X: state,A3: set_state,B2: set_state] :
( ~ ( member_state @ X @ A3 )
=> ( ( ord_le2494988322063910608_state @ A3 @ ( insert_state @ X @ B2 ) )
= ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_770_insert__mono,axiom,
! [C3: set_state,D2: set_state,A: state] :
( ( ord_le2494988322063910608_state @ C3 @ D2 )
=> ( ord_le2494988322063910608_state @ ( insert_state @ A @ C3 ) @ ( insert_state @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_771_rangeI,axiom,
! [F: state > state,X: state] : ( member_state @ ( F @ X ) @ ( image_state_state @ F @ top_top_set_state ) ) ).
% rangeI
thf(fact_772_range__eqI,axiom,
! [B: state,F: state > state,X: state] :
( ( B
= ( F @ X ) )
=> ( member_state @ B @ ( image_state_state @ F @ top_top_set_state ) ) ) ).
% range_eqI
thf(fact_773_empty__not__UNIV,axiom,
bot_bot_set_state != top_top_set_state ).
% empty_not_UNIV
thf(fact_774_insert__UNIV,axiom,
! [X: state] :
( ( insert_state @ X @ top_top_set_state )
= top_top_set_state ) ).
% insert_UNIV
thf(fact_775_sep__algebra_Oup__closed__plus__UNIV,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( sep_up_closed_state @ Plus @ ( sep_add_set_state @ Plus @ A3 @ top_top_set_state ) ) ) ).
% sep_algebra.up_closed_plus_UNIV
thf(fact_776_PartialSA_Osub__bigger,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( greater_set @ A3 @ B2 ) ) ).
% PartialSA.sub_bigger
thf(fact_777_sep__algebra_Oadd__set__asso,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state,C3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_add_set_state @ Plus @ ( sep_add_set_state @ Plus @ A3 @ B2 ) @ C3 )
= ( sep_add_set_state @ Plus @ A3 @ ( sep_add_set_state @ Plus @ B2 @ C3 ) ) ) ) ).
% sep_algebra.add_set_asso
thf(fact_778_sep__algebra_Oadd__set__commm,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_add_set_state @ Plus @ A3 @ B2 )
= ( sep_add_set_state @ Plus @ B2 @ A3 ) ) ) ).
% sep_algebra.add_set_commm
thf(fact_779_UNIV__option__conv,axiom,
( top_to7666338855062656496_state
= ( insert_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) ) ).
% UNIV_option_conv
thf(fact_780_subset__singleton__iff,axiom,
! [X8: set_state,A: state] :
( ( ord_le2494988322063910608_state @ X8 @ ( insert_state @ A @ bot_bot_set_state ) )
= ( ( X8 = bot_bot_set_state )
| ( X8
= ( insert_state @ A @ bot_bot_set_state ) ) ) ) ).
% subset_singleton_iff
thf(fact_781_subset__singletonD,axiom,
! [A3: set_state,X: state] :
( ( ord_le2494988322063910608_state @ A3 @ ( insert_state @ X @ bot_bot_set_state ) )
=> ( ( A3 = bot_bot_set_state )
| ( A3
= ( insert_state @ X @ bot_bot_set_state ) ) ) ) ).
% subset_singletonD
thf(fact_782_notin__range__Some,axiom,
! [X: option_state] :
( ( ~ ( member_option_state @ X @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) )
= ( X = none_state ) ) ).
% notin_range_Some
thf(fact_783_sep__algebra_Osub__bigger,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 ) ) ) ).
% sep_algebra.sub_bigger
thf(fact_784_sep__algebra_Oempty__set__sum,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_add_set_state @ Plus @ bot_bot_set_state @ A3 )
= bot_bot_set_state ) ) ).
% sep_algebra.empty_set_sum
thf(fact_785_sep__algebra_Ox__elem__set__product,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( member_state @ X @ ( sep_add_set_state @ Plus @ A3 @ B2 ) )
= ( ? [A4: state,B3: state] :
( ( member_state @ A4 @ A3 )
& ( member_state @ B3 @ B2 )
& ( ( some_state @ X )
= ( Plus @ A4 @ B3 ) ) ) ) ) ) ).
% sep_algebra.x_elem_set_product
thf(fact_786_sep__algebra_Oadd__set__elem,axiom,
! [Plus: state > state > option_state,Core: state > state,Phi: state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( member_state @ Phi @ ( sep_add_set_state @ Plus @ A3 @ B2 ) )
= ( ? [A4: state,B3: state] :
( ( ( some_state @ Phi )
= ( Plus @ A4 @ B3 ) )
& ( member_state @ A4 @ A3 )
& ( member_state @ B3 @ B2 ) ) ) ) ) ).
% sep_algebra.add_set_elem
thf(fact_787_add__set__def,axiom,
( add_set
= ( sep_add_set_state @ plus ) ) ).
% add_set_def
thf(fact_788_sep__algebra_Oup__closed__sum,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_up_closed_state @ Plus @ A3 )
=> ( sep_up_closed_state @ Plus @ ( sep_add_set_state @ Plus @ A3 @ B2 ) ) ) ) ).
% sep_algebra.up_closed_sum
thf(fact_789_sep__algebra_Obigger__set,axiom,
! [Plus: state > state > option_state,Core: state > state,A6: set_state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_gr7105985528888466643_state @ Plus @ A6 @ A3 )
=> ( sep_gr7105985528888466643_state @ Plus @ ( sep_add_set_state @ Plus @ A6 @ B2 ) @ ( sep_add_set_state @ Plus @ A3 @ B2 ) ) ) ) ).
% sep_algebra.bigger_set
thf(fact_790_sep__algebra_Oequiv__stable__sum,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state,C3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_equiv_state @ Plus @ A3 @ B2 )
=> ( sep_equiv_state @ Plus @ ( sep_add_set_state @ Plus @ A3 @ C3 ) @ ( sep_add_set_state @ Plus @ B2 @ C3 ) ) ) ) ).
% sep_algebra.equiv_stable_sum
thf(fact_791_range__eq__singletonD,axiom,
! [F: state > state,A: state,X: state] :
( ( ( image_state_state @ F @ top_top_set_state )
= ( insert_state @ A @ bot_bot_set_state ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_792_sep__algebra_Oup__closed__bigger__subset,axiom,
! [Plus: state > state > option_state,Core: state > state,B2: set_state,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_up_closed_state @ Plus @ B2 )
=> ( ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ) ) ).
% sep_algebra.up_closed_bigger_subset
thf(fact_793_sep__algebra_Oequiv__up__closed__subset,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state,C3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_up_closed_state @ Plus @ A3 )
=> ( ( sep_equiv_state @ Plus @ B2 @ C3 )
=> ( ( ord_le2494988322063910608_state @ B2 @ A3 )
= ( ord_le2494988322063910608_state @ C3 @ A3 ) ) ) ) ) ).
% sep_algebra.equiv_up_closed_subset
thf(fact_794_PartialSA_Oup__closed__bigger__subset,axiom,
! [B2: set_state,A3: set_state] :
( ( sep_up_closed_state @ plus @ B2 )
=> ( ( greater_set @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ) ).
% PartialSA.up_closed_bigger_subset
thf(fact_795_insert__subsetI,axiom,
! [X: state,A3: set_state,X8: set_state] :
( ( member_state @ X @ A3 )
=> ( ( ord_le2494988322063910608_state @ X8 @ A3 )
=> ( ord_le2494988322063910608_state @ ( insert_state @ X @ X8 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_796_subset__emptyI,axiom,
! [A3: set_state] :
( ! [X4: state] :
~ ( member_state @ X4 @ A3 )
=> ( ord_le2494988322063910608_state @ A3 @ bot_bot_set_state ) ) ).
% subset_emptyI
thf(fact_797_all__subset__image,axiom,
! [F: state > state,A3: set_state,P: set_state > $o] :
( ( ! [B5: set_state] :
( ( ord_le2494988322063910608_state @ B5 @ ( image_state_state @ F @ A3 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_state] :
( ( ord_le2494988322063910608_state @ B5 @ A3 )
=> ( P @ ( image_state_state @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_798_surj__def,axiom,
! [F: state > state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
= ( ! [Y4: state] :
? [X3: state] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_799_surjI,axiom,
! [G: state > state,F: state > state] :
( ! [X4: state] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_state_state @ G @ top_top_set_state )
= top_top_set_state ) ) ).
% surjI
thf(fact_800_surjE,axiom,
! [F: state > state,Y3: state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ~ ! [X4: state] :
( Y3
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_801_surjD,axiom,
! [F: state > state,Y3: state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ? [X4: state] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_802_PartialSA_Osmaller__pure__sum__smaller,axiom,
! [Y3: state,A: state,B: state,X: state] :
( ( greater @ Y3 @ A )
=> ( ( greater @ Y3 @ B )
=> ( ( ( some_state @ X )
= ( plus @ A @ B ) )
=> ( ( sep_pure_state @ plus @ B )
=> ( greater @ Y3 @ X ) ) ) ) ) ).
% PartialSA.smaller_pure_sum_smaller
thf(fact_803_sep__algebra_Opure_Ocong,axiom,
sep_pure_state = sep_pure_state ).
% sep_algebra.pure.cong
thf(fact_804_top__set__def,axiom,
( top_top_set_state
= ( collect_state @ top_top_state_o ) ) ).
% top_set_def
thf(fact_805_top__empty__eq,axiom,
( top_top_state_o
= ( ^ [X3: state] : ( member_state @ X3 @ top_top_set_state ) ) ) ).
% top_empty_eq
thf(fact_806_PartialSA_Opure__def,axiom,
! [A: state] :
( ( sep_pure_state @ plus @ A )
= ( ( some_state @ A )
= ( plus @ A @ A ) ) ) ).
% PartialSA.pure_def
thf(fact_807_PartialSA_Opure__stable,axiom,
! [A: state,B: state,C: state] :
( ( sep_pure_state @ plus @ A )
=> ( ( sep_pure_state @ plus @ B )
=> ( ( ( some_state @ C )
= ( plus @ A @ B ) )
=> ( sep_pure_state @ plus @ C ) ) ) ) ).
% PartialSA.pure_stable
thf(fact_808_PartialSA_Opure__smaller,axiom,
! [A: state,B: state] :
( ( sep_pure_state @ plus @ A )
=> ( ( greater @ A @ B )
=> ( sep_pure_state @ plus @ B ) ) ) ).
% PartialSA.pure_smaller
thf(fact_809_sep__algebra_Opure__def,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_pure_state @ Plus @ A )
= ( ( some_state @ A )
= ( Plus @ A @ A ) ) ) ) ).
% sep_algebra.pure_def
thf(fact_810_sep__algebra_Opure__stable,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_pure_state @ Plus @ A )
=> ( ( sep_pure_state @ Plus @ B )
=> ( ( ( some_state @ C )
= ( Plus @ A @ B ) )
=> ( sep_pure_state @ Plus @ C ) ) ) ) ) ).
% sep_algebra.pure_stable
thf(fact_811_PartialSA_Omono__pruner__def,axiom,
! [P4: state > $o] :
( ( packag2456304381420842418_state @ plus @ P4 )
= ( ! [Phi3: state,Phi6: state,R3: state] :
( ( ( sep_pure_state @ plus @ R3 )
& ( P4 @ Phi6 )
& ( ( some_state @ Phi3 )
= ( plus @ Phi6 @ R3 ) ) )
=> ( P4 @ Phi3 ) ) ) ) ).
% PartialSA.mono_pruner_def
thf(fact_812_sep__algebra_Omax__projection__prop__pure__core,axiom,
! [Plus: state > state > option_state,Core: state > state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( sep_ma8214210560313151521_state @ Plus @ ( sep_pure_state @ Plus ) @ Core ) ) ).
% sep_algebra.max_projection_prop_pure_core
thf(fact_813_PartialSA_Ointuitionistic__def,axiom,
! [A3: state > $o] :
( ( packag8361946002163212404_state @ plus @ A3 )
= ( ! [Phi3: state,Phi6: state] :
( ( ( greater @ Phi3 @ Phi6 )
& ( A3 @ Phi6 ) )
=> ( A3 @ Phi3 ) ) ) ) ).
% PartialSA.intuitionistic_def
thf(fact_814_PartialSA_Ominus__sum,axiom,
! [A: state,B: state,C: state,X: state] :
( ( ( some_state @ A )
= ( plus @ B @ C ) )
=> ( ( greater @ X @ A )
=> ( ( minus @ X @ A )
= ( minus @ ( minus @ X @ B ) @ C ) ) ) ) ).
% PartialSA.minus_sum
thf(fact_815_PartialSA_Ominus__some,axiom,
! [A: state,B: state] :
( ( greater @ A @ B )
=> ( ( some_state @ A )
= ( plus @ B @ ( minus @ A @ B ) ) ) ) ).
% PartialSA.minus_some
thf(fact_816_PartialSA_Ominus__bigger,axiom,
! [X: state,A: state,B: state] :
( ( ( some_state @ X )
= ( plus @ A @ B ) )
=> ( greater @ ( minus @ X @ A ) @ B ) ) ).
% PartialSA.minus_bigger
thf(fact_817_sep__algebra_Omax__projection__prop_Ocong,axiom,
sep_ma8214210560313151521_state = sep_ma8214210560313151521_state ).
% sep_algebra.max_projection_prop.cong
thf(fact_818_PartialSA_Ominus__default,axiom,
! [B: state,A: state] :
( ~ ( greater @ B @ A )
=> ( ( minus @ B @ A )
= B ) ) ).
% PartialSA.minus_default
thf(fact_819_PartialSA_Ominus__smaller,axiom,
! [X: state,A: state] :
( ( greater @ X @ A )
=> ( greater @ X @ ( minus @ X @ A ) ) ) ).
% PartialSA.minus_smaller
thf(fact_820_PartialSA_Ogreater__minus__trans,axiom,
! [Y3: state,X: state,A: state] :
( ( greater @ Y3 @ X )
=> ( ( greater @ X @ A )
=> ( greater @ ( minus @ Y3 @ A ) @ ( minus @ X @ A ) ) ) ) ).
% PartialSA.greater_minus_trans
thf(fact_821_PartialSA_Ompp__prop,axiom,
! [P: state > $o,F: state > state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P @ F )
=> ( P @ ( F @ X ) ) ) ).
% PartialSA.mpp_prop
thf(fact_822_PartialSA_Ompp__invo,axiom,
! [P: state > $o,F: state > state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P @ F )
=> ( ( F @ ( F @ X ) )
= ( F @ X ) ) ) ).
% PartialSA.mpp_invo
thf(fact_823_sep__algebra_Ompp__invo,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
=> ( ( F @ ( F @ X ) )
= ( F @ X ) ) ) ) ).
% sep_algebra.mpp_invo
thf(fact_824_sep__algebra_Ompp__prop,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
=> ( P @ ( F @ X ) ) ) ) ).
% sep_algebra.mpp_prop
thf(fact_825_PartialSA_Omax__projection__propE_I3_J,axiom,
! [P: state > $o,F: state > state,P4: state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P @ F )
=> ( ( ( P @ P4 )
& ( greater @ X @ P4 ) )
=> ( greater @ ( F @ X ) @ P4 ) ) ) ).
% PartialSA.max_projection_propE(3)
thf(fact_826_PartialSA_OmppI,axiom,
! [P: state > $o,F: state > state,A: state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P @ F )
=> ( ( greater @ A @ X )
=> ( ( P @ X )
=> ( ( greater @ X @ ( F @ A ) )
=> ( X
= ( F @ A ) ) ) ) ) ) ).
% PartialSA.mppI
thf(fact_827_PartialSA_Ompp__mono,axiom,
! [P: state > $o,F: state > state,A: state,B: state] :
( ( sep_ma8214210560313151521_state @ plus @ P @ F )
=> ( ( greater @ A @ B )
=> ( greater @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% PartialSA.mpp_mono
thf(fact_828_PartialSA_Ompp__smaller,axiom,
! [P: state > $o,F: state > state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P @ F )
=> ( greater @ X @ ( F @ X ) ) ) ).
% PartialSA.mpp_smaller
thf(fact_829_PartialSA_Omax__projection__propI,axiom,
! [F: state > state,P: state > $o] :
( ! [X4: state] : ( greater @ X4 @ ( F @ X4 ) )
=> ( ! [X4: state] : ( P @ ( F @ X4 ) )
=> ( ! [X4: state,P5: state] :
( ( ( P @ P5 )
& ( greater @ X4 @ P5 ) )
=> ( greater @ ( F @ X4 ) @ P5 ) )
=> ( sep_ma8214210560313151521_state @ plus @ P @ F ) ) ) ) ).
% PartialSA.max_projection_propI
thf(fact_830_PartialSA_Omax__projection__prop__def,axiom,
! [P: state > $o,F: state > state] :
( ( sep_ma8214210560313151521_state @ plus @ P @ F )
= ( ! [X3: state] :
( ( greater @ X3 @ ( F @ X3 ) )
& ( P @ ( F @ X3 ) )
& ! [P6: state] :
( ( ( P @ P6 )
& ( greater @ X3 @ P6 ) )
=> ( greater @ ( F @ X3 ) @ P6 ) ) ) ) ) ).
% PartialSA.max_projection_prop_def
thf(fact_831_PartialSA_Oprove__last__completeness,axiom,
! [A7: state,A: state,Nf1: state,F2: state] :
( ( greater @ A7 @ A )
=> ( ( ( some_state @ A )
= ( plus @ Nf1 @ F2 ) )
=> ( greater @ ( minus @ A7 @ Nf1 ) @ F2 ) ) ) ).
% PartialSA.prove_last_completeness
thf(fact_832_PartialSA_Ominus__and__plus,axiom,
! [Omega: state,Omega2: state,R4: state,A: state] :
( ( ( some_state @ Omega )
= ( plus @ Omega2 @ R4 ) )
=> ( ( greater @ Omega2 @ A )
=> ( ( some_state @ ( minus @ Omega @ A ) )
= ( plus @ ( minus @ Omega2 @ A ) @ R4 ) ) ) ) ).
% PartialSA.minus_and_plus
thf(fact_833_PartialSA_Ominus__equiv__def,axiom,
! [B: state,A: state] :
( ( greater @ B @ A )
=> ( ( ( some_state @ B )
= ( plus @ A @ ( minus @ B @ A ) ) )
& ( greater @ ( minus @ B @ A ) @ ( core @ B ) ) ) ) ).
% PartialSA.minus_equiv_def
thf(fact_834_PartialSA_OminusI,axiom,
! [B: state,A: state,X: state] :
( ( ( some_state @ B )
= ( plus @ A @ X ) )
=> ( ( greater @ X @ ( core @ B ) )
=> ( X
= ( minus @ B @ A ) ) ) ) ).
% PartialSA.minusI
thf(fact_835_sep__algebra_Obigger__singleton,axiom,
! [Plus: state > state > option_state,Core: state > state,Phi2: state,Phi: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ Phi2 @ Phi )
=> ( sep_gr7105985528888466643_state @ Plus @ ( insert_state @ Phi2 @ bot_bot_set_state ) @ ( insert_state @ Phi @ bot_bot_set_state ) ) ) ) ).
% sep_algebra.bigger_singleton
thf(fact_836_subset__Compl__singleton,axiom,
! [A3: set_state,B: state] :
( ( ord_le2494988322063910608_state @ A3 @ ( uminus472742206872269241_state @ ( insert_state @ B @ bot_bot_set_state ) ) )
= ( ~ ( member_state @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_837_GreatestI2__order,axiom,
! [P: set_state > $o,X: set_state,Q: set_state > $o] :
( ( P @ X )
=> ( ! [Y: set_state] :
( ( P @ Y )
=> ( ord_le2494988322063910608_state @ Y @ X ) )
=> ( ! [X4: set_state] :
( ( P @ X4 )
=> ( ! [Y6: set_state] :
( ( P @ Y6 )
=> ( ord_le2494988322063910608_state @ Y6 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_2642746146112740183_state @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_838_Compl__iff,axiom,
! [C: state,A3: set_state] :
( ( member_state @ C @ ( uminus472742206872269241_state @ A3 ) )
= ( ~ ( member_state @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_839_ComplI,axiom,
! [C: state,A3: set_state] :
( ~ ( member_state @ C @ A3 )
=> ( member_state @ C @ ( uminus472742206872269241_state @ A3 ) ) ) ).
% ComplI
thf(fact_840_Compl__subset__Compl__iff,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ A3 ) @ ( uminus472742206872269241_state @ B2 ) )
= ( ord_le2494988322063910608_state @ B2 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_841_Compl__anti__mono,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ B2 ) @ ( uminus472742206872269241_state @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_842_sep__algebra_Ogreater_Ocong,axiom,
sep_greater_state = sep_greater_state ).
% sep_algebra.greater.cong
thf(fact_843_ComplD,axiom,
! [C: state,A3: set_state] :
( ( member_state @ C @ ( uminus472742206872269241_state @ A3 ) )
=> ~ ( member_state @ C @ A3 ) ) ).
% ComplD
thf(fact_844_sep__algebra_Osucc__antisym,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ A @ B )
=> ( ( sep_greater_state @ Plus @ B @ A )
=> ( A = B ) ) ) ) ).
% sep_algebra.succ_antisym
thf(fact_845_sep__algebra_Osucc__trans,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ A @ B )
=> ( ( sep_greater_state @ Plus @ B @ C )
=> ( sep_greater_state @ Plus @ A @ C ) ) ) ) ).
% sep_algebra.succ_trans
thf(fact_846_sep__algebra_Osucc__refl,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( sep_greater_state @ Plus @ A @ A ) ) ).
% sep_algebra.succ_refl
thf(fact_847_PartialSA_Osep__algebra__axioms,axiom,
sep_algebra_state @ plus @ core ).
% PartialSA.sep_algebra_axioms
thf(fact_848_PartialSA_Ocore__mono,axiom,
! [A: state,B: state] :
( ( greater @ A @ B )
=> ( greater @ ( core @ A ) @ ( core @ B ) ) ) ).
% PartialSA.core_mono
thf(fact_849_PartialSA_Ominus__core__weaker,axiom,
! [A: state,B: state] :
( ( core @ ( minus @ A @ B ) )
= ( minus @ ( core @ A ) @ ( core @ B ) ) ) ).
% PartialSA.minus_core_weaker
thf(fact_850_PartialSA_Ominus__core,axiom,
! [A: state,B: state] :
( ( core @ ( minus @ A @ B ) )
= ( core @ A ) ) ).
% PartialSA.minus_core
thf(fact_851_sep__algebra_Obigger__sum__smaller,axiom,
! [Plus: state > state > option_state,Core: state > state,C: state,A: state,B: state,A7: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ C )
= ( Plus @ A @ B ) )
=> ( ( sep_greater_state @ Plus @ A @ A7 )
=> ? [B7: state] :
( ( sep_greater_state @ Plus @ B7 @ B )
& ( ( some_state @ C )
= ( Plus @ A7 @ B7 ) ) ) ) ) ) ).
% sep_algebra.bigger_sum_smaller
thf(fact_852_sep__algebra_Osmaller__than__core,axiom,
! [Plus: state > state > option_state,Core: state > state,Y3: state,X: state,Z: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ Y3 @ X )
=> ( ( ( some_state @ Z )
= ( Plus @ X @ ( Core @ Y3 ) ) )
=> ( ( Core @ Z )
= ( Core @ Y3 ) ) ) ) ) ).
% sep_algebra.smaller_than_core
thf(fact_853_sep__algebra_Oaddition__bigger,axiom,
! [Plus: state > state > option_state,Core: state > state,A7: state,A: state,X7: state,B: state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ A7 @ A )
=> ( ( ( some_state @ X7 )
= ( Plus @ A7 @ B ) )
=> ( ( ( some_state @ X )
= ( Plus @ A @ B ) )
=> ( sep_greater_state @ Plus @ X7 @ X ) ) ) ) ) ).
% sep_algebra.addition_bigger
thf(fact_854_sep__algebra_Ogreater__equiv,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ A @ B )
= ( ? [C5: state] :
( ( some_state @ A )
= ( Plus @ C5 @ B ) ) ) ) ) ).
% sep_algebra.greater_equiv
thf(fact_855_sep__algebra_Ominus__unique,axiom,
! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X: state,Y3: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( ( some_state @ B )
= ( Plus @ A @ X ) )
& ( sep_greater_state @ Plus @ X @ ( Core @ B ) ) )
=> ( ( ( ( some_state @ B )
= ( Plus @ A @ Y3 ) )
& ( sep_greater_state @ Plus @ Y3 @ ( Core @ B ) ) )
=> ( X = Y3 ) ) ) ) ).
% sep_algebra.minus_unique
thf(fact_856_sep__algebra_Ominus__exists,axiom,
! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ B @ A )
=> ? [X4: state] :
( ( ( some_state @ B )
= ( Plus @ A @ X4 ) )
& ( sep_greater_state @ Plus @ X4 @ ( Core @ B ) ) ) ) ) ).
% sep_algebra.minus_exists
thf(fact_857_sep__algebra_Oextract__core,axiom,
! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( ( some_state @ B )
= ( Plus @ A @ X ) )
& ( sep_greater_state @ Plus @ X @ ( Core @ B ) ) )
=> ( ( Core @ X )
= ( Core @ B ) ) ) ) ).
% sep_algebra.extract_core
thf(fact_858_sep__algebra_Ogreater__def,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ A @ B )
= ( ? [C5: state] :
( ( some_state @ A )
= ( Plus @ B @ C5 ) ) ) ) ) ).
% sep_algebra.greater_def
thf(fact_859_greater__def,axiom,
( greater
= ( sep_greater_state @ plus ) ) ).
% greater_def
thf(fact_860_core__max,axiom,
! [X: state,C: state] :
( ( ( some_state @ X )
= ( plus @ X @ C ) )
=> ? [R: state] :
( ( some_state @ ( core @ X ) )
= ( plus @ C @ R ) ) ) ).
% core_max
thf(fact_861_core__sum,axiom,
! [C: state,A: state,B: state] :
( ( ( some_state @ C )
= ( plus @ A @ B ) )
=> ( ( some_state @ ( core @ C ) )
= ( plus @ ( core @ A ) @ ( core @ B ) ) ) ) ).
% core_sum
thf(fact_862_cancellative,axiom,
! [A: state,B: state,X: state,Y3: state] :
( ( ( some_state @ A )
= ( plus @ B @ X ) )
=> ( ( ( some_state @ A )
= ( plus @ B @ Y3 ) )
=> ( ( ( core @ X )
= ( core @ Y3 ) )
=> ( X = Y3 ) ) ) ) ).
% cancellative
thf(fact_863_core__is__pure,axiom,
! [X: state] :
( ( some_state @ ( core @ X ) )
= ( plus @ ( core @ X ) @ ( core @ X ) ) ) ).
% core_is_pure
thf(fact_864_core__is__smaller,axiom,
( some_state
= ( ^ [X3: state] : ( plus @ X3 @ ( core @ X3 ) ) ) ) ).
% core_is_smaller
thf(fact_865_sep__algebra_Opure__smaller,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_pure_state @ Plus @ A )
=> ( ( sep_greater_state @ Plus @ A @ B )
=> ( sep_pure_state @ Plus @ B ) ) ) ) ).
% sep_algebra.pure_smaller
thf(fact_866_Compl__UNIV__eq,axiom,
( ( uminus472742206872269241_state @ top_top_set_state )
= bot_bot_set_state ) ).
% Compl_UNIV_eq
thf(fact_867_Compl__empty__eq,axiom,
( ( uminus472742206872269241_state @ bot_bot_set_state )
= top_top_set_state ) ).
% Compl_empty_eq
thf(fact_868_sep__algebra_Oup__closed__def,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_up_closed_state @ Plus @ A3 )
= ( ! [Phi3: state] :
( ? [X3: state] :
( ( member_state @ X3 @ A3 )
& ( sep_greater_state @ Plus @ Phi3 @ X3 ) )
=> ( member_state @ Phi3 @ A3 ) ) ) ) ) ).
% sep_algebra.up_closed_def
thf(fact_869_sep__algebra_Oup__closedI,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ! [Phi4: state,Phi5: state] :
( ( ( sep_greater_state @ Plus @ Phi4 @ Phi5 )
& ( member_state @ Phi5 @ A3 ) )
=> ( member_state @ Phi4 @ A3 ) )
=> ( sep_up_closed_state @ Plus @ A3 ) ) ) ).
% sep_algebra.up_closedI
thf(fact_870_sep__algebra_Omax__projection__propE_I3_J,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,P4: state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
=> ( ( ( P @ P4 )
& ( sep_greater_state @ Plus @ X @ P4 ) )
=> ( sep_greater_state @ Plus @ ( F @ X ) @ P4 ) ) ) ) ).
% sep_algebra.max_projection_propE(3)
thf(fact_871_sep__algebra_OmppI,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
=> ( ( sep_greater_state @ Plus @ A @ X )
=> ( ( P @ X )
=> ( ( sep_greater_state @ Plus @ X @ ( F @ A ) )
=> ( X
= ( F @ A ) ) ) ) ) ) ) ).
% sep_algebra.mppI
thf(fact_872_sep__algebra_Ompp__mono,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
=> ( ( sep_greater_state @ Plus @ A @ B )
=> ( sep_greater_state @ Plus @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% sep_algebra.mpp_mono
thf(fact_873_sep__algebra_Ompp__smaller,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
=> ( sep_greater_state @ Plus @ X @ ( F @ X ) ) ) ) ).
% sep_algebra.mpp_smaller
thf(fact_874_sep__algebra_Omax__projection__propI,axiom,
! [Plus: state > state > option_state,Core: state > state,F: state > state,P: state > $o] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ! [X4: state] : ( sep_greater_state @ Plus @ X4 @ ( F @ X4 ) )
=> ( ! [X4: state] : ( P @ ( F @ X4 ) )
=> ( ! [X4: state,P5: state] :
( ( ( P @ P5 )
& ( sep_greater_state @ Plus @ X4 @ P5 ) )
=> ( sep_greater_state @ Plus @ ( F @ X4 ) @ P5 ) )
=> ( sep_ma8214210560313151521_state @ Plus @ P @ F ) ) ) ) ) ).
% sep_algebra.max_projection_propI
thf(fact_875_sep__algebra_Omax__projection__prop__def,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
= ( ! [X3: state] :
( ( sep_greater_state @ Plus @ X3 @ ( F @ X3 ) )
& ( P @ ( F @ X3 ) )
& ! [P6: state] :
( ( ( P @ P6 )
& ( sep_greater_state @ Plus @ X3 @ P6 ) )
=> ( sep_greater_state @ Plus @ ( F @ X3 ) @ P6 ) ) ) ) ) ) ).
% sep_algebra.max_projection_prop_def
thf(fact_876_subset__Compl__self__eq,axiom,
! [A3: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ ( uminus472742206872269241_state @ A3 ) )
= ( A3 = bot_bot_set_state ) ) ).
% subset_Compl_self_eq
thf(fact_877_sep__algebra_Ogreater__set__def,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ? [Y4: state] :
( ( member_state @ Y4 @ B2 )
& ( sep_greater_state @ Plus @ X3 @ Y4 ) ) ) ) ) ) ).
% sep_algebra.greater_set_def
thf(fact_878_sep__algebra_Ogreater__setI,axiom,
! [Plus: state > state > option_state,Core: state > state,A3: set_state,B2: set_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ! [A2: state] :
( ( member_state @ A2 @ A3 )
=> ? [X5: state] :
( ( member_state @ X5 @ B2 )
& ( sep_greater_state @ Plus @ A2 @ X5 ) ) )
=> ( sep_gr7105985528888466643_state @ Plus @ A3 @ B2 ) ) ) ).
% sep_algebra.greater_setI
thf(fact_879_empty__in__Fpow,axiom,
! [A3: set_state] : ( member_set_state @ bot_bot_set_state @ ( finite_Fpow_state @ A3 ) ) ).
% empty_in_Fpow
thf(fact_880_sep__algebra_Omono__propI,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ! [X4: state,Y: state] :
( ( ( sep_greater_state @ Plus @ Y @ X4 )
& ( P @ X4 ) )
=> ( P @ Y ) )
=> ( sep_mono_prop_state @ Plus @ P ) ) ) ).
% sep_algebra.mono_propI
thf(fact_881_sep__algebra_Omono__prop__def,axiom,
! [Plus: state > state > option_state,Core: state > state,P: state > $o] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_mono_prop_state @ Plus @ P )
= ( ! [X3: state,Y4: state] :
( ( ( sep_greater_state @ Plus @ Y4 @ X3 )
& ( P @ X3 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% sep_algebra.mono_prop_def
thf(fact_882_PartialSA_Omax__projection__prop__pure__core,axiom,
sep_ma8214210560313151521_state @ plus @ ( sep_pure_state @ plus ) @ core ).
% PartialSA.max_projection_prop_pure_core
thf(fact_883_sep__algebra_Osmaller__pure__sum__smaller,axiom,
! [Plus: state > state > option_state,Core: state > state,Y3: state,A: state,B: state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ Y3 @ A )
=> ( ( sep_greater_state @ Plus @ Y3 @ B )
=> ( ( ( some_state @ X )
= ( Plus @ A @ B ) )
=> ( ( sep_pure_state @ Plus @ B )
=> ( sep_greater_state @ Plus @ Y3 @ X ) ) ) ) ) ) ).
% sep_algebra.smaller_pure_sum_smaller
thf(fact_884_surj__Compl__image__subset,axiom,
! [F: state > state,A3: set_state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ ( image_state_state @ F @ A3 ) ) @ ( image_state_state @ F @ ( uminus472742206872269241_state @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_885_PartialSA_Oextract__core,axiom,
! [B: state,A: state,X: state] :
( ( ( ( some_state @ B )
= ( plus @ A @ X ) )
& ( greater @ X @ ( core @ B ) ) )
=> ( ( core @ X )
= ( core @ B ) ) ) ).
% PartialSA.extract_core
thf(fact_886_PartialSA_Ominus__exists,axiom,
! [B: state,A: state] :
( ( greater @ B @ A )
=> ? [X4: state] :
( ( ( some_state @ B )
= ( plus @ A @ X4 ) )
& ( greater @ X4 @ ( core @ B ) ) ) ) ).
% PartialSA.minus_exists
thf(fact_887_PartialSA_Ominus__unique,axiom,
! [B: state,A: state,X: state,Y3: state] :
( ( ( ( some_state @ B )
= ( plus @ A @ X ) )
& ( greater @ X @ ( core @ B ) ) )
=> ( ( ( ( some_state @ B )
= ( plus @ A @ Y3 ) )
& ( greater @ Y3 @ ( core @ B ) ) )
=> ( X = Y3 ) ) ) ).
% PartialSA.minus_unique
thf(fact_888_PartialSA_Osmaller__than__core,axiom,
! [Y3: state,X: state,Z: state] :
( ( greater @ Y3 @ X )
=> ( ( ( some_state @ Z )
= ( plus @ X @ ( core @ Y3 ) ) )
=> ( ( core @ Z )
= ( core @ Y3 ) ) ) ) ).
% PartialSA.smaller_than_core
thf(fact_889_PartialSA_Obigger__core__sum__defined,axiom,
! [A: state,B: state] :
( ( greater @ ( core @ A ) @ B )
=> ( ( some_state @ A )
= ( plus @ A @ B ) ) ) ).
% PartialSA.bigger_core_sum_defined
thf(fact_890_PartialSA_Ogreater__than__sum__exists,axiom,
! [A: state,B: state,B1: state,B22: state] :
( ( greater @ A @ B )
=> ( ( ( some_state @ B )
= ( plus @ B1 @ B22 ) )
=> ? [R: state] :
( ( ( some_state @ A )
= ( plus @ R @ B22 ) )
& ( greater @ ( core @ R ) @ ( core @ A ) )
& ( greater @ R @ B1 ) ) ) ) ).
% PartialSA.greater_than_sum_exists
thf(fact_891_PartialSA_Ominus__equiv__def__any__elem,axiom,
! [X: state,A: state,B: state] :
( ( ( some_state @ X )
= ( plus @ A @ B ) )
=> ( ( some_state @ ( minus @ X @ A ) )
= ( plus @ B @ ( core @ X ) ) ) ) ).
% PartialSA.minus_equiv_def_any_elem
thf(fact_892_Greatest__equality,axiom,
! [P: set_state > $o,X: set_state] :
( ( P @ X )
=> ( ! [Y: set_state] :
( ( P @ Y )
=> ( ord_le2494988322063910608_state @ Y @ X ) )
=> ( ( order_2642746146112740183_state @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_893_boolean__algebra_Ocompl__one,axiom,
( ( uminus472742206872269241_state @ top_top_set_state )
= bot_bot_set_state ) ).
% boolean_algebra.compl_one
thf(fact_894_boolean__algebra_Ocompl__zero,axiom,
( ( uminus472742206872269241_state @ bot_bot_set_state )
= top_top_set_state ) ).
% boolean_algebra.compl_zero
thf(fact_895_PartialSA_Omono__pure__cond__def,axiom,
! [B: state > $o] :
( ( packag6595153354952283300_state @ plus @ core @ B )
= ( ! [Phi6: state] :
( ( B @ Phi6 )
= ( B @ ( core @ Phi6 ) ) )
& ! [Phi3: state,Phi6: state,R3: state] :
( ( ( sep_pure_state @ plus @ R3 )
& ( ( some_state @ Phi3 )
= ( plus @ Phi6 @ R3 ) )
& ~ ( B @ Phi6 ) )
=> ~ ( B @ Phi3 ) ) ) ) ).
% PartialSA.mono_pure_cond_def
thf(fact_896_PartialSA_Omono__pure__condI,axiom,
! [B: state > $o] :
( ! [Phi5: state] :
( ( B @ Phi5 )
= ( B @ ( core @ Phi5 ) ) )
=> ( ! [Phi5: state,Phi4: state,R: state] :
( ( ( sep_pure_state @ plus @ R )
& ( ( some_state @ Phi4 )
= ( plus @ Phi5 @ R ) )
& ~ ( B @ Phi5 ) )
=> ~ ( B @ Phi4 ) )
=> ( packag6595153354952283300_state @ plus @ core @ B ) ) ) ).
% PartialSA.mono_pure_condI
thf(fact_897_PartialSA_Obigger__the,axiom,
! [A: state,X7: state,Y3: state,X: state] :
( ( ( some_state @ A )
= ( plus @ X7 @ Y3 ) )
=> ( ( greater @ X7 @ X )
=> ( greater @ ( the_state @ ( plus @ ( core @ A ) @ X7 ) ) @ ( the_state @ ( plus @ ( core @ A ) @ X ) ) ) ) ) ).
% PartialSA.bigger_the
thf(fact_898_Pow__UNIV,axiom,
( ( pow_state @ top_top_set_state )
= top_to5262587396890829782_state ) ).
% Pow_UNIV
thf(fact_899_Pow__empty,axiom,
( ( pow_state @ bot_bot_set_state )
= ( insert_set_state @ bot_bot_set_state @ bot_bo2271482359692755898_state ) ) ).
% Pow_empty
thf(fact_900_Pow__singleton__iff,axiom,
! [X8: set_state,Y7: set_state] :
( ( ( pow_state @ X8 )
= ( insert_set_state @ Y7 @ bot_bo2271482359692755898_state ) )
= ( ( X8 = bot_bot_set_state )
& ( Y7 = bot_bot_set_state ) ) ) ).
% Pow_singleton_iff
thf(fact_901_PowI,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( member_set_state @ A3 @ ( pow_state @ B2 ) ) ) ).
% PowI
thf(fact_902_Pow__iff,axiom,
! [A3: set_state,B2: set_state] :
( ( member_set_state @ A3 @ ( pow_state @ B2 ) )
= ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).
% Pow_iff
thf(fact_903_option_Ocollapse,axiom,
! [Option: option_state] :
( ( Option != none_state )
=> ( ( some_state @ ( the_state @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_904_Pow__bottom,axiom,
! [B2: set_state] : ( member_set_state @ bot_bot_set_state @ ( pow_state @ B2 ) ) ).
% Pow_bottom
thf(fact_905_PowD,axiom,
! [A3: set_state,B2: set_state] :
( ( member_set_state @ A3 @ ( pow_state @ B2 ) )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).
% PowD
thf(fact_906_option_Osel,axiom,
! [X2: state] :
( ( the_state @ ( some_state @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_907_option_Oexpand,axiom,
! [Option: option_state,Option2: option_state] :
( ( ( Option = none_state )
= ( Option2 = none_state ) )
=> ( ( ( Option != none_state )
=> ( ( Option2 != none_state )
=> ( ( the_state @ Option )
= ( the_state @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_908_option_Oexhaust__sel,axiom,
! [Option: option_state] :
( ( Option != none_state )
=> ( Option
= ( some_state @ ( the_state @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_909_Pow__mono,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ord_le5175021213330142598_state @ ( pow_state @ A3 ) @ ( pow_state @ B2 ) ) ) ).
% Pow_mono
thf(fact_910_option_Oset__sel,axiom,
! [A: option_state] :
( ( A != none_state )
=> ( member_state @ ( the_state @ A ) @ ( set_option_state2 @ A ) ) ) ).
% option.set_sel
thf(fact_911_PartialSA_Omono__pure__cond__conj,axiom,
! [Pc: state > $o,B: state > $o] :
( ( packag6595153354952283300_state @ plus @ core @ Pc )
=> ( ( packag6595153354952283300_state @ plus @ core @ B )
=> ( packag6595153354952283300_state @ plus @ core @ ( packag1330488657391168505_state @ Pc @ B ) ) ) ) ).
% PartialSA.mono_pure_cond_conj
thf(fact_912_inj__image__Compl__subset,axiom,
! [F: state > state,A3: set_state] :
( ( inj_on_state_state @ F @ top_top_set_state )
=> ( ord_le2494988322063910608_state @ ( image_state_state @ F @ ( uminus472742206872269241_state @ A3 ) ) @ ( uminus472742206872269241_state @ ( image_state_state @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_913_minus__def,axiom,
( minus
= ( sep_minus_state @ plus @ core ) ) ).
% minus_def
thf(fact_914_inj__Some,axiom,
! [A3: set_state] : ( inj_on3577428053172332983_state @ some_state @ A3 ) ).
% inj_Some
thf(fact_915_sep__algebra_Ominus_Ocong,axiom,
sep_minus_state = sep_minus_state ).
% sep_algebra.minus.cong
thf(fact_916_PartialSA_Obool__conj__def,axiom,
( packag1330488657391168505_state
= ( ^ [A4: state > $o,B3: state > $o,X3: state] :
( ( A4 @ X3 )
& ( B3 @ X3 ) ) ) ) ).
% PartialSA.bool_conj_def
thf(fact_917_sep__algebra_Ominus__core__weaker,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( Core @ ( sep_minus_state @ Plus @ Core @ A @ B ) )
= ( sep_minus_state @ Plus @ Core @ ( Core @ A ) @ ( Core @ B ) ) ) ) ).
% sep_algebra.minus_core_weaker
thf(fact_918_sep__algebra_Ominus__core,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( Core @ ( sep_minus_state @ Plus @ Core @ A @ B ) )
= ( Core @ A ) ) ) ).
% sep_algebra.minus_core
thf(fact_919_range__ex1__eq,axiom,
! [F: state > state,B: state] :
( ( inj_on_state_state @ F @ top_top_set_state )
=> ( ( member_state @ B @ ( image_state_state @ F @ top_top_set_state ) )
= ( ? [X3: state] :
( ( B
= ( F @ X3 ) )
& ! [Y4: state] :
( ( B
= ( F @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_920_inj__image__mem__iff,axiom,
! [F: state > state,A: state,A3: set_state] :
( ( inj_on_state_state @ F @ top_top_set_state )
=> ( ( member_state @ ( F @ A ) @ ( image_state_state @ F @ A3 ) )
= ( member_state @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_921_inj__on__image__mem__iff,axiom,
! [F: state > state,B2: set_state,A: state,A3: set_state] :
( ( inj_on_state_state @ F @ B2 )
=> ( ( member_state @ A @ B2 )
=> ( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( member_state @ ( F @ A ) @ ( image_state_state @ F @ A3 ) )
= ( member_state @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_922_inj__img__insertE,axiom,
! [F: state > state,A3: set_state,X: state,B2: set_state] :
( ( inj_on_state_state @ F @ A3 )
=> ( ~ ( member_state @ X @ B2 )
=> ( ( ( insert_state @ X @ B2 )
= ( image_state_state @ F @ A3 ) )
=> ~ ! [X9: state,A8: set_state] :
( ~ ( member_state @ X9 @ A8 )
=> ( ( A3
= ( insert_state @ X9 @ A8 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_state_state @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_923_sep__algebra_Ominus__equiv__def__any__elem,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ X )
= ( Plus @ A @ B ) )
=> ( ( some_state @ ( sep_minus_state @ Plus @ Core @ X @ A ) )
= ( Plus @ B @ ( Core @ X ) ) ) ) ) ).
% sep_algebra.minus_equiv_def_any_elem
thf(fact_924_sep__algebra_Ominus__default,axiom,
! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ~ ( sep_greater_state @ Plus @ B @ A )
=> ( ( sep_minus_state @ Plus @ Core @ B @ A )
= B ) ) ) ).
% sep_algebra.minus_default
thf(fact_925_sep__algebra_Ominus__smaller,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ X @ A )
=> ( sep_greater_state @ Plus @ X @ ( sep_minus_state @ Plus @ Core @ X @ A ) ) ) ) ).
% sep_algebra.minus_smaller
thf(fact_926_sep__algebra_Ogreater__minus__trans,axiom,
! [Plus: state > state > option_state,Core: state > state,Y3: state,X: state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ Y3 @ X )
=> ( ( sep_greater_state @ Plus @ X @ A )
=> ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ Y3 @ A ) @ ( sep_minus_state @ Plus @ Core @ X @ A ) ) ) ) ) ).
% sep_algebra.greater_minus_trans
thf(fact_927_inj__image__subset__iff,axiom,
! [F: state > state,A3: set_state,B2: set_state] :
( ( inj_on_state_state @ F @ top_top_set_state )
=> ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A3 ) @ ( image_state_state @ F @ B2 ) )
= ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_928_sep__algebra_Ominus__equiv__def,axiom,
! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( sep_greater_state @ Plus @ B @ A )
=> ( ( ( some_state @ B )
= ( Plus @ A @ ( sep_minus_state @ Plus @ Core @ B @ A ) ) )
& ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ B @ A ) @ ( Core @ B ) ) ) ) ) ).
% sep_algebra.minus_equiv_def
thf(fact_929_sep__algebra_Ominus__and__plus,axiom,
! [Plus: state > state > option_state,Core: state > state,Omega: state,Omega2: state,R4: state,A: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ Omega )
= ( Plus @ Omega2 @ R4 ) )
=> ( ( sep_greater_state @ Plus @ Omega2 @ A )
=> ( ( some_state @ ( sep_minus_state @ Plus @ Core @ Omega @ A ) )
= ( Plus @ ( sep_minus_state @ Plus @ Core @ Omega2 @ A ) @ R4 ) ) ) ) ) ).
% sep_algebra.minus_and_plus
thf(fact_930_sep__algebra_Ominus__bigger,axiom,
! [Plus: state > state > option_state,Core: state > state,X: state,A: state,B: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ X )
= ( Plus @ A @ B ) )
=> ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ X @ A ) @ B ) ) ) ).
% sep_algebra.minus_bigger
thf(fact_931_sep__algebra_Ominus__sum,axiom,
! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ A )
= ( Plus @ B @ C ) )
=> ( ( sep_greater_state @ Plus @ X @ A )
=> ( ( sep_minus_state @ Plus @ Core @ X @ A )
= ( sep_minus_state @ Plus @ Core @ ( sep_minus_state @ Plus @ Core @ X @ B ) @ C ) ) ) ) ) ).
% sep_algebra.minus_sum
thf(fact_932_sep__algebra_OminusI,axiom,
! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X: state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ( ( some_state @ B )
= ( Plus @ A @ X ) )
=> ( ( sep_greater_state @ Plus @ X @ ( Core @ B ) )
=> ( X
= ( sep_minus_state @ Plus @ Core @ B @ A ) ) ) ) ) ).
% sep_algebra.minusI
thf(fact_933_inj__on__iff__surj,axiom,
! [A3: set_state,A6: set_state] :
( ( A3 != bot_bot_set_state )
=> ( ( ? [F3: state > state] :
( ( inj_on_state_state @ F3 @ A3 )
& ( ord_le2494988322063910608_state @ ( image_state_state @ F3 @ A3 ) @ A6 ) ) )
= ( ? [G2: state > state] :
( ( image_state_state @ G2 @ A6 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_934_subset__image__inj,axiom,
! [S: set_state,F: state > state,T2: set_state] :
( ( ord_le2494988322063910608_state @ S @ ( image_state_state @ F @ T2 ) )
= ( ? [U: set_state] :
( ( ord_le2494988322063910608_state @ U @ T2 )
& ( inj_on_state_state @ F @ U )
& ( S
= ( image_state_state @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_935_inj__on__insert,axiom,
! [F: state > state,A: state,A3: set_state] :
( ( inj_on_state_state @ F @ ( insert_state @ A @ A3 ) )
= ( ( inj_on_state_state @ F @ A3 )
& ~ ( member_state @ ( F @ A ) @ ( image_state_state @ F @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ A @ bot_bot_set_state ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_936_the__inv__into__into,axiom,
! [F: state > state,A3: set_state,X: state,B2: set_state] :
( ( inj_on_state_state @ F @ A3 )
=> ( ( member_state @ X @ ( image_state_state @ F @ A3 ) )
=> ( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( member_state @ ( the_in3035302284364921129_state @ A3 @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_937_DiffI,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ A3 )
=> ( ~ ( member_state @ C @ B2 )
=> ( member_state @ C @ ( minus_3933957440811877961_state @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_938_Diff__iff,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( minus_3933957440811877961_state @ A3 @ B2 ) )
= ( ( member_state @ C @ A3 )
& ~ ( member_state @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_939_Diff__cancel,axiom,
! [A3: set_state] :
( ( minus_3933957440811877961_state @ A3 @ A3 )
= bot_bot_set_state ) ).
% Diff_cancel
thf(fact_940_empty__Diff,axiom,
! [A3: set_state] :
( ( minus_3933957440811877961_state @ bot_bot_set_state @ A3 )
= bot_bot_set_state ) ).
% empty_Diff
thf(fact_941_Diff__empty,axiom,
! [A3: set_state] :
( ( minus_3933957440811877961_state @ A3 @ bot_bot_set_state )
= A3 ) ).
% Diff_empty
thf(fact_942_insert__Diff1,axiom,
! [X: state,B2: set_state,A3: set_state] :
( ( member_state @ X @ B2 )
=> ( ( minus_3933957440811877961_state @ ( insert_state @ X @ A3 ) @ B2 )
= ( minus_3933957440811877961_state @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_943_Diff__insert0,axiom,
! [X: state,A3: set_state,B2: set_state] :
( ~ ( member_state @ X @ A3 )
=> ( ( minus_3933957440811877961_state @ A3 @ ( insert_state @ X @ B2 ) )
= ( minus_3933957440811877961_state @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_944_Diff__UNIV,axiom,
! [A3: set_state] :
( ( minus_3933957440811877961_state @ A3 @ top_top_set_state )
= bot_bot_set_state ) ).
% Diff_UNIV
thf(fact_945_Diff__eq__empty__iff,axiom,
! [A3: set_state,B2: set_state] :
( ( ( minus_3933957440811877961_state @ A3 @ B2 )
= bot_bot_set_state )
= ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_946_insert__Diff__single,axiom,
! [A: state,A3: set_state] :
( ( insert_state @ A @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ A @ bot_bot_set_state ) ) )
= ( insert_state @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_947_double__diff,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( ord_le2494988322063910608_state @ B2 @ C3 )
=> ( ( minus_3933957440811877961_state @ B2 @ ( minus_3933957440811877961_state @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_948_Diff__subset,axiom,
! [A3: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_949_Diff__mono,axiom,
! [A3: set_state,C3: set_state,D2: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ C3 )
=> ( ( ord_le2494988322063910608_state @ D2 @ B2 )
=> ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A3 @ B2 ) @ ( minus_3933957440811877961_state @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_950_insert__Diff__if,axiom,
! [X: state,B2: set_state,A3: set_state] :
( ( ( member_state @ X @ B2 )
=> ( ( minus_3933957440811877961_state @ ( insert_state @ X @ A3 ) @ B2 )
= ( minus_3933957440811877961_state @ A3 @ B2 ) ) )
& ( ~ ( member_state @ X @ B2 )
=> ( ( minus_3933957440811877961_state @ ( insert_state @ X @ A3 ) @ B2 )
= ( insert_state @ X @ ( minus_3933957440811877961_state @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_951_DiffE,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( minus_3933957440811877961_state @ A3 @ B2 ) )
=> ~ ( ( member_state @ C @ A3 )
=> ( member_state @ C @ B2 ) ) ) ).
% DiffE
thf(fact_952_DiffD1,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( minus_3933957440811877961_state @ A3 @ B2 ) )
=> ( member_state @ C @ A3 ) ) ).
% DiffD1
thf(fact_953_DiffD2,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( minus_3933957440811877961_state @ A3 @ B2 ) )
=> ~ ( member_state @ C @ B2 ) ) ).
% DiffD2
thf(fact_954_diff__shunt__var,axiom,
! [X: set_state,Y3: set_state] :
( ( ( minus_3933957440811877961_state @ X @ Y3 )
= bot_bot_set_state )
= ( ord_le2494988322063910608_state @ X @ Y3 ) ) ).
% diff_shunt_var
thf(fact_955_Diff__insert__absorb,axiom,
! [X: state,A3: set_state] :
( ~ ( member_state @ X @ A3 )
=> ( ( minus_3933957440811877961_state @ ( insert_state @ X @ A3 ) @ ( insert_state @ X @ bot_bot_set_state ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_956_Diff__insert2,axiom,
! [A3: set_state,A: state,B2: set_state] :
( ( minus_3933957440811877961_state @ A3 @ ( insert_state @ A @ B2 ) )
= ( minus_3933957440811877961_state @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ A @ bot_bot_set_state ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_957_insert__Diff,axiom,
! [A: state,A3: set_state] :
( ( member_state @ A @ A3 )
=> ( ( insert_state @ A @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ A @ bot_bot_set_state ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_958_Diff__insert,axiom,
! [A3: set_state,A: state,B2: set_state] :
( ( minus_3933957440811877961_state @ A3 @ ( insert_state @ A @ B2 ) )
= ( minus_3933957440811877961_state @ ( minus_3933957440811877961_state @ A3 @ B2 ) @ ( insert_state @ A @ bot_bot_set_state ) ) ) ).
% Diff_insert
thf(fact_959_subset__Diff__insert,axiom,
! [A3: set_state,B2: set_state,X: state,C3: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ ( minus_3933957440811877961_state @ B2 @ ( insert_state @ X @ C3 ) ) )
= ( ( ord_le2494988322063910608_state @ A3 @ ( minus_3933957440811877961_state @ B2 @ C3 ) )
& ~ ( member_state @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_960_Compl__eq__Diff__UNIV,axiom,
( uminus472742206872269241_state
= ( minus_3933957440811877961_state @ top_top_set_state ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_961_Diff__single__insert,axiom,
! [A3: set_state,X: state,B2: set_state] :
( ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ X @ bot_bot_set_state ) ) @ B2 )
=> ( ord_le2494988322063910608_state @ A3 @ ( insert_state @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_962_subset__insert__iff,axiom,
! [A3: set_state,X: state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ ( insert_state @ X @ B2 ) )
= ( ( ( member_state @ X @ A3 )
=> ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ X @ bot_bot_set_state ) ) @ B2 ) )
& ( ~ ( member_state @ X @ A3 )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_963_Compl__insert,axiom,
! [X: state,A3: set_state] :
( ( uminus472742206872269241_state @ ( insert_state @ X @ A3 ) )
= ( minus_3933957440811877961_state @ ( uminus472742206872269241_state @ A3 ) @ ( insert_state @ X @ bot_bot_set_state ) ) ) ).
% Compl_insert
thf(fact_964_in__image__insert__iff,axiom,
! [B2: set_set_state,X: state,A3: set_state] :
( ! [C6: set_state] :
( ( member_set_state @ C6 @ B2 )
=> ~ ( member_state @ X @ C6 ) )
=> ( ( member_set_state @ A3 @ ( image_2476256681063834599_state @ ( insert_state @ X ) @ B2 ) )
= ( ( member_state @ X @ A3 )
& ( member_set_state @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ X @ bot_bot_set_state ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_965_remove__def,axiom,
( remove_state
= ( ^ [X3: state,A5: set_state] : ( minus_3933957440811877961_state @ A5 @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) ).
% remove_def
thf(fact_966_fun__upd__image,axiom,
! [X: state,A3: set_state,F: state > state,Y3: state] :
( ( ( member_state @ X @ A3 )
=> ( ( image_state_state @ ( fun_upd_state_state @ F @ X @ Y3 ) @ A3 )
= ( insert_state @ Y3 @ ( image_state_state @ F @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ X @ bot_bot_set_state ) ) ) ) ) )
& ( ~ ( member_state @ X @ A3 )
=> ( ( image_state_state @ ( fun_upd_state_state @ F @ X @ Y3 ) @ A3 )
= ( image_state_state @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_967_pairwise__alt,axiom,
( pairwise_state
= ( ^ [R5: state > state > $o,S2: set_state] :
! [X3: state] :
( ( member_state @ X3 @ S2 )
=> ! [Y4: state] :
( ( member_state @ Y4 @ ( minus_3933957440811877961_state @ S2 @ ( insert_state @ X3 @ bot_bot_set_state ) ) )
=> ( R5 @ X3 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_968_psubset__insert__iff,axiom,
! [A3: set_state,X: state,B2: set_state] :
( ( ord_less_set_state @ A3 @ ( insert_state @ X @ B2 ) )
= ( ( ( member_state @ X @ B2 )
=> ( ord_less_set_state @ A3 @ B2 ) )
& ( ~ ( member_state @ X @ B2 )
=> ( ( ( member_state @ X @ A3 )
=> ( ord_less_set_state @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ X @ bot_bot_set_state ) ) @ B2 ) )
& ( ~ ( member_state @ X @ A3 )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_969_finite__remove__induct,axiom,
! [B2: set_state,P: set_state > $o] :
( ( finite_finite_state @ B2 )
=> ( ( P @ bot_bot_set_state )
=> ( ! [A9: set_state] :
( ( finite_finite_state @ A9 )
=> ( ( A9 != bot_bot_set_state )
=> ( ( ord_le2494988322063910608_state @ A9 @ B2 )
=> ( ! [X5: state] :
( ( member_state @ X5 @ A9 )
=> ( P @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X5 @ bot_bot_set_state ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_970_member__remove,axiom,
! [X: state,Y3: state,A3: set_state] :
( ( member_state @ X @ ( remove_state @ Y3 @ A3 ) )
= ( ( member_state @ X @ A3 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_971_finite__insert,axiom,
! [A: state,A3: set_state] :
( ( finite_finite_state @ ( insert_state @ A @ A3 ) )
= ( finite_finite_state @ A3 ) ) ).
% finite_insert
thf(fact_972_psubsetI,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_state @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_973_finite__Diff__insert,axiom,
! [A3: set_state,A: state,B2: set_state] :
( ( finite_finite_state @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ A @ B2 ) ) )
= ( finite_finite_state @ ( minus_3933957440811877961_state @ A3 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_974_finite__option__UNIV,axiom,
( ( finite3180955649987104801_state @ top_to7666338855062656496_state )
= ( finite_finite_state @ top_top_set_state ) ) ).
% finite_option_UNIV
thf(fact_975_infinite__countable__subset,axiom,
! [S: set_state] :
( ~ ( finite_finite_state @ S )
=> ? [F4: nat > state] :
( ( inj_on_nat_state @ F4 @ top_top_set_nat )
& ( ord_le2494988322063910608_state @ ( image_nat_state @ F4 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_976_infinite__iff__countable__subset,axiom,
! [S: set_state] :
( ( ~ ( finite_finite_state @ S ) )
= ( ? [F3: nat > state] :
( ( inj_on_nat_state @ F3 @ top_top_set_nat )
& ( ord_le2494988322063910608_state @ ( image_nat_state @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_977_psubset__imp__ex__mem,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_less_set_state @ A3 @ B2 )
=> ? [B6: state] : ( member_state @ B6 @ ( minus_3933957440811877961_state @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_978_pairwiseI,axiom,
! [S: set_state,R6: state > state > $o] :
( ! [X4: state,Y: state] :
( ( member_state @ X4 @ S )
=> ( ( member_state @ Y @ S )
=> ( ( X4 != Y )
=> ( R6 @ X4 @ Y ) ) ) )
=> ( pairwise_state @ R6 @ S ) ) ).
% pairwiseI
thf(fact_979_pairwiseD,axiom,
! [R6: state > state > $o,S: set_state,X: state,Y3: state] :
( ( pairwise_state @ R6 @ S )
=> ( ( member_state @ X @ S )
=> ( ( member_state @ Y3 @ S )
=> ( ( X != Y3 )
=> ( R6 @ X @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_980_psubsetD,axiom,
! [A3: set_state,B2: set_state,C: state] :
( ( ord_less_set_state @ A3 @ B2 )
=> ( ( member_state @ C @ A3 )
=> ( member_state @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_981_psubsetE,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_less_set_state @ A3 @ B2 )
=> ~ ( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_982_psubset__eq,axiom,
( ord_less_set_state
= ( ^ [A5: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_983_psubset__imp__subset,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_less_set_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_984_psubset__subset__trans,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( ord_less_set_state @ A3 @ B2 )
=> ( ( ord_le2494988322063910608_state @ B2 @ C3 )
=> ( ord_less_set_state @ A3 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_985_subset__not__subset__eq,axiom,
( ord_less_set_state
= ( ^ [A5: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A5 @ B5 )
& ~ ( ord_le2494988322063910608_state @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_986_subset__psubset__trans,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( ord_less_set_state @ B2 @ C3 )
=> ( ord_less_set_state @ A3 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_987_subset__iff__psubset__eq,axiom,
( ord_le2494988322063910608_state
= ( ^ [A5: set_state,B5: set_state] :
( ( ord_less_set_state @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_988_not__psubset__empty,axiom,
! [A3: set_state] :
~ ( ord_less_set_state @ A3 @ bot_bot_set_state ) ).
% not_psubset_empty
thf(fact_989_finite_OinsertI,axiom,
! [A3: set_state,A: state] :
( ( finite_finite_state @ A3 )
=> ( finite_finite_state @ ( insert_state @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_990_top_Oextremum__strict,axiom,
! [A: set_state] :
~ ( ord_less_set_state @ top_top_set_state @ A ) ).
% top.extremum_strict
thf(fact_991_top_Onot__eq__extremum,axiom,
! [A: set_state] :
( ( A != top_top_set_state )
= ( ord_less_set_state @ A @ top_top_set_state ) ) ).
% top.not_eq_extremum
thf(fact_992_bot_Onot__eq__extremum,axiom,
! [A: set_state] :
( ( A != bot_bot_set_state )
= ( ord_less_set_state @ bot_bot_set_state @ A ) ) ).
% bot.not_eq_extremum
thf(fact_993_bot_Oextremum__strict,axiom,
! [A: set_state] :
~ ( ord_less_set_state @ A @ bot_bot_set_state ) ).
% bot.extremum_strict
thf(fact_994_finite_OemptyI,axiom,
finite_finite_state @ bot_bot_set_state ).
% finite.emptyI
thf(fact_995_infinite__imp__nonempty,axiom,
! [S: set_state] :
( ~ ( finite_finite_state @ S )
=> ( S != bot_bot_set_state ) ) ).
% infinite_imp_nonempty
thf(fact_996_leD,axiom,
! [Y3: set_state,X: set_state] :
( ( ord_le2494988322063910608_state @ Y3 @ X )
=> ~ ( ord_less_set_state @ X @ Y3 ) ) ).
% leD
thf(fact_997_nless__le,axiom,
! [A: set_state,B: set_state] :
( ( ~ ( ord_less_set_state @ A @ B ) )
= ( ~ ( ord_le2494988322063910608_state @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_998_antisym__conv1,axiom,
! [X: set_state,Y3: set_state] :
( ~ ( ord_less_set_state @ X @ Y3 )
=> ( ( ord_le2494988322063910608_state @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_999_antisym__conv2,axiom,
! [X: set_state,Y3: set_state] :
( ( ord_le2494988322063910608_state @ X @ Y3 )
=> ( ( ~ ( ord_less_set_state @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_1000_less__le__not__le,axiom,
( ord_less_set_state
= ( ^ [X3: set_state,Y4: set_state] :
( ( ord_le2494988322063910608_state @ X3 @ Y4 )
& ~ ( ord_le2494988322063910608_state @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1001_order_Oorder__iff__strict,axiom,
( ord_le2494988322063910608_state
= ( ^ [A4: set_state,B3: set_state] :
( ( ord_less_set_state @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1002_order_Ostrict__iff__order,axiom,
( ord_less_set_state
= ( ^ [A4: set_state,B3: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1003_order_Ostrict__trans1,axiom,
! [A: set_state,B: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_less_set_state @ B @ C )
=> ( ord_less_set_state @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1004_order_Ostrict__trans2,axiom,
! [A: set_state,B: set_state,C: set_state] :
( ( ord_less_set_state @ A @ B )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ord_less_set_state @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1005_order_Ostrict__iff__not,axiom,
( ord_less_set_state
= ( ^ [A4: set_state,B3: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B3 )
& ~ ( ord_le2494988322063910608_state @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1006_dual__order_Oorder__iff__strict,axiom,
( ord_le2494988322063910608_state
= ( ^ [B3: set_state,A4: set_state] :
( ( ord_less_set_state @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1007_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_state
= ( ^ [B3: set_state,A4: set_state] :
( ( ord_le2494988322063910608_state @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1008_dual__order_Ostrict__trans1,axiom,
! [B: set_state,A: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ B @ A )
=> ( ( ord_less_set_state @ C @ B )
=> ( ord_less_set_state @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1009_dual__order_Ostrict__trans2,axiom,
! [B: set_state,A: set_state,C: set_state] :
( ( ord_less_set_state @ B @ A )
=> ( ( ord_le2494988322063910608_state @ C @ B )
=> ( ord_less_set_state @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1010_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_state
= ( ^ [B3: set_state,A4: set_state] :
( ( ord_le2494988322063910608_state @ B3 @ A4 )
& ~ ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1011_order_Ostrict__implies__order,axiom,
! [A: set_state,B: set_state] :
( ( ord_less_set_state @ A @ B )
=> ( ord_le2494988322063910608_state @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1012_dual__order_Ostrict__implies__order,axiom,
! [B: set_state,A: set_state] :
( ( ord_less_set_state @ B @ A )
=> ( ord_le2494988322063910608_state @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1013_order__le__less,axiom,
( ord_le2494988322063910608_state
= ( ^ [X3: set_state,Y4: set_state] :
( ( ord_less_set_state @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1014_order__less__le,axiom,
( ord_less_set_state
= ( ^ [X3: set_state,Y4: set_state] :
( ( ord_le2494988322063910608_state @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1015_order__less__imp__le,axiom,
! [X: set_state,Y3: set_state] :
( ( ord_less_set_state @ X @ Y3 )
=> ( ord_le2494988322063910608_state @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1016_order__le__neq__trans,axiom,
! [A: set_state,B: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_state @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1017_order__neq__le__trans,axiom,
! [A: set_state,B: set_state] :
( ( A != B )
=> ( ( ord_le2494988322063910608_state @ A @ B )
=> ( ord_less_set_state @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1018_order__le__less__trans,axiom,
! [X: set_state,Y3: set_state,Z: set_state] :
( ( ord_le2494988322063910608_state @ X @ Y3 )
=> ( ( ord_less_set_state @ Y3 @ Z )
=> ( ord_less_set_state @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1019_order__less__le__trans,axiom,
! [X: set_state,Y3: set_state,Z: set_state] :
( ( ord_less_set_state @ X @ Y3 )
=> ( ( ord_le2494988322063910608_state @ Y3 @ Z )
=> ( ord_less_set_state @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1020_order__le__less__subst2,axiom,
! [A: set_state,B: set_state,F: set_state > set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A @ B )
=> ( ( ord_less_set_state @ ( F @ B ) @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_state @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1021_order__less__le__subst1,axiom,
! [A: set_state,F: set_state > set_state,B: set_state,C: set_state] :
( ( ord_less_set_state @ A @ ( F @ B ) )
=> ( ( ord_le2494988322063910608_state @ B @ C )
=> ( ! [X4: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X4 @ Y )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_state @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1022_order__le__imp__less__or__eq,axiom,
! [X: set_state,Y3: set_state] :
( ( ord_le2494988322063910608_state @ X @ Y3 )
=> ( ( ord_less_set_state @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1023_finite__induct__select,axiom,
! [S: set_state,P: set_state > $o] :
( ( finite_finite_state @ S )
=> ( ( P @ bot_bot_set_state )
=> ( ! [T3: set_state] :
( ( ord_less_set_state @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: state] :
( ( member_state @ X5 @ ( minus_3933957440811877961_state @ S @ T3 ) )
& ( P @ ( insert_state @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1024_pairwise__empty,axiom,
! [P: state > state > $o] : ( pairwise_state @ P @ bot_bot_set_state ) ).
% pairwise_empty
thf(fact_1025_pairwise__subset,axiom,
! [P: state > state > $o,S: set_state,T2: set_state] :
( ( pairwise_state @ P @ S )
=> ( ( ord_le2494988322063910608_state @ T2 @ S )
=> ( pairwise_state @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_1026_pairwise__mono,axiom,
! [P: state > state > $o,A3: set_state,Q: state > state > $o,B2: set_state] :
( ( pairwise_state @ P @ A3 )
=> ( ! [X4: state,Y: state] :
( ( P @ X4 @ Y )
=> ( Q @ X4 @ Y ) )
=> ( ( ord_le2494988322063910608_state @ B2 @ A3 )
=> ( pairwise_state @ Q @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_1027_pairwise__insert,axiom,
! [R4: state > state > $o,X: state,S3: set_state] :
( ( pairwise_state @ R4 @ ( insert_state @ X @ S3 ) )
= ( ! [Y4: state] :
( ( ( member_state @ Y4 @ S3 )
& ( Y4 != X ) )
=> ( ( R4 @ X @ Y4 )
& ( R4 @ Y4 @ X ) ) )
& ( pairwise_state @ R4 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1028_finite__range__Some,axiom,
( ( finite3180955649987104801_state @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) )
= ( finite_finite_state @ top_top_set_state ) ) ).
% finite_range_Some
thf(fact_1029_finite__has__maximal,axiom,
! [A3: set_set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ? [X4: set_state] :
( ( member_set_state @ X4 @ A3 )
& ! [Xa2: set_state] :
( ( member_set_state @ Xa2 @ A3 )
=> ( ( ord_le2494988322063910608_state @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1030_finite__has__minimal,axiom,
! [A3: set_set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ? [X4: set_state] :
( ( member_set_state @ X4 @ A3 )
& ! [Xa2: set_state] :
( ( member_set_state @ Xa2 @ A3 )
=> ( ( ord_le2494988322063910608_state @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1031_finite__subset__image,axiom,
! [B2: set_state,F: state > state,A3: set_state] :
( ( finite_finite_state @ B2 )
=> ( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A3 ) )
=> ? [C6: set_state] :
( ( ord_le2494988322063910608_state @ C6 @ A3 )
& ( finite_finite_state @ C6 )
& ( B2
= ( image_state_state @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1032_ex__finite__subset__image,axiom,
! [F: state > state,A3: set_state,P: set_state > $o] :
( ( ? [B5: set_state] :
( ( finite_finite_state @ B5 )
& ( ord_le2494988322063910608_state @ B5 @ ( image_state_state @ F @ A3 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_state] :
( ( finite_finite_state @ B5 )
& ( ord_le2494988322063910608_state @ B5 @ A3 )
& ( P @ ( image_state_state @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1033_all__finite__subset__image,axiom,
! [F: state > state,A3: set_state,P: set_state > $o] :
( ( ! [B5: set_state] :
( ( ( finite_finite_state @ B5 )
& ( ord_le2494988322063910608_state @ B5 @ ( image_state_state @ F @ A3 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_state] :
( ( ( finite_finite_state @ B5 )
& ( ord_le2494988322063910608_state @ B5 @ A3 ) )
=> ( P @ ( image_state_state @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1034_finite_Ocases,axiom,
! [A: set_state] :
( ( finite_finite_state @ A )
=> ( ( A != bot_bot_set_state )
=> ~ ! [A9: set_state] :
( ? [A2: state] :
( A
= ( insert_state @ A2 @ A9 ) )
=> ~ ( finite_finite_state @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1035_finite_Osimps,axiom,
( finite_finite_state
= ( ^ [A4: set_state] :
( ( A4 = bot_bot_set_state )
| ? [A5: set_state,B3: state] :
( ( A4
= ( insert_state @ B3 @ A5 ) )
& ( finite_finite_state @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1036_finite__induct,axiom,
! [F5: set_state,P: set_state > $o] :
( ( finite_finite_state @ F5 )
=> ( ( P @ bot_bot_set_state )
=> ( ! [X4: state,F6: set_state] :
( ( finite_finite_state @ F6 )
=> ( ~ ( member_state @ X4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_state @ X4 @ F6 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_1037_finite__ne__induct,axiom,
! [F5: set_state,P: set_state > $o] :
( ( finite_finite_state @ F5 )
=> ( ( F5 != bot_bot_set_state )
=> ( ! [X4: state] : ( P @ ( insert_state @ X4 @ bot_bot_set_state ) )
=> ( ! [X4: state,F6: set_state] :
( ( finite_finite_state @ F6 )
=> ( ( F6 != bot_bot_set_state )
=> ( ~ ( member_state @ X4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_state @ X4 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1038_infinite__finite__induct,axiom,
! [P: set_state > $o,A3: set_state] :
( ! [A9: set_state] :
( ~ ( finite_finite_state @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_state )
=> ( ! [X4: state,F6: set_state] :
( ( finite_finite_state @ F6 )
=> ( ~ ( member_state @ X4 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_state @ X4 @ F6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1039_pairwise__singleton,axiom,
! [P: state > state > $o,A3: state] : ( pairwise_state @ P @ ( insert_state @ A3 @ bot_bot_set_state ) ) ).
% pairwise_singleton
thf(fact_1040_finite__subset__induct,axiom,
! [F5: set_state,A3: set_state,P: set_state > $o] :
( ( finite_finite_state @ F5 )
=> ( ( ord_le2494988322063910608_state @ F5 @ A3 )
=> ( ( P @ bot_bot_set_state )
=> ( ! [A2: state,F6: set_state] :
( ( finite_finite_state @ F6 )
=> ( ( member_state @ A2 @ A3 )
=> ( ~ ( member_state @ A2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_state @ A2 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1041_finite__subset__induct_H,axiom,
! [F5: set_state,A3: set_state,P: set_state > $o] :
( ( finite_finite_state @ F5 )
=> ( ( ord_le2494988322063910608_state @ F5 @ A3 )
=> ( ( P @ bot_bot_set_state )
=> ( ! [A2: state,F6: set_state] :
( ( finite_finite_state @ F6 )
=> ( ( member_state @ A2 @ A3 )
=> ( ( ord_le2494988322063910608_state @ F6 @ A3 )
=> ( ~ ( member_state @ A2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_state @ A2 @ F6 ) ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1042_finite__UNIV__surj__inj,axiom,
! [F: state > state] :
( ( finite_finite_state @ top_top_set_state )
=> ( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( inj_on_state_state @ F @ top_top_set_state ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1043_finite__UNIV__inj__surj,axiom,
! [F: state > state] :
( ( finite_finite_state @ top_top_set_state )
=> ( ( inj_on_state_state @ F @ top_top_set_state )
=> ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1044_finite__surj__inj,axiom,
! [A3: set_state,F: state > state] :
( ( finite_finite_state @ A3 )
=> ( ( ord_le2494988322063910608_state @ A3 @ ( image_state_state @ F @ A3 ) )
=> ( inj_on_state_state @ F @ A3 ) ) ) ).
% finite_surj_inj
thf(fact_1045_endo__inj__surj,axiom,
! [A3: set_state,F: state > state] :
( ( finite_finite_state @ A3 )
=> ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A3 ) @ A3 )
=> ( ( inj_on_state_state @ F @ A3 )
=> ( ( image_state_state @ F @ A3 )
= A3 ) ) ) ) ).
% endo_inj_surj
thf(fact_1046_finite__empty__induct,axiom,
! [A3: set_state,P: set_state > $o] :
( ( finite_finite_state @ A3 )
=> ( ( P @ A3 )
=> ( ! [A2: state,A9: set_state] :
( ( finite_finite_state @ A9 )
=> ( ( member_state @ A2 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ A2 @ bot_bot_set_state ) ) ) ) ) )
=> ( P @ bot_bot_set_state ) ) ) ) ).
% finite_empty_induct
thf(fact_1047_infinite__coinduct,axiom,
! [X8: set_state > $o,A3: set_state] :
( ( X8 @ A3 )
=> ( ! [A9: set_state] :
( ( X8 @ A9 )
=> ? [X5: state] :
( ( member_state @ X5 @ A9 )
& ( ( X8 @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X5 @ bot_bot_set_state ) ) )
| ~ ( finite_finite_state @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X5 @ bot_bot_set_state ) ) ) ) ) )
=> ~ ( finite_finite_state @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_1048_infinite__remove,axiom,
! [S: set_state,A: state] :
( ~ ( finite_finite_state @ S )
=> ~ ( finite_finite_state @ ( minus_3933957440811877961_state @ S @ ( insert_state @ A @ bot_bot_set_state ) ) ) ) ).
% infinite_remove
thf(fact_1049_remove__induct,axiom,
! [P: set_state > $o,B2: set_state] :
( ( P @ bot_bot_set_state )
=> ( ( ~ ( finite_finite_state @ B2 )
=> ( P @ B2 ) )
=> ( ! [A9: set_state] :
( ( finite_finite_state @ A9 )
=> ( ( A9 != bot_bot_set_state )
=> ( ( ord_le2494988322063910608_state @ A9 @ B2 )
=> ( ! [X5: state] :
( ( member_state @ X5 @ A9 )
=> ( P @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X5 @ bot_bot_set_state ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1050_image__map__upd,axiom,
! [X: state,A3: set_state,M2: state > option_state,Y3: state] :
( ~ ( member_state @ X @ A3 )
=> ( ( image_6076465424260689483_state @ ( fun_up8843634000204221123_state @ M2 @ X @ ( some_state @ Y3 ) ) @ A3 )
= ( image_6076465424260689483_state @ M2 @ A3 ) ) ) ).
% image_map_upd
thf(fact_1051_finite__range__updI,axiom,
! [F: state > option_state,A: state,B: state] :
( ( finite3180955649987104801_state @ ( image_6076465424260689483_state @ F @ top_top_set_state ) )
=> ( finite3180955649987104801_state @ ( image_6076465424260689483_state @ ( fun_up8843634000204221123_state @ F @ A @ ( some_state @ B ) ) @ top_top_set_state ) ) ) ).
% finite_range_updI
thf(fact_1052_cofinite__bot,axiom,
( ( cofinite_state = bot_bot_filter_state )
= ( finite_finite_state @ top_top_set_state ) ) ).
% cofinite_bot
thf(fact_1053_restrict__upd__same,axiom,
! [M2: state > option_state,X: state,Y3: state] :
( ( restri2287918369865870758_state @ ( fun_up8843634000204221123_state @ M2 @ X @ ( some_state @ Y3 ) ) @ ( uminus472742206872269241_state @ ( insert_state @ X @ bot_bot_set_state ) ) )
= ( restri2287918369865870758_state @ M2 @ ( uminus472742206872269241_state @ ( insert_state @ X @ bot_bot_set_state ) ) ) ) ).
% restrict_upd_same
thf(fact_1054_fun__upd__None__restrict,axiom,
! [X: state,D2: set_state,M2: state > option_state] :
( ( ( member_state @ X @ D2 )
=> ( ( fun_up8843634000204221123_state @ ( restri2287918369865870758_state @ M2 @ D2 ) @ X @ none_state )
= ( restri2287918369865870758_state @ M2 @ ( minus_3933957440811877961_state @ D2 @ ( insert_state @ X @ bot_bot_set_state ) ) ) ) )
& ( ~ ( member_state @ X @ D2 )
=> ( ( fun_up8843634000204221123_state @ ( restri2287918369865870758_state @ M2 @ D2 ) @ X @ none_state )
= ( restri2287918369865870758_state @ M2 @ D2 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_1055_restrict__out,axiom,
! [X: state,A3: set_state,M2: state > option_state] :
( ~ ( member_state @ X @ A3 )
=> ( ( restri2287918369865870758_state @ M2 @ A3 @ X )
= none_state ) ) ).
% restrict_out
thf(fact_1056_restrict__map__to__empty,axiom,
! [M2: state > option_state] :
( ( restri2287918369865870758_state @ M2 @ bot_bot_set_state )
= ( ^ [X3: state] : none_state ) ) ).
% restrict_map_to_empty
thf(fact_1057_restrict__map__def,axiom,
( restri2287918369865870758_state
= ( ^ [M3: state > option_state,A5: set_state,X3: state] : ( if_option_state @ ( member_state @ X3 @ A5 ) @ ( M3 @ X3 ) @ none_state ) ) ) ).
% restrict_map_def
thf(fact_1058_restrict__complement__singleton__eq,axiom,
! [F: state > option_state,X: state] :
( ( restri2287918369865870758_state @ F @ ( uminus472742206872269241_state @ ( insert_state @ X @ bot_bot_set_state ) ) )
= ( fun_up8843634000204221123_state @ F @ X @ none_state ) ) ).
% restrict_complement_singleton_eq
thf(fact_1059_dom__fun__upd,axiom,
! [Y3: option_state,F: state > option_state,X: state] :
( ( ( Y3 = none_state )
=> ( ( dom_state_state @ ( fun_up8843634000204221123_state @ F @ X @ Y3 ) )
= ( minus_3933957440811877961_state @ ( dom_state_state @ F ) @ ( insert_state @ X @ bot_bot_set_state ) ) ) )
& ( ( Y3 != none_state )
=> ( ( dom_state_state @ ( fun_up8843634000204221123_state @ F @ X @ Y3 ) )
= ( insert_state @ X @ ( dom_state_state @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_1060_top_Oordering__top__axioms,axiom,
orderi2162033302644881791_state @ ord_le2494988322063910608_state @ ord_less_set_state @ top_top_set_state ).
% top.ordering_top_axioms
thf(fact_1061_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ( ord_le5175021213330142598_state @ A3 @ B2 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ( finite4951987536711252743_state @ B2 )
=> ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A3 ) @ ( lattic1454283544731368441_state @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1062_dom__eq__empty__conv,axiom,
! [F: state > option_state] :
( ( ( dom_state_state @ F )
= bot_bot_set_state )
= ( F
= ( ^ [X3: state] : none_state ) ) ) ).
% dom_eq_empty_conv
thf(fact_1063_fun__upd__None__if__notin__dom,axiom,
! [K: state,M2: state > option_state] :
( ~ ( member_state @ K @ ( dom_state_state @ M2 ) )
=> ( ( fun_up8843634000204221123_state @ M2 @ K @ none_state )
= M2 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_1064_domD,axiom,
! [A: state,M2: state > option_state] :
( ( member_state @ A @ ( dom_state_state @ M2 ) )
=> ? [B6: state] :
( ( M2 @ A )
= ( some_state @ B6 ) ) ) ).
% domD
thf(fact_1065_domI,axiom,
! [M2: state > option_state,A: state,B: state] :
( ( ( M2 @ A )
= ( some_state @ B ) )
=> ( member_state @ A @ ( dom_state_state @ M2 ) ) ) ).
% domI
thf(fact_1066_domIff,axiom,
! [A: state,M2: state > option_state] :
( ( member_state @ A @ ( dom_state_state @ M2 ) )
= ( ( M2 @ A )
!= none_state ) ) ).
% domIff
thf(fact_1067_insert__dom,axiom,
! [F: state > option_state,X: state,Y3: state] :
( ( ( F @ X )
= ( some_state @ Y3 ) )
=> ( ( insert_state @ X @ ( dom_state_state @ F ) )
= ( dom_state_state @ F ) ) ) ).
% insert_dom
thf(fact_1068_finite__map__freshness,axiom,
! [F: state > option_state] :
( ( finite_finite_state @ ( dom_state_state @ F ) )
=> ( ~ ( finite_finite_state @ top_top_set_state )
=> ? [X4: state] :
( ( F @ X4 )
= none_state ) ) ) ).
% finite_map_freshness
thf(fact_1069_dom__minus,axiom,
! [F: state > option_state,X: state,A3: set_state] :
( ( ( F @ X )
= none_state )
=> ( ( minus_3933957440811877961_state @ ( dom_state_state @ F ) @ ( insert_state @ X @ A3 ) )
= ( minus_3933957440811877961_state @ ( dom_state_state @ F ) @ A3 ) ) ) ).
% dom_minus
thf(fact_1070_Sup__fin_OboundedE,axiom,
! [A3: set_set_state,X: set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A3 ) @ X )
=> ! [A10: set_state] :
( ( member_set_state @ A10 @ A3 )
=> ( ord_le2494988322063910608_state @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1071_Sup__fin_OboundedI,axiom,
! [A3: set_set_state,X: set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ! [A2: set_state] :
( ( member_set_state @ A2 @ A3 )
=> ( ord_le2494988322063910608_state @ A2 @ X ) )
=> ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1072_Sup__fin_Obounded__iff,axiom,
! [A3: set_set_state,X: set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A3 ) @ X )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( ord_le2494988322063910608_state @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1073_UnCI,axiom,
! [C: state,B2: set_state,A3: set_state] :
( ( ~ ( member_state @ C @ B2 )
=> ( member_state @ C @ A3 ) )
=> ( member_state @ C @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_1074_Un__iff,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( sup_sup_set_state @ A3 @ B2 ) )
= ( ( member_state @ C @ A3 )
| ( member_state @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_1075_Un__empty,axiom,
! [A3: set_state,B2: set_state] :
( ( ( sup_sup_set_state @ A3 @ B2 )
= bot_bot_set_state )
= ( ( A3 = bot_bot_set_state )
& ( B2 = bot_bot_set_state ) ) ) ).
% Un_empty
thf(fact_1076_Un__subset__iff,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A3 @ B2 ) @ C3 )
= ( ( ord_le2494988322063910608_state @ A3 @ C3 )
& ( ord_le2494988322063910608_state @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_1077_Un__insert__left,axiom,
! [A: state,B2: set_state,C3: set_state] :
( ( sup_sup_set_state @ ( insert_state @ A @ B2 ) @ C3 )
= ( insert_state @ A @ ( sup_sup_set_state @ B2 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_1078_Un__insert__right,axiom,
! [A3: set_state,A: state,B2: set_state] :
( ( sup_sup_set_state @ A3 @ ( insert_state @ A @ B2 ) )
= ( insert_state @ A @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1079_Un__empty__left,axiom,
! [B2: set_state] :
( ( sup_sup_set_state @ bot_bot_set_state @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1080_Un__empty__right,axiom,
! [A3: set_state] :
( ( sup_sup_set_state @ A3 @ bot_bot_set_state )
= A3 ) ).
% Un_empty_right
thf(fact_1081_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_state] :
( ( sup_sup_set_state @ X @ bot_bot_set_state )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1082_subset__Un__eq,axiom,
( ord_le2494988322063910608_state
= ( ^ [A5: set_state,B5: set_state] :
( ( sup_sup_set_state @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1083_subset__UnE,axiom,
! [C3: set_state,A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ C3 @ ( sup_sup_set_state @ A3 @ B2 ) )
=> ~ ! [A8: set_state] :
( ( ord_le2494988322063910608_state @ A8 @ A3 )
=> ! [B8: set_state] :
( ( ord_le2494988322063910608_state @ B8 @ B2 )
=> ( C3
!= ( sup_sup_set_state @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1084_Un__absorb2,axiom,
! [B2: set_state,A3: set_state] :
( ( ord_le2494988322063910608_state @ B2 @ A3 )
=> ( ( sup_sup_set_state @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_1085_Un__absorb1,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( sup_sup_set_state @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1086_Un__upper2,axiom,
! [B2: set_state,A3: set_state] : ( ord_le2494988322063910608_state @ B2 @ ( sup_sup_set_state @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_1087_Un__upper1,axiom,
! [A3: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ A3 @ ( sup_sup_set_state @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_1088_Un__least,axiom,
! [A3: set_state,C3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ C3 )
=> ( ( ord_le2494988322063910608_state @ B2 @ C3 )
=> ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A3 @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_1089_Un__mono,axiom,
! [A3: set_state,C3: set_state,B2: set_state,D2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ C3 )
=> ( ( ord_le2494988322063910608_state @ B2 @ D2 )
=> ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A3 @ B2 ) @ ( sup_sup_set_state @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1090_UnE,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( sup_sup_set_state @ A3 @ B2 ) )
=> ( ~ ( member_state @ C @ A3 )
=> ( member_state @ C @ B2 ) ) ) ).
% UnE
thf(fact_1091_UnI1,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ A3 )
=> ( member_state @ C @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_1092_UnI2,axiom,
! [C: state,B2: set_state,A3: set_state] :
( ( member_state @ C @ B2 )
=> ( member_state @ C @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_1093_Un__UNIV__right,axiom,
! [A3: set_state] :
( ( sup_sup_set_state @ A3 @ top_top_set_state )
= top_top_set_state ) ).
% Un_UNIV_right
thf(fact_1094_Un__UNIV__left,axiom,
! [B2: set_state] :
( ( sup_sup_set_state @ top_top_set_state @ B2 )
= top_top_set_state ) ).
% Un_UNIV_left
thf(fact_1095_singleton__Un__iff,axiom,
! [X: state,A3: set_state,B2: set_state] :
( ( ( insert_state @ X @ bot_bot_set_state )
= ( sup_sup_set_state @ A3 @ B2 ) )
= ( ( ( A3 = bot_bot_set_state )
& ( B2
= ( insert_state @ X @ bot_bot_set_state ) ) )
| ( ( A3
= ( insert_state @ X @ bot_bot_set_state ) )
& ( B2 = bot_bot_set_state ) )
| ( ( A3
= ( insert_state @ X @ bot_bot_set_state ) )
& ( B2
= ( insert_state @ X @ bot_bot_set_state ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1096_Un__singleton__iff,axiom,
! [A3: set_state,B2: set_state,X: state] :
( ( ( sup_sup_set_state @ A3 @ B2 )
= ( insert_state @ X @ bot_bot_set_state ) )
= ( ( ( A3 = bot_bot_set_state )
& ( B2
= ( insert_state @ X @ bot_bot_set_state ) ) )
| ( ( A3
= ( insert_state @ X @ bot_bot_set_state ) )
& ( B2 = bot_bot_set_state ) )
| ( ( A3
= ( insert_state @ X @ bot_bot_set_state ) )
& ( B2
= ( insert_state @ X @ bot_bot_set_state ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1097_insert__is__Un,axiom,
( insert_state
= ( ^ [A4: state] : ( sup_sup_set_state @ ( insert_state @ A4 @ bot_bot_set_state ) ) ) ) ).
% insert_is_Un
thf(fact_1098_Diff__subset__conv,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A3 @ B2 ) @ C3 )
= ( ord_le2494988322063910608_state @ A3 @ ( sup_sup_set_state @ B2 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1099_Diff__partition,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( sup_sup_set_state @ A3 @ ( minus_3933957440811877961_state @ B2 @ A3 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_1100_Compl__partition2,axiom,
! [A3: set_state] :
( ( sup_sup_set_state @ ( uminus472742206872269241_state @ A3 ) @ A3 )
= top_top_set_state ) ).
% Compl_partition2
thf(fact_1101_Compl__partition,axiom,
! [A3: set_state] :
( ( sup_sup_set_state @ A3 @ ( uminus472742206872269241_state @ A3 ) )
= top_top_set_state ) ).
% Compl_partition
thf(fact_1102_Pow__insert,axiom,
! [A: state,A3: set_state] :
( ( pow_state @ ( insert_state @ A @ A3 ) )
= ( sup_su4188871578264421970_state @ ( pow_state @ A3 ) @ ( image_2476256681063834599_state @ ( insert_state @ A ) @ ( pow_state @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_1103_sup__bot__left,axiom,
! [X: set_state] :
( ( sup_sup_set_state @ bot_bot_set_state @ X )
= X ) ).
% sup_bot_left
thf(fact_1104_sup__bot__right,axiom,
! [X: set_state] :
( ( sup_sup_set_state @ X @ bot_bot_set_state )
= X ) ).
% sup_bot_right
thf(fact_1105_bot__eq__sup__iff,axiom,
! [X: set_state,Y3: set_state] :
( ( bot_bot_set_state
= ( sup_sup_set_state @ X @ Y3 ) )
= ( ( X = bot_bot_set_state )
& ( Y3 = bot_bot_set_state ) ) ) ).
% bot_eq_sup_iff
thf(fact_1106_sup__bot_Oright__neutral,axiom,
! [A: set_state] :
( ( sup_sup_set_state @ A @ bot_bot_set_state )
= A ) ).
% sup_bot.right_neutral
thf(fact_1107_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_state,B: set_state] :
( ( bot_bot_set_state
= ( sup_sup_set_state @ A @ B ) )
= ( ( A = bot_bot_set_state )
& ( B = bot_bot_set_state ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1108_sup__bot_Oleft__neutral,axiom,
! [A: set_state] :
( ( sup_sup_set_state @ bot_bot_set_state @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1109_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_state,B: set_state] :
( ( ( sup_sup_set_state @ A @ B )
= bot_bot_set_state )
= ( ( A = bot_bot_set_state )
& ( B = bot_bot_set_state ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1110_sup__eq__bot__iff,axiom,
! [X: set_state,Y3: set_state] :
( ( ( sup_sup_set_state @ X @ Y3 )
= bot_bot_set_state )
= ( ( X = bot_bot_set_state )
& ( Y3 = bot_bot_set_state ) ) ) ).
% sup_eq_bot_iff
thf(fact_1111_UNION__fun__upd,axiom,
! [A3: state > set_state,I: state,B2: set_state,J: set_state] :
( ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ ( fun_up837824997938104489_state @ A3 @ I @ B2 ) @ J ) )
= ( sup_sup_set_state @ ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ A3 @ ( minus_3933957440811877961_state @ J @ ( insert_state @ I @ bot_bot_set_state ) ) ) ) @ ( if_set_state @ ( member_state @ I @ J ) @ B2 @ bot_bot_set_state ) ) ) ).
% UNION_fun_upd
thf(fact_1112_Inf__fin__le__Sup__fin,axiom,
! [A3: set_set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ord_le2494988322063910608_state @ ( lattic4879230916095660051_state @ A3 ) @ ( lattic1454283544731368441_state @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1113_IntI,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ A3 )
=> ( ( member_state @ C @ B2 )
=> ( member_state @ C @ ( inf_inf_set_state @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_1114_Int__iff,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( inf_inf_set_state @ A3 @ B2 ) )
= ( ( member_state @ C @ A3 )
& ( member_state @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1115_UnionI,axiom,
! [X8: set_state,C3: set_set_state,A3: state] :
( ( member_set_state @ X8 @ C3 )
=> ( ( member_state @ A3 @ X8 )
=> ( member_state @ A3 @ ( comple4352483261711748803_state @ C3 ) ) ) ) ).
% UnionI
thf(fact_1116_Union__iff,axiom,
! [A3: state,C3: set_set_state] :
( ( member_state @ A3 @ ( comple4352483261711748803_state @ C3 ) )
= ( ? [X3: set_state] :
( ( member_set_state @ X3 @ C3 )
& ( member_state @ A3 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_1117_inf__bot__right,axiom,
! [X: set_state] :
( ( inf_inf_set_state @ X @ bot_bot_set_state )
= bot_bot_set_state ) ).
% inf_bot_right
thf(fact_1118_inf__bot__left,axiom,
! [X: set_state] :
( ( inf_inf_set_state @ bot_bot_set_state @ X )
= bot_bot_set_state ) ).
% inf_bot_left
thf(fact_1119_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_state] :
( ( inf_inf_set_state @ bot_bot_set_state @ X )
= bot_bot_set_state ) ).
% boolean_algebra.conj_zero_left
thf(fact_1120_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_state] :
( ( inf_inf_set_state @ X @ bot_bot_set_state )
= bot_bot_set_state ) ).
% boolean_algebra.conj_zero_right
thf(fact_1121_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_state] :
( ( ( comple4352483261711748803_state @ A3 )
= bot_bot_set_state )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( X3 = bot_bot_set_state ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1122_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_state] :
( ( bot_bot_set_state
= ( comple4352483261711748803_state @ A3 ) )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( X3 = bot_bot_set_state ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1123_Int__UNIV,axiom,
! [A3: set_state,B2: set_state] :
( ( ( inf_inf_set_state @ A3 @ B2 )
= top_top_set_state )
= ( ( A3 = top_top_set_state )
& ( B2 = top_top_set_state ) ) ) ).
% Int_UNIV
thf(fact_1124_Int__subset__iff,axiom,
! [C3: set_state,A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ C3 @ ( inf_inf_set_state @ A3 @ B2 ) )
= ( ( ord_le2494988322063910608_state @ C3 @ A3 )
& ( ord_le2494988322063910608_state @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1125_Int__insert__left__if0,axiom,
! [A: state,C3: set_state,B2: set_state] :
( ~ ( member_state @ A @ C3 )
=> ( ( inf_inf_set_state @ ( insert_state @ A @ B2 ) @ C3 )
= ( inf_inf_set_state @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1126_Int__insert__left__if1,axiom,
! [A: state,C3: set_state,B2: set_state] :
( ( member_state @ A @ C3 )
=> ( ( inf_inf_set_state @ ( insert_state @ A @ B2 ) @ C3 )
= ( insert_state @ A @ ( inf_inf_set_state @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1127_insert__inter__insert,axiom,
! [A: state,A3: set_state,B2: set_state] :
( ( inf_inf_set_state @ ( insert_state @ A @ A3 ) @ ( insert_state @ A @ B2 ) )
= ( insert_state @ A @ ( inf_inf_set_state @ A3 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_1128_Int__insert__right__if0,axiom,
! [A: state,A3: set_state,B2: set_state] :
( ~ ( member_state @ A @ A3 )
=> ( ( inf_inf_set_state @ A3 @ ( insert_state @ A @ B2 ) )
= ( inf_inf_set_state @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1129_Int__insert__right__if1,axiom,
! [A: state,A3: set_state,B2: set_state] :
( ( member_state @ A @ A3 )
=> ( ( inf_inf_set_state @ A3 @ ( insert_state @ A @ B2 ) )
= ( insert_state @ A @ ( inf_inf_set_state @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1130_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_state] :
( ( inf_inf_set_state @ X @ ( uminus472742206872269241_state @ X ) )
= bot_bot_set_state ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1131_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_state] :
( ( inf_inf_set_state @ ( uminus472742206872269241_state @ X ) @ X )
= bot_bot_set_state ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1132_inf__compl__bot__right,axiom,
! [X: set_state,Y3: set_state] :
( ( inf_inf_set_state @ X @ ( inf_inf_set_state @ Y3 @ ( uminus472742206872269241_state @ X ) ) )
= bot_bot_set_state ) ).
% inf_compl_bot_right
thf(fact_1133_inf__compl__bot__left2,axiom,
! [X: set_state,Y3: set_state] :
( ( inf_inf_set_state @ X @ ( inf_inf_set_state @ ( uminus472742206872269241_state @ X ) @ Y3 ) )
= bot_bot_set_state ) ).
% inf_compl_bot_left2
thf(fact_1134_inf__compl__bot__left1,axiom,
! [X: set_state,Y3: set_state] :
( ( inf_inf_set_state @ ( uminus472742206872269241_state @ X ) @ ( inf_inf_set_state @ X @ Y3 ) )
= bot_bot_set_state ) ).
% inf_compl_bot_left1
thf(fact_1135_Sup__empty,axiom,
( ( comple4352483261711748803_state @ bot_bo2271482359692755898_state )
= bot_bot_set_state ) ).
% Sup_empty
thf(fact_1136_Sup__UNIV,axiom,
( ( comple4352483261711748803_state @ top_to5262587396890829782_state )
= top_top_set_state ) ).
% Sup_UNIV
thf(fact_1137_insert__disjoint_I1_J,axiom,
! [A: state,A3: set_state,B2: set_state] :
( ( ( inf_inf_set_state @ ( insert_state @ A @ A3 ) @ B2 )
= bot_bot_set_state )
= ( ~ ( member_state @ A @ B2 )
& ( ( inf_inf_set_state @ A3 @ B2 )
= bot_bot_set_state ) ) ) ).
% insert_disjoint(1)
thf(fact_1138_insert__disjoint_I2_J,axiom,
! [A: state,A3: set_state,B2: set_state] :
( ( bot_bot_set_state
= ( inf_inf_set_state @ ( insert_state @ A @ A3 ) @ B2 ) )
= ( ~ ( member_state @ A @ B2 )
& ( bot_bot_set_state
= ( inf_inf_set_state @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1139_disjoint__insert_I1_J,axiom,
! [B2: set_state,A: state,A3: set_state] :
( ( ( inf_inf_set_state @ B2 @ ( insert_state @ A @ A3 ) )
= bot_bot_set_state )
= ( ~ ( member_state @ A @ B2 )
& ( ( inf_inf_set_state @ B2 @ A3 )
= bot_bot_set_state ) ) ) ).
% disjoint_insert(1)
thf(fact_1140_disjoint__insert_I2_J,axiom,
! [A3: set_state,B: state,B2: set_state] :
( ( bot_bot_set_state
= ( inf_inf_set_state @ A3 @ ( insert_state @ B @ B2 ) ) )
= ( ~ ( member_state @ B @ A3 )
& ( bot_bot_set_state
= ( inf_inf_set_state @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1141_Diff__disjoint,axiom,
! [A3: set_state,B2: set_state] :
( ( inf_inf_set_state @ A3 @ ( minus_3933957440811877961_state @ B2 @ A3 ) )
= bot_bot_set_state ) ).
% Diff_disjoint
thf(fact_1142_Compl__disjoint2,axiom,
! [A3: set_state] :
( ( inf_inf_set_state @ ( uminus472742206872269241_state @ A3 ) @ A3 )
= bot_bot_set_state ) ).
% Compl_disjoint2
thf(fact_1143_Compl__disjoint,axiom,
! [A3: set_state] :
( ( inf_inf_set_state @ A3 @ ( uminus472742206872269241_state @ A3 ) )
= bot_bot_set_state ) ).
% Compl_disjoint
thf(fact_1144_inf__cancel__left1,axiom,
! [X: set_state,A: set_state,B: set_state] :
( ( inf_inf_set_state @ ( inf_inf_set_state @ X @ A ) @ ( inf_inf_set_state @ ( uminus472742206872269241_state @ X ) @ B ) )
= bot_bot_set_state ) ).
% inf_cancel_left1
thf(fact_1145_inf__cancel__left2,axiom,
! [X: set_state,A: set_state,B: set_state] :
( ( inf_inf_set_state @ ( inf_inf_set_state @ ( uminus472742206872269241_state @ X ) @ A ) @ ( inf_inf_set_state @ X @ B ) )
= bot_bot_set_state ) ).
% inf_cancel_left2
thf(fact_1146_Int__Diff__disjoint,axiom,
! [A3: set_state,B2: set_state] :
( ( inf_inf_set_state @ ( inf_inf_set_state @ A3 @ B2 ) @ ( minus_3933957440811877961_state @ A3 @ B2 ) )
= bot_bot_set_state ) ).
% Int_Diff_disjoint
thf(fact_1147_Diff__triv,axiom,
! [A3: set_state,B2: set_state] :
( ( ( inf_inf_set_state @ A3 @ B2 )
= bot_bot_set_state )
=> ( ( minus_3933957440811877961_state @ A3 @ B2 )
= A3 ) ) ).
% Diff_triv
thf(fact_1148_Un__Int__assoc__eq,axiom,
! [A3: set_state,B2: set_state,C3: set_state] :
( ( ( sup_sup_set_state @ ( inf_inf_set_state @ A3 @ B2 ) @ C3 )
= ( inf_inf_set_state @ A3 @ ( sup_sup_set_state @ B2 @ C3 ) ) )
= ( ord_le2494988322063910608_state @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1149_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_state,A: set_state] :
( ( ( inf_inf_set_state @ ( comple4352483261711748803_state @ B2 ) @ A )
= bot_bot_set_state )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ B2 )
=> ( ( inf_inf_set_state @ X3 @ A )
= bot_bot_set_state ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1150_disjoint__iff__not__equal,axiom,
! [A3: set_state,B2: set_state] :
( ( ( inf_inf_set_state @ A3 @ B2 )
= bot_bot_set_state )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ! [Y4: state] :
( ( member_state @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1151_Int__empty__right,axiom,
! [A3: set_state] :
( ( inf_inf_set_state @ A3 @ bot_bot_set_state )
= bot_bot_set_state ) ).
% Int_empty_right
thf(fact_1152_Int__empty__left,axiom,
! [B2: set_state] :
( ( inf_inf_set_state @ bot_bot_set_state @ B2 )
= bot_bot_set_state ) ).
% Int_empty_left
thf(fact_1153_disjoint__iff,axiom,
! [A3: set_state,B2: set_state] :
( ( ( inf_inf_set_state @ A3 @ B2 )
= bot_bot_set_state )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ~ ( member_state @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1154_Int__emptyI,axiom,
! [A3: set_state,B2: set_state] :
( ! [X4: state] :
( ( member_state @ X4 @ A3 )
=> ~ ( member_state @ X4 @ B2 ) )
=> ( ( inf_inf_set_state @ A3 @ B2 )
= bot_bot_set_state ) ) ).
% Int_emptyI
thf(fact_1155_insert__partition,axiom,
! [X: set_state,F5: set_set_state] :
( ~ ( member_set_state @ X @ F5 )
=> ( ! [X4: set_state] :
( ( member_set_state @ X4 @ ( insert_set_state @ X @ F5 ) )
=> ! [Xa3: set_state] :
( ( member_set_state @ Xa3 @ ( insert_set_state @ X @ F5 ) )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf_set_state @ X4 @ Xa3 )
= bot_bot_set_state ) ) ) )
=> ( ( inf_inf_set_state @ X @ ( comple4352483261711748803_state @ F5 ) )
= bot_bot_set_state ) ) ) ).
% insert_partition
thf(fact_1156_Union__disjoint,axiom,
! [C3: set_set_state,A3: set_state] :
( ( ( inf_inf_set_state @ ( comple4352483261711748803_state @ C3 ) @ A3 )
= bot_bot_set_state )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ C3 )
=> ( ( inf_inf_set_state @ X3 @ A3 )
= bot_bot_set_state ) ) ) ) ).
% Union_disjoint
thf(fact_1157_Union__empty__conv,axiom,
! [A3: set_set_state] :
( ( ( comple4352483261711748803_state @ A3 )
= bot_bot_set_state )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( X3 = bot_bot_set_state ) ) ) ) ).
% Union_empty_conv
thf(fact_1158_empty__Union__conv,axiom,
! [A3: set_set_state] :
( ( bot_bot_set_state
= ( comple4352483261711748803_state @ A3 ) )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( X3 = bot_bot_set_state ) ) ) ) ).
% empty_Union_conv
thf(fact_1159_Union__subsetI,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ! [X4: set_state] :
( ( member_set_state @ X4 @ A3 )
=> ? [Y6: set_state] :
( ( member_set_state @ Y6 @ B2 )
& ( ord_le2494988322063910608_state @ X4 @ Y6 ) ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1160_Union__upper,axiom,
! [B2: set_state,A3: set_set_state] :
( ( member_set_state @ B2 @ A3 )
=> ( ord_le2494988322063910608_state @ B2 @ ( comple4352483261711748803_state @ A3 ) ) ) ).
% Union_upper
thf(fact_1161_Union__least,axiom,
! [A3: set_set_state,C3: set_state] :
( ! [X10: set_state] :
( ( member_set_state @ X10 @ A3 )
=> ( ord_le2494988322063910608_state @ X10 @ C3 ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ C3 ) ) ).
% Union_least
thf(fact_1162_Int__Collect__mono,axiom,
! [A3: set_state,B2: set_state,P: state > $o,Q: state > $o] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ! [X4: state] :
( ( member_state @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ ( collect_state @ P ) ) @ ( inf_inf_set_state @ B2 @ ( collect_state @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1163_Int__greatest,axiom,
! [C3: set_state,A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ C3 @ A3 )
=> ( ( ord_le2494988322063910608_state @ C3 @ B2 )
=> ( ord_le2494988322063910608_state @ C3 @ ( inf_inf_set_state @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1164_Int__absorb2,axiom,
! [A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ( inf_inf_set_state @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_1165_Int__absorb1,axiom,
! [B2: set_state,A3: set_state] :
( ( ord_le2494988322063910608_state @ B2 @ A3 )
=> ( ( inf_inf_set_state @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1166_Int__lower2,axiom,
! [A3: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1167_Int__lower1,axiom,
! [A3: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_1168_Int__mono,axiom,
! [A3: set_state,C3: set_state,B2: set_state,D2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ C3 )
=> ( ( ord_le2494988322063910608_state @ B2 @ D2 )
=> ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ ( inf_inf_set_state @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1169_Int__insert__left,axiom,
! [A: state,C3: set_state,B2: set_state] :
( ( ( member_state @ A @ C3 )
=> ( ( inf_inf_set_state @ ( insert_state @ A @ B2 ) @ C3 )
= ( insert_state @ A @ ( inf_inf_set_state @ B2 @ C3 ) ) ) )
& ( ~ ( member_state @ A @ C3 )
=> ( ( inf_inf_set_state @ ( insert_state @ A @ B2 ) @ C3 )
= ( inf_inf_set_state @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1170_Int__insert__right,axiom,
! [A: state,A3: set_state,B2: set_state] :
( ( ( member_state @ A @ A3 )
=> ( ( inf_inf_set_state @ A3 @ ( insert_state @ A @ B2 ) )
= ( insert_state @ A @ ( inf_inf_set_state @ A3 @ B2 ) ) ) )
& ( ~ ( member_state @ A @ A3 )
=> ( ( inf_inf_set_state @ A3 @ ( insert_state @ A @ B2 ) )
= ( inf_inf_set_state @ A3 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1171_Union__Int__subset,axiom,
! [A3: set_set_state,B2: set_set_state] : ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ ( inf_in8939913482472430008_state @ A3 @ B2 ) ) @ ( inf_inf_set_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_1172_Sup__inter__less__eq,axiom,
! [A3: set_set_state,B2: set_set_state] : ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ ( inf_in8939913482472430008_state @ A3 @ B2 ) ) @ ( inf_inf_set_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1173_UnionE,axiom,
! [A3: state,C3: set_set_state] :
( ( member_state @ A3 @ ( comple4352483261711748803_state @ C3 ) )
=> ~ ! [X10: set_state] :
( ( member_state @ A3 @ X10 )
=> ~ ( member_set_state @ X10 @ C3 ) ) ) ).
% UnionE
thf(fact_1174_IntE,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( inf_inf_set_state @ A3 @ B2 ) )
=> ~ ( ( member_state @ C @ A3 )
=> ~ ( member_state @ C @ B2 ) ) ) ).
% IntE
thf(fact_1175_IntD1,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( inf_inf_set_state @ A3 @ B2 ) )
=> ( member_state @ C @ A3 ) ) ).
% IntD1
thf(fact_1176_IntD2,axiom,
! [C: state,A3: set_state,B2: set_state] :
( ( member_state @ C @ ( inf_inf_set_state @ A3 @ B2 ) )
=> ( member_state @ C @ B2 ) ) ).
% IntD2
thf(fact_1177_Sup__upper2,axiom,
! [U2: set_state,A3: set_set_state,V: set_state] :
( ( member_set_state @ U2 @ A3 )
=> ( ( ord_le2494988322063910608_state @ V @ U2 )
=> ( ord_le2494988322063910608_state @ V @ ( comple4352483261711748803_state @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_1178_Sup__le__iff,axiom,
! [A3: set_set_state,B: set_state] :
( ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ B )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( ord_le2494988322063910608_state @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1179_Sup__upper,axiom,
! [X: set_state,A3: set_set_state] :
( ( member_set_state @ X @ A3 )
=> ( ord_le2494988322063910608_state @ X @ ( comple4352483261711748803_state @ A3 ) ) ) ).
% Sup_upper
thf(fact_1180_Sup__least,axiom,
! [A3: set_set_state,Z: set_state] :
( ! [X4: set_state] :
( ( member_set_state @ X4 @ A3 )
=> ( ord_le2494988322063910608_state @ X4 @ Z ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_1181_Sup__mono,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ! [A2: set_state] :
( ( member_set_state @ A2 @ A3 )
=> ? [X5: set_state] :
( ( member_set_state @ X5 @ B2 )
& ( ord_le2494988322063910608_state @ A2 @ X5 ) ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ).
% Sup_mono
thf(fact_1182_Sup__eqI,axiom,
! [A3: set_set_state,X: set_state] :
( ! [Y: set_state] :
( ( member_set_state @ Y @ A3 )
=> ( ord_le2494988322063910608_state @ Y @ X ) )
=> ( ! [Y: set_state] :
( ! [Z3: set_state] :
( ( member_set_state @ Z3 @ A3 )
=> ( ord_le2494988322063910608_state @ Z3 @ Y ) )
=> ( ord_le2494988322063910608_state @ X @ Y ) )
=> ( ( comple4352483261711748803_state @ A3 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1183_Int__UNIV__left,axiom,
! [B2: set_state] :
( ( inf_inf_set_state @ top_top_set_state @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1184_Int__UNIV__right,axiom,
! [A3: set_state] :
( ( inf_inf_set_state @ A3 @ top_top_set_state )
= A3 ) ).
% Int_UNIV_right
thf(fact_1185_Sup__subset__mono,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ( ord_le5175021213330142598_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_1186_less__eq__Sup,axiom,
! [A3: set_set_state,U2: set_state] :
( ! [V2: set_state] :
( ( member_set_state @ V2 @ A3 )
=> ( ord_le2494988322063910608_state @ U2 @ V2 ) )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ord_le2494988322063910608_state @ U2 @ ( comple4352483261711748803_state @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_1187_SUP__eq,axiom,
! [A3: set_state,B2: set_state,F: state > set_state,G: state > set_state] :
( ! [I2: state] :
( ( member_state @ I2 @ A3 )
=> ? [X5: state] :
( ( member_state @ X5 @ B2 )
& ( ord_le2494988322063910608_state @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: state] :
( ( member_state @ J2 @ B2 )
=> ? [X5: state] :
( ( member_state @ X5 @ A3 )
& ( ord_le2494988322063910608_state @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ F @ A3 ) )
= ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1188_Union__mono,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ( ord_le5175021213330142598_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ).
% Union_mono
thf(fact_1189_Union__empty,axiom,
( ( comple4352483261711748803_state @ bot_bo2271482359692755898_state )
= bot_bot_set_state ) ).
% Union_empty
thf(fact_1190_Union__UNIV,axiom,
( ( comple4352483261711748803_state @ top_to5262587396890829782_state )
= top_top_set_state ) ).
% Union_UNIV
thf(fact_1191_SUP__eq__iff,axiom,
! [I3: set_state,C: set_state,F: state > set_state] :
( ( I3 != bot_bot_set_state )
=> ( ! [I2: state] :
( ( member_state @ I2 @ I3 )
=> ( ord_le2494988322063910608_state @ C @ ( F @ I2 ) ) )
=> ( ( ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ F @ I3 ) )
= C )
= ( ! [X3: state] :
( ( member_state @ X3 @ I3 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1192_inf__shunt,axiom,
! [X: set_state,Y3: set_state] :
( ( ( inf_inf_set_state @ X @ Y3 )
= bot_bot_set_state )
= ( ord_le2494988322063910608_state @ X @ ( uminus472742206872269241_state @ Y3 ) ) ) ).
% inf_shunt
thf(fact_1193_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_state,X: set_state,Y3: set_state] :
( ( ( inf_inf_set_state @ A @ X )
= bot_bot_set_state )
=> ( ( ( sup_sup_set_state @ A @ X )
= top_top_set_state )
=> ( ( ( inf_inf_set_state @ A @ Y3 )
= bot_bot_set_state )
=> ( ( ( sup_sup_set_state @ A @ Y3 )
= top_top_set_state )
=> ( X = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1194_disjoint__eq__subset__Compl,axiom,
! [A3: set_state,B2: set_state] :
( ( ( inf_inf_set_state @ A3 @ B2 )
= bot_bot_set_state )
= ( ord_le2494988322063910608_state @ A3 @ ( uminus472742206872269241_state @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1195_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_state,X: set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ( ord_le2494988322063910608_state @ X @ ( lattic4879230916095660051_state @ A3 ) )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ A3 )
=> ( ord_le2494988322063910608_state @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1196_Inf__fin_OboundedI,axiom,
! [A3: set_set_state,X: set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ! [A2: set_state] :
( ( member_set_state @ A2 @ A3 )
=> ( ord_le2494988322063910608_state @ X @ A2 ) )
=> ( ord_le2494988322063910608_state @ X @ ( lattic4879230916095660051_state @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1197_Inf__fin_OboundedE,axiom,
! [A3: set_set_state,X: set_state] :
( ( finite4951987536711252743_state @ A3 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ( ord_le2494988322063910608_state @ X @ ( lattic4879230916095660051_state @ A3 ) )
=> ! [A10: set_state] :
( ( member_set_state @ A10 @ A3 )
=> ( ord_le2494988322063910608_state @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1198_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_state,Y3: set_state] :
( ( ( inf_inf_set_state @ X @ Y3 )
= bot_bot_set_state )
=> ( ( ( sup_sup_set_state @ X @ Y3 )
= top_top_set_state )
=> ( ( uminus472742206872269241_state @ X )
= Y3 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1199_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ( ord_le5175021213330142598_state @ A3 @ B2 )
=> ( ( A3 != bot_bo2271482359692755898_state )
=> ( ( finite4951987536711252743_state @ B2 )
=> ( ord_le2494988322063910608_state @ ( lattic4879230916095660051_state @ B2 ) @ ( lattic4879230916095660051_state @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1200_Sup__SUP__eq,axiom,
( comple1664268212576242562tate_o
= ( ^ [S2: set_state_o,X3: state] : ( member_state @ X3 @ ( comple4352483261711748803_state @ ( image_7376656169852520768_state @ collect_state @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1201_cSup__least,axiom,
! [X8: set_set_state,Z: set_state] :
( ( X8 != bot_bo2271482359692755898_state )
=> ( ! [X4: set_state] :
( ( member_set_state @ X4 @ X8 )
=> ( ord_le2494988322063910608_state @ X4 @ Z ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ X8 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1202_cSup__eq__non__empty,axiom,
! [X8: set_set_state,A: set_state] :
( ( X8 != bot_bo2271482359692755898_state )
=> ( ! [X4: set_state] :
( ( member_set_state @ X4 @ X8 )
=> ( ord_le2494988322063910608_state @ X4 @ A ) )
=> ( ! [Y: set_state] :
( ! [X5: set_state] :
( ( member_set_state @ X5 @ X8 )
=> ( ord_le2494988322063910608_state @ X5 @ Y ) )
=> ( ord_le2494988322063910608_state @ A @ Y ) )
=> ( ( comple4352483261711748803_state @ X8 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1203_cSUP__least,axiom,
! [A3: set_state,F: state > set_state,M: set_state] :
( ( A3 != bot_bot_set_state )
=> ( ! [X4: state] :
( ( member_state @ X4 @ A3 )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ M ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ F @ A3 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1204_map__add__upd__left,axiom,
! [M2: state,E2: state > option_state,E1: state > option_state,U1: state] :
( ~ ( member_state @ M2 @ ( dom_state_state @ E2 ) )
=> ( ( map_add_state_state @ ( fun_up8843634000204221123_state @ E1 @ M2 @ ( some_state @ U1 ) ) @ E2 )
= ( fun_up8843634000204221123_state @ ( map_add_state_state @ E1 @ E2 ) @ M2 @ ( some_state @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_1205_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea5298875108296682874_state @ inf_inf_set_state @ sup_sup_set_state @ uminus472742206872269241_state @ bot_bot_set_state @ top_top_set_state ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1206_graph__map__upd,axiom,
! [M2: option_state > option_option_state,K: option_state,V: option_state] :
( ( graph_628825510059920660_state @ ( fun_up8433615654909047651_state @ M2 @ K @ ( some_option_state @ V ) ) )
= ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ K @ V ) @ ( graph_628825510059920660_state @ ( fun_up8433615654909047651_state @ M2 @ K @ none_option_state ) ) ) ) ).
% graph_map_upd
thf(fact_1207_in__graphD,axiom,
! [K: option_state,V: option_state,M2: option_state > option_option_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ K @ V ) @ ( graph_628825510059920660_state @ M2 ) )
=> ( ( M2 @ K )
= ( some_option_state @ V ) ) ) ).
% in_graphD
thf(fact_1208_in__graphI,axiom,
! [M2: option_state > option_option_state,K: option_state,V: option_state] :
( ( ( M2 @ K )
= ( some_option_state @ V ) )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ K @ V ) @ ( graph_628825510059920660_state @ M2 ) ) ) ).
% in_graphI
thf(fact_1209_compatible__options_Ocases,axiom,
! [X: produc3142500478612311029_state] :
( ! [A2: state,B6: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ A2 ) @ ( some_state @ B6 ) ) )
=> ( ! [Uv2: option_state] :
( X
!= ( produc9160152616361873709_state @ none_state @ Uv2 ) )
=> ~ ! [Uu2: option_state] :
( X
!= ( produc9160152616361873709_state @ Uu2 @ none_state ) ) ) ) ).
% compatible_options.cases
thf(fact_1210_subrelI,axiom,
! [R4: set_Pr1688445902015331925_state,S3: set_Pr1688445902015331925_state] :
( ! [X4: option_state,Y: option_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X4 @ Y ) @ R4 )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X4 @ Y ) @ S3 ) )
=> ( ord_le6423325748750870005_state @ R4 @ S3 ) ) ).
% subrelI
thf(fact_1211_PartialSA_Osplus_Ocases,axiom,
! [X: produc3142500478612311029_state] :
( ! [Uu2: option_state] :
( X
!= ( produc9160152616361873709_state @ none_state @ Uu2 ) )
=> ( ! [V2: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ V2 ) @ none_state ) )
=> ~ ! [A2: state,B6: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ A2 ) @ ( some_state @ B6 ) ) ) ) ) ).
% PartialSA.splus.cases
thf(fact_1212_sep__algebra_Osplus_Ocases,axiom,
! [Plus: state > state > option_state,Core: state > state,X: produc3142500478612311029_state] :
( ( sep_algebra_state @ Plus @ Core )
=> ( ! [Uu2: option_state] :
( X
!= ( produc9160152616361873709_state @ none_state @ Uu2 ) )
=> ( ! [V2: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ V2 ) @ none_state ) )
=> ~ ! [A2: state,B6: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ A2 ) @ ( some_state @ B6 ) ) ) ) ) ) ).
% sep_algebra.splus.cases
thf(fact_1213_graph__restrictD_I2_J,axiom,
! [K: option_state,V: option_state,M2: option_state > option_option_state,A3: set_option_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ K @ V ) @ ( graph_628825510059920660_state @ ( restri6741197069250724038_state @ M2 @ A3 ) ) )
=> ( ( M2 @ K )
= ( some_option_state @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_1214_Field__insert,axiom,
! [A: option_state,B: option_state,R4: set_Pr1688445902015331925_state] :
( ( field_option_state @ ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ A @ B ) @ R4 ) )
= ( sup_su99422728725954156_state @ ( insert_option_state @ A @ ( insert_option_state @ B @ bot_bo710180891245420500_state ) ) @ ( field_option_state @ R4 ) ) ) ).
% Field_insert
thf(fact_1215_Field__insert,axiom,
! [A: state,B: state,R4: set_Pr1785066336555260981_state] :
( ( field_state @ ( insert7525286303735658661_state @ ( produc544653723139640077_state @ A @ B ) @ R4 ) )
= ( sup_sup_set_state @ ( insert_state @ A @ ( insert_state @ B @ bot_bot_set_state ) ) @ ( field_state @ R4 ) ) ) ).
% Field_insert
thf(fact_1216_Field__empty,axiom,
( ( field_state @ bot_bo9041262728264437921_state )
= bot_bot_set_state ) ).
% Field_empty
thf(fact_1217_FieldI2,axiom,
! [I: state,J3: state,R6: set_Pr1785066336555260981_state] :
( ( member753036827967488894_state @ ( produc544653723139640077_state @ I @ J3 ) @ R6 )
=> ( member_state @ J3 @ ( field_state @ R6 ) ) ) ).
% FieldI2
thf(fact_1218_FieldI2,axiom,
! [I: option_state,J3: option_state,R6: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I @ J3 ) @ R6 )
=> ( member_option_state @ J3 @ ( field_option_state @ R6 ) ) ) ).
% FieldI2
thf(fact_1219_FieldI1,axiom,
! [I: state,J3: state,R6: set_Pr1785066336555260981_state] :
( ( member753036827967488894_state @ ( produc544653723139640077_state @ I @ J3 ) @ R6 )
=> ( member_state @ I @ ( field_state @ R6 ) ) ) ).
% FieldI1
thf(fact_1220_FieldI1,axiom,
! [I: option_state,J3: option_state,R6: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I @ J3 ) @ R6 )
=> ( member_option_state @ I @ ( field_option_state @ R6 ) ) ) ).
% FieldI1
thf(fact_1221_cSup__mono,axiom,
! [B2: set_set_state,A3: set_set_state] :
( ( B2 != bot_bo2271482359692755898_state )
=> ( ( condit2486170202803819276_state @ A3 )
=> ( ! [B6: set_state] :
( ( member_set_state @ B6 @ B2 )
=> ? [X5: set_state] :
( ( member_set_state @ X5 @ A3 )
& ( ord_le2494988322063910608_state @ B6 @ X5 ) ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ B2 ) @ ( comple4352483261711748803_state @ A3 ) ) ) ) ) ).
% cSup_mono
thf(fact_1222_cSup__le__iff,axiom,
! [S: set_set_state,A: set_state] :
( ( S != bot_bo2271482359692755898_state )
=> ( ( condit2486170202803819276_state @ S )
=> ( ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ S ) @ A )
= ( ! [X3: set_state] :
( ( member_set_state @ X3 @ S )
=> ( ord_le2494988322063910608_state @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1223_mono__Field,axiom,
! [R4: set_Pr1785066336555260981_state,S3: set_Pr1785066336555260981_state] :
( ( ord_le2777189432094499797_state @ R4 @ S3 )
=> ( ord_le2494988322063910608_state @ ( field_state @ R4 ) @ ( field_state @ S3 ) ) ) ).
% mono_Field
thf(fact_1224_bdd__above_OI2,axiom,
! [A3: set_state,F: state > set_state,M: set_state] :
( ! [X4: state] :
( ( member_state @ X4 @ A3 )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ M ) )
=> ( condit2486170202803819276_state @ ( image_4774290769506072625_state @ F @ A3 ) ) ) ).
% bdd_above.I2
thf(fact_1225_cSUP__upper,axiom,
! [X: state,A3: set_state,F: state > set_state] :
( ( member_state @ X @ A3 )
=> ( ( condit2486170202803819276_state @ ( image_4774290769506072625_state @ F @ A3 ) )
=> ( ord_le2494988322063910608_state @ ( F @ X ) @ ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ F @ A3 ) ) ) ) ) ).
% cSUP_upper
thf(fact_1226_cSUP__upper2,axiom,
! [F: state > set_state,A3: set_state,X: state,U2: set_state] :
( ( condit2486170202803819276_state @ ( image_4774290769506072625_state @ F @ A3 ) )
=> ( ( member_state @ X @ A3 )
=> ( ( ord_le2494988322063910608_state @ U2 @ ( F @ X ) )
=> ( ord_le2494988322063910608_state @ U2 @ ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ F @ A3 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1227_cSUP__le__iff,axiom,
! [A3: set_state,F: state > set_state,U2: set_state] :
( ( A3 != bot_bot_set_state )
=> ( ( condit2486170202803819276_state @ ( image_4774290769506072625_state @ F @ A3 ) )
=> ( ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ F @ A3 ) ) @ U2 )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ( ord_le2494988322063910608_state @ ( F @ X3 ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1228_cSup__subset__mono,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ( A3 != bot_bo2271482359692755898_state )
=> ( ( condit2486170202803819276_state @ B2 )
=> ( ( ord_le5175021213330142598_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1229_cSUP__subset__mono,axiom,
! [A3: set_state,G: state > set_state,B2: set_state,F: state > set_state] :
( ( A3 != bot_bot_set_state )
=> ( ( condit2486170202803819276_state @ ( image_4774290769506072625_state @ G @ B2 ) )
=> ( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ! [X4: state] :
( ( member_state @ X4 @ A3 )
=> ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ F @ A3 ) ) @ ( comple4352483261711748803_state @ ( image_4774290769506072625_state @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1230_cSup__inter__less__eq,axiom,
! [A3: set_set_state,B2: set_set_state] :
( ( condit2486170202803819276_state @ A3 )
=> ( ( condit2486170202803819276_state @ B2 )
=> ( ( ( inf_in8939913482472430008_state @ A3 @ B2 )
!= bot_bo2271482359692755898_state )
=> ( ord_le2494988322063910608_state @ ( comple4352483261711748803_state @ ( inf_in8939913482472430008_state @ A3 @ B2 ) ) @ ( sup_sup_set_state @ ( comple4352483261711748803_state @ A3 ) @ ( comple4352483261711748803_state @ B2 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1231_Range__insert,axiom,
! [A: option_state,B: option_state,R4: set_Pr1688445902015331925_state] :
( ( range_718955123782594827_state @ ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ A @ B ) @ R4 ) )
= ( insert_option_state @ B @ ( range_718955123782594827_state @ R4 ) ) ) ).
% Range_insert
thf(fact_1232_refl__on__singleton,axiom,
! [X: option_state] : ( refl_on_option_state @ ( insert_option_state @ X @ bot_bo710180891245420500_state ) @ ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ X @ X ) @ bot_bo1080640394036989633_state ) ) ).
% refl_on_singleton
thf(fact_1233_refl__on__singleton,axiom,
! [X: state] : ( refl_on_state @ ( insert_state @ X @ bot_bot_set_state ) @ ( insert7525286303735658661_state @ ( produc544653723139640077_state @ X @ X ) @ bot_bo9041262728264437921_state ) ) ).
% refl_on_singleton
thf(fact_1234_reflD,axiom,
! [R4: set_Pr1688445902015331925_state,A: option_state] :
( ( refl_on_option_state @ top_to7666338855062656496_state @ R4 )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A ) @ R4 ) ) ).
% reflD
thf(fact_1235_reflD,axiom,
! [R4: set_Pr1785066336555260981_state,A: state] :
( ( refl_on_state @ top_top_set_state @ R4 )
=> ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ A ) @ R4 ) ) ).
% reflD
thf(fact_1236_reflI,axiom,
! [R4: set_Pr1688445902015331925_state] :
( ! [X4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X4 @ X4 ) @ R4 )
=> ( refl_on_option_state @ top_to7666338855062656496_state @ R4 ) ) ).
% reflI
thf(fact_1237_reflI,axiom,
! [R4: set_Pr1785066336555260981_state] :
( ! [X4: state] : ( member753036827967488894_state @ ( produc544653723139640077_state @ X4 @ X4 ) @ R4 )
=> ( refl_on_state @ top_top_set_state @ R4 ) ) ).
% reflI
thf(fact_1238_refl__onD2,axiom,
! [A3: set_state,R4: set_Pr1785066336555260981_state,X: state,Y3: state] :
( ( refl_on_state @ A3 @ R4 )
=> ( ( member753036827967488894_state @ ( produc544653723139640077_state @ X @ Y3 ) @ R4 )
=> ( member_state @ Y3 @ A3 ) ) ) ).
% refl_onD2
thf(fact_1239_refl__onD2,axiom,
! [A3: set_option_state,R4: set_Pr1688445902015331925_state,X: option_state,Y3: option_state] :
( ( refl_on_option_state @ A3 @ R4 )
=> ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R4 )
=> ( member_option_state @ Y3 @ A3 ) ) ) ).
% refl_onD2
thf(fact_1240_refl__onD1,axiom,
! [A3: set_state,R4: set_Pr1785066336555260981_state,X: state,Y3: state] :
( ( refl_on_state @ A3 @ R4 )
=> ( ( member753036827967488894_state @ ( produc544653723139640077_state @ X @ Y3 ) @ R4 )
=> ( member_state @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_1241_refl__onD1,axiom,
! [A3: set_option_state,R4: set_Pr1688445902015331925_state,X: option_state,Y3: option_state] :
( ( refl_on_option_state @ A3 @ R4 )
=> ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R4 )
=> ( member_option_state @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_1242_Range__iff,axiom,
! [A: option_state,R4: set_Pr1688445902015331925_state] :
( ( member_option_state @ A @ ( range_718955123782594827_state @ R4 ) )
= ( ? [Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Y4 @ A ) @ R4 ) ) ) ).
% Range_iff
thf(fact_1243_refl__onD,axiom,
! [A3: set_state,R4: set_Pr1785066336555260981_state,A: state] :
( ( refl_on_state @ A3 @ R4 )
=> ( ( member_state @ A @ A3 )
=> ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1244_refl__onD,axiom,
! [A3: set_option_state,R4: set_Pr1688445902015331925_state,A: option_state] :
( ( refl_on_option_state @ A3 @ R4 )
=> ( ( member_option_state @ A @ A3 )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1245_RangeE,axiom,
! [B: option_state,R4: set_Pr1688445902015331925_state] :
( ( member_option_state @ B @ ( range_718955123782594827_state @ R4 ) )
=> ~ ! [A2: option_state] :
~ ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A2 @ B ) @ R4 ) ) ).
% RangeE
thf(fact_1246_Range_Ointros,axiom,
! [A: option_state,B: option_state,R4: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ R4 )
=> ( member_option_state @ B @ ( range_718955123782594827_state @ R4 ) ) ) ).
% Range.intros
thf(fact_1247_Range_Osimps,axiom,
! [A: option_state,R4: set_Pr1688445902015331925_state] :
( ( member_option_state @ A @ ( range_718955123782594827_state @ R4 ) )
= ( ? [A4: option_state,B3: option_state] :
( ( A = B3 )
& ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A4 @ B3 ) @ R4 ) ) ) ) ).
% Range.simps
thf(fact_1248_Range_Ocases,axiom,
! [A: option_state,R4: set_Pr1688445902015331925_state] :
( ( member_option_state @ A @ ( range_718955123782594827_state @ R4 ) )
=> ~ ! [A2: option_state] :
~ ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A2 @ A ) @ R4 ) ) ).
% Range.cases
thf(fact_1249_refl__on__empty,axiom,
refl_on_state @ bot_bot_set_state @ bot_bo9041262728264437921_state ).
% refl_on_empty
thf(fact_1250_linear__order__on__singleton,axiom,
! [X: option_state] : ( order_7502633587729487893_state @ ( insert_option_state @ X @ bot_bo710180891245420500_state ) @ ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ X @ X ) @ bot_bo1080640394036989633_state ) ) ).
% linear_order_on_singleton
thf(fact_1251_linear__order__on__singleton,axiom,
! [X: state] : ( order_7174064491916216133_state @ ( insert_state @ X @ bot_bot_set_state ) @ ( insert7525286303735658661_state @ ( produc544653723139640077_state @ X @ X ) @ bot_bo9041262728264437921_state ) ) ).
% linear_order_on_singleton
thf(fact_1252_vimageI,axiom,
! [F: state > state,A: state,B: state,B2: set_state] :
( ( ( F @ A )
= B )
=> ( ( member_state @ B @ B2 )
=> ( member_state @ A @ ( vimage_state_state @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_1253_vimage__eq,axiom,
! [A: state,F: state > state,B2: set_state] :
( ( member_state @ A @ ( vimage_state_state @ F @ B2 ) )
= ( member_state @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_1254_vimage__UNIV,axiom,
! [F: state > state] :
( ( vimage_state_state @ F @ top_top_set_state )
= top_top_set_state ) ).
% vimage_UNIV
thf(fact_1255_vimage__empty,axiom,
! [F: state > state] :
( ( vimage_state_state @ F @ bot_bot_set_state )
= bot_bot_set_state ) ).
% vimage_empty
thf(fact_1256_image__subset__iff__subset__vimage,axiom,
! [F: state > state,A3: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A3 ) @ B2 )
= ( ord_le2494988322063910608_state @ A3 @ ( vimage_state_state @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1257_vimage__subsetD,axiom,
! [F: state > state,B2: set_state,A3: set_state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ( ord_le2494988322063910608_state @ ( vimage_state_state @ F @ B2 ) @ A3 )
=> ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A3 ) ) ) ) ).
% vimage_subsetD
thf(fact_1258_surj__vimage__empty,axiom,
! [F: state > state,A3: set_state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ( ( vimage_state_state @ F @ A3 )
= bot_bot_set_state )
= ( A3 = bot_bot_set_state ) ) ) ).
% surj_vimage_empty
thf(fact_1259_finite__vimageD,axiom,
! [H: state > state,F5: set_state] :
( ( finite_finite_state @ ( vimage_state_state @ H @ F5 ) )
=> ( ( ( image_state_state @ H @ top_top_set_state )
= top_top_set_state )
=> ( finite_finite_state @ F5 ) ) ) ).
% finite_vimageD
thf(fact_1260_surj__image__vimage__eq,axiom,
! [F: state > state,A3: set_state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ( image_state_state @ F @ ( vimage_state_state @ F @ A3 ) )
= A3 ) ) ).
% surj_image_vimage_eq
thf(fact_1261_lnear__order__on__empty,axiom,
order_7174064491916216133_state @ bot_bot_set_state @ bot_bo9041262728264437921_state ).
% lnear_order_on_empty
thf(fact_1262_vimage__singleton__eq,axiom,
! [A: state,F: state > state,B: state] :
( ( member_state @ A @ ( vimage_state_state @ F @ ( insert_state @ B @ bot_bot_set_state ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_1263_vimageD,axiom,
! [A: state,F: state > state,A3: set_state] :
( ( member_state @ A @ ( vimage_state_state @ F @ A3 ) )
=> ( member_state @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_1264_vimageE,axiom,
! [A: state,F: state > state,B2: set_state] :
( ( member_state @ A @ ( vimage_state_state @ F @ B2 ) )
=> ( member_state @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_1265_vimageI2,axiom,
! [F: state > state,A: state,A3: set_state] :
( ( member_state @ ( F @ A ) @ A3 )
=> ( member_state @ A @ ( vimage_state_state @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_1266_vimage__mono,axiom,
! [A3: set_state,B2: set_state,F: state > state] :
( ( ord_le2494988322063910608_state @ A3 @ B2 )
=> ( ord_le2494988322063910608_state @ ( vimage_state_state @ F @ A3 ) @ ( vimage_state_state @ F @ B2 ) ) ) ).
% vimage_mono
thf(fact_1267_subset__vimage__iff,axiom,
! [A3: set_state,F: state > state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ ( vimage_state_state @ F @ B2 ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ A3 )
=> ( member_state @ ( F @ X3 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1268_finite__vimageD_H,axiom,
! [F: state > state,A3: set_state] :
( ( finite_finite_state @ ( vimage_state_state @ F @ A3 ) )
=> ( ( ord_le2494988322063910608_state @ A3 @ ( image_state_state @ F @ top_top_set_state ) )
=> ( finite_finite_state @ A3 ) ) ) ).
% finite_vimageD'
thf(fact_1269_vimage__subsetI,axiom,
! [F: state > state,B2: set_state,A3: set_state] :
( ( inj_on_state_state @ F @ top_top_set_state )
=> ( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A3 ) )
=> ( ord_le2494988322063910608_state @ ( vimage_state_state @ F @ B2 ) @ A3 ) ) ) ).
% vimage_subsetI
thf(fact_1270_card__vimage__inj,axiom,
! [F: state > state,A3: set_state] :
( ( inj_on_state_state @ F @ top_top_set_state )
=> ( ( ord_le2494988322063910608_state @ A3 @ ( image_state_state @ F @ top_top_set_state ) )
=> ( ( finite_card_state @ ( vimage_state_state @ F @ A3 ) )
= ( finite_card_state @ A3 ) ) ) ) ).
% card_vimage_inj
thf(fact_1271_Linear__order__Well__order__iff,axiom,
! [R4: set_Pr1688445902015331925_state] :
( ( order_7502633587729487893_state @ ( field_option_state @ R4 ) @ R4 )
=> ( ( order_4461428600198971116_state @ ( field_option_state @ R4 ) @ R4 )
= ( ! [A5: set_option_state] :
( ( ord_le7116032884704190368_state @ A5 @ ( field_option_state @ R4 ) )
=> ( ( A5 != bot_bo710180891245420500_state )
=> ? [X3: option_state] :
( ( member_option_state @ X3 @ A5 )
& ! [Y4: option_state] :
( ( member_option_state @ Y4 @ A5 )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ R4 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1272_Linear__order__Well__order__iff,axiom,
! [R4: set_Pr1785066336555260981_state] :
( ( order_7174064491916216133_state @ ( field_state @ R4 ) @ R4 )
=> ( ( order_3972865417222356124_state @ ( field_state @ R4 ) @ R4 )
= ( ! [A5: set_state] :
( ( ord_le2494988322063910608_state @ A5 @ ( field_state @ R4 ) )
=> ( ( A5 != bot_bot_set_state )
=> ? [X3: state] :
( ( member_state @ X3 @ A5 )
& ! [Y4: state] :
( ( member_state @ Y4 @ A5 )
=> ( member753036827967488894_state @ ( produc544653723139640077_state @ X3 @ Y4 ) @ R4 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1273_card__Diff1__le,axiom,
! [A3: set_state,X: state] : ( ord_less_eq_nat @ ( finite_card_state @ ( minus_3933957440811877961_state @ A3 @ ( insert_state @ X @ bot_bot_set_state ) ) ) @ ( finite_card_state @ A3 ) ) ).
% card_Diff1_le
thf(fact_1274_well__order__on__empty,axiom,
order_3972865417222356124_state @ bot_bot_set_state @ bot_bo9041262728264437921_state ).
% well_order_on_empty
thf(fact_1275_card__insert__le,axiom,
! [A3: set_state,X: state] : ( ord_less_eq_nat @ ( finite_card_state @ A3 ) @ ( finite_card_state @ ( insert_state @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_1276_card__bij__eq,axiom,
! [F: state > state,A3: set_state,B2: set_state,G: state > state] :
( ( inj_on_state_state @ F @ A3 )
=> ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A3 ) @ B2 )
=> ( ( inj_on_state_state @ G @ B2 )
=> ( ( ord_le2494988322063910608_state @ ( image_state_state @ G @ B2 ) @ A3 )
=> ( ( finite_finite_state @ A3 )
=> ( ( finite_finite_state @ B2 )
=> ( ( finite_card_state @ A3 )
= ( finite_card_state @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
% Helper facts (5)
thf(help_If_2_1_If_001t__Set__Oset_It__PartialHeapSA__Ostate_J_T,axiom,
! [X: set_state,Y3: set_state] :
( ( if_set_state @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__PartialHeapSA__Ostate_J_T,axiom,
! [X: set_state,Y3: set_state] :
( ( if_set_state @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
! [X: option_state,Y3: option_state] :
( ( if_option_state @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
! [X: option_state,Y3: option_state] :
( ( if_option_state @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_state @ x @ b ).
%------------------------------------------------------------------------------