TPTP Problem File: SLH0483^1.p

%------------------------------------------------------------------------------
% 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_00362_014184__7801228_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2154 ( 557 unt; 872 typ;   0 def)
%            Number of atoms       : 4933 (2064 equ;   0 cnn)
%            Maximal formula atoms :   23 (   3 avg)
%            Number of connectives : 17574 ( 494   ~;  45   |; 813   &;14157   @)
%                                         (   0 <=>;2065  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   8 avg)
%            Number of types       :  125 ( 124 usr)
%            Number of type conns  : 4023 (4023   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  751 ( 748 usr;  35 con; 0-11 aty)
%            Number of variables   : 5706 ( 756   ^;4463   !; 487   ?;5706   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:09:07.038
%------------------------------------------------------------------------------
% Could-be-implicit typings (124)
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    bNF_Gr947463461346416350_state: set_state > ( state > state ) > ( state > produc3882902684156697774_state ) > set_Pr4231240917283394591_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__PartialHeapSA__Ostate_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__PartialHeapSA__Ostate_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr5554302884217825927t_char: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > nat ) > ( produc3882902684156697774_state > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr4913868530397957098_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > option_state ) > ( produc3882902684156697774_state > option_state ) > set_Pr1688445902015331925_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
    bNF_Gr5413576055295359779_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > state ) > ( produc3882902684156697774_state > state > state ) > set_Pr2391701600042252046_state ).

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    bNF_Gr5432878666621363808_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > prat ) > ( produc3882902684156697774_state > state ) > set_Pr6039890582625207499_state ).

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thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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    bNF_Gr919661598790814775_state: set_Pr2391701600042252046_state > set_Pr1688445902015331925_state > ( produc3882902684156697774_state > option_state ) > set_Pr4846723933414538759_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr5630821380644201398_state: set_Pr4231240917283394591_state > set_Pr1688445902015331925_state > ( produc7272280185086996969_state > option_state ) > set_Pr7975217055296730899_state ).

thf(sy_c_Basic__BNFs_Ofsts_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
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thf(sy_c_Basic__BNFs_Ofsts_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_Basic__BNFs_Ofsts_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Basic__BNFs_Ofsts_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Basic__BNFs_Ofsts_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Basic__BNFs_Osnds_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
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thf(sy_c_Basic__BNFs_Osnds_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_Basic__BNFs_Osnds_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Basic__BNFs_Osnds_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Basic__BNFs_Osnds_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_CombinableWands_OR,type,
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thf(sy_c_CombinableWands_Obinary,type,
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thf(sy_c_CombinableWands_Obinary__mask,type,
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thf(sy_c_CombinableWands_Ocombinable,type,
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thf(sy_c_CombinableWands_Ocomp__min__mask,type,
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thf(sy_c_CombinableWands_Ocwand,type,
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thf(sy_c_CombinableWands_Ointuitionistic,type,
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thf(sy_c_CombinableWands_Omultiply,type,
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thf(sy_c_CombinableWands_Omultiply__rel,type,
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thf(sy_c_CombinableWands_Omultiply__sem__assertion,type,
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thf(sy_c_CombinableWands_Omultiply__sem__assertion__rel,type,
    multip8073008153270241703on_rel: produc6251377953716854561_state > produc6251377953716854561_state > $o ).

thf(sy_c_CombinableWands_Oscalable,type,
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thf(sy_c_CombinableWands_Oscaled,type,
    scaled: state > set_state ).

thf(sy_c_CombinableWands_Owand,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
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thf(sy_c_Fun_Oid_001t__Mask__Oval,type,
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thf(sy_c_Fun_Oid_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
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thf(sy_c_Fun__Def_OTHE__default_001t__Mask__Oval,type,
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thf(sy_c_Fun__Def_OTHE__default_001t__PartialHeapSA__Ostate,type,
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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__PartialHeapSA__Ostate,type,
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thf(sy_c_If_001t__PosRat__Oprat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Mask__Oval_Mt__Product____Type__Oprod_It__Mask__Oval_M_062_It__Mask__Oval_Mt__Mask__Oval_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
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thf(sy_c_Map_Omap__add_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
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thf(sy_c_PackageLogic_Oaassertion_OAStar_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    packag3677137384183703351on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val ) > aasser1163550960519286933on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    packag2518615827549365843_state: ( produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state ) > aasser3179708585289803787_state > produc3882902684156697774_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    packag4865960244449834244_state: ( produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state ) > aasser7019605857572776418_state > produc7272280185086996969_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat__rel_001t__Mask__Oval,type,
    package_sat_rel_val: produc1707155752992360906al_val > produc1707155752992360906al_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat__rel_001t__PartialHeapSA__Ostate,type,
    packag1918864394588125018_state: produc2756261550280135502_state > produc2756261550280135502_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Ovalid__package__set_001t__Mask__Oval,type,
    packag7517340229427718612et_val: ( val > val > option_val ) > ( val > val ) > val > set_Pr4723104092121316667al_val > val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Ovalid__package__set_001t__PartialHeapSA__Ostate,type,
    packag3877571396724326311_state: ( state > state > option_state ) > ( state > state ) > state > set_Pr4231240917283394591_state > state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__Mask__Oval,type,
    packag5732365195153136185on_val: ( val > val > option_val ) > ( val > val ) > aassertion_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__PartialHeapSA__Ostate,type,
    packag7298478212905792770_state: ( state > state > option_state ) > ( state > state ) > aassertion_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    packag5541956234368469604on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > aasser1163550960519286933on_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    packag8517968027831007846_state: ( produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state ) > ( produc3882902684156697774_state > produc3882902684156697774_state ) > aasser3179708585289803787_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    packag6254535061403973169_state: ( produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state ) > ( produc7272280185086996969_state > produc7272280185086996969_state ) > aasser7019605857572776418_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion__rel_001t__PartialHeapSA__Ostate,type,
    packag4834522500328176237_state: aassertion_state > aassertion_state > $o ).

thf(sy_c_PartialHeapSA_Oadd__set,type,
    add_set: set_state > set_state > set_state ).

thf(sy_c_PartialHeapSA_Oadd__states,type,
    add_states: produc9003781588162623004on_val > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_PartialHeapSA_Oadd__states__rel,type,
    add_states_rel: produc1317171644805857433on_val > produc1317171644805857433on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible,type,
    compatible: produc9003781588162623004on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__heaps,type,
    compatible_heaps: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options_001t__Mask__Oval,type,
    compat7209469420100717861ns_val: option_val > option_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options_001t__PartialHeapSA__Ostate,type,
    compat2278460363914054422_state: option_state > option_state > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__Mask__Oval,type,
    compat8294034345482142370el_val: produc5122021904355189735on_val > produc5122021904355189735on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__PartialHeapSA__Ostate,type,
    compat1844074344818064729_state: produc3142500478612311029_state > produc3142500478612311029_state > $o ).

thf(sy_c_PartialHeapSA_Ocore,type,
    core: state > state ).

thf(sy_c_PartialHeapSA_Odefined,type,
    defined: state > state > $o ).

thf(sy_c_PartialHeapSA_Oempty__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
    empty_7512026568161686585ar_val: produc1457211279475724562t_char > option_val ).

thf(sy_c_PartialHeapSA_Oget__h,type,
    get_h: state > produc1457211279475724562t_char > option_val ).

thf(sy_c_PartialHeapSA_Oget__h__rel,type,
    get_h_rel: state > state > $o ).

thf(sy_c_PartialHeapSA_Oget__m,type,
    get_m: state > produc1457211279475724562t_char > prat ).

thf(sy_c_PartialHeapSA_Oget__m__rel,type,
    get_m_rel: state > state > $o ).

thf(sy_c_PartialHeapSA_Ogreater,type,
    greater: state > state > $o ).

thf(sy_c_PartialHeapSA_Ogreater__set,type,
    greater_set: set_state > set_state > $o ).

thf(sy_c_PartialHeapSA_Olarger__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
    larger1532547405891354081ar_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Ominus,type,
    minus: state > state > state ).

thf(sy_c_PartialHeapSA_Oplus,type,
    plus: state > state > option_state ).

thf(sy_c_PartialHeapSA_Ostable,type,
    stable: state > $o ).

thf(sy_c_PartialHeapSA_Ostate_OAbs__state,type,
    abs_state: produc9003781588162623004on_val > state ).

thf(sy_c_PartialHeapSA_Ostate_ORep__state,type,
    rep_state: state > produc9003781588162623004on_val ).

thf(sy_c_PartialHeapSA_Ounit,type,
    unit: state ).

thf(sy_c_PartialHeapSA_Ovalid__heap,type,
    valid_heap: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Ovalid__state,type,
    valid_state: produc9003781588162623004on_val > $o ).

thf(sy_c_PartialHeapSA_Ovalid__state__rel,type,
    valid_state_rel: produc9003781588162623004on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_PosRat_Ocomp__one,type,
    comp_one: prat > prat ).

thf(sy_c_PosRat_Ohalf,type,
    half: prat ).

thf(sy_c_PosRat_Opadd,type,
    padd: prat > prat > prat ).

thf(sy_c_PosRat_Opdiv,type,
    pdiv: prat > prat > prat ).

thf(sy_c_PosRat_Opgt,type,
    pgt: prat > prat > $o ).

thf(sy_c_PosRat_Opgte,type,
    pgte: prat > prat > $o ).

thf(sy_c_PosRat_Opinv,type,
    pinv: prat > prat ).

thf(sy_c_PosRat_Opmax,type,
    pmax: prat > prat > prat ).

thf(sy_c_PosRat_Opmin,type,
    pmin: prat > prat > prat ).

thf(sy_c_PosRat_Opmult,type,
    pmult: prat > prat > prat ).

thf(sy_c_PosRat_Opnone,type,
    pnone: prat ).

thf(sy_c_PosRat_Oppos,type,
    ppos: prat > $o ).

thf(sy_c_PosRat_Opwrite,type,
    pwrite: prat ).

thf(sy_c_Product__Type_OPair_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
    produc5362444129537099455_state: ( state > state ) > ( state > state ) > produc948624807956558599_state ).

thf(sy_c_Product__Type_OPair_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_001t__PartialHeapSA__Ostate,type,
    produc4737362649269059174_state: ( state > state ) > state > produc5876769945872133550_state ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    produc8008623296994167373on_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > produc1757238238866670357on_val ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    produc1897429902272487136r_prat: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > produc7776263825685906734r_prat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    produc74620467203698198on_val: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    produc1164665353665699927r_prat: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > produc3336119888008106215r_prat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    produc2957999048406202538t_char: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > produc5803078220529002682t_char ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    produc2591414526589101846t_char: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > produc2489117125269924006t_char ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    produc1606739095562851931t_char: list_char > list_char > produc6859378861088059299t_char ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    produc8702345860417625034ar_nat: list_char > nat > produc7606469099237612818ar_nat ).

thf(sy_c_Product__Type_OPair_001t__Mask__Oval_001_062_It__Mask__Oval_Mt__Mask__Oval_J,type,
    produc303288909735261391al_val: val > ( val > val ) > produc5503543764249810261al_val ).

thf(sy_c_Product__Type_OPair_001t__Mask__Oval_001t__Product____Type__Oprod_It__Mask__Oval_M_062_It__Mask__Oval_Mt__Mask__Oval_J_J,type,
    produc7187740677493149823al_val: val > produc5503543764249810261al_val > produc2072440392815185925al_val ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    produc120197956887798346t_char: nat > list_char > produc1457211279475724562t_char ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    produc904852209897547607on_val: option_val > option_val > produc5122021904355189735on_val ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    produc9160152616361873709_state: option_state > option_state > produc3142500478612311029_state ).

thf(sy_c_Product__Type_OPair_001t__PackageLogic__Oaassertion_It__Mask__Oval_J_001t__Mask__Oval,type,
    produc1969469836606013754al_val: aassertion_val > val > produc1707155752992360906al_val ).

thf(sy_c_Product__Type_OPair_001t__PackageLogic__Oaassertion_It__PartialHeapSA__Ostate_J_001t__PartialHeapSA__Ostate,type,
    produc6654088501289988998_state: aassertion_state > state > produc2756261550280135502_state ).

thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
    produc8632743443909064678_state: state > ( state > state ) > produc3882902684156697774_state ).

thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    produc544653723139640077_state: state > state > produc8023240190789890773_state ).

thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001t__PosRat__Oprat,type,
    produc5862297150877942435e_prat: state > prat > produc7334356684927367531e_prat ).

thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    produc5886875603596089627_state: state > produc3882902684156697774_state > produc7272280185086996969_state ).

thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    produc5194695453622026300_state: state > produc7272280185086996969_state > produc4357056439811976324_state ).

thf(sy_c_Product__Type_OPair_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
    produc6205719116230455971_state: prat > state > produc4844402304972151915_state ).

thf(sy_c_Product__Type_OPair_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
    produc7183142570155660857t_prat: prat > prat > produc2235545006120619777t_prat ).

thf(sy_c_Product__Type_OPair_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    produc9080284787853255769_state: prat > set_state > produc6251377953716854561_state ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_001_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    produc6949276712527400972on_val: produc9003781588162623004on_val > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > produc5221327116631553236on_val ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    produc4958730763592889809on_val: produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ).

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thf(sy_c_Product__Type_Oprod_Oswap_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Product__Type_Oprod_Oswap_001t__Set__Oset_It__PartialHeapSA__Ostate_J_001t__PosRat__Oprat,type,
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thf(sy_c_Product__Type_Oproduct_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Relation_OField_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Relation_OField_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Relation_OField_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
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thf(sy_c_Relation_OId__on_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Relation_Oconversep_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Relation_Oconversep_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_Relation_Oconversep_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Relation_Oconversep_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Relation_Oconversep_001t__PartialHeapSA__Ostate_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
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thf(sy_c_Relation_Oconversep_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Relation_Oconversep_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Relation_Orefl__on_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Relation_Orefl__on_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Oadd__set_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Oadd__set_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Odefined_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Odefined_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Oequiv_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Ogreater_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Ogreater_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Omax__projection__prop_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Ominus_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Ominus_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Omono__prop_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Opure_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Opure_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Osetify_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Osplus_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Osplus_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Osplus__rel_001t__Mask__Oval,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Osplus__rel_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Ounder_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Oup__closed_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_SepAlgebra_Osep__algebra_Oupper__closure_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Set_OCollect_001t__Mask__Oval,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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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__092_060phi_062____,type,
    phi: state ).

thf(sy_v_a____,type,
    a2: state ).

thf(sy_v_hl____,type,
    hl: produc1457211279475724562t_char ).

thf(sy_v_p____,type,
    p: prat ).

thf(sy_v_w____,type,
    w: state ).

thf(sy_v_x____,type,
    x: state ).

% Relevant facts (1273)
thf(fact_0_assms_I2_J,axiom,
    intuitionistic @ b ).

% assms(2)
thf(fact_1_assms_I1_J,axiom,
    binary @ a ).

% assms(1)
thf(fact_2_True,axiom,
    ( ( get_m @ a2 @ hl )
    = pwrite ) ).

% True
thf(fact_3_False,axiom,
    ~ ( scalable @ w @ a2 ) ).

% False
thf(fact_4__092_060open_062ppos_Ap_092_060close_062,axiom,
    ppos @ p ).

% \<open>ppos p\<close>
thf(fact_5__092_060open_062pgt_A_Ipadd_A_Iget__m_A_Imultiply_Ap_Aw_J_Ahl_J_A_Iget__m_Aa_Ahl_J_J_Apwrite_092_060close_062,axiom,
    pgt @ ( padd @ ( get_m @ ( multiply @ p @ w ) @ hl ) @ ( get_m @ a2 @ hl ) ) @ pwrite ).

% \<open>pgt (padd (get_m (multiply p w) hl) (get_m a hl)) pwrite\<close>
thf(fact_6__092_060open_062_092_060phi_062_A_092_060in_062_Ascaled_Aw_092_060close_062,axiom,
    member_state @ phi @ ( scaled @ w ) ).

% \<open>\<phi> \<in> scaled w\<close>
thf(fact_7__092_060open_062get__m_Aw_Ahl_A_092_060noteq_062_Apnone_092_060close_062,axiom,
    ( ( get_m @ w @ hl )
   != pnone ) ).

% \<open>get_m w hl \<noteq> pnone\<close>
thf(fact_8__092_060open_062multiply_Ap_Aw_A_124_D_124_Aa_092_060close_062,axiom,
    defined @ ( multiply @ p @ w ) @ a2 ).

% \<open>multiply p w |#| a\<close>
thf(fact_9__092_060open_062pgte_Apwrite_Ap_092_060close_062,axiom,
    pgte @ pwrite @ p ).

% \<open>pgte pwrite p\<close>
thf(fact_10__092_060open_062ppos_A_Iget__m_A_Imultiply_Ap_Aw_J_Ahl_J_092_060close_062,axiom,
    ppos @ ( get_m @ ( multiply @ p @ w ) @ hl ) ).

% \<open>ppos (get_m (multiply p w) hl)\<close>
thf(fact_11__092_060open_062a_A_124_D_124_A_092_060phi_062_092_060close_062,axiom,
    defined @ a2 @ phi ).

% \<open>a |#| \<phi>\<close>
thf(fact_12_upper__valid,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( ( valid_mask @ B )
          & ( valid_mask @ C ) ) ) ) ).

% upper_valid
thf(fact_13_upper__valid__aux,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( valid_mask @ B ) ) ) ).

% upper_valid_aux
thf(fact_14_calculation,axiom,
    ppos @ ( get_m @ w @ hl ) ).

% calculation
thf(fact_15__092_060open_062get__m_A_Imultiply_Ap_A_092_060phi_062_J_A_061_Amultiply__mask_Ap_A_Iget__m_A_092_060phi_062_J_092_060close_062,axiom,
    ( ( get_m @ ( multiply @ p @ phi ) )
    = ( multiply_mask @ p @ ( get_m @ phi ) ) ) ).

% \<open>get_m (multiply p \<phi>) = multiply_mask p (get_m \<phi>)\<close>
thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060phi_062_O_A_092_060lbrakk_062_092_060phi_062_A_092_060in_062_Ascaled_Aw_059_Aa_A_124_D_124_A_092_060phi_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Phi: state] :
        ( ( member_state @ Phi @ ( scaled @ w ) )
       => ~ ( defined @ a2 @ Phi ) ) ).

% \<open>\<And>thesis. (\<And>\<phi>. \<lbrakk>\<phi> \<in> scaled w; a |#| \<phi>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_17__092_060open_062w_A_092_060in_062_Awand_AA_AB_092_060close_062,axiom,
    member_state @ w @ ( wand @ a @ b ) ).

% \<open>w \<in> wand A B\<close>
thf(fact_18_compatible__same__mask,axiom,
    ! [A: produc1457211279475724562t_char > prat,W: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ ( add_ma6586698345353345811t_char @ A @ W ) )
     => ( W
        = ( comp_min_mask @ A @ W ) ) ) ).

% compatible_same_mask
thf(fact_19_valid__mask__add__comp__min,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( valid_mask @ B )
       => ( valid_mask @ ( add_ma6586698345353345811t_char @ ( comp_min_mask @ B @ A ) @ B ) ) ) ) ).

% valid_mask_add_comp_min
thf(fact_20_add__masks__asso,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( add_ma6586698345353345811t_char @ ( add_ma6586698345353345811t_char @ A @ B ) @ C )
      = ( add_ma6586698345353345811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) ) ) ).

% add_masks_asso
thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_O_A_092_060lbrakk_062ppos_Ap_059_Apgte_Apwrite_Ap_059_Amultiply_Ap_Aw_A_124_D_124_Aa_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [P: prat] :
        ( ( ppos @ P )
       => ( ( pgte @ pwrite @ P )
         => ~ ( defined @ ( multiply @ P @ w ) @ a2 ) ) ) ).

% \<open>\<And>thesis. (\<And>p. \<lbrakk>ppos p; pgte pwrite p; multiply p w |#| a\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_22_valid__mult,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,P2: prat] :
      ( ( valid_mask @ Pi )
     => ( ( pgte @ pwrite @ P2 )
       => ( valid_mask @ ( multiply_mask @ P2 @ Pi ) ) ) ) ).

% valid_mult
thf(fact_23_mult__write__mask,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ pwrite @ Pi )
      = Pi ) ).

% mult_write_mask
thf(fact_24_scalable__def,axiom,
    ( scalable
    = ( ^ [W2: state,A2: state] :
        ! [X: state] :
          ( ( member_state @ X @ ( scaled @ W2 ) )
         => ~ ( defined @ A2 @ X ) ) ) ) ).

% scalable_def
thf(fact_25_non__scalable__instantiate,axiom,
    ! [W: state,A: state] :
      ( ~ ( scalable @ W @ A )
     => ? [P: prat] :
          ( ( ppos @ P )
          & ( pgte @ pwrite @ P )
          & ( defined @ A @ ( multiply @ P @ W ) ) ) ) ).

% non_scalable_instantiate
thf(fact_26_w__in__scaled,axiom,
    ! [W: state] : ( member_state @ W @ ( scaled @ W ) ) ).

% w_in_scaled
thf(fact_27_mult__add__states,axiom,
    ! [A: prat,B: prat,Pi: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ ( padd @ A @ B ) @ Pi )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ Pi ) @ ( multiply_mask @ B @ Pi ) ) ) ).

% mult_add_states
thf(fact_28_mult__distr__masks,axiom,
    ! [A: prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ B ) @ ( multiply_mask @ A @ C ) ) ) ).

% mult_distr_masks
thf(fact_29_get__m__smaller__mask,axiom,
    ! [P2: prat,A: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( get_m @ ( multiply @ P2 @ A ) )
        = ( multiply_mask @ P2 @ ( get_m @ A ) ) ) ) ).

% get_m_smaller_mask
thf(fact_30_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( padd @ ( Pi2 @ Hl ) @ ( Pi3 @ Hl ) ) ) ) ).

% add_masks.simps
thf(fact_31_add__masks_Oelims,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( padd @ ( X2 @ Xb ) @ ( Xa @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_32_add__masks__cancellative,axiom,
    ! [B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ B @ C )
        = ( add_ma6586698345353345811t_char @ B @ D ) )
     => ( C = D ) ) ).

% add_masks_cancellative
thf(fact_33_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B2 @ A2 ) ) ) ).

% add_masks_comm
thf(fact_34_pgte__pgt,axiom,
    ! [A: prat,B: prat,C: prat,D: prat] :
      ( ( pgt @ A @ B )
     => ( ( pgte @ C @ D )
       => ( pgt @ ( padd @ A @ C ) @ ( padd @ B @ D ) ) ) ) ).

% pgte_pgt
thf(fact_35_valid__sum_I1_J,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,Ma: prat,Mb: prat] :
      ( ( valid_mask @ A )
     => ( ( valid_mask @ B )
       => ( ( upper_bounded @ A @ Ma )
         => ( ( upper_bounded @ B @ Mb )
           => ( ( pgte @ pwrite @ ( padd @ Ma @ Mb ) )
             => ( valid_mask @ ( add_ma6586698345353345811t_char @ A @ B ) ) ) ) ) ) ) ).

% valid_sum(1)
thf(fact_36_valid__sum_I2_J,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,Ma: prat,Mb: prat] :
      ( ( valid_mask @ A )
     => ( ( valid_mask @ B )
       => ( ( upper_bounded @ A @ Ma )
         => ( ( upper_bounded @ B @ Mb )
           => ( ( pgte @ pwrite @ ( padd @ Ma @ Mb ) )
             => ( upper_bounded @ ( add_ma6586698345353345811t_char @ A @ B ) @ ( padd @ Ma @ Mb ) ) ) ) ) ) ) ).

% valid_sum(2)
thf(fact_37_not__pgte__charact,axiom,
    ! [A: prat,B: prat] :
      ( ( ~ ( pgte @ A @ B ) )
      = ( pgt @ B @ A ) ) ).

% not_pgte_charact
thf(fact_38_pgt__implies__pgte,axiom,
    ! [A: prat,B: prat] :
      ( ( pgt @ A @ B )
     => ( pgte @ A @ B ) ) ).

% pgt_implies_pgte
thf(fact_39_ppos__eq__pnone,axiom,
    ( ppos
    = ( ^ [P3: prat] : ( P3 != pnone ) ) ) ).

% ppos_eq_pnone
thf(fact_40_padd__zero,axiom,
    ! [A: prat,B: prat] :
      ( ( pnone
        = ( padd @ A @ B ) )
      = ( ( A = pnone )
        & ( B = pnone ) ) ) ).

% padd_zero
thf(fact_41_ppos__add,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ppos @ ( padd @ A @ B ) ) ) ).

% ppos_add
thf(fact_42_r0,axiom,
    ( ( ( get_m @ ( r @ a2 @ w ) )
      = ( comp_min_mask @ ( get_m @ a2 ) @ ( get_m @ w ) ) )
    & ( ( get_h @ ( r @ a2 @ w ) )
      = ( get_h @ w ) ) ) ).

% r0
thf(fact_43_greater__sum__both,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( pgte @ A @ ( padd @ B @ C ) )
     => ? [A1: prat,A22: prat] :
          ( ( A
            = ( padd @ A1 @ A22 ) )
          & ( pgte @ A1 @ B )
          & ( pgte @ A22 @ C ) ) ) ).

% greater_sum_both
thf(fact_44_p__greater__exists,axiom,
    ( pgte
    = ( ^ [A2: prat,B2: prat] :
        ? [R: prat] :
          ( A2
          = ( padd @ B2 @ R ) ) ) ) ).

% p_greater_exists
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: produc3882902684156697774_state,P4: produc3882902684156697774_state > $o] :
      ( ( member6715010751053357271_state @ A @ ( collec394671971739345689_state @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
    ! [A: produc7272280185086996969_state,P4: produc7272280185086996969_state > $o] :
      ( ( member7461423308196221312_state @ A @ ( collec906845016635587262_state @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A: produc9003781588162623004on_val,P4: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ A @ ( collec7475294502966882545on_val @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: state,P4: state > $o] :
      ( ( member_state @ A @ ( collect_state @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
    ! [A3: set_Pr2391701600042252046_state] :
      ( ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A3: set_Pr4231240917283394591_state] :
      ( ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A3: set_Pr7649655576376376530on_val] :
      ( ( collec7475294502966882545on_val
        @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A3: set_state] :
      ( ( collect_state
        @ ^ [X: state] : ( member_state @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_53_Collect__cong,axiom,
    ! [P4: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ! [X3: produc7272280185086996969_state] :
          ( ( P4 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec906845016635587262_state @ P4 )
        = ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_cong
thf(fact_54_Collect__cong,axiom,
    ! [P4: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ! [X3: produc9003781588162623004on_val] :
          ( ( P4 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec7475294502966882545on_val @ P4 )
        = ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_cong
thf(fact_55_Collect__cong,axiom,
    ! [P4: state > $o,Q: state > $o] :
      ( ! [X3: state] :
          ( ( P4 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_state @ P4 )
        = ( collect_state @ Q ) ) ) ).

% Collect_cong
thf(fact_56_upper__bounded__bigger,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,A: prat,B: prat] :
      ( ( upper_bounded @ Pi @ A )
     => ( ( pgte @ B @ A )
       => ( upper_bounded @ Pi @ B ) ) ) ).

% upper_bounded_bigger
thf(fact_57_upper__bounded__def,axiom,
    ( upper_bounded
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,P3: prat] :
        ! [Hl: produc1457211279475724562t_char] : ( pgte @ P3 @ ( Pi3 @ Hl ) ) ) ) ).

% upper_bounded_def
thf(fact_58_upper__boundedI,axiom,
    ! [P2: prat,Pi: produc1457211279475724562t_char > prat] :
      ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ P2 @ ( Pi @ Hl2 ) )
     => ( upper_bounded @ Pi @ P2 ) ) ).

% upper_boundedI
thf(fact_59_R__compatible__same,axiom,
    ! [A: state,W: state] :
      ( ( defined @ A @ W )
     => ( ( r @ A @ W )
        = W ) ) ).

% R_compatible_same
thf(fact_60_non__scalable__R__charact,axiom,
    ! [W: state,A: state] :
      ( ~ ( scalable @ W @ A )
     => ( ( ( get_m @ ( r @ A @ W ) )
          = ( comp_min_mask @ ( get_m @ A ) @ ( get_m @ W ) ) )
        & ( ( get_h @ ( r @ A @ W ) )
          = ( get_h @ W ) ) ) ) ).

% non_scalable_R_charact
thf(fact_61_pgte__antisym,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pgte @ B @ A )
       => ( A = B ) ) ) ).

% pgte_antisym
thf(fact_62_padd__cancellative,axiom,
    ! [A: prat,X2: prat,B: prat,Y: prat] :
      ( ( A
        = ( padd @ X2 @ B ) )
     => ( ( A
          = ( padd @ Y @ B ) )
       => ( X2 = Y ) ) ) ).

% padd_cancellative
thf(fact_63_padd__comm,axiom,
    ( padd
    = ( ^ [A2: prat,B2: prat] : ( padd @ B2 @ A2 ) ) ) ).

% padd_comm
thf(fact_64_padd__asso,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( padd @ ( padd @ A @ B ) @ C )
      = ( padd @ A @ ( padd @ B @ C ) ) ) ).

% padd_asso
thf(fact_65_get__h__multiply,axiom,
    ! [P2: prat,X2: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( get_h @ ( multiply @ P2 @ X2 ) )
        = ( get_h @ X2 ) ) ) ).

% get_h_multiply
thf(fact_66_padd__comp__one,axiom,
    ! [X2: prat] :
      ( ( pgte @ pwrite @ X2 )
     => ( ( padd @ X2 @ ( comp_one @ X2 ) )
        = pwrite ) ) ).

% padd_comp_one
thf(fact_67_state__ext,axiom,
    ! [A: state,B: state] :
      ( ( ( get_h @ A )
        = ( get_h @ B ) )
     => ( ( ( get_m @ A )
          = ( get_m @ B ) )
       => ( A = B ) ) ) ).

% state_ext
thf(fact_68_half__between__0__1,axiom,
    ( ( ppos @ half )
    & ( pgt @ pwrite @ half ) ) ).

% half_between_0_1
thf(fact_69_valid__multiply,axiom,
    ! [A: produc1457211279475724562t_char > prat,Ma: prat,P2: prat] :
      ( ( valid_mask @ A )
     => ( ( upper_bounded @ A @ Ma )
       => ( ( pgte @ pwrite @ ( pmult @ Ma @ P2 ) )
         => ( valid_mask @ ( multiply_mask @ P2 @ A ) ) ) ) ) ).

% valid_multiply
thf(fact_70_asm1,axiom,
    ( ( member_state @ a2 @ a )
    & ( ( some_state @ x )
      = ( plus @ ( r @ a2 @ w ) @ a2 ) ) ) ).

% asm1
thf(fact_71_sum__coeff,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ( padd @ ( pdiv @ A @ ( padd @ A @ B ) ) @ ( pdiv @ B @ ( padd @ A @ B ) ) )
          = pwrite ) ) ) ).

% sum_coeff
thf(fact_72_pdiv__smaller,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ A )
       => ( pgte @ pwrite @ ( pdiv @ B @ A ) ) ) ) ).

% pdiv_smaller
thf(fact_73_binary__mask__def,axiom,
    ( binary_mask
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,L: produc1457211279475724562t_char] :
          ( if_prat
          @ ( ( Pi3 @ L )
            = pwrite )
          @ pwrite
          @ pnone ) ) ) ).

% binary_mask_def
thf(fact_74_compatibleI,axiom,
    ! [A: state,B: state] :
      ( ( compatible_heaps @ ( get_h @ A ) @ ( get_h @ B ) )
     => ( ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ A ) @ ( get_m @ B ) ) )
       => ( defined @ A @ B ) ) ) ).

% compatibleI
thf(fact_75_get__m__smaller,axiom,
    ! [P2: prat,A: state,Hl3: produc1457211279475724562t_char] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( get_m @ ( multiply @ P2 @ A ) @ Hl3 )
        = ( pmult @ P2 @ ( get_m @ A @ Hl3 ) ) ) ) ).

% get_m_smaller
thf(fact_76_half__plus__half,axiom,
    ( ( padd @ half @ half )
    = pwrite ) ).

% half_plus_half
thf(fact_77_pmult__comm,axiom,
    ( pmult
    = ( ^ [A2: prat,B2: prat] : ( pmult @ B2 @ A2 ) ) ) ).

% pmult_comm
thf(fact_78_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_79_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_80_commutative,axiom,
    ( plus
    = ( ^ [A2: state,B2: state] : ( plus @ B2 @ A2 ) ) ) ).

% commutative
thf(fact_81_PartialSA_Odefined__sum__move,axiom,
    ! [A: state,B: state,X2: state,Y: state,A4: state] :
      ( ( defined @ A @ B )
     => ( ( ( some_state @ B )
          = ( plus @ X2 @ Y ) )
       => ( ( ( some_state @ A4 )
            = ( plus @ A @ X2 ) )
         => ( defined @ A4 @ Y ) ) ) ) ).

% PartialSA.defined_sum_move
thf(fact_82_wandI,axiom,
    ! [A3: set_state,W: state,B3: set_state] :
      ( ! [A5: state,X3: state] :
          ( ( ( member_state @ A5 @ A3 )
            & ( ( some_state @ X3 )
              = ( plus @ A5 @ W ) ) )
         => ( member_state @ X3 @ B3 ) )
     => ( member_state @ W @ ( wand @ A3 @ B3 ) ) ) ).

% wandI
thf(fact_83_plus__charact_I1_J,axiom,
    ! [A: state,B: state,X2: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X2 ) )
     => ( ( get_m @ X2 )
        = ( add_ma6586698345353345811t_char @ ( get_m @ A ) @ ( get_m @ B ) ) ) ) ).

% plus_charact(1)
thf(fact_84_pmult__pdiv__cancel,axiom,
    ! [A: prat,X2: prat] :
      ( ( ppos @ A )
     => ( ( pmult @ A @ ( pdiv @ X2 @ A ) )
        = X2 ) ) ).

% pmult_pdiv_cancel
thf(fact_85_pmult__order,axiom,
    ! [A: prat,B: prat,P2: prat] :
      ( ( pgte @ A @ B )
     => ( pgte @ ( pmult @ P2 @ A ) @ ( pmult @ B @ P2 ) ) ) ).

% pmult_order
thf(fact_86_pmult__special_I1_J,axiom,
    ! [X2: prat] :
      ( ( pmult @ pwrite @ X2 )
      = X2 ) ).

% pmult_special(1)
thf(fact_87_pmult__padd,axiom,
    ! [A: prat,B: prat,X2: prat,C: prat,Y: prat] :
      ( ( pmult @ A @ ( padd @ ( pmult @ B @ X2 ) @ ( pmult @ C @ Y ) ) )
      = ( padd @ ( pmult @ ( pmult @ A @ B ) @ X2 ) @ ( pmult @ ( pmult @ A @ C ) @ Y ) ) ) ).

% pmult_padd
thf(fact_88_pmult__distr,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( pmult @ A @ ( padd @ B @ C ) )
      = ( padd @ ( pmult @ A @ B ) @ ( pmult @ A @ C ) ) ) ).

% pmult_distr
thf(fact_89_pmult__ppos,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ppos @ ( pmult @ A @ B ) ) ) ) ).

% pmult_ppos
thf(fact_90_pmult__special_I2_J,axiom,
    ! [X2: prat] :
      ( ( pmult @ pnone @ X2 )
      = pnone ) ).

% pmult_special(2)
thf(fact_91_multiply__mask__def,axiom,
    ( multiply_mask
    = ( ^ [P3: prat,Pi3: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( pmult @ P3 @ ( Pi3 @ Hl ) ) ) ) ).

% multiply_mask_def
thf(fact_92_multiply__smaller__pwrite,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ pwrite @ A )
     => ( ( pgte @ pwrite @ B )
       => ( pgte @ pwrite @ ( pmult @ A @ B ) ) ) ) ).

% multiply_smaller_pwrite
thf(fact_93_upper__bounded__smaller,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,A: prat,P2: prat] :
      ( ( upper_bounded @ Pi @ A )
     => ( upper_bounded @ ( multiply_mask @ P2 @ Pi ) @ ( pmult @ P2 @ A ) ) ) ).

% upper_bounded_smaller
thf(fact_94_padd__one__ineq__sum,axiom,
    ! [A: prat,B: prat,X2: prat,Aa: prat,Bb: prat] :
      ( ( ( padd @ A @ B )
        = pwrite )
     => ( ( pgte @ X2 @ Aa )
       => ( ( pgte @ X2 @ Bb )
         => ( pgte @ X2 @ ( padd @ ( pmult @ A @ Aa ) @ ( pmult @ B @ Bb ) ) ) ) ) ) ).

% padd_one_ineq_sum
thf(fact_95_multiply__twice,axiom,
    ! [A: prat,B: prat,X2: state] :
      ( ( ( pgte @ pwrite @ A )
        & ( pgte @ pwrite @ B ) )
     => ( ( multiply @ A @ ( multiply @ B @ X2 ) )
        = ( multiply @ ( pmult @ A @ B ) @ X2 ) ) ) ).

% multiply_twice
thf(fact_96_in__cwand,axiom,
    ! [A3: set_state,W: state,B3: set_state] :
      ( ! [A5: state,X3: state] :
          ( ( ( member_state @ A5 @ A3 )
            & ( ( some_state @ X3 )
              = ( plus @ ( r @ A5 @ W ) @ A5 ) ) )
         => ( member_state @ X3 @ B3 ) )
     => ( member_state @ W @ ( cwand @ A3 @ B3 ) ) ) ).

% in_cwand
thf(fact_97_pinv__double__half,axiom,
    ! [P2: prat] :
      ( ( ppos @ P2 )
     => ( ( pmult @ half @ ( pinv @ P2 ) )
        = ( pinv @ ( padd @ P2 @ P2 ) ) ) ) ).

% pinv_double_half
thf(fact_98_option_Oinject,axiom,
    ! [X22: state,Y2: state] :
      ( ( ( some_state @ X22 )
        = ( some_state @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_99_option_Oinject,axiom,
    ! [X22: val,Y2: val] :
      ( ( ( some_val @ X22 )
        = ( some_val @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_100__092_060open_062Abs__state_A_Ibinary__mask_A_Iget__m_Aa_J_M_Aget__h_Aa_J_A_092_060in_062_AA_092_060close_062,axiom,
    member_state @ ( abs_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ a2 ) ) @ ( get_h @ a2 ) ) ) @ a ).

% \<open>Abs_state (binary_mask (get_m a), get_h a) \<in> A\<close>
thf(fact_101_plus__ab__defined,axiom,
    ! [Phi2: state,Phi3: state] :
      ( ( ( plus @ Phi2 @ Phi3 )
       != none_state )
      = ( ( compatible_heaps @ ( get_h @ Phi2 ) @ ( get_h @ Phi3 ) )
        & ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ Phi2 ) @ ( get_m @ Phi3 ) ) ) ) ) ).

% plus_ab_defined
thf(fact_102_pinv__pmult__ok,axiom,
    ! [P2: prat] :
      ( ( ppos @ P2 )
     => ( ( pmult @ P2 @ ( pinv @ P2 ) )
        = pwrite ) ) ).

% pinv_pmult_ok
thf(fact_103_comp__min__mask__def,axiom,
    ( comp_min_mask
    = ( ^ [B2: produc1457211279475724562t_char > prat,A2: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( pmin @ ( A2 @ Hl ) @ ( comp_one @ ( B2 @ Hl ) ) ) ) ) ).

% comp_min_mask_def
thf(fact_104_pinv__def,axiom,
    ( pinv
    = ( pdiv @ pwrite ) ) ).

% pinv_def
thf(fact_105_pinv__inverts,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ B )
       => ( pgte @ ( pinv @ B ) @ ( pinv @ A ) ) ) ) ).

% pinv_inverts
thf(fact_106_valid__mask__full__mask,axiom,
    valid_mask @ full_mask ).

% valid_mask_full_mask
thf(fact_107_not__Some__eq,axiom,
    ! [X2: option_state] :
      ( ( ! [Y3: state] :
            ( X2
           != ( some_state @ Y3 ) ) )
      = ( X2 = none_state ) ) ).

% not_Some_eq
thf(fact_108_not__Some__eq,axiom,
    ! [X2: option_val] :
      ( ( ! [Y3: val] :
            ( X2
           != ( some_val @ Y3 ) ) )
      = ( X2 = none_val ) ) ).

% not_Some_eq
thf(fact_109_not__None__eq,axiom,
    ! [X2: option_state] :
      ( ( X2 != none_state )
      = ( ? [Y3: state] :
            ( X2
            = ( some_state @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_110_not__None__eq,axiom,
    ! [X2: option_val] :
      ( ( X2 != none_val )
      = ( ? [Y3: val] :
            ( X2
            = ( some_val @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_111_valid__state_Ocases,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( X2
         != ( produc74620467203698198on_val @ Pi4 @ H ) ) ).

% valid_state.cases
thf(fact_112_pmin__comm,axiom,
    ( pmin
    = ( ^ [A2: prat,B2: prat] : ( pmin @ B2 @ A2 ) ) ) ).

% pmin_comm
thf(fact_113_combine__options__cases,axiom,
    ! [X2: option_state,P4: option_state > option_state > $o,Y: option_state] :
      ( ( ( X2 = none_state )
       => ( P4 @ X2 @ Y ) )
     => ( ( ( Y = none_state )
         => ( P4 @ X2 @ Y ) )
       => ( ! [A5: state,B4: state] :
              ( ( X2
                = ( some_state @ A5 ) )
             => ( ( Y
                  = ( some_state @ B4 ) )
               => ( P4 @ X2 @ Y ) ) )
         => ( P4 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_114_combine__options__cases,axiom,
    ! [X2: option_state,P4: option_state > option_val > $o,Y: option_val] :
      ( ( ( X2 = none_state )
       => ( P4 @ X2 @ Y ) )
     => ( ( ( Y = none_val )
         => ( P4 @ X2 @ Y ) )
       => ( ! [A5: state,B4: val] :
              ( ( X2
                = ( some_state @ A5 ) )
             => ( ( Y
                  = ( some_val @ B4 ) )
               => ( P4 @ X2 @ Y ) ) )
         => ( P4 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_115_combine__options__cases,axiom,
    ! [X2: option_val,P4: option_val > option_state > $o,Y: option_state] :
      ( ( ( X2 = none_val )
       => ( P4 @ X2 @ Y ) )
     => ( ( ( Y = none_state )
         => ( P4 @ X2 @ Y ) )
       => ( ! [A5: val,B4: state] :
              ( ( X2
                = ( some_val @ A5 ) )
             => ( ( Y
                  = ( some_state @ B4 ) )
               => ( P4 @ X2 @ Y ) ) )
         => ( P4 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_116_combine__options__cases,axiom,
    ! [X2: option_val,P4: option_val > option_val > $o,Y: option_val] :
      ( ( ( X2 = none_val )
       => ( P4 @ X2 @ Y ) )
     => ( ( ( Y = none_val )
         => ( P4 @ X2 @ Y ) )
       => ( ! [A5: val,B4: val] :
              ( ( X2
                = ( some_val @ A5 ) )
             => ( ( Y
                  = ( some_val @ B4 ) )
               => ( P4 @ X2 @ Y ) ) )
         => ( P4 @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_117_split__option__all,axiom,
    ( ( ^ [P5: option_state > $o] :
        ! [X4: option_state] : ( P5 @ X4 ) )
    = ( ^ [P6: option_state > $o] :
          ( ( P6 @ none_state )
          & ! [X: state] : ( P6 @ ( some_state @ X ) ) ) ) ) ).

% split_option_all
thf(fact_118_split__option__all,axiom,
    ( ( ^ [P5: option_val > $o] :
        ! [X4: option_val] : ( P5 @ X4 ) )
    = ( ^ [P6: option_val > $o] :
          ( ( P6 @ none_val )
          & ! [X: val] : ( P6 @ ( some_val @ X ) ) ) ) ) ).

% split_option_all
thf(fact_119_split__option__ex,axiom,
    ( ( ^ [P5: option_state > $o] :
        ? [X4: option_state] : ( P5 @ X4 ) )
    = ( ^ [P6: option_state > $o] :
          ( ( P6 @ none_state )
          | ? [X: state] : ( P6 @ ( some_state @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_120_split__option__ex,axiom,
    ( ( ^ [P5: option_val > $o] :
        ? [X4: option_val] : ( P5 @ X4 ) )
    = ( ^ [P6: option_val > $o] :
          ( ( P6 @ none_val )
          | ? [X: val] : ( P6 @ ( some_val @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_121_option_Oexhaust,axiom,
    ! [Y: option_state] :
      ( ( Y != none_state )
     => ~ ! [X23: state] :
            ( Y
           != ( some_state @ X23 ) ) ) ).

% option.exhaust
thf(fact_122_option_Oexhaust,axiom,
    ! [Y: option_val] :
      ( ( Y != none_val )
     => ~ ! [X23: val] :
            ( Y
           != ( some_val @ X23 ) ) ) ).

% option.exhaust
thf(fact_123_option_OdiscI,axiom,
    ! [Option: option_state,X22: state] :
      ( ( Option
        = ( some_state @ X22 ) )
     => ( Option != none_state ) ) ).

% option.discI
thf(fact_124_option_OdiscI,axiom,
    ! [Option: option_val,X22: val] :
      ( ( Option
        = ( some_val @ X22 ) )
     => ( Option != none_val ) ) ).

% option.discI
thf(fact_125_option_Odistinct_I1_J,axiom,
    ! [X22: state] :
      ( none_state
     != ( some_state @ X22 ) ) ).

% option.distinct(1)
thf(fact_126_option_Odistinct_I1_J,axiom,
    ! [X22: val] :
      ( none_val
     != ( some_val @ X22 ) ) ).

% option.distinct(1)
thf(fact_127_compatible__heapsI,axiom,
    ! [H2: produc1457211279475724562t_char > option_val,H3: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char,A5: val,B4: val] :
          ( ( ( H2 @ Hl2 )
            = ( some_val @ A5 ) )
         => ( ( ( H3 @ Hl2 )
              = ( some_val @ B4 ) )
           => ( A5 = B4 ) ) )
     => ( compatible_heaps @ H2 @ H3 ) ) ).

% compatible_heapsI
thf(fact_128_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_129_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_130_pmin__greater,axiom,
    ! [A: prat,B: prat] : ( pgte @ A @ ( pmin @ A @ B ) ) ).

% pmin_greater
thf(fact_131_pmin__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmin @ A @ B )
        = B ) ) ).

% pmin_is
thf(fact_132_PartialSA_Odefined__def,axiom,
    ( defined
    = ( ^ [A2: state,B2: state] :
          ( ( plus @ A2 @ B2 )
         != none_state ) ) ) ).

% PartialSA.defined_def
thf(fact_133_pinv__pwrite,axiom,
    ( ( pinv @ pwrite )
    = pwrite ) ).

% pinv_pwrite
thf(fact_134_multiply_Oelims,axiom,
    ! [X2: prat,Xa: state,Y: state] :
      ( ( ( multiply @ X2 @ Xa )
        = Y )
     => ( Y
        = ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X2 @ ( get_m @ Xa ) ) @ ( get_h @ Xa ) ) ) ) ) ).

% multiply.elims
thf(fact_135_multiply_Osimps,axiom,
    ( multiply
    = ( ^ [P3: prat,Phi4: state] : ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ P3 @ ( get_m @ Phi4 ) ) @ ( get_h @ Phi4 ) ) ) ) ) ).

% multiply.simps
thf(fact_136_ppos__inv,axiom,
    ! [P2: prat] :
      ( ( ppos @ P2 )
     => ( ppos @ ( pinv @ P2 ) ) ) ).

% ppos_inv
thf(fact_137_binary__def,axiom,
    ( binary
    = ( ^ [A6: set_state] :
        ! [X: state] :
          ( ( member_state @ X @ A6 )
         => ( member_state @ ( abs_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ X ) ) @ ( get_h @ X ) ) ) @ A6 ) ) ) ) ).

% binary_def
thf(fact_138_R__def,axiom,
    ( r
    = ( ^ [A2: state,W2: state] : ( if_state @ ( scalable @ W2 @ A2 ) @ W2 @ ( abs_state @ ( produc74620467203698198on_val @ ( comp_min_mask @ ( get_m @ A2 ) @ ( get_m @ W2 ) ) @ ( get_h @ W2 ) ) ) ) ) ) ).

% R_def
thf(fact_139_multiply__valid,axiom,
    ! [P2: prat,Phi2: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( valid_state @ ( produc74620467203698198on_val @ ( multiply_mask @ P2 @ ( get_m @ Phi2 ) ) @ ( get_h @ Phi2 ) ) ) ) ).

% multiply_valid
thf(fact_140_old_Oprod_Oinject,axiom,
    ! [A: option_state,B: option_state,A4: option_state,B5: option_state] :
      ( ( ( produc9160152616361873709_state @ A @ B )
        = ( produc9160152616361873709_state @ A4 @ B5 ) )
      = ( ( A = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_141_old_Oprod_Oinject,axiom,
    ! [A: prat,B: state,A4: prat,B5: state] :
      ( ( ( produc6205719116230455971_state @ A @ B )
        = ( produc6205719116230455971_state @ A4 @ B5 ) )
      = ( ( A = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_142_old_Oprod_Oinject,axiom,
    ! [A: prat,B: set_state,A4: prat,B5: set_state] :
      ( ( ( produc9080284787853255769_state @ A @ B )
        = ( produc9080284787853255769_state @ A4 @ B5 ) )
      = ( ( A = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_143_old_Oprod_Oinject,axiom,
    ! [A: nat,B: list_char,A4: nat,B5: list_char] :
      ( ( ( produc120197956887798346t_char @ A @ B )
        = ( produc120197956887798346t_char @ A4 @ B5 ) )
      = ( ( A = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_144_old_Oprod_Oinject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > prat,B5: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A4 @ B5 ) )
      = ( ( A = A4 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_145_prod_Oinject,axiom,
    ! [X1: option_state,X22: option_state,Y1: option_state,Y2: option_state] :
      ( ( ( produc9160152616361873709_state @ X1 @ X22 )
        = ( produc9160152616361873709_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_146_prod_Oinject,axiom,
    ! [X1: prat,X22: state,Y1: prat,Y2: state] :
      ( ( ( produc6205719116230455971_state @ X1 @ X22 )
        = ( produc6205719116230455971_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_147_prod_Oinject,axiom,
    ! [X1: prat,X22: set_state,Y1: prat,Y2: set_state] :
      ( ( ( produc9080284787853255769_state @ X1 @ X22 )
        = ( produc9080284787853255769_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_148_prod_Oinject,axiom,
    ! [X1: nat,X22: list_char,Y1: nat,Y2: list_char] :
      ( ( ( produc120197956887798346t_char @ X1 @ X22 )
        = ( produc120197956887798346t_char @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_149_prod_Oinject,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val,Y1: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ X1 @ X22 )
        = ( produc74620467203698198on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_150_valid__bin,axiom,
    ! [A: state] : ( valid_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ A ) ) @ ( get_h @ A ) ) ) ).

% valid_bin
thf(fact_151_empty__heap__def,axiom,
    ( empty_7512026568161686585ar_val
    = ( ^ [Hl: produc1457211279475724562t_char] : none_val ) ) ).

% empty_heap_def
thf(fact_152_get__h__m,axiom,
    ( rep_state
    = ( ^ [X: state] : ( produc74620467203698198on_val @ ( get_m @ X ) @ ( get_h @ X ) ) ) ) ).

% get_h_m
thf(fact_153_pmin__pmax,axiom,
    ! [X2: prat,A: prat,B: prat] :
      ( ( pgte @ X2 @ ( pmin @ A @ B ) )
     => ( X2
        = ( pmin @ ( pmax @ X2 @ A ) @ ( pmax @ X2 @ B ) ) ) ) ).

% pmin_pmax
thf(fact_154_compatible__options_Oelims_I1_J,axiom,
    ! [X2: option_state,Xa: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X2 @ Xa )
        = Y )
     => ( ! [A5: state] :
            ( ( X2
              = ( some_state @ A5 ) )
           => ! [B4: state] :
                ( ( Xa
                  = ( some_state @ B4 ) )
               => ( Y
                  = ( A5 != B4 ) ) ) )
       => ( ( ( X2 = none_state )
           => ~ Y )
         => ~ ( ( Xa = none_state )
             => ~ Y ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_155_compatible__options_Oelims_I1_J,axiom,
    ! [X2: option_val,Xa: option_val,Y: $o] :
      ( ( ( compat7209469420100717861ns_val @ X2 @ Xa )
        = Y )
     => ( ! [A5: val] :
            ( ( X2
              = ( some_val @ A5 ) )
           => ! [B4: val] :
                ( ( Xa
                  = ( some_val @ B4 ) )
               => ( Y
                  = ( A5 != B4 ) ) ) )
       => ( ( ( X2 = none_val )
           => ~ Y )
         => ~ ( ( Xa = none_val )
             => ~ Y ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_156_compatible__options_Oelims_I2_J,axiom,
    ! [X2: option_state,Xa: option_state] :
      ( ( compat2278460363914054422_state @ X2 @ Xa )
     => ( ! [A5: state] :
            ( ( X2
              = ( some_state @ A5 ) )
           => ! [B4: state] :
                ( ( Xa
                  = ( some_state @ B4 ) )
               => ( A5 != B4 ) ) )
       => ( ( X2 != none_state )
         => ( Xa = none_state ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_157_compatible__options_Oelims_I2_J,axiom,
    ! [X2: option_val,Xa: option_val] :
      ( ( compat7209469420100717861ns_val @ X2 @ Xa )
     => ( ! [A5: val] :
            ( ( X2
              = ( some_val @ A5 ) )
           => ! [B4: val] :
                ( ( Xa
                  = ( some_val @ B4 ) )
               => ( A5 != B4 ) ) )
       => ( ( X2 != none_val )
         => ( Xa = none_val ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_158_PartialSA_Osplus_Oelims,axiom,
    ! [X2: option_state,Xa: option_state,Y: option_state] :
      ( ( ( sep_splus_state @ plus @ X2 @ Xa )
        = Y )
     => ( ( ( X2 = none_state )
         => ( Y != none_state ) )
       => ( ( ? [V: state] :
                ( X2
                = ( some_state @ V ) )
           => ( ( Xa = none_state )
             => ( Y != none_state ) ) )
         => ~ ! [A5: state] :
                ( ( X2
                  = ( some_state @ A5 ) )
               => ! [B4: state] :
                    ( ( Xa
                      = ( some_state @ B4 ) )
                   => ( Y
                     != ( plus @ A5 @ B4 ) ) ) ) ) ) ) ).

% PartialSA.splus.elims
thf(fact_159_pmax__comm,axiom,
    ( pmax
    = ( ^ [A2: prat,B2: prat] : ( pmax @ B2 @ A2 ) ) ) ).

% pmax_comm
thf(fact_160_Rep__state__inject,axiom,
    ! [X2: state,Y: state] :
      ( ( ( rep_state @ X2 )
        = ( rep_state @ Y ) )
      = ( X2 = Y ) ) ).

% Rep_state_inject
thf(fact_161_add__states_Ocases,axiom,
    ! [X2: produc1317171644805857433on_val] :
      ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val,Pi5: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
          ( X2
         != ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi4 @ H ) @ ( produc74620467203698198on_val @ Pi5 @ H4 ) ) ) ).

% add_states.cases
thf(fact_162_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_163_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_164_valid__stateI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ( valid_mask @ Pi )
     => ( ! [Hl2: produc1457211279475724562t_char] :
            ( ( ppos @ ( Pi @ Hl2 ) )
           => ( ( H2 @ Hl2 )
             != none_val ) )
       => ( valid_state @ ( produc74620467203698198on_val @ Pi @ H2 ) ) ) ) ).

% valid_stateI
thf(fact_165_Rep__state__inverse,axiom,
    ! [X2: state] :
      ( ( abs_state @ ( rep_state @ X2 ) )
      = X2 ) ).

% Rep_state_inverse
thf(fact_166_compatible__options_Ocases,axiom,
    ! [X2: produc3142500478612311029_state] :
      ( ! [A5: state,B4: state] :
          ( X2
         != ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) )
     => ( ! [Uv: option_state] :
            ( X2
           != ( produc9160152616361873709_state @ none_state @ Uv ) )
       => ~ ! [Uu: option_state] :
              ( X2
             != ( produc9160152616361873709_state @ Uu @ none_state ) ) ) ) ).

% compatible_options.cases
thf(fact_167_compatible__options_Ocases,axiom,
    ! [X2: produc5122021904355189735on_val] :
      ( ! [A5: val,B4: val] :
          ( X2
         != ( produc904852209897547607on_val @ ( some_val @ A5 ) @ ( some_val @ B4 ) ) )
     => ( ! [Uv: option_val] :
            ( X2
           != ( produc904852209897547607on_val @ none_val @ Uv ) )
       => ~ ! [Uu: option_val] :
              ( X2
             != ( produc904852209897547607on_val @ Uu @ none_val ) ) ) ) ).

% compatible_options.cases
thf(fact_168_compatible__options_Oelims_I3_J,axiom,
    ! [X2: option_state,Xa: option_state] :
      ( ~ ( compat2278460363914054422_state @ X2 @ Xa )
     => ~ ! [A5: state] :
            ( ( X2
              = ( some_state @ A5 ) )
           => ! [B4: state] :
                ( ( Xa
                  = ( some_state @ B4 ) )
               => ( A5 = B4 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_169_compatible__options_Oelims_I3_J,axiom,
    ! [X2: option_val,Xa: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X2 @ Xa )
     => ~ ! [A5: val] :
            ( ( X2
              = ( some_val @ A5 ) )
           => ! [B4: val] :
                ( ( Xa
                  = ( some_val @ B4 ) )
               => ( A5 = B4 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_170_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_171_compatible__options_Osimps_I1_J,axiom,
    ! [A: val,B: val] :
      ( ( compat7209469420100717861ns_val @ ( some_val @ A ) @ ( some_val @ B ) )
      = ( A = B ) ) ).

% compatible_options.simps(1)
thf(fact_172_pmax__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmax @ A @ B )
        = A ) ) ).

% pmax_is
thf(fact_173_pmax__smaller,axiom,
    ! [A: prat,B: prat] : ( pgte @ ( pmax @ A @ B ) @ A ) ).

% pmax_smaller
thf(fact_174_pmax__is__smaller,axiom,
    ! [X2: prat,A: prat,B: prat] :
      ( ( pgte @ X2 @ A )
     => ( ( pgte @ X2 @ B )
       => ( pgte @ X2 @ ( pmax @ A @ B ) ) ) ) ).

% pmax_is_smaller
thf(fact_175_compatible__options_Osimps_I2_J,axiom,
    ! [Uv2: option_state] : ( compat2278460363914054422_state @ none_state @ Uv2 ) ).

% compatible_options.simps(2)
thf(fact_176_compatible__options_Osimps_I2_J,axiom,
    ! [Uv2: option_val] : ( compat7209469420100717861ns_val @ none_val @ Uv2 ) ).

% compatible_options.simps(2)
thf(fact_177_compatible__options_Osimps_I3_J,axiom,
    ! [Uu2: option_state] : ( compat2278460363914054422_state @ Uu2 @ none_state ) ).

% compatible_options.simps(3)
thf(fact_178_compatible__options_Osimps_I3_J,axiom,
    ! [Uu2: option_val] : ( compat7209469420100717861ns_val @ Uu2 @ none_val ) ).

% compatible_options.simps(3)
thf(fact_179_PartialSA_Osplus_Ocases,axiom,
    ! [X2: produc3142500478612311029_state] :
      ( ! [Uu: option_state] :
          ( X2
         != ( produc9160152616361873709_state @ none_state @ Uu ) )
     => ( ! [V: state] :
            ( X2
           != ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) )
       => ~ ! [A5: state,B4: state] :
              ( X2
             != ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) ) ) ) ).

% PartialSA.splus.cases
thf(fact_180_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_181_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_182_compatible__heaps__def,axiom,
    ( compatible_heaps
    = ( ^ [H5: produc1457211279475724562t_char > option_val,H6: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char] : ( compat7209469420100717861ns_val @ ( H5 @ Hl ) @ ( H6 @ Hl ) ) ) ) ).

% compatible_heaps_def
thf(fact_183_PartialSA_Osplus_Osimps_I1_J,axiom,
    ! [Uu2: option_state] :
      ( ( sep_splus_state @ plus @ none_state @ Uu2 )
      = none_state ) ).

% PartialSA.splus.simps(1)
thf(fact_184_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3142500478612311029_state] :
      ~ ! [A5: option_state,B4: option_state] :
          ( Y
         != ( produc9160152616361873709_state @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_185_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4844402304972151915_state] :
      ~ ! [A5: prat,B4: state] :
          ( Y
         != ( produc6205719116230455971_state @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_186_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6251377953716854561_state] :
      ~ ! [A5: prat,B4: set_state] :
          ( Y
         != ( produc9080284787853255769_state @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_187_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1457211279475724562t_char] :
      ~ ! [A5: nat,B4: list_char] :
          ( Y
         != ( produc120197956887798346t_char @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_188_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ~ ! [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc74620467203698198on_val @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_189_surj__pair,axiom,
    ! [P2: produc3142500478612311029_state] :
    ? [X3: option_state,Y4: option_state] :
      ( P2
      = ( produc9160152616361873709_state @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_190_surj__pair,axiom,
    ! [P2: produc4844402304972151915_state] :
    ? [X3: prat,Y4: state] :
      ( P2
      = ( produc6205719116230455971_state @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_191_surj__pair,axiom,
    ! [P2: produc6251377953716854561_state] :
    ? [X3: prat,Y4: set_state] :
      ( P2
      = ( produc9080284787853255769_state @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_192_surj__pair,axiom,
    ! [P2: produc1457211279475724562t_char] :
    ? [X3: nat,Y4: list_char] :
      ( P2
      = ( produc120197956887798346t_char @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_193_surj__pair,axiom,
    ! [P2: produc9003781588162623004on_val] :
    ? [X3: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
      ( P2
      = ( produc74620467203698198on_val @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_194_prod__cases,axiom,
    ! [P4: produc3142500478612311029_state > $o,P2: produc3142500478612311029_state] :
      ( ! [A5: option_state,B4: option_state] : ( P4 @ ( produc9160152616361873709_state @ A5 @ B4 ) )
     => ( P4 @ P2 ) ) ).

% prod_cases
thf(fact_195_prod__cases,axiom,
    ! [P4: produc4844402304972151915_state > $o,P2: produc4844402304972151915_state] :
      ( ! [A5: prat,B4: state] : ( P4 @ ( produc6205719116230455971_state @ A5 @ B4 ) )
     => ( P4 @ P2 ) ) ).

% prod_cases
thf(fact_196_prod__cases,axiom,
    ! [P4: produc6251377953716854561_state > $o,P2: produc6251377953716854561_state] :
      ( ! [A5: prat,B4: set_state] : ( P4 @ ( produc9080284787853255769_state @ A5 @ B4 ) )
     => ( P4 @ P2 ) ) ).

% prod_cases
thf(fact_197_prod__cases,axiom,
    ! [P4: produc1457211279475724562t_char > $o,P2: produc1457211279475724562t_char] :
      ( ! [A5: nat,B4: list_char] : ( P4 @ ( produc120197956887798346t_char @ A5 @ B4 ) )
     => ( P4 @ P2 ) ) ).

% prod_cases
thf(fact_198_prod__cases,axiom,
    ! [P4: produc9003781588162623004on_val > $o,P2: produc9003781588162623004on_val] :
      ( ! [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] : ( P4 @ ( produc74620467203698198on_val @ A5 @ B4 ) )
     => ( P4 @ P2 ) ) ).

% prod_cases
thf(fact_199_Pair__inject,axiom,
    ! [A: option_state,B: option_state,A4: option_state,B5: option_state] :
      ( ( ( produc9160152616361873709_state @ A @ B )
        = ( produc9160152616361873709_state @ A4 @ B5 ) )
     => ~ ( ( A = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_200_Pair__inject,axiom,
    ! [A: prat,B: state,A4: prat,B5: state] :
      ( ( ( produc6205719116230455971_state @ A @ B )
        = ( produc6205719116230455971_state @ A4 @ B5 ) )
     => ~ ( ( A = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_201_Pair__inject,axiom,
    ! [A: prat,B: set_state,A4: prat,B5: set_state] :
      ( ( ( produc9080284787853255769_state @ A @ B )
        = ( produc9080284787853255769_state @ A4 @ B5 ) )
     => ~ ( ( A = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_202_Pair__inject,axiom,
    ! [A: nat,B: list_char,A4: nat,B5: list_char] :
      ( ( ( produc120197956887798346t_char @ A @ B )
        = ( produc120197956887798346t_char @ A4 @ B5 ) )
     => ~ ( ( A = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_203_Pair__inject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > prat,B5: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A4 @ B5 ) )
     => ~ ( ( A = A4 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_204_valid__state__decompose,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ A @ B ) @ H2 ) )
     => ( valid_state @ ( produc74620467203698198on_val @ A @ H2 ) ) ) ).

% valid_state_decompose
thf(fact_205_PartialSA_Osplus_Osimps_I2_J,axiom,
    ! [V2: state] :
      ( ( sep_splus_state @ plus @ ( some_state @ V2 ) @ none_state )
      = none_state ) ).

% PartialSA.splus.simps(2)
thf(fact_206_valid__mask_Oelims_I3_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X2 )
     => ~ ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X2 @ Hl2 ) )
          & ! [F: list_char] :
              ( ( X2 @ ( produc120197956887798346t_char @ null_nat @ F ) )
              = pnone ) ) ) ).

% valid_mask.elims(3)
thf(fact_207_valid__mask_Oelims_I2_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X2 )
     => ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X2 @ Hl4 ) )
        & ! [F2: list_char] :
            ( ( X2 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
            = pnone ) ) ) ).

% valid_mask.elims(2)
thf(fact_208_valid__mask_Oelims_I1_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X2 )
        = Y )
     => ( Y
        = ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X2 @ Hl ) )
          & ! [F3: list_char] :
              ( ( X2 @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
              = pnone ) ) ) ) ).

% valid_mask.elims(1)
thf(fact_209_valid__mask_Osimps,axiom,
    ( valid_mask
    = ( ^ [Pi3: produc1457211279475724562t_char > prat] :
          ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi3 @ Hl ) )
          & ! [F3: list_char] :
              ( ( Pi3 @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
              = pnone ) ) ) ) ).

% valid_mask.simps
thf(fact_210_valid__maskI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] :
      ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi @ Hl2 ) )
     => ( ! [F: list_char] :
            ( ( Pi @ ( produc120197956887798346t_char @ null_nat @ F ) )
            = pnone )
       => ( valid_mask @ Pi ) ) ) ).

% valid_maskI
thf(fact_211_plus__def,axiom,
    ( plus
    = ( ^ [A2: state,B2: state] : ( if_option_state @ ( compatible @ ( rep_state @ A2 ) @ ( rep_state @ B2 ) ) @ ( some_state @ ( abs_state @ ( add_states @ ( rep_state @ A2 ) @ ( rep_state @ B2 ) ) ) ) @ none_state ) ) ) ).

% plus_def
thf(fact_212_valid__state_Osimps,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ Pi @ H2 ) )
      = ( ( valid_mask @ Pi )
        & ( valid_heap @ Pi @ H2 ) ) ) ).

% valid_state.simps
thf(fact_213_valid__state_Oelims_I1_J,axiom,
    ! [X2: produc9003781588162623004on_val,Y: $o] :
      ( ( ( valid_state @ X2 )
        = Y )
     => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X2
              = ( produc74620467203698198on_val @ Pi4 @ H ) )
           => ( Y
              = ( ~ ( ( valid_mask @ Pi4 )
                    & ( valid_heap @ Pi4 @ H ) ) ) ) ) ) ).

% valid_state.elims(1)
thf(fact_214_valid__heap__def,axiom,
    ( valid_heap
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,H5: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi3 @ Hl ) )
         => ( ( H5 @ Hl )
           != none_val ) ) ) ) ).

% valid_heap_def
thf(fact_215_valid__heapI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi @ Hl2 ) )
         => ( ( H2 @ Hl2 )
           != none_val ) )
     => ( valid_heap @ Pi @ H2 ) ) ).

% valid_heapI
thf(fact_216_valid__state_Oelims_I3_J,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X2 )
     => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X2
              = ( produc74620467203698198on_val @ Pi4 @ H ) )
           => ( ( valid_mask @ Pi4 )
              & ( valid_heap @ Pi4 @ H ) ) ) ) ).

% valid_state.elims(3)
thf(fact_217_valid__state_Oelims_I2_J,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ( ( valid_state @ X2 )
     => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X2
              = ( produc74620467203698198on_val @ Pi4 @ H ) )
           => ~ ( ( valid_mask @ Pi4 )
                & ( valid_heap @ Pi4 @ H ) ) ) ) ).

% valid_state.elims(2)
thf(fact_218_valid__null__def,axiom,
    ( valid_null
    = ( ^ [Pi3: produc1457211279475724562t_char > prat] :
        ! [F3: list_char] :
          ( ( Pi3 @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
          = pnone ) ) ) ).

% valid_null_def
thf(fact_219_multiply_Opelims,axiom,
    ! [X2: prat,Xa: state,Y: state] :
      ( ( ( multiply @ X2 @ Xa )
        = Y )
     => ( ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X2 @ Xa ) )
       => ~ ( ( Y
              = ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X2 @ ( get_m @ Xa ) ) @ ( get_h @ Xa ) ) ) )
           => ~ ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X2 @ Xa ) ) ) ) ) ).

% multiply.pelims
thf(fact_220_full__mask__def,axiom,
    ( full_mask
    = ( ^ [Hl: produc1457211279475724562t_char] :
          ( if_prat
          @ ( ( produc6204653154488278070t_char @ Hl )
            = null_nat )
          @ pnone
          @ pwrite ) ) ) ).

% full_mask_def
thf(fact_221_PartialHeapSA_Ostable__def,axiom,
    ( stable
    = ( ^ [Phi4: state] :
        ! [Hl: produc1457211279475724562t_char] :
          ( ( ppos @ ( get_m @ Phi4 @ Hl ) )
          = ( ( get_h @ Phi4 @ Hl )
           != none_val ) ) ) ) ).

% PartialHeapSA.stable_def
thf(fact_222_stableI,axiom,
    ! [Phi2: state] :
      ( ! [Hl2: produc1457211279475724562t_char] :
          ( ( ppos @ ( get_m @ Phi2 @ Hl2 ) )
          = ( ( get_h @ Phi2 @ Hl2 )
           != none_val ) )
     => ( stable @ Phi2 ) ) ).

% stableI
thf(fact_223_valid__mask_Opelims_I1_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X2 )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X2 )
       => ~ ( ( Y
              = ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X2 @ Hl ) )
                & ! [F3: list_char] :
                    ( ( X2 @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
                    = pnone ) ) )
           => ~ ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X2 ) ) ) ) ).

% valid_mask.pelims(1)
thf(fact_224_valid__mask_Opelims_I2_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X2 )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X2 )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X2 )
           => ~ ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X2 @ Hl4 ) )
                & ! [F2: list_char] :
                    ( ( X2 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
                    = pnone ) ) ) ) ) ).

% valid_mask.pelims(2)
thf(fact_225_valid__mask_Opelims_I3_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X2 )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X2 )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X2 )
           => ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X2 @ Hl2 ) )
              & ! [F: list_char] :
                  ( ( X2 @ ( produc120197956887798346t_char @ null_nat @ F ) )
                  = pnone ) ) ) ) ) ).

% valid_mask.pelims(3)
thf(fact_226_multiply_Ocases,axiom,
    ! [X2: produc4844402304972151915_state] :
      ~ ! [P: prat,Phi: state] :
          ( X2
         != ( produc6205719116230455971_state @ P @ Phi ) ) ).

% multiply.cases
thf(fact_227_fst__conv,axiom,
    ! [X1: option_state,X22: option_state] :
      ( ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_228_fst__conv,axiom,
    ! [X1: prat,X22: state] :
      ( ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_229_fst__conv,axiom,
    ! [X1: prat,X22: set_state] :
      ( ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_230_fst__conv,axiom,
    ! [X1: nat,X22: list_char] :
      ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_231_fst__conv,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] :
      ( ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_232_fst__eqD,axiom,
    ! [X2: option_state,Y: option_state,A: option_state] :
      ( ( ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_233_fst__eqD,axiom,
    ! [X2: prat,Y: state,A: prat] :
      ( ( ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_234_fst__eqD,axiom,
    ! [X2: prat,Y: set_state,A: prat] :
      ( ( ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_235_fst__eqD,axiom,
    ! [X2: nat,Y: list_char,A: nat] :
      ( ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_236_fst__eqD,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > prat] :
      ( ( ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_237_stable__sum,axiom,
    ! [A: state,B: state,X2: state] :
      ( ( stable @ A )
     => ( ( stable @ B )
       => ( ( ( some_state @ X2 )
            = ( plus @ A @ B ) )
         => ( stable @ X2 ) ) ) ) ).

% stable_sum
thf(fact_238_add__masks__equiv__valid__null,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( valid_null @ ( add_ma6586698345353345811t_char @ A @ B ) )
      = ( ( valid_null @ A )
        & ( valid_null @ B ) ) ) ).

% add_masks_equiv_valid_null
thf(fact_239_eq__fst__iff,axiom,
    ! [A: option_state,P2: produc3142500478612311029_state] :
      ( ( A
        = ( produc3900939990612242713_state @ P2 ) )
      = ( ? [B2: option_state] :
            ( P2
            = ( produc9160152616361873709_state @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_240_eq__fst__iff,axiom,
    ! [A: prat,P2: produc4844402304972151915_state] :
      ( ( A
        = ( produc6030790797199838863_state @ P2 ) )
      = ( ? [B2: state] :
            ( P2
            = ( produc6205719116230455971_state @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_241_eq__fst__iff,axiom,
    ! [A: prat,P2: produc6251377953716854561_state] :
      ( ( A
        = ( produc7279835715735703621_state @ P2 ) )
      = ( ? [B2: set_state] :
            ( P2
            = ( produc9080284787853255769_state @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_242_eq__fst__iff,axiom,
    ! [A: nat,P2: produc1457211279475724562t_char] :
      ( ( A
        = ( produc6204653154488278070t_char @ P2 ) )
      = ( ? [B2: list_char] :
            ( P2
            = ( produc120197956887798346t_char @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_243_eq__fst__iff,axiom,
    ! [A: produc1457211279475724562t_char > prat,P2: produc9003781588162623004on_val] :
      ( ( A
        = ( produc5239619271231976746on_val @ P2 ) )
      = ( ? [B2: produc1457211279475724562t_char > option_val] :
            ( P2
            = ( produc74620467203698198on_val @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_244_fstI,axiom,
    ! [X2: produc3142500478612311029_state,Y: option_state,Z: option_state] :
      ( ( X2
        = ( produc9160152616361873709_state @ Y @ Z ) )
     => ( ( produc3900939990612242713_state @ X2 )
        = Y ) ) ).

% fstI
thf(fact_245_fstI,axiom,
    ! [X2: produc4844402304972151915_state,Y: prat,Z: state] :
      ( ( X2
        = ( produc6205719116230455971_state @ Y @ Z ) )
     => ( ( produc6030790797199838863_state @ X2 )
        = Y ) ) ).

% fstI
thf(fact_246_fstI,axiom,
    ! [X2: produc6251377953716854561_state,Y: prat,Z: set_state] :
      ( ( X2
        = ( produc9080284787853255769_state @ Y @ Z ) )
     => ( ( produc7279835715735703621_state @ X2 )
        = Y ) ) ).

% fstI
thf(fact_247_fstI,axiom,
    ! [X2: produc1457211279475724562t_char,Y: nat,Z: list_char] :
      ( ( X2
        = ( produc120197956887798346t_char @ Y @ Z ) )
     => ( ( produc6204653154488278070t_char @ X2 )
        = Y ) ) ).

% fstI
thf(fact_248_fstI,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > option_val] :
      ( ( X2
        = ( produc74620467203698198on_val @ Y @ Z ) )
     => ( ( produc5239619271231976746on_val @ X2 )
        = Y ) ) ).

% fstI
thf(fact_249_PartialSA_Ox__elem__set__product__splus,axiom,
    ! [X2: state,A3: set_state,B3: set_state] :
      ( ( member_state @ X2 @ ( add_set @ A3 @ B3 ) )
      = ( ? [A2: state,B2: state] :
            ( ( member_state @ A2 @ A3 )
            & ( member_state @ B2 @ B3 )
            & ( ( some_state @ X2 )
              = ( sep_splus_state @ plus @ ( some_state @ A2 ) @ ( some_state @ B2 ) ) ) ) ) ) ).

% PartialSA.x_elem_set_product_splus
thf(fact_250_same__fstI,axiom,
    ! [P4: prat > $o,X2: prat,Y5: state,Y: state,R2: prat > set_Pr1785066336555260981_state] :
      ( ( P4 @ X2 )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ Y5 @ Y ) @ ( R2 @ X2 ) )
       => ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ ( produc6205719116230455971_state @ X2 @ Y ) ) @ ( same_fst_prat_state @ P4 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_251_same__fstI,axiom,
    ! [P4: prat > $o,X2: prat,Y5: set_state,Y: set_state,R2: prat > set_Pr5618112136499017889_state] :
      ( ( P4 @ X2 )
     => ( ( member1976142906092959210_state @ ( produc1834047914073684089_state @ Y5 @ Y ) @ ( R2 @ X2 ) )
       => ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ X2 @ Y5 ) @ ( produc9080284787853255769_state @ X2 @ Y ) ) @ ( same_f5333510722715586990_state @ P4 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_252_same__fstI,axiom,
    ! [P4: nat > $o,X2: nat,Y5: list_char,Y: list_char,R2: nat > set_Pr4480099295215233283t_char] :
      ( ( P4 @ X2 )
     => ( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ Y5 @ Y ) @ ( R2 @ X2 ) )
       => ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ ( produc120197956887798346t_char @ X2 @ Y ) ) @ ( same_f652031872680891423t_char @ P4 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_253_same__fstI,axiom,
    ! [P4: ( produc1457211279475724562t_char > prat ) > $o,X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val,R2: ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val] :
      ( ( P4 @ X2 )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y5 @ Y ) @ ( R2 @ X2 ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ ( produc74620467203698198on_val @ X2 @ Y ) ) @ ( same_f8138205353876899841on_val @ P4 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_254_same__fstI,axiom,
    ! [P4: option_state > $o,X2: option_state,Y5: option_state,Y: option_state,R2: option_state > set_Pr1688445902015331925_state] :
      ( ( P4 @ X2 )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Y5 @ Y ) @ ( R2 @ X2 ) )
       => ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ X2 @ Y5 ) @ ( produc9160152616361873709_state @ X2 @ Y ) ) @ ( same_f2704544105346032002_state @ P4 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_255_add__states_Osimps,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val,Pi6: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ( add_states @ ( produc74620467203698198on_val @ Pi @ H2 ) @ ( produc74620467203698198on_val @ Pi6 @ H3 ) )
      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi6 ) @ ( map_ad2908576761495157028ar_val @ H2 @ H3 ) ) ) ).

% add_states.simps
thf(fact_256_add__states_Oelims,axiom,
    ! [X2: produc9003781588162623004on_val,Xa: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X2 @ Xa )
        = Y )
     => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X2
              = ( produc74620467203698198on_val @ Pi4 @ H ) )
           => ! [Pi5: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
                ( ( Xa
                  = ( produc74620467203698198on_val @ Pi5 @ H4 ) )
               => ( Y
                 != ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi4 @ Pi5 ) @ ( map_ad2908576761495157028ar_val @ H @ H4 ) ) ) ) ) ) ).

% add_states.elims
thf(fact_257_add__masks_Opelims,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X2 @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( padd @ ( X2 @ Xb ) @ ( Xa @ Xb ) ) )
           => ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X2 @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_258_PartialSA_Oadd__set__asso,axiom,
    ! [A3: set_state,B3: set_state,C2: set_state] :
      ( ( add_set @ ( add_set @ A3 @ B3 ) @ C2 )
      = ( add_set @ A3 @ ( add_set @ B3 @ C2 ) ) ) ).

% PartialSA.add_set_asso
thf(fact_259_PartialSA_Oadd__set__commm,axiom,
    ( add_set
    = ( ^ [A6: set_state,B6: set_state] : ( add_set @ B6 @ A6 ) ) ) ).

% PartialSA.add_set_commm
thf(fact_260_compatible__heaps__comm,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( map_ad2908576761495157028ar_val @ A @ B )
        = ( map_ad2908576761495157028ar_val @ B @ A ) ) ) ).

% compatible_heaps_comm
thf(fact_261_compatible__heaps__sum,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( compatible_heaps @ A @ C )
       => ( compatible_heaps @ A @ ( map_ad2908576761495157028ar_val @ B @ C ) ) ) ) ).

% compatible_heaps_sum
thf(fact_262_empty__heap__neutral,axiom,
    ! [A: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ A @ empty_7512026568161686585ar_val )
      = A ) ).

% empty_heap_neutral
thf(fact_263_get__m_Osimps,axiom,
    ( get_m
    = ( ^ [A2: state] : ( produc5239619271231976746on_val @ ( rep_state @ A2 ) ) ) ) ).

% get_m.simps
thf(fact_264_get__m_Oelims,axiom,
    ! [X2: state,Y: produc1457211279475724562t_char > prat] :
      ( ( ( get_m @ X2 )
        = Y )
     => ( Y
        = ( produc5239619271231976746on_val @ ( rep_state @ X2 ) ) ) ) ).

% get_m.elims
thf(fact_265_get__pre_I2_J,axiom,
    ( get_m
    = ( ^ [X: state] : ( produc5239619271231976746on_val @ ( rep_state @ X ) ) ) ) ).

% get_pre(2)
thf(fact_266_PartialSA_Ox__elem__set__product,axiom,
    ! [X2: state,A3: set_state,B3: set_state] :
      ( ( member_state @ X2 @ ( add_set @ A3 @ B3 ) )
      = ( ? [A2: state,B2: state] :
            ( ( member_state @ A2 @ A3 )
            & ( member_state @ B2 @ B3 )
            & ( ( some_state @ X2 )
              = ( plus @ A2 @ B2 ) ) ) ) ) ).

% PartialSA.x_elem_set_product
thf(fact_267_PartialSA_Oadd__set__elem,axiom,
    ! [Phi2: state,A3: set_state,B3: set_state] :
      ( ( member_state @ Phi2 @ ( add_set @ A3 @ B3 ) )
      = ( ? [A2: state,B2: state] :
            ( ( ( some_state @ Phi2 )
              = ( plus @ A2 @ B2 ) )
            & ( member_state @ A2 @ A3 )
            & ( member_state @ B2 @ B3 ) ) ) ) ).

% PartialSA.add_set_elem
thf(fact_268_plus__charact_I2_J,axiom,
    ! [A: state,B: state,X2: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X2 ) )
     => ( ( get_h @ X2 )
        = ( map_ad2908576761495157028ar_val @ ( get_h @ A ) @ ( get_h @ B ) ) ) ) ).

% plus_charact(2)
thf(fact_269_map__add__eq__empty__iff,axiom,
    ! [F4: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( ( map_ad2908576761495157028ar_val @ F4 @ G )
        = ( ^ [X: produc1457211279475724562t_char] : none_val ) )
      = ( ( F4
          = ( ^ [X: produc1457211279475724562t_char] : none_val ) )
        & ( G
          = ( ^ [X: produc1457211279475724562t_char] : none_val ) ) ) ) ).

% map_add_eq_empty_iff
thf(fact_270_map__add__None,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = none_val )
      = ( ( ( N @ K )
          = none_val )
        & ( ( M @ K )
          = none_val ) ) ) ).

% map_add_None
thf(fact_271_map__add__find__right,axiom,
    ! [N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,Xx: val,M: produc1457211279475724562t_char > option_val] :
      ( ( ( N @ K )
        = ( some_val @ Xx ) )
     => ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ Xx ) ) ) ).

% map_add_find_right
thf(fact_272_map__add__SomeD,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X2: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X2 ) )
     => ( ( ( N @ K )
          = ( some_val @ X2 ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X2 ) ) ) ) ) ).

% map_add_SomeD
thf(fact_273_map__add__Some__iff,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X2: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X2 ) )
      = ( ( ( N @ K )
          = ( some_val @ X2 ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X2 ) ) ) ) ) ).

% map_add_Some_iff
thf(fact_274_add__states_Opelims,axiom,
    ! [X2: produc9003781588162623004on_val,Xa: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X2 @ Xa )
        = Y )
     => ( ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ X2 @ Xa ) )
       => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X2
                = ( produc74620467203698198on_val @ Pi4 @ H ) )
             => ! [Pi5: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
                  ( ( Xa
                    = ( produc74620467203698198on_val @ Pi5 @ H4 ) )
                 => ( ( Y
                      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi4 @ Pi5 ) @ ( map_ad2908576761495157028ar_val @ H @ H4 ) ) )
                   => ~ ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi4 @ H ) @ ( produc74620467203698198on_val @ Pi5 @ H4 ) ) ) ) ) ) ) ) ).

% add_states.pelims
thf(fact_275_compatibleI__old,axiom,
    ! [Phi2: produc9003781588162623004on_val,Phi3: produc9003781588162623004on_val] :
      ( ! [Hl2: produc1457211279475724562t_char,X3: val,Y4: val] :
          ( ( ( ( produc2319605628138516840on_val @ Phi2 @ Hl2 )
              = ( some_val @ X3 ) )
            & ( ( produc2319605628138516840on_val @ Phi3 @ Hl2 )
              = ( some_val @ Y4 ) ) )
         => ( X3 = Y4 ) )
     => ( ( valid_mask @ ( add_ma6586698345353345811t_char @ ( produc5239619271231976746on_val @ Phi2 ) @ ( produc5239619271231976746on_val @ Phi3 ) ) )
       => ( compatible @ Phi2 @ Phi3 ) ) ) ).

% compatibleI_old
thf(fact_276_compatible__def,axiom,
    ( compatible
    = ( ^ [Phi4: produc9003781588162623004on_val,Phi5: produc9003781588162623004on_val] :
          ( ( compatible_heaps @ ( produc2319605628138516840on_val @ Phi4 ) @ ( produc2319605628138516840on_val @ Phi5 ) )
          & ( valid_mask @ ( add_ma6586698345353345811t_char @ ( produc5239619271231976746on_val @ Phi4 ) @ ( produc5239619271231976746on_val @ Phi5 ) ) ) ) ) ) ).

% compatible_def
thf(fact_277_prod_Ocollapse,axiom,
    ! [Prod: produc3142500478612311029_state] :
      ( ( produc9160152616361873709_state @ ( produc3900939990612242713_state @ Prod ) @ ( produc9117690093182453595_state @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_278_prod_Ocollapse,axiom,
    ! [Prod: produc4844402304972151915_state] :
      ( ( produc6205719116230455971_state @ ( produc6030790797199838863_state @ Prod ) @ ( produc8138835347214057169_state @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_279_prod_Ocollapse,axiom,
    ! [Prod: produc6251377953716854561_state] :
      ( ( produc9080284787853255769_state @ ( produc7279835715735703621_state @ Prod ) @ ( produc4944973370551335047_state @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_280_prod_Ocollapse,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_281_prod_Ocollapse,axiom,
    ! [Prod: produc9003781588162623004on_val] :
      ( ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_282_snd__conv,axiom,
    ! [X1: option_state,X22: option_state] :
      ( ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_283_snd__conv,axiom,
    ! [X1: prat,X22: state] :
      ( ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_284_snd__conv,axiom,
    ! [X1: prat,X22: set_state] :
      ( ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_285_snd__conv,axiom,
    ! [X1: nat,X22: list_char] :
      ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_286_snd__conv,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] :
      ( ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_287_snd__eqD,axiom,
    ! [X2: option_state,Y: option_state,A: option_state] :
      ( ( ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_288_snd__eqD,axiom,
    ! [X2: prat,Y: state,A: state] :
      ( ( ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_289_snd__eqD,axiom,
    ! [X2: prat,Y: set_state,A: set_state] :
      ( ( ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_290_snd__eqD,axiom,
    ! [X2: nat,Y: list_char,A: list_char] :
      ( ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_291_snd__eqD,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > option_val] :
      ( ( ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_292_sndI,axiom,
    ! [X2: produc3142500478612311029_state,Y: option_state,Z: option_state] :
      ( ( X2
        = ( produc9160152616361873709_state @ Y @ Z ) )
     => ( ( produc9117690093182453595_state @ X2 )
        = Z ) ) ).

% sndI
thf(fact_293_sndI,axiom,
    ! [X2: produc4844402304972151915_state,Y: prat,Z: state] :
      ( ( X2
        = ( produc6205719116230455971_state @ Y @ Z ) )
     => ( ( produc8138835347214057169_state @ X2 )
        = Z ) ) ).

% sndI
thf(fact_294_sndI,axiom,
    ! [X2: produc6251377953716854561_state,Y: prat,Z: set_state] :
      ( ( X2
        = ( produc9080284787853255769_state @ Y @ Z ) )
     => ( ( produc4944973370551335047_state @ X2 )
        = Z ) ) ).

% sndI
thf(fact_295_sndI,axiom,
    ! [X2: produc1457211279475724562t_char,Y: nat,Z: list_char] :
      ( ( X2
        = ( produc120197956887798346t_char @ Y @ Z ) )
     => ( ( produc1900778367047502200t_char @ X2 )
        = Z ) ) ).

% sndI
thf(fact_296_sndI,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > option_val] :
      ( ( X2
        = ( produc74620467203698198on_val @ Y @ Z ) )
     => ( ( produc2319605628138516840on_val @ X2 )
        = Z ) ) ).

% sndI
thf(fact_297_eq__snd__iff,axiom,
    ! [B: option_state,P2: produc3142500478612311029_state] :
      ( ( B
        = ( produc9117690093182453595_state @ P2 ) )
      = ( ? [A2: option_state] :
            ( P2
            = ( produc9160152616361873709_state @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_298_eq__snd__iff,axiom,
    ! [B: state,P2: produc4844402304972151915_state] :
      ( ( B
        = ( produc8138835347214057169_state @ P2 ) )
      = ( ? [A2: prat] :
            ( P2
            = ( produc6205719116230455971_state @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_299_eq__snd__iff,axiom,
    ! [B: set_state,P2: produc6251377953716854561_state] :
      ( ( B
        = ( produc4944973370551335047_state @ P2 ) )
      = ( ? [A2: prat] :
            ( P2
            = ( produc9080284787853255769_state @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_300_eq__snd__iff,axiom,
    ! [B: list_char,P2: produc1457211279475724562t_char] :
      ( ( B
        = ( produc1900778367047502200t_char @ P2 ) )
      = ( ? [A2: nat] :
            ( P2
            = ( produc120197956887798346t_char @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_301_eq__snd__iff,axiom,
    ! [B: produc1457211279475724562t_char > option_val,P2: produc9003781588162623004on_val] :
      ( ( B
        = ( produc2319605628138516840on_val @ P2 ) )
      = ( ? [A2: produc1457211279475724562t_char > prat] :
            ( P2
            = ( produc74620467203698198on_val @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_302_prod__eq__iff,axiom,
    ( ( ^ [Y6: produc1457211279475724562t_char,Z2: produc1457211279475724562t_char] : ( Y6 = Z2 ) )
    = ( ^ [S: produc1457211279475724562t_char,T: produc1457211279475724562t_char] :
          ( ( ( produc6204653154488278070t_char @ S )
            = ( produc6204653154488278070t_char @ T ) )
          & ( ( produc1900778367047502200t_char @ S )
            = ( produc1900778367047502200t_char @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_303_prod__eq__iff,axiom,
    ( ( ^ [Y6: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] : ( Y6 = Z2 ) )
    = ( ^ [S: produc9003781588162623004on_val,T: produc9003781588162623004on_val] :
          ( ( ( produc5239619271231976746on_val @ S )
            = ( produc5239619271231976746on_val @ T ) )
          & ( ( produc2319605628138516840on_val @ S )
            = ( produc2319605628138516840on_val @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_304_prod__eqI,axiom,
    ! [P2: produc1457211279475724562t_char,Q2: produc1457211279475724562t_char] :
      ( ( ( produc6204653154488278070t_char @ P2 )
        = ( produc6204653154488278070t_char @ Q2 ) )
     => ( ( ( produc1900778367047502200t_char @ P2 )
          = ( produc1900778367047502200t_char @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_305_prod__eqI,axiom,
    ! [P2: produc9003781588162623004on_val,Q2: produc9003781588162623004on_val] :
      ( ( ( produc5239619271231976746on_val @ P2 )
        = ( produc5239619271231976746on_val @ Q2 ) )
     => ( ( ( produc2319605628138516840on_val @ P2 )
          = ( produc2319605628138516840on_val @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_306_prod_Oexpand,axiom,
    ! [Prod: produc1457211279475724562t_char,Prod2: produc1457211279475724562t_char] :
      ( ( ( ( produc6204653154488278070t_char @ Prod )
          = ( produc6204653154488278070t_char @ Prod2 ) )
        & ( ( produc1900778367047502200t_char @ Prod )
          = ( produc1900778367047502200t_char @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_307_prod_Oexpand,axiom,
    ! [Prod: produc9003781588162623004on_val,Prod2: produc9003781588162623004on_val] :
      ( ( ( ( produc5239619271231976746on_val @ Prod )
          = ( produc5239619271231976746on_val @ Prod2 ) )
        & ( ( produc2319605628138516840on_val @ Prod )
          = ( produc2319605628138516840on_val @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_308_surjective__pairing,axiom,
    ! [T2: produc3142500478612311029_state] :
      ( T2
      = ( produc9160152616361873709_state @ ( produc3900939990612242713_state @ T2 ) @ ( produc9117690093182453595_state @ T2 ) ) ) ).

% surjective_pairing
thf(fact_309_surjective__pairing,axiom,
    ! [T2: produc4844402304972151915_state] :
      ( T2
      = ( produc6205719116230455971_state @ ( produc6030790797199838863_state @ T2 ) @ ( produc8138835347214057169_state @ T2 ) ) ) ).

% surjective_pairing
thf(fact_310_surjective__pairing,axiom,
    ! [T2: produc6251377953716854561_state] :
      ( T2
      = ( produc9080284787853255769_state @ ( produc7279835715735703621_state @ T2 ) @ ( produc4944973370551335047_state @ T2 ) ) ) ).

% surjective_pairing
thf(fact_311_surjective__pairing,axiom,
    ! [T2: produc1457211279475724562t_char] :
      ( T2
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ T2 ) @ ( produc1900778367047502200t_char @ T2 ) ) ) ).

% surjective_pairing
thf(fact_312_surjective__pairing,axiom,
    ! [T2: produc9003781588162623004on_val] :
      ( T2
      = ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ T2 ) @ ( produc2319605628138516840on_val @ T2 ) ) ) ).

% surjective_pairing
thf(fact_313_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3142500478612311029_state] :
      ( Prod
      = ( produc9160152616361873709_state @ ( produc3900939990612242713_state @ Prod ) @ ( produc9117690093182453595_state @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_314_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4844402304972151915_state] :
      ( Prod
      = ( produc6205719116230455971_state @ ( produc6030790797199838863_state @ Prod ) @ ( produc8138835347214057169_state @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_315_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6251377953716854561_state] :
      ( Prod
      = ( produc9080284787853255769_state @ ( produc7279835715735703621_state @ Prod ) @ ( produc4944973370551335047_state @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_316_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( Prod
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_317_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9003781588162623004on_val] :
      ( Prod
      = ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_318_get__h_Osimps,axiom,
    ( get_h
    = ( ^ [A2: state] : ( produc2319605628138516840on_val @ ( rep_state @ A2 ) ) ) ) ).

% get_h.simps
thf(fact_319_get__h_Oelims,axiom,
    ! [X2: state,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( get_h @ X2 )
        = Y )
     => ( Y
        = ( produc2319605628138516840on_val @ ( rep_state @ X2 ) ) ) ) ).

% get_h.elims
thf(fact_320_get__pre_I1_J,axiom,
    ( get_h
    = ( ^ [X: state] : ( produc2319605628138516840on_val @ ( rep_state @ X ) ) ) ) ).

% get_pre(1)
thf(fact_321_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P4: option_state > option_state > $o,X2: option_state,Y: option_state,A: produc3142500478612311029_state] :
      ( ( P4 @ X2 @ Y )
     => ( ( A
          = ( produc9160152616361873709_state @ X2 @ Y ) )
       => ( P4 @ ( produc3900939990612242713_state @ A ) @ ( produc9117690093182453595_state @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_322_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P4: prat > state > $o,X2: prat,Y: state,A: produc4844402304972151915_state] :
      ( ( P4 @ X2 @ Y )
     => ( ( A
          = ( produc6205719116230455971_state @ X2 @ Y ) )
       => ( P4 @ ( produc6030790797199838863_state @ A ) @ ( produc8138835347214057169_state @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_323_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P4: prat > set_state > $o,X2: prat,Y: set_state,A: produc6251377953716854561_state] :
      ( ( P4 @ X2 @ Y )
     => ( ( A
          = ( produc9080284787853255769_state @ X2 @ Y ) )
       => ( P4 @ ( produc7279835715735703621_state @ A ) @ ( produc4944973370551335047_state @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_324_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P4: nat > list_char > $o,X2: nat,Y: list_char,A: produc1457211279475724562t_char] :
      ( ( P4 @ X2 @ Y )
     => ( ( A
          = ( produc120197956887798346t_char @ X2 @ Y ) )
       => ( P4 @ ( produc6204653154488278070t_char @ A ) @ ( produc1900778367047502200t_char @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_325_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,X2: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A: produc9003781588162623004on_val] :
      ( ( P4 @ X2 @ Y )
     => ( ( A
          = ( produc74620467203698198on_val @ X2 @ Y ) )
       => ( P4 @ ( produc5239619271231976746on_val @ A ) @ ( produc2319605628138516840on_val @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_326_conjI__realizer,axiom,
    ! [P4: option_state > $o,P2: option_state,Q: option_state > $o,Q2: option_state] :
      ( ( P4 @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P4 @ ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ P2 @ Q2 ) ) )
          & ( Q @ ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_327_conjI__realizer,axiom,
    ! [P4: prat > $o,P2: prat,Q: state > $o,Q2: state] :
      ( ( P4 @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P4 @ ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ P2 @ Q2 ) ) )
          & ( Q @ ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_328_conjI__realizer,axiom,
    ! [P4: prat > $o,P2: prat,Q: set_state > $o,Q2: set_state] :
      ( ( P4 @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P4 @ ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ P2 @ Q2 ) ) )
          & ( Q @ ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_329_conjI__realizer,axiom,
    ! [P4: nat > $o,P2: nat,Q: list_char > $o,Q2: list_char] :
      ( ( P4 @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P4 @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ P2 @ Q2 ) ) )
          & ( Q @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_330_conjI__realizer,axiom,
    ! [P4: ( produc1457211279475724562t_char > prat ) > $o,P2: produc1457211279475724562t_char > prat,Q: ( produc1457211279475724562t_char > option_val ) > $o,Q2: produc1457211279475724562t_char > option_val] :
      ( ( P4 @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P4 @ ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ P2 @ Q2 ) ) )
          & ( Q @ ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_331_exI__realizer,axiom,
    ! [P4: option_state > option_state > $o,Y: option_state,X2: option_state] :
      ( ( P4 @ Y @ X2 )
     => ( P4 @ ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ X2 @ Y ) ) @ ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_332_exI__realizer,axiom,
    ! [P4: state > prat > $o,Y: state,X2: prat] :
      ( ( P4 @ Y @ X2 )
     => ( P4 @ ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ X2 @ Y ) ) @ ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_333_exI__realizer,axiom,
    ! [P4: set_state > prat > $o,Y: set_state,X2: prat] :
      ( ( P4 @ Y @ X2 )
     => ( P4 @ ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ X2 @ Y ) ) @ ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_334_exI__realizer,axiom,
    ! [P4: list_char > nat > $o,Y: list_char,X2: nat] :
      ( ( P4 @ Y @ X2 )
     => ( P4 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X2 @ Y ) ) @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_335_exI__realizer,axiom,
    ! [P4: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > $o,Y: produc1457211279475724562t_char > option_val,X2: produc1457211279475724562t_char > prat] :
      ( ( P4 @ Y @ X2 )
     => ( P4 @ ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X2 @ Y ) ) @ ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_336_prod_Oswap__def,axiom,
    ( produc8425362190318354125_state
    = ( ^ [P3: produc3142500478612311029_state] : ( produc9160152616361873709_state @ ( produc9117690093182453595_state @ P3 ) @ ( produc3900939990612242713_state @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_337_prod_Oswap__def,axiom,
    ( produc6343331096166649923e_prat
    = ( ^ [P3: produc7334356684927367531e_prat] : ( produc6205719116230455971_state @ ( produc7795413381861543633e_prat @ P3 ) @ ( produc5687368831847325327e_prat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_338_prod_Oswap__def,axiom,
    ( produc7016766326345971705e_prat
    = ( ^ [P3: produc2654212508281724193e_prat] : ( produc9080284787853255769_state @ ( produc1175801883329052295e_prat @ P3 ) @ ( produc3510664228513420869e_prat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_339_prod_Oswap__def,axiom,
    ( produc4410195085741832554ar_nat
    = ( ^ [P3: produc7606469099237612818ar_nat] : ( produc120197956887798346t_char @ ( produc1259554233722553080ar_nat @ P3 ) @ ( produc5563429021163328950ar_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_340_prod_Oswap__def,axiom,
    ( produc5819413633627552064r_prat
    = ( ^ [P3: produc7776263825685906734r_prat] : ( produc74620467203698198on_val @ ( produc4142415063207305778r_prat @ P3 ) @ ( produc7062428706300765684r_prat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_341_prod_Oswap__def,axiom,
    ( produc5051419219066781674t_char
    = ( ^ [P3: produc1457211279475724562t_char] : ( produc8702345860417625034ar_nat @ ( produc1900778367047502200t_char @ P3 ) @ ( produc6204653154488278070t_char @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_342_prod_Oswap__def,axiom,
    ( produc3996604198558763126on_val
    = ( ^ [P3: produc9003781588162623004on_val] : ( produc1897429902272487136r_prat @ ( produc2319605628138516840on_val @ P3 ) @ ( produc5239619271231976746on_val @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_343_snd__swap,axiom,
    ! [X2: produc1457211279475724562t_char] :
      ( ( produc1259554233722553080ar_nat @ ( produc5051419219066781674t_char @ X2 ) )
      = ( produc6204653154488278070t_char @ X2 ) ) ).

% snd_swap
thf(fact_344_snd__swap,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ( ( produc4142415063207305778r_prat @ ( produc3996604198558763126on_val @ X2 ) )
      = ( produc5239619271231976746on_val @ X2 ) ) ).

% snd_swap
thf(fact_345_snd__swap,axiom,
    ! [X2: produc7776263825685906734r_prat] :
      ( ( produc2319605628138516840on_val @ ( produc5819413633627552064r_prat @ X2 ) )
      = ( produc7062428706300765684r_prat @ X2 ) ) ).

% snd_swap
thf(fact_346_fst__swap,axiom,
    ! [X2: produc7606469099237612818ar_nat] :
      ( ( produc6204653154488278070t_char @ ( produc4410195085741832554ar_nat @ X2 ) )
      = ( produc1259554233722553080ar_nat @ X2 ) ) ).

% fst_swap
thf(fact_347_fst__swap,axiom,
    ! [X2: produc7776263825685906734r_prat] :
      ( ( produc5239619271231976746on_val @ ( produc5819413633627552064r_prat @ X2 ) )
      = ( produc4142415063207305778r_prat @ X2 ) ) ).

% fst_swap
thf(fact_348_fst__swap,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ( ( produc7062428706300765684r_prat @ ( produc3996604198558763126on_val @ X2 ) )
      = ( produc2319605628138516840on_val @ X2 ) ) ).

% fst_swap
thf(fact_349_sep__algebra_Osplus_Ocong,axiom,
    sep_splus_state = sep_splus_state ).

% sep_algebra.splus.cong
thf(fact_350_swap__simp,axiom,
    ! [X2: state,Y: prat] :
      ( ( produc6343331096166649923e_prat @ ( produc5862297150877942435e_prat @ X2 @ Y ) )
      = ( produc6205719116230455971_state @ Y @ X2 ) ) ).

% swap_simp
thf(fact_351_swap__simp,axiom,
    ! [X2: set_state,Y: prat] :
      ( ( produc7016766326345971705e_prat @ ( produc5311113300630973017e_prat @ X2 @ Y ) )
      = ( produc9080284787853255769_state @ Y @ X2 ) ) ).

% swap_simp
thf(fact_352_swap__simp,axiom,
    ! [X2: list_char,Y: nat] :
      ( ( produc4410195085741832554ar_nat @ ( produc8702345860417625034ar_nat @ X2 @ Y ) )
      = ( produc120197956887798346t_char @ Y @ X2 ) ) ).

% swap_simp
thf(fact_353_swap__simp,axiom,
    ! [X2: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > prat] :
      ( ( produc5819413633627552064r_prat @ ( produc1897429902272487136r_prat @ X2 @ Y ) )
      = ( produc74620467203698198on_val @ Y @ X2 ) ) ).

% swap_simp
thf(fact_354_swap__simp,axiom,
    ! [X2: option_state,Y: option_state] :
      ( ( produc8425362190318354125_state @ ( produc9160152616361873709_state @ X2 @ Y ) )
      = ( produc9160152616361873709_state @ Y @ X2 ) ) ).

% swap_simp
thf(fact_355_swap__simp,axiom,
    ! [X2: prat,Y: state] :
      ( ( produc6686753061519163459_state @ ( produc6205719116230455971_state @ X2 @ Y ) )
      = ( produc5862297150877942435e_prat @ Y @ X2 ) ) ).

% swap_simp
thf(fact_356_swap__simp,axiom,
    ! [X2: prat,Y: set_state] :
      ( ( produc1562565776713478649_state @ ( produc9080284787853255769_state @ X2 @ Y ) )
      = ( produc5311113300630973017e_prat @ Y @ X2 ) ) ).

% swap_simp
thf(fact_357_swap__simp,axiom,
    ! [X2: nat,Y: list_char] :
      ( ( produc5051419219066781674t_char @ ( produc120197956887798346t_char @ X2 @ Y ) )
      = ( produc8702345860417625034ar_nat @ Y @ X2 ) ) ).

% swap_simp
thf(fact_358_swap__simp,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( produc3996604198558763126on_val @ ( produc74620467203698198on_val @ X2 @ Y ) )
      = ( produc1897429902272487136r_prat @ Y @ X2 ) ) ).

% swap_simp
thf(fact_359_valid__pre__unit,axiom,
    valid_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ).

% valid_pre_unit
thf(fact_360_pre__get__h_Oelims,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( pre_ge1486375753531218896on_val @ X2 )
        = Y )
     => ( Y
        = ( produc2319605628138516840on_val @ X2 ) ) ) ).

% pre_get_h.elims
thf(fact_361_pre__get__h_Osimps,axiom,
    pre_ge1486375753531218896on_val = produc2319605628138516840on_val ).

% pre_get_h.simps
thf(fact_362_pre__get__m_Oelims,axiom,
    ! [X2: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X2 )
        = Y )
     => ( Y
        = ( produc6204653154488278070t_char @ X2 ) ) ) ).

% pre_get_m.elims
thf(fact_363_pre__get__m_Oelims,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat] :
      ( ( ( pre_ge8481574585653477131on_val @ X2 )
        = Y )
     => ( Y
        = ( produc5239619271231976746on_val @ X2 ) ) ) ).

% pre_get_m.elims
thf(fact_364_pre__get__m_Osimps,axiom,
    pre_ge2740421702909910421t_char = produc6204653154488278070t_char ).

% pre_get_m.simps
thf(fact_365_pre__get__m_Osimps,axiom,
    pre_ge8481574585653477131on_val = produc5239619271231976746on_val ).

% pre_get_m.simps
thf(fact_366_valid__larger__mask,axiom,
    ( valid_mask
    = ( greate7680731506090375811t_char @ full_mask ) ) ).

% valid_larger_mask
thf(fact_367_greater__mask__properties_I1_J,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] : ( greate7680731506090375811t_char @ Pi @ Pi ) ).

% greater_mask_properties(1)
thf(fact_368_greater__mask__properties_I2_J,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ A @ B )
        & ( greate7680731506090375811t_char @ B @ C ) )
     => ( greate7680731506090375811t_char @ A @ C ) ) ).

% greater_mask_properties(2)
thf(fact_369_greater__mask__properties_I3_J,axiom,
    ! [Pi6: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ Pi6 @ Pi )
        & ( greate7680731506090375811t_char @ Pi @ Pi6 ) )
     => ( Pi = Pi6 ) ) ).

% greater_mask_properties(3)
thf(fact_370_greater__maskI,axiom,
    ! [Pi6: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
      ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ ( Pi6 @ Hl2 ) @ ( Pi @ Hl2 ) )
     => ( greate7680731506090375811t_char @ Pi6 @ Pi ) ) ).

% greater_maskI
thf(fact_371_greater__mask__equiv__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat] :
        ! [Hl: produc1457211279475724562t_char] : ( pgte @ ( Pi2 @ Hl ) @ ( Pi3 @ Hl ) ) ) ) ).

% greater_mask_equiv_def
thf(fact_372_const__sum__mask__greater,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ A @ B )
        = ( add_ma6586698345353345811t_char @ C @ D ) )
     => ( ( greate7680731506090375811t_char @ A @ C )
       => ( greate7680731506090375811t_char @ D @ B ) ) ) ).

% const_sum_mask_greater
thf(fact_373_greater__mask__decomp,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( greate7680731506090375811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
     => ? [A1: produc1457211279475724562t_char > prat,A22: produc1457211279475724562t_char > prat] :
          ( ( A
            = ( add_ma6586698345353345811t_char @ A1 @ A22 ) )
          & ( greate7680731506090375811t_char @ A1 @ B )
          & ( greate7680731506090375811t_char @ A22 @ C ) ) ) ).

% greater_mask_decomp
thf(fact_374_greater__mask__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat] :
        ? [R: produc1457211279475724562t_char > prat] :
          ( Pi2
          = ( add_ma6586698345353345811t_char @ Pi3 @ R ) ) ) ) ).

% greater_mask_def
thf(fact_375_mult__greater,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,P2: prat] :
      ( ( greate7680731506090375811t_char @ A @ B )
     => ( greate7680731506090375811t_char @ ( multiply_mask @ P2 @ A ) @ ( multiply_mask @ P2 @ B ) ) ) ).

% mult_greater
thf(fact_376_valid__empty,axiom,
    valid_mask @ empty_3446695950879338768t_char ).

% valid_empty
thf(fact_377_minus__empty,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] :
      ( Pi
      = ( add_ma6586698345353345811t_char @ Pi @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_378_empty__mask_Oelims,axiom,
    ! [X2: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X2 )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_379_empty__mask_Osimps,axiom,
    ( empty_3446695950879338768t_char
    = ( ^ [Hl: produc1457211279475724562t_char] : pnone ) ) ).

% empty_mask.simps
thf(fact_380_pre__get__h_Opelims,axiom,
    ! [X2: produc3142500478612311029_state,Y: option_state] :
      ( ( ( pre_ge6337690187194669811_state @ X2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ pre_ge4642282138912335602_state @ X2 )
       => ~ ( ( Y
              = ( produc9117690093182453595_state @ X2 ) )
           => ~ ( accp_P7183347322223220350_state @ pre_ge4642282138912335602_state @ X2 ) ) ) ) ).

% pre_get_h.pelims
thf(fact_381_pre__get__h_Opelims,axiom,
    ! [X2: produc4844402304972151915_state,Y: state] :
      ( ( ( pre_get_h_prat_state @ X2 )
        = Y )
     => ( ( accp_P6233459008282737652_state @ pre_ge6006653645540518248_state @ X2 )
       => ~ ( ( Y
              = ( produc8138835347214057169_state @ X2 ) )
           => ~ ( accp_P6233459008282737652_state @ pre_ge6006653645540518248_state @ X2 ) ) ) ) ).

% pre_get_h.pelims
thf(fact_382_pre__get__h_Opelims,axiom,
    ! [X2: produc6251377953716854561_state,Y: set_state] :
      ( ( ( pre_ge1447473398360597535_state @ X2 )
        = Y )
     => ( ( accp_P6554740477754186410_state @ pre_ge3291400433338533662_state @ X2 )
       => ~ ( ( Y
              = ( produc4944973370551335047_state @ X2 ) )
           => ~ ( accp_P6554740477754186410_state @ pre_ge3291400433338533662_state @ X2 ) ) ) ) ).

% pre_get_h.pelims
thf(fact_383_pre__get__h_Opelims,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( pre_ge1486375753531218896on_val @ X2 )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ pre_ge6140436789258014353on_val @ X2 )
       => ~ ( ( Y
              = ( produc2319605628138516840on_val @ X2 ) )
           => ~ ( accp_P4405556307444930259on_val @ pre_ge6140436789258014353on_val @ X2 ) ) ) ) ).

% pre_get_h.pelims
thf(fact_384_pre__get__m_Opelims,axiom,
    ! [X2: produc3142500478612311029_state,Y: option_state] :
      ( ( ( pre_ge4435996478124457464_state @ X2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ pre_ge9112490230592956973_state @ X2 )
       => ~ ( ( Y
              = ( produc3900939990612242713_state @ X2 ) )
           => ~ ( accp_P7183347322223220350_state @ pre_ge9112490230592956973_state @ X2 ) ) ) ) ).

% pre_get_m.pelims
thf(fact_385_pre__get__m_Opelims,axiom,
    ! [X2: produc4844402304972151915_state,Y: prat] :
      ( ( ( pre_get_m_prat_state @ X2 )
        = Y )
     => ( ( accp_P6233459008282737652_state @ pre_ge6835382860682234787_state @ X2 )
       => ~ ( ( Y
              = ( produc6030790797199838863_state @ X2 ) )
           => ~ ( accp_P6233459008282737652_state @ pre_ge6835382860682234787_state @ X2 ) ) ) ) ).

% pre_get_m.pelims
thf(fact_386_pre__get__m_Opelims,axiom,
    ! [X2: produc6251377953716854561_state,Y: prat] :
      ( ( ( pre_ge1435879308799382564_state @ X2 )
        = Y )
     => ( ( accp_P6554740477754186410_state @ pre_ge4013444835548200281_state @ X2 )
       => ~ ( ( Y
              = ( produc7279835715735703621_state @ X2 ) )
           => ~ ( accp_P6554740477754186410_state @ pre_ge4013444835548200281_state @ X2 ) ) ) ) ).

% pre_get_m.pelims
thf(fact_387_pre__get__m_Opelims,axiom,
    ! [X2: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X2 )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X2 )
       => ~ ( ( Y
              = ( produc6204653154488278070t_char @ X2 ) )
           => ~ ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X2 ) ) ) ) ).

% pre_get_m.pelims
thf(fact_388_pre__get__m_Opelims,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat] :
      ( ( ( pre_ge8481574585653477131on_val @ X2 )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ pre_ge4462515927930180886on_val @ X2 )
       => ~ ( ( Y
              = ( produc5239619271231976746on_val @ X2 ) )
           => ~ ( accp_P4405556307444930259on_val @ pre_ge4462515927930180886on_val @ X2 ) ) ) ) ).

% pre_get_m.pelims
thf(fact_389_empty__mask_Opelims,axiom,
    ! [X2: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X2 )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X2 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X2 ) ) ) ) ).

% empty_mask.pelims
thf(fact_390_empty__mask_Opelims,axiom,
    ! [X2: produc3142500478612311029_state,Y: prat] :
      ( ( ( empty_5858998862663000819_state @ X2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ empty_3683758330361983676_state @ X2 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P7183347322223220350_state @ empty_3683758330361983676_state @ X2 ) ) ) ) ).

% empty_mask.pelims
thf(fact_391_empty__mask_Opelims,axiom,
    ! [X2: produc4844402304972151915_state,Y: prat] :
      ( ( ( empty_3913122416104905065_state @ X2 )
        = Y )
     => ( ( accp_P6233459008282737652_state @ empty_5283331674940636722_state @ X2 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P6233459008282737652_state @ empty_5283331674940636722_state @ X2 ) ) ) ) ).

% empty_mask.pelims
thf(fact_392_empty__mask_Opelims,axiom,
    ! [X2: produc6251377953716854561_state,Y: prat] :
      ( ( ( empty_5541330147480151071_state @ X2 )
        = Y )
     => ( ( accp_P6554740477754186410_state @ empty_7304051301540881640_state @ X2 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P6554740477754186410_state @ empty_7304051301540881640_state @ X2 ) ) ) ) ).

% empty_mask.pelims
thf(fact_393_empty__mask_Opelims,axiom,
    ! [X2: produc9003781588162623004on_val,Y: prat] :
      ( ( ( empty_6102255041883331742on_val @ X2 )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ empty_5847456986941130773on_val @ X2 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P4405556307444930259on_val @ empty_5847456986941130773on_val @ X2 ) ) ) ) ).

% empty_mask.pelims
thf(fact_394_empty__mask_Opelims,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: prat] :
      ( ( ( empty_6616431780912394577r_prat @ X2 )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X2 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X2 ) ) ) ) ).

% empty_mask.pelims
thf(fact_395_unit__def,axiom,
    ( unit
    = ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ) ) ).

% unit_def
thf(fact_396_greater__mask__uniI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Hl3: produc1457211279475724562t_char,R3: prat] :
      ( ( pgte @ ( Pi @ Hl3 ) @ R3 )
     => ( greate7680731506090375811t_char @ Pi @ ( uni_ma6578582744724537101t_char @ Hl3 @ R3 ) ) ) ).

% greater_mask_uniI
thf(fact_397_greater__mask__uni__equiv,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Hl3: produc1457211279475724562t_char,R3: prat] :
      ( ( greate7680731506090375811t_char @ Pi @ ( uni_ma6578582744724537101t_char @ Hl3 @ R3 ) )
      = ( pgte @ ( Pi @ Hl3 ) @ R3 ) ) ).

% greater_mask_uni_equiv
thf(fact_398_core__def,axiom,
    ( core
    = ( ^ [Phi4: state] : ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ ( get_h @ Phi4 ) ) ) ) ) ).

% core_def
thf(fact_399_PartialSA_Ounit__core,axiom,
    ( ( core @ unit )
    = unit ) ).

% PartialSA.unit_core
thf(fact_400_core__defined_I1_J,axiom,
    ! [Phi2: state] :
      ( ( get_h @ ( core @ Phi2 ) )
      = ( get_h @ Phi2 ) ) ).

% core_defined(1)
thf(fact_401_stable__unit,axiom,
    stable @ unit ).

% stable_unit
thf(fact_402_core__is__smaller,axiom,
    ( some_state
    = ( ^ [X: state] : ( plus @ X @ ( core @ X ) ) ) ) ).

% core_is_smaller
thf(fact_403_core__is__pure,axiom,
    ! [X2: state] :
      ( ( some_state @ ( core @ X2 ) )
      = ( plus @ ( core @ X2 ) @ ( core @ X2 ) ) ) ).

% core_is_pure
thf(fact_404_cancellative,axiom,
    ! [A: state,B: state,X2: state,Y: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ X2 ) )
     => ( ( ( some_state @ A )
          = ( plus @ B @ Y ) )
       => ( ( ( core @ X2 )
            = ( core @ Y ) )
         => ( X2 = Y ) ) ) ) ).

% cancellative
thf(fact_405_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_406_core__max,axiom,
    ! [X2: state,C: state] :
      ( ( ( some_state @ X2 )
        = ( plus @ X2 @ C ) )
     => ? [R4: state] :
          ( ( some_state @ ( core @ X2 ) )
          = ( plus @ C @ R4 ) ) ) ).

% core_max
thf(fact_407_core__defined_I2_J,axiom,
    ! [Phi2: state] :
      ( ( get_m @ ( core @ Phi2 ) )
      = empty_3446695950879338768t_char ) ).

% core_defined(2)
thf(fact_408_unit__neutral,axiom,
    ( some_state
    = ( ^ [A2: state] : ( plus @ A2 @ unit ) ) ) ).

% unit_neutral
thf(fact_409_unit__charact_I2_J,axiom,
    ( ( get_m @ unit )
    = empty_3446695950879338768t_char ) ).

% unit_charact(2)
thf(fact_410_unit__charact_I1_J,axiom,
    ( ( get_h @ unit )
    = empty_7512026568161686585ar_val ) ).

% unit_charact(1)
thf(fact_411_get__h_Opelims,axiom,
    ! [X2: state,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( get_h @ X2 )
        = Y )
     => ( ( accp_state @ get_h_rel @ X2 )
       => ~ ( ( Y
              = ( produc2319605628138516840on_val @ ( rep_state @ X2 ) ) )
           => ~ ( accp_state @ get_h_rel @ X2 ) ) ) ) ).

% get_h.pelims
thf(fact_412_get__m_Opelims,axiom,
    ! [X2: state,Y: produc1457211279475724562t_char > prat] :
      ( ( ( get_m @ X2 )
        = Y )
     => ( ( accp_state @ get_m_rel @ X2 )
       => ~ ( ( Y
              = ( produc5239619271231976746on_val @ ( rep_state @ X2 ) ) )
           => ~ ( accp_state @ get_m_rel @ X2 ) ) ) ) ).

% get_m.pelims
thf(fact_413_PartialSA_Osplus_Opelims,axiom,
    ! [X2: option_state,Xa: option_state,Y: option_state] :
      ( ( ( sep_splus_state @ plus @ X2 @ Xa )
        = Y )
     => ( ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ X2 @ Xa ) )
       => ( ( ( X2 = none_state )
           => ( ( Y = none_state )
             => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) ) )
         => ( ! [V: state] :
                ( ( X2
                  = ( some_state @ V ) )
               => ( ( Xa = none_state )
                 => ( ( Y = none_state )
                   => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) ) ) ) )
           => ~ ! [A5: state] :
                  ( ( X2
                    = ( some_state @ A5 ) )
                 => ! [B4: state] :
                      ( ( Xa
                        = ( some_state @ B4 ) )
                     => ( ( Y
                          = ( plus @ A5 @ B4 ) )
                       => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) ) ) ) ) ) ) ) ) ).

% PartialSA.splus.pelims
thf(fact_414_valid__state_Opelims_I3_J,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X2 )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X2 )
       => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X2
                = ( produc74620467203698198on_val @ Pi4 @ H ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi4 @ H ) )
               => ( ( valid_mask @ Pi4 )
                  & ( valid_heap @ Pi4 @ H ) ) ) ) ) ) ).

% valid_state.pelims(3)
thf(fact_415_valid__state_Opelims_I2_J,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ( ( valid_state @ X2 )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X2 )
       => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X2
                = ( produc74620467203698198on_val @ Pi4 @ H ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi4 @ H ) )
               => ~ ( ( valid_mask @ Pi4 )
                    & ( valid_heap @ Pi4 @ H ) ) ) ) ) ) ).

% valid_state.pelims(2)
thf(fact_416_valid__state_Opelims_I1_J,axiom,
    ! [X2: produc9003781588162623004on_val,Y: $o] :
      ( ( ( valid_state @ X2 )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X2 )
       => ~ ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X2
                = ( produc74620467203698198on_val @ Pi4 @ H ) )
             => ( ( Y
                  = ( ( valid_mask @ Pi4 )
                    & ( valid_heap @ Pi4 @ H ) ) )
               => ~ ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi4 @ H ) ) ) ) ) ) ).

% valid_state.pelims(1)
thf(fact_417_compatible__options_Opelims_I2_J,axiom,
    ! [X2: option_state,Xa: option_state] :
      ( ( compat2278460363914054422_state @ X2 @ Xa )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ Xa ) )
       => ( ! [A5: state] :
              ( ( X2
                = ( some_state @ A5 ) )
             => ! [B4: state] :
                  ( ( Xa
                    = ( some_state @ B4 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) )
                   => ( A5 != B4 ) ) ) )
         => ( ( ( X2 = none_state )
             => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) )
           => ~ ( ( Xa = none_state )
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ none_state ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_418_compatible__options_Opelims_I2_J,axiom,
    ! [X2: option_val,Xa: option_val] :
      ( ( compat7209469420100717861ns_val @ X2 @ Xa )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X2 @ Xa ) )
       => ( ! [A5: val] :
              ( ( X2
                = ( some_val @ A5 ) )
             => ! [B4: val] :
                  ( ( Xa
                    = ( some_val @ B4 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A5 ) @ ( some_val @ B4 ) ) )
                   => ( A5 != B4 ) ) ) )
         => ( ( ( X2 = none_val )
             => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) )
           => ~ ( ( Xa = none_val )
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X2 @ none_val ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_419_compatible__options_Opelims_I3_J,axiom,
    ! [X2: option_state,Xa: option_state] :
      ( ~ ( compat2278460363914054422_state @ X2 @ Xa )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ Xa ) )
       => ~ ! [A5: state] :
              ( ( X2
                = ( some_state @ A5 ) )
             => ! [B4: state] :
                  ( ( Xa
                    = ( some_state @ B4 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) )
                   => ( A5 = B4 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_420_compatible__options_Opelims_I3_J,axiom,
    ! [X2: option_val,Xa: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X2 @ Xa )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X2 @ Xa ) )
       => ~ ! [A5: val] :
              ( ( X2
                = ( some_val @ A5 ) )
             => ! [B4: val] :
                  ( ( Xa
                    = ( some_val @ B4 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A5 ) @ ( some_val @ B4 ) ) )
                   => ( A5 = B4 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_421_compatible__options_Opelims_I1_J,axiom,
    ! [X2: option_state,Xa: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X2 @ Xa )
        = Y )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ Xa ) )
       => ( ! [A5: state] :
              ( ( X2
                = ( some_state @ A5 ) )
             => ! [B4: state] :
                  ( ( Xa
                    = ( some_state @ B4 ) )
                 => ( ( Y
                      = ( A5 = B4 ) )
                   => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) ) ) ) )
         => ( ( ( X2 = none_state )
             => ( Y
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) ) )
           => ~ ( ( Xa = none_state )
               => ( Y
                 => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ none_state ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_422_compatible__options_Opelims_I1_J,axiom,
    ! [X2: option_val,Xa: option_val,Y: $o] :
      ( ( ( compat7209469420100717861ns_val @ X2 @ Xa )
        = Y )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X2 @ Xa ) )
       => ( ! [A5: val] :
              ( ( X2
                = ( some_val @ A5 ) )
             => ! [B4: val] :
                  ( ( Xa
                    = ( some_val @ B4 ) )
                 => ( ( Y
                      = ( A5 = B4 ) )
                   => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A5 ) @ ( some_val @ B4 ) ) ) ) ) )
         => ( ( ( X2 = none_val )
             => ( Y
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) ) )
           => ~ ( ( Xa = none_val )
               => ( Y
                 => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X2 @ none_val ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_423_add__acc__uni__mask,axiom,
    ( add_ac6043276672051819257t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char,P3: prat] : ( add_ma6586698345353345811t_char @ Pi3 @ ( uni_ma6578582744724537101t_char @ Hl @ P3 ) ) ) ) ).

% add_acc_uni_mask
thf(fact_424_sep__algebra_Osplus_Opelims,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: option_val,Xa: option_val,Y: option_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( sep_splus_val @ Plus @ X2 @ Xa )
          = Y )
       => ( ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ X2 @ Xa ) )
         => ( ( ( X2 = none_val )
             => ( ( Y = none_val )
               => ~ ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) ) )
           => ( ! [V: val] :
                  ( ( X2
                    = ( some_val @ V ) )
                 => ( ( Xa = none_val )
                   => ( ( Y = none_val )
                     => ~ ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ ( some_val @ V ) @ none_val ) ) ) ) )
             => ~ ! [A5: val] :
                    ( ( X2
                      = ( some_val @ A5 ) )
                   => ! [B4: val] :
                        ( ( Xa
                          = ( some_val @ B4 ) )
                       => ( ( Y
                            = ( Plus @ A5 @ B4 ) )
                         => ~ ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ ( some_val @ A5 ) @ ( some_val @ B4 ) ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.pelims
thf(fact_425_sep__algebra_Osplus_Opelims,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: option_state,Xa: option_state,Y: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( sep_splus_state @ Plus @ X2 @ Xa )
          = Y )
       => ( ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ X2 @ Xa ) )
         => ( ( ( X2 = none_state )
             => ( ( Y = none_state )
               => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) ) )
           => ( ! [V: state] :
                  ( ( X2
                    = ( some_state @ V ) )
                 => ( ( Xa = none_state )
                   => ( ( Y = none_state )
                     => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) ) ) ) )
             => ~ ! [A5: state] :
                    ( ( X2
                      = ( some_state @ A5 ) )
                   => ! [B4: state] :
                        ( ( Xa
                          = ( some_state @ B4 ) )
                       => ( ( Y
                            = ( Plus @ A5 @ B4 ) )
                         => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.pelims
thf(fact_426_PartialSA_Opackage__logic__axioms,axiom,
    package_logic_state @ plus @ core @ unit @ stable ).

% PartialSA.package_logic_axioms
thf(fact_427_larger__heaps__sum__ineq,axiom,
    ! [A4: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > option_val,B5: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A4 @ A )
     => ( ( larger1532547405891354081ar_val @ B5 @ B )
       => ( ( compatible_heaps @ A4 @ B5 )
         => ( larger1532547405891354081ar_val @ ( map_ad2908576761495157028ar_val @ A4 @ B5 ) @ ( map_ad2908576761495157028ar_val @ A @ B ) ) ) ) ) ).

% larger_heaps_sum_ineq
thf(fact_428_PartialSA_Ominus__equiv__def__any__elem,axiom,
    ! [X2: state,A: state,B: state] :
      ( ( ( some_state @ X2 )
        = ( plus @ A @ B ) )
     => ( ( some_state @ ( minus @ X2 @ A ) )
        = ( plus @ B @ ( core @ X2 ) ) ) ) ).

% PartialSA.minus_equiv_def_any_elem
thf(fact_429_larger__heap__anti,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ B @ A )
       => ( A = B ) ) ) ).

% larger_heap_anti
thf(fact_430_larger__heap__refl,axiom,
    ! [H2: produc1457211279475724562t_char > option_val] : ( larger1532547405891354081ar_val @ H2 @ H2 ) ).

% larger_heap_refl
thf(fact_431_larger__heap__trans,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ B @ C )
       => ( larger1532547405891354081ar_val @ A @ C ) ) ) ).

% larger_heap_trans
thf(fact_432_sep__algebra_Ocore__is__smaller,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( some_val @ X2 )
        = ( Plus @ X2 @ ( Core @ X2 ) ) ) ) ).

% sep_algebra.core_is_smaller
thf(fact_433_sep__algebra_Ocore__is__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ X2 )
        = ( Plus @ X2 @ ( Core @ X2 ) ) ) ) ).

% sep_algebra.core_is_smaller
thf(fact_434_sep__algebra_Ocore__is__pure,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( some_val @ ( Core @ X2 ) )
        = ( Plus @ ( Core @ X2 ) @ ( Core @ X2 ) ) ) ) ).

% sep_algebra.core_is_pure
thf(fact_435_sep__algebra_Ocore__is__pure,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ ( Core @ X2 ) )
        = ( Plus @ ( Core @ X2 ) @ ( Core @ X2 ) ) ) ) ).

% sep_algebra.core_is_pure
thf(fact_436_sep__algebra_Ocancellative,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,X2: val,Y: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ A )
          = ( Plus @ B @ X2 ) )
       => ( ( ( some_val @ A )
            = ( Plus @ B @ Y ) )
         => ( ( ( Core @ X2 )
              = ( Core @ Y ) )
           => ( X2 = Y ) ) ) ) ) ).

% sep_algebra.cancellative
thf(fact_437_sep__algebra_Ocancellative,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,X2: state,Y: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B @ X2 ) )
       => ( ( ( some_state @ A )
            = ( Plus @ B @ Y ) )
         => ( ( ( Core @ X2 )
              = ( Core @ Y ) )
           => ( X2 = Y ) ) ) ) ) ).

% sep_algebra.cancellative
thf(fact_438_sep__algebra_Opositivity,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( Plus @ A @ B )
          = ( some_val @ C ) )
       => ( ( ( some_val @ C )
            = ( Plus @ C @ C ) )
         => ( ( some_val @ A )
            = ( Plus @ A @ A ) ) ) ) ) ).

% sep_algebra.positivity
thf(fact_439_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_440_sep__algebra_Ocore__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,C: val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ C )
          = ( Plus @ A @ B ) )
       => ( ( some_val @ ( Core @ C ) )
          = ( Plus @ ( Core @ A ) @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.core_sum
thf(fact_441_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_442_sep__algebra_Ocore__max,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ X2 )
          = ( Plus @ X2 @ C ) )
       => ? [R4: val] :
            ( ( some_val @ ( Core @ X2 ) )
            = ( Plus @ C @ R4 ) ) ) ) ).

% sep_algebra.core_max
thf(fact_443_sep__algebra_Ocore__max,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X2 )
          = ( Plus @ X2 @ C ) )
       => ? [R4: state] :
            ( ( some_state @ ( Core @ X2 ) )
            = ( Plus @ C @ R4 ) ) ) ) ).

% sep_algebra.core_max
thf(fact_444_sep__algebra_Oasso1,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,Ab: val,C: val,Bc: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B )
            = ( some_val @ Ab ) )
          & ( ( Plus @ B @ C )
            = ( some_val @ Bc ) ) )
       => ( ( Plus @ Ab @ C )
          = ( Plus @ A @ Bc ) ) ) ) ).

% sep_algebra.asso1
thf(fact_445_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_446_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_447_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_448_PartialSA_Ominus__core,axiom,
    ! [A: state,B: state] :
      ( ( core @ ( minus @ A @ B ) )
      = ( core @ A ) ) ).

% PartialSA.minus_core
thf(fact_449_PartialSA_Ominus__core__weaker,axiom,
    ! [A: state,B: state] :
      ( ( core @ ( minus @ A @ B ) )
      = ( minus @ ( core @ A ) @ ( core @ B ) ) ) ).

% PartialSA.minus_core_weaker
thf(fact_450_sep__algebra_Oasso2,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,Ab: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B )
            = ( some_val @ Ab ) )
          & ( ( Plus @ B @ C )
            = none_val ) )
       => ( ( Plus @ Ab @ C )
          = none_val ) ) ) ).

% sep_algebra.asso2
thf(fact_451_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_452_sep__algebra_Oasso3,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val,Bc: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( Plus @ A @ B )
          = none_val )
       => ( ( ( Plus @ B @ C )
            = ( some_val @ Bc ) )
         => ( ( Plus @ A @ Bc )
            = none_val ) ) ) ) ).

% sep_algebra.asso3
thf(fact_453_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_454_sep__algebra_Ointro,axiom,
    ! [Plus: val > val > option_val,Core: val > val] :
      ( ! [A5: val,B4: val] :
          ( ( Plus @ A5 @ B4 )
          = ( Plus @ B4 @ A5 ) )
     => ( ! [A5: val,B4: val,Ab2: val,C3: val,Bc2: val] :
            ( ( ( ( Plus @ A5 @ B4 )
                = ( some_val @ Ab2 ) )
              & ( ( Plus @ B4 @ C3 )
                = ( some_val @ Bc2 ) ) )
           => ( ( Plus @ Ab2 @ C3 )
              = ( Plus @ A5 @ Bc2 ) ) )
       => ( ! [A5: val,B4: val,Ab2: val,C3: val] :
              ( ( ( ( Plus @ A5 @ B4 )
                  = ( some_val @ Ab2 ) )
                & ( ( Plus @ B4 @ C3 )
                  = none_val ) )
             => ( ( Plus @ Ab2 @ C3 )
                = none_val ) )
         => ( ! [X3: val] :
                ( ( some_val @ X3 )
                = ( Plus @ X3 @ ( Core @ X3 ) ) )
           => ( ! [X3: val] :
                  ( ( some_val @ ( Core @ X3 ) )
                  = ( Plus @ ( Core @ X3 ) @ ( Core @ X3 ) ) )
             => ( ! [X3: val,C3: val] :
                    ( ( ( some_val @ X3 )
                      = ( Plus @ X3 @ C3 ) )
                   => ? [R5: val] :
                        ( ( some_val @ ( Core @ X3 ) )
                        = ( Plus @ C3 @ R5 ) ) )
               => ( ! [C3: val,A5: val,B4: val] :
                      ( ( ( some_val @ C3 )
                        = ( Plus @ A5 @ B4 ) )
                     => ( ( some_val @ ( Core @ C3 ) )
                        = ( Plus @ ( Core @ A5 ) @ ( Core @ B4 ) ) ) )
                 => ( ! [A5: val,B4: val,C3: val] :
                        ( ( ( Plus @ A5 @ B4 )
                          = ( some_val @ C3 ) )
                       => ( ( ( some_val @ C3 )
                            = ( Plus @ C3 @ C3 ) )
                         => ( ( some_val @ A5 )
                            = ( Plus @ A5 @ A5 ) ) ) )
                   => ( ! [A5: val,B4: val,X3: val,Y4: val] :
                          ( ( ( some_val @ A5 )
                            = ( Plus @ B4 @ X3 ) )
                         => ( ( ( some_val @ A5 )
                              = ( Plus @ B4 @ Y4 ) )
                           => ( ( ( Core @ X3 )
                                = ( Core @ Y4 ) )
                             => ( X3 = Y4 ) ) ) )
                     => ( sep_algebra_val @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.intro
thf(fact_455_sep__algebra_Ointro,axiom,
    ! [Plus: state > state > option_state,Core: state > state] :
      ( ! [A5: state,B4: state] :
          ( ( Plus @ A5 @ B4 )
          = ( Plus @ B4 @ A5 ) )
     => ( ! [A5: state,B4: state,Ab2: state,C3: state,Bc2: state] :
            ( ( ( ( Plus @ A5 @ B4 )
                = ( some_state @ Ab2 ) )
              & ( ( Plus @ B4 @ C3 )
                = ( some_state @ Bc2 ) ) )
           => ( ( Plus @ Ab2 @ C3 )
              = ( Plus @ A5 @ Bc2 ) ) )
       => ( ! [A5: state,B4: state,Ab2: state,C3: state] :
              ( ( ( ( Plus @ A5 @ B4 )
                  = ( some_state @ Ab2 ) )
                & ( ( Plus @ B4 @ C3 )
                  = none_state ) )
             => ( ( Plus @ Ab2 @ C3 )
                = none_state ) )
         => ( ! [X3: state] :
                ( ( some_state @ X3 )
                = ( Plus @ X3 @ ( Core @ X3 ) ) )
           => ( ! [X3: state] :
                  ( ( some_state @ ( Core @ X3 ) )
                  = ( Plus @ ( Core @ X3 ) @ ( Core @ X3 ) ) )
             => ( ! [X3: state,C3: state] :
                    ( ( ( some_state @ X3 )
                      = ( Plus @ X3 @ C3 ) )
                   => ? [R5: state] :
                        ( ( some_state @ ( Core @ X3 ) )
                        = ( Plus @ C3 @ R5 ) ) )
               => ( ! [C3: state,A5: state,B4: state] :
                      ( ( ( some_state @ C3 )
                        = ( Plus @ A5 @ B4 ) )
                     => ( ( some_state @ ( Core @ C3 ) )
                        = ( Plus @ ( Core @ A5 ) @ ( Core @ B4 ) ) ) )
                 => ( ! [A5: state,B4: state,C3: state] :
                        ( ( ( Plus @ A5 @ B4 )
                          = ( some_state @ C3 ) )
                       => ( ( ( some_state @ C3 )
                            = ( Plus @ C3 @ C3 ) )
                         => ( ( some_state @ A5 )
                            = ( Plus @ A5 @ A5 ) ) ) )
                   => ( ! [A5: state,B4: state,X3: state,Y4: state] :
                          ( ( ( some_state @ A5 )
                            = ( Plus @ B4 @ X3 ) )
                         => ( ( ( some_state @ A5 )
                              = ( Plus @ B4 @ Y4 ) )
                           => ( ( ( Core @ X3 )
                                = ( Core @ Y4 ) )
                             => ( X3 = Y4 ) ) ) )
                     => ( sep_algebra_state @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.intro
thf(fact_456_sep__algebra__def,axiom,
    ( sep_algebra_val
    = ( ^ [Plus2: val > val > option_val,Core2: val > val] :
          ( ! [A2: val,B2: val] :
              ( ( Plus2 @ A2 @ B2 )
              = ( Plus2 @ B2 @ A2 ) )
          & ! [A2: val,B2: val,Ab3: val,C4: val,Bc3: val] :
              ( ( ( ( Plus2 @ A2 @ B2 )
                  = ( some_val @ Ab3 ) )
                & ( ( Plus2 @ B2 @ C4 )
                  = ( some_val @ Bc3 ) ) )
             => ( ( Plus2 @ Ab3 @ C4 )
                = ( Plus2 @ A2 @ Bc3 ) ) )
          & ! [A2: val,B2: val,Ab3: val,C4: val] :
              ( ( ( ( Plus2 @ A2 @ B2 )
                  = ( some_val @ Ab3 ) )
                & ( ( Plus2 @ B2 @ C4 )
                  = none_val ) )
             => ( ( Plus2 @ Ab3 @ C4 )
                = none_val ) )
          & ! [X: val] :
              ( ( some_val @ X )
              = ( Plus2 @ X @ ( Core2 @ X ) ) )
          & ! [X: val] :
              ( ( some_val @ ( Core2 @ X ) )
              = ( Plus2 @ ( Core2 @ X ) @ ( Core2 @ X ) ) )
          & ! [X: val,C4: val] :
              ( ( ( some_val @ X )
                = ( Plus2 @ X @ C4 ) )
             => ? [R: val] :
                  ( ( some_val @ ( Core2 @ X ) )
                  = ( Plus2 @ C4 @ R ) ) )
          & ! [C4: val,A2: val,B2: val] :
              ( ( ( some_val @ C4 )
                = ( Plus2 @ A2 @ B2 ) )
             => ( ( some_val @ ( Core2 @ C4 ) )
                = ( Plus2 @ ( Core2 @ A2 ) @ ( Core2 @ B2 ) ) ) )
          & ! [A2: val,B2: val,C4: val] :
              ( ( ( Plus2 @ A2 @ B2 )
                = ( some_val @ C4 ) )
             => ( ( ( some_val @ C4 )
                  = ( Plus2 @ C4 @ C4 ) )
               => ( ( some_val @ A2 )
                  = ( Plus2 @ A2 @ A2 ) ) ) )
          & ! [A2: val,B2: val,X: val,Y3: val] :
              ( ( ( some_val @ A2 )
                = ( Plus2 @ B2 @ X ) )
             => ( ( ( some_val @ A2 )
                  = ( Plus2 @ B2 @ Y3 ) )
               => ( ( ( Core2 @ X )
                    = ( Core2 @ Y3 ) )
                 => ( X = Y3 ) ) ) ) ) ) ) ).

% sep_algebra_def
thf(fact_457_sep__algebra__def,axiom,
    ( sep_algebra_state
    = ( ^ [Plus2: state > state > option_state,Core2: state > state] :
          ( ! [A2: state,B2: state] :
              ( ( Plus2 @ A2 @ B2 )
              = ( Plus2 @ B2 @ A2 ) )
          & ! [A2: state,B2: state,Ab3: state,C4: state,Bc3: state] :
              ( ( ( ( Plus2 @ A2 @ B2 )
                  = ( some_state @ Ab3 ) )
                & ( ( Plus2 @ B2 @ C4 )
                  = ( some_state @ Bc3 ) ) )
             => ( ( Plus2 @ Ab3 @ C4 )
                = ( Plus2 @ A2 @ Bc3 ) ) )
          & ! [A2: state,B2: state,Ab3: state,C4: state] :
              ( ( ( ( Plus2 @ A2 @ B2 )
                  = ( some_state @ Ab3 ) )
                & ( ( Plus2 @ B2 @ C4 )
                  = none_state ) )
             => ( ( Plus2 @ Ab3 @ C4 )
                = none_state ) )
          & ! [X: state] :
              ( ( some_state @ X )
              = ( Plus2 @ X @ ( Core2 @ X ) ) )
          & ! [X: state] :
              ( ( some_state @ ( Core2 @ X ) )
              = ( Plus2 @ ( Core2 @ X ) @ ( Core2 @ X ) ) )
          & ! [X: state,C4: state] :
              ( ( ( some_state @ X )
                = ( Plus2 @ X @ C4 ) )
             => ? [R: state] :
                  ( ( some_state @ ( Core2 @ X ) )
                  = ( Plus2 @ C4 @ R ) ) )
          & ! [C4: state,A2: state,B2: state] :
              ( ( ( some_state @ C4 )
                = ( Plus2 @ A2 @ B2 ) )
             => ( ( some_state @ ( Core2 @ C4 ) )
                = ( Plus2 @ ( Core2 @ A2 ) @ ( Core2 @ B2 ) ) ) )
          & ! [A2: state,B2: state,C4: state] :
              ( ( ( Plus2 @ A2 @ B2 )
                = ( some_state @ C4 ) )
             => ( ( ( some_state @ C4 )
                  = ( Plus2 @ C4 @ C4 ) )
               => ( ( some_state @ A2 )
                  = ( Plus2 @ A2 @ A2 ) ) ) )
          & ! [A2: state,B2: state,X: state,Y3: state] :
              ( ( ( some_state @ A2 )
                = ( Plus2 @ B2 @ X ) )
             => ( ( ( some_state @ A2 )
                  = ( Plus2 @ B2 @ Y3 ) )
               => ( ( ( Core2 @ X )
                    = ( Core2 @ Y3 ) )
                 => ( X = Y3 ) ) ) ) ) ) ) ).

% sep_algebra_def
thf(fact_458_PartialSA_Osep__algebra__axioms,axiom,
    sep_algebra_state @ plus @ core ).

% PartialSA.sep_algebra_axioms
thf(fact_459_larger__heap__def,axiom,
    ( larger1532547405891354081ar_val
    = ( ^ [H6: produc1457211279475724562t_char > option_val,H5: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char,X: val] :
          ( ( ( H5 @ Hl )
            = ( some_val @ X ) )
         => ( ( H6 @ Hl )
            = ( some_val @ X ) ) ) ) ) ).

% larger_heap_def
thf(fact_460_larger__heapI,axiom,
    ! [H2: produc1457211279475724562t_char > option_val,H3: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char,X3: val] :
          ( ( ( H2 @ Hl2 )
            = ( some_val @ X3 ) )
         => ( ( H3 @ Hl2 )
            = ( some_val @ X3 ) ) )
     => ( larger1532547405891354081ar_val @ H3 @ H2 ) ) ).

% larger_heapI
thf(fact_461_sep__algebra_Osplus__develop,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val,D: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ A )
          = ( Plus @ B @ C ) )
       => ( ( Plus @ A @ D )
          = ( sep_splus_val @ Plus @ ( sep_splus_val @ Plus @ ( some_val @ B ) @ ( some_val @ C ) ) @ ( some_val @ D ) ) ) ) ) ).

% sep_algebra.splus_develop
thf(fact_462_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_463_sep__algebra_Osplus_Osimps_I3_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_splus_val @ Plus @ ( some_val @ A ) @ ( some_val @ B ) )
        = ( Plus @ A @ B ) ) ) ).

% sep_algebra.splus.simps(3)
thf(fact_464_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_465_larger__compatible__sum__heaps,axiom,
    ! [A: produc1457211279475724562t_char > option_val,X2: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ X2 )
     => ( ( larger1532547405891354081ar_val @ B @ Y )
       => ( ( compatible_heaps @ A @ B )
         => ( compatible_heaps @ X2 @ Y ) ) ) ) ).

% larger_compatible_sum_heaps
thf(fact_466_larger__heap__comp,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( compatible_heaps @ A @ C )
       => ( compatible_heaps @ B @ C ) ) ) ).

% larger_heap_comp
thf(fact_467_sep__algebra_Osplus_Osimps_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Uu2: option_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_splus_val @ Plus @ none_val @ Uu2 )
        = none_val ) ) ).

% sep_algebra.splus.simps(1)
thf(fact_468_sep__algebra_Osplus_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Uu2: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ none_state @ Uu2 )
        = none_state ) ) ).

% sep_algebra.splus.simps(1)
thf(fact_469_larger__heap__plus,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ A @ C )
       => ( larger1532547405891354081ar_val @ A @ ( map_ad2908576761495157028ar_val @ B @ C ) ) ) ) ).

% larger_heap_plus
thf(fact_470_sep__algebra_Osplus_Osimps_I2_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,V2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_splus_val @ Plus @ ( some_val @ V2 ) @ none_val )
        = none_val ) ) ).

% sep_algebra.splus.simps(2)
thf(fact_471_sep__algebra_Osplus_Osimps_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,V2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( some_state @ V2 ) @ none_state )
        = none_state ) ) ).

% sep_algebra.splus.simps(2)
thf(fact_472_sep__algebra_Osplus_Oelims,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: option_val,Xa: option_val,Y: option_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( sep_splus_val @ Plus @ X2 @ Xa )
          = Y )
       => ( ( ( X2 = none_val )
           => ( Y != none_val ) )
         => ( ( ? [V: val] :
                  ( X2
                  = ( some_val @ V ) )
             => ( ( Xa = none_val )
               => ( Y != none_val ) ) )
           => ~ ! [A5: val] :
                  ( ( X2
                    = ( some_val @ A5 ) )
                 => ! [B4: val] :
                      ( ( Xa
                        = ( some_val @ B4 ) )
                     => ( Y
                       != ( Plus @ A5 @ B4 ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.elims
thf(fact_473_sep__algebra_Osplus_Oelims,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: option_state,Xa: option_state,Y: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( sep_splus_state @ Plus @ X2 @ Xa )
          = Y )
       => ( ( ( X2 = none_state )
           => ( Y != none_state ) )
         => ( ( ? [V: state] :
                  ( X2
                  = ( some_state @ V ) )
             => ( ( Xa = none_state )
               => ( Y != none_state ) ) )
           => ~ ! [A5: state] :
                  ( ( X2
                    = ( some_state @ A5 ) )
                 => ! [B4: state] :
                      ( ( Xa
                        = ( some_state @ B4 ) )
                     => ( Y
                       != ( Plus @ A5 @ B4 ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.elims
thf(fact_474_sep__algebra_Osplus_Ocases,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: produc5122021904355189735on_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ! [Uu: option_val] :
            ( X2
           != ( produc904852209897547607on_val @ none_val @ Uu ) )
       => ( ! [V: val] :
              ( X2
             != ( produc904852209897547607on_val @ ( some_val @ V ) @ none_val ) )
         => ~ ! [A5: val,B4: val] :
                ( X2
               != ( produc904852209897547607on_val @ ( some_val @ A5 ) @ ( some_val @ B4 ) ) ) ) ) ) ).

% sep_algebra.splus.cases
thf(fact_475_sep__algebra_Osplus_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: produc3142500478612311029_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [Uu: option_state] :
            ( X2
           != ( produc9160152616361873709_state @ none_state @ Uu ) )
       => ( ! [V: state] :
              ( X2
             != ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) )
         => ~ ! [A5: state,B4: state] :
                ( X2
               != ( produc9160152616361873709_state @ ( some_state @ A5 ) @ ( some_state @ B4 ) ) ) ) ) ) ).

% sep_algebra.splus.cases
thf(fact_476_package__logic_Ostable__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val,X2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( Stable @ A )
       => ( ( Stable @ B )
         => ( ( ( some_val @ X2 )
              = ( Plus @ A @ B ) )
           => ( Stable @ X2 ) ) ) ) ) ).

% package_logic.stable_sum
thf(fact_477_package__logic_Ostable__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state,X2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( Stable @ A )
       => ( ( Stable @ B )
         => ( ( ( some_state @ X2 )
              = ( Plus @ A @ B ) )
           => ( Stable @ X2 ) ) ) ) ) ).

% package_logic.stable_sum
thf(fact_478_package__logic_Ounit__neutral,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( some_val @ A )
        = ( Plus @ A @ Unit ) ) ) ).

% package_logic.unit_neutral
thf(fact_479_package__logic_Ounit__neutral,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( some_state @ A )
        = ( Plus @ A @ Unit ) ) ) ).

% package_logic.unit_neutral
thf(fact_480_greaterI,axiom,
    ! [A: state,B: state] :
      ( ( larger1532547405891354081ar_val @ ( get_h @ A ) @ ( get_h @ B ) )
     => ( ( greate7680731506090375811t_char @ ( get_m @ A ) @ ( get_m @ B ) )
       => ( greater @ A @ B ) ) ) ).

% greaterI
thf(fact_481_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,X2: produc7272280185086996969_state,A3: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ X2 @ ( sep_ad3908482799505143387_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc7272280185086996969_state,B2: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ A2 @ A3 )
              & ( member7461423308196221312_state @ B2 @ B3 )
              & ( ( some_P6297086604321900740_state @ X2 )
                = ( sep_sp9219505543755390023_state @ Plus @ ( some_P6297086604321900740_state @ A2 ) @ ( some_P6297086604321900740_state @ B2 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_482_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,X2: produc9003781588162623004on_val,A3: set_Pr7649655576376376530on_val,B3: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ X2 @ ( sep_ad8320153009520438926on_val @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ A2 @ A3 )
              & ( member2879030624002546611on_val @ B2 @ B3 )
              & ( ( some_P2581650809551911671on_val @ X2 )
                = ( sep_sp4146280900173761658on_val @ Plus @ ( some_P2581650809551911671on_val @ A2 ) @ ( some_P2581650809551911671on_val @ B2 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_483_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,X2: produc3882902684156697774_state,A3: set_Pr2391701600042252046_state,B3: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ X2 @ ( sep_ad8203089552910907580_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc3882902684156697774_state,B2: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ A2 @ A3 )
              & ( member6715010751053357271_state @ B2 @ B3 )
              & ( ( some_P5383513176685068947_state @ X2 )
                = ( sep_sp7584349535955199440_state @ Plus @ ( some_P5383513176685068947_state @ A2 ) @ ( some_P5383513176685068947_state @ B2 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_484_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: val,A3: set_val,B3: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( member_val @ X2 @ ( sep_add_set_val @ Plus @ A3 @ B3 ) )
        = ( ? [A2: val,B2: val] :
              ( ( member_val @ A2 @ A3 )
              & ( member_val @ B2 @ B3 )
              & ( ( some_val @ X2 )
                = ( sep_splus_val @ Plus @ ( some_val @ A2 ) @ ( some_val @ B2 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_485_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A3: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ X2 @ ( sep_add_set_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: state,B2: state] :
              ( ( member_state @ A2 @ A3 )
              & ( member_state @ B2 @ B3 )
              & ( ( some_state @ X2 )
                = ( sep_splus_state @ Plus @ ( some_state @ A2 ) @ ( some_state @ B2 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_486_PartialSA_OminusI,axiom,
    ! [B: state,A: state,X2: state] :
      ( ( ( some_state @ B )
        = ( plus @ A @ X2 ) )
     => ( ( greater @ X2 @ ( core @ B ) )
       => ( X2
          = ( minus @ B @ A ) ) ) ) ).

% PartialSA.minusI
thf(fact_487_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_488_PartialSA_Osucc__antisym,axiom,
    ! [A: state,B: state] :
      ( ( greater @ A @ B )
     => ( ( greater @ B @ A )
       => ( A = B ) ) ) ).

% PartialSA.succ_antisym
thf(fact_489_PartialSA_Osucc__trans,axiom,
    ! [A: state,B: state,C: state] :
      ( ( greater @ A @ B )
     => ( ( greater @ B @ C )
       => ( greater @ A @ C ) ) ) ).

% PartialSA.succ_trans
thf(fact_490_PartialSA_Osucc__refl,axiom,
    ! [A: state] : ( greater @ A @ A ) ).

% PartialSA.succ_refl
thf(fact_491_PartialSA_Ocore__mono,axiom,
    ! [A: state,B: state] :
      ( ( greater @ A @ B )
     => ( greater @ ( core @ A ) @ ( core @ B ) ) ) ).

% PartialSA.core_mono
thf(fact_492_PartialSA_Ogreater__minus__trans,axiom,
    ! [Y: state,X2: state,A: state] :
      ( ( greater @ Y @ X2 )
     => ( ( greater @ X2 @ A )
       => ( greater @ ( minus @ Y @ A ) @ ( minus @ X2 @ A ) ) ) ) ).

% PartialSA.greater_minus_trans
thf(fact_493_PartialSA_Ominus__smaller,axiom,
    ! [X2: state,A: state] :
      ( ( greater @ X2 @ A )
     => ( greater @ X2 @ ( minus @ X2 @ A ) ) ) ).

% PartialSA.minus_smaller
thf(fact_494_PartialSA_Ominus__default,axiom,
    ! [B: state,A: state] :
      ( ~ ( greater @ B @ A )
     => ( ( minus @ B @ A )
        = B ) ) ).

% PartialSA.minus_default
thf(fact_495_PartialSA_Ounit__smaller,axiom,
    ! [Phi2: state] : ( greater @ Phi2 @ unit ) ).

% PartialSA.unit_smaller
thf(fact_496_PartialSA_Osmaller__compatible,axiom,
    ! [A4: state,B: state,A: state] :
      ( ( defined @ A4 @ B )
     => ( ( greater @ A4 @ A )
       => ( defined @ A @ B ) ) ) ).

% PartialSA.smaller_compatible
thf(fact_497_R__smaller,axiom,
    ! [W: state,A: state] : ( greater @ W @ ( r @ A @ W ) ) ).

% R_smaller
thf(fact_498_PartialSA_Obigger__sum,axiom,
    ! [Phi2: state,A: state,B: state,Phi3: state] :
      ( ( ( some_state @ Phi2 )
        = ( plus @ A @ B ) )
     => ( ( greater @ Phi3 @ Phi2 )
       => ? [B7: state] :
            ( ( greater @ B7 @ B )
            & ( ( some_state @ Phi3 )
              = ( plus @ A @ B7 ) ) ) ) ) ).

% PartialSA.bigger_sum
thf(fact_499_PartialSA_Ogreater__def,axiom,
    ( greater
    = ( ^ [A2: state,B2: state] :
        ? [C4: state] :
          ( ( some_state @ A2 )
          = ( plus @ B2 @ C4 ) ) ) ) ).

% PartialSA.greater_def
thf(fact_500_PartialSA_Ogreater__equiv,axiom,
    ( greater
    = ( ^ [A2: state,B2: state] :
        ? [C4: state] :
          ( ( some_state @ A2 )
          = ( plus @ C4 @ B2 ) ) ) ) ).

% PartialSA.greater_equiv
thf(fact_501_PartialSA_Oaddition__bigger,axiom,
    ! [A4: state,A: state,X5: state,B: state,X2: state] :
      ( ( greater @ A4 @ A )
     => ( ( ( some_state @ X5 )
          = ( plus @ A4 @ B ) )
       => ( ( ( some_state @ X2 )
            = ( plus @ A @ B ) )
         => ( greater @ X5 @ X2 ) ) ) ) ).

% PartialSA.addition_bigger
thf(fact_502_PartialSA_Obigger__sum__smaller,axiom,
    ! [C: state,A: state,B: state,A4: state] :
      ( ( ( some_state @ C )
        = ( plus @ A @ B ) )
     => ( ( greater @ A @ A4 )
       => ? [B7: state] :
            ( ( greater @ B7 @ B )
            & ( ( some_state @ C )
              = ( plus @ A4 @ B7 ) ) ) ) ) ).

% PartialSA.bigger_sum_smaller
thf(fact_503_intuitionistic__def,axiom,
    ( intuitionistic
    = ( ^ [A6: set_state] :
        ! [A2: state,B2: state] :
          ( ( ( greater @ A2 @ B2 )
            & ( member_state @ B2 @ A6 ) )
         => ( member_state @ A2 @ A6 ) ) ) ) ).

% intuitionistic_def
thf(fact_504_PartialSA_Osmaller__compatible__core,axiom,
    ! [Y: state,X2: state] :
      ( ( greater @ Y @ X2 )
     => ( defined @ X2 @ ( core @ Y ) ) ) ).

% PartialSA.smaller_compatible_core
thf(fact_505_larger__implies__greater__mask__hl,axiom,
    ! [A: state,B: state,Hl3: produc1457211279475724562t_char] :
      ( ( greater @ A @ B )
     => ( pgte @ ( get_m @ A @ Hl3 ) @ ( get_m @ B @ Hl3 ) ) ) ).

% larger_implies_greater_mask_hl
thf(fact_506_larger__implies__larger__heap,axiom,
    ! [A: state,B: state] :
      ( ( greater @ A @ B )
     => ( larger1532547405891354081ar_val @ ( get_h @ A ) @ ( get_h @ B ) ) ) ).

% larger_implies_larger_heap
thf(fact_507_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,Phi2: produc7272280185086996969_state,A3: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ Phi2 @ ( sep_ad3908482799505143387_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc7272280185086996969_state,B2: produc7272280185086996969_state] :
              ( ( ( some_P6297086604321900740_state @ Phi2 )
                = ( Plus @ A2 @ B2 ) )
              & ( member7461423308196221312_state @ A2 @ A3 )
              & ( member7461423308196221312_state @ B2 @ B3 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_508_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,Phi2: produc9003781588162623004on_val,A3: set_Pr7649655576376376530on_val,B3: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ Phi2 @ ( sep_ad8320153009520438926on_val @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
              ( ( ( some_P2581650809551911671on_val @ Phi2 )
                = ( Plus @ A2 @ B2 ) )
              & ( member2879030624002546611on_val @ A2 @ A3 )
              & ( member2879030624002546611on_val @ B2 @ B3 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_509_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,Phi2: produc3882902684156697774_state,A3: set_Pr2391701600042252046_state,B3: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ Phi2 @ ( sep_ad8203089552910907580_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc3882902684156697774_state,B2: produc3882902684156697774_state] :
              ( ( ( some_P5383513176685068947_state @ Phi2 )
                = ( Plus @ A2 @ B2 ) )
              & ( member6715010751053357271_state @ A2 @ A3 )
              & ( member6715010751053357271_state @ B2 @ B3 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_510_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Phi2: val,A3: set_val,B3: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( member_val @ Phi2 @ ( sep_add_set_val @ Plus @ A3 @ B3 ) )
        = ( ? [A2: val,B2: val] :
              ( ( ( some_val @ Phi2 )
                = ( Plus @ A2 @ B2 ) )
              & ( member_val @ A2 @ A3 )
              & ( member_val @ B2 @ B3 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_511_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Phi2: state,A3: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ Phi2 @ ( sep_add_set_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: state,B2: state] :
              ( ( ( some_state @ Phi2 )
                = ( Plus @ A2 @ B2 ) )
              & ( member_state @ A2 @ A3 )
              & ( member_state @ B2 @ B3 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_512_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,X2: produc7272280185086996969_state,A3: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ X2 @ ( sep_ad3908482799505143387_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc7272280185086996969_state,B2: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ A2 @ A3 )
              & ( member7461423308196221312_state @ B2 @ B3 )
              & ( ( some_P6297086604321900740_state @ X2 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_513_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,X2: produc9003781588162623004on_val,A3: set_Pr7649655576376376530on_val,B3: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ X2 @ ( sep_ad8320153009520438926on_val @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ A2 @ A3 )
              & ( member2879030624002546611on_val @ B2 @ B3 )
              & ( ( some_P2581650809551911671on_val @ X2 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_514_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,X2: produc3882902684156697774_state,A3: set_Pr2391701600042252046_state,B3: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ X2 @ ( sep_ad8203089552910907580_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: produc3882902684156697774_state,B2: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ A2 @ A3 )
              & ( member6715010751053357271_state @ B2 @ B3 )
              & ( ( some_P5383513176685068947_state @ X2 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_515_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: val,A3: set_val,B3: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( member_val @ X2 @ ( sep_add_set_val @ Plus @ A3 @ B3 ) )
        = ( ? [A2: val,B2: val] :
              ( ( member_val @ A2 @ A3 )
              & ( member_val @ B2 @ B3 )
              & ( ( some_val @ X2 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_516_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A3: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ X2 @ ( sep_add_set_state @ Plus @ A3 @ B3 ) )
        = ( ? [A2: state,B2: state] :
              ( ( member_state @ A2 @ A3 )
              & ( member_state @ B2 @ B3 )
              & ( ( some_state @ X2 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_517_add__set__def,axiom,
    ( add_set
    = ( sep_add_set_state @ plus ) ) ).

% add_set_def
thf(fact_518_PartialSA_Oextract__core,axiom,
    ! [B: state,A: state,X2: state] :
      ( ( ( ( some_state @ B )
          = ( plus @ A @ X2 ) )
        & ( greater @ X2 @ ( core @ B ) ) )
     => ( ( core @ X2 )
        = ( core @ B ) ) ) ).

% PartialSA.extract_core
thf(fact_519_PartialSA_Ominus__exists,axiom,
    ! [B: state,A: state] :
      ( ( greater @ B @ A )
     => ? [X3: state] :
          ( ( ( some_state @ B )
            = ( plus @ A @ X3 ) )
          & ( greater @ X3 @ ( core @ B ) ) ) ) ).

% PartialSA.minus_exists
thf(fact_520_PartialSA_Ominus__unique,axiom,
    ! [B: state,A: state,X2: state,Y: state] :
      ( ( ( ( some_state @ B )
          = ( plus @ A @ X2 ) )
        & ( greater @ X2 @ ( core @ B ) ) )
     => ( ( ( ( some_state @ B )
            = ( plus @ A @ Y ) )
          & ( greater @ Y @ ( core @ B ) ) )
       => ( X2 = Y ) ) ) ).

% PartialSA.minus_unique
thf(fact_521_PartialSA_Osmaller__than__core,axiom,
    ! [Y: state,X2: state,Z: state] :
      ( ( greater @ Y @ X2 )
     => ( ( ( some_state @ Z )
          = ( plus @ X2 @ ( core @ Y ) ) )
       => ( ( core @ Z )
          = ( core @ Y ) ) ) ) ).

% PartialSA.smaller_than_core
thf(fact_522_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_523_PartialSA_Ogreater__than__sum__exists,axiom,
    ! [A: state,B: state,B1: state,B22: state] :
      ( ( greater @ A @ B )
     => ( ( ( some_state @ B )
          = ( plus @ B1 @ B22 ) )
       => ? [R4: state] :
            ( ( ( some_state @ A )
              = ( plus @ R4 @ B22 ) )
            & ( greater @ ( core @ R4 ) @ ( core @ A ) )
            & ( greater @ R4 @ B1 ) ) ) ) ).

% PartialSA.greater_than_sum_exists
thf(fact_524_PartialSA_Ominus__sum,axiom,
    ! [A: state,B: state,C: state,X2: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ C ) )
     => ( ( greater @ X2 @ A )
       => ( ( minus @ X2 @ A )
          = ( minus @ ( minus @ X2 @ B ) @ C ) ) ) ) ).

% PartialSA.minus_sum
thf(fact_525_PartialSA_Ominus__some,axiom,
    ! [A: state,B: state] :
      ( ( greater @ A @ B )
     => ( ( some_state @ A )
        = ( plus @ B @ ( minus @ A @ B ) ) ) ) ).

% PartialSA.minus_some
thf(fact_526_PartialSA_Ominus__bigger,axiom,
    ! [X2: state,A: state,B: state] :
      ( ( ( some_state @ X2 )
        = ( plus @ A @ B ) )
     => ( greater @ ( minus @ X2 @ A ) @ B ) ) ).

% PartialSA.minus_bigger
thf(fact_527_PartialSA_Ominus__and__plus,axiom,
    ! [Omega: state,Omega2: state,R3: state,A: state] :
      ( ( ( some_state @ Omega )
        = ( plus @ Omega2 @ R3 ) )
     => ( ( greater @ Omega2 @ A )
       => ( ( some_state @ ( minus @ Omega @ A ) )
          = ( plus @ ( minus @ Omega2 @ A ) @ R3 ) ) ) ) ).

% PartialSA.minus_and_plus
thf(fact_528_PartialSA_Oprove__last__completeness,axiom,
    ! [A4: state,A: state,Nf1: state,F22: state] :
      ( ( greater @ A4 @ A )
     => ( ( ( some_state @ A )
          = ( plus @ Nf1 @ F22 ) )
       => ( greater @ ( minus @ A4 @ Nf1 ) @ F22 ) ) ) ).

% PartialSA.prove_last_completeness
thf(fact_529_multiply__order,axiom,
    ! [P2: prat,A: state,B: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( greater @ A @ B )
       => ( greater @ ( multiply @ P2 @ A ) @ ( multiply @ P2 @ B ) ) ) ) ).

% multiply_order
thf(fact_530_PartialSA_Opackage__rhs__connection__instantiate,axiom,
    ! [Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state,A: state,U: state,T3: state > state] :
      ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
     => ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
       => ~ ! [Au: state] :
              ( ( ( some_state @ Au )
                = ( plus @ A @ U ) )
             => ~ ( ( defined @ Au @ ( minus @ ( T3 @ F5 ) @ ( T3 @ F4 ) ) )
                 => ? [A7: state,U2: state] :
                      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A7 @ ( produc8632743443909064678_state @ U2 @ T3 ) ) @ S3 )
                      & ( greater @ ( core @ A7 ) @ ( core @ A ) )
                      & ( ( plus @ Au @ ( minus @ ( T3 @ F5 ) @ ( T3 @ F4 ) ) )
                        = ( plus @ A7 @ U2 ) )
                      & ( greater @ U2 @ U )
                      & ( packag8070727096299822599_state @ plus @ core @ Pc @ A3 @ A7 @ U2 @ U ) ) ) ) ) ) ).

% PartialSA.package_rhs_connection_instantiate
thf(fact_531_PartialSA_Opackage__rhs__connectionI,axiom,
    ! [Phi2: state,F4: state,Phi3: state,F5: state,S2: set_Pr4231240917283394591_state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state] :
      ( ( ( plus @ Phi2 @ F4 )
        = ( plus @ Phi3 @ F5 ) )
     => ( ( stable @ F5 )
       => ( ( defined @ Phi2 @ F4 )
         => ( ( greater @ F5 @ F4 )
           => ( ! [A5: state,U3: state,T4: state > state] :
                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
                 => ? [Au2: state] :
                      ( ( ( some_state @ Au2 )
                        = ( plus @ A5 @ U3 ) )
                      & ( ( defined @ Au2 @ ( minus @ ( T4 @ F5 ) @ ( T4 @ F4 ) ) )
                       => ? [A8: state,U4: state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A8 @ ( produc8632743443909064678_state @ U4 @ T4 ) ) @ S3 )
                            & ( greater @ ( core @ A8 ) @ ( core @ A5 ) )
                            & ( ( plus @ Au2 @ ( minus @ ( T4 @ F5 ) @ ( T4 @ F4 ) ) )
                              = ( plus @ A8 @ U4 ) )
                            & ( greater @ U4 @ U3 )
                            & ( packag8070727096299822599_state @ plus @ core @ Pc @ A3 @ A8 @ U4 @ U3 ) ) ) ) )
             => ( packag2054414653763826108_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 ) ) ) ) ) ) ).

% PartialSA.package_rhs_connectionI
thf(fact_532_PartialSA_Omono__transformer__def,axiom,
    ! [T3: state > state] :
      ( ( packag4817583915586052229_state @ plus @ unit @ T3 )
      = ( ! [Phi4: state,Phi5: state] :
            ( ( greater @ Phi5 @ Phi4 )
           => ( greater @ ( T3 @ Phi5 ) @ ( T3 @ Phi4 ) ) )
        & ( ( T3 @ unit )
          = unit ) ) ) ).

% PartialSA.mono_transformer_def
thf(fact_533_in__multiply__sem,axiom,
    ! [X2: state,P2: prat,A3: set_state] :
      ( ( member_state @ X2 @ ( multip8064567061438756306ertion @ P2 @ A3 ) )
     => ? [X3: state] :
          ( ( member_state @ X3 @ A3 )
          & ( greater @ X2 @ ( multiply @ P2 @ X3 ) ) ) ) ).

% in_multiply_sem
thf(fact_534_PartialSA_Obigger__the,axiom,
    ! [A: state,X5: state,Y: state,X2: state] :
      ( ( ( some_state @ A )
        = ( plus @ X5 @ Y ) )
     => ( ( greater @ X5 @ X2 )
       => ( greater @ ( the_state @ ( plus @ ( core @ A ) @ X5 ) ) @ ( the_state @ ( plus @ ( core @ A ) @ X2 ) ) ) ) ) ).

% PartialSA.bigger_the
thf(fact_535_PartialSA_Ovalid__package__setI,axiom,
    ! [S2: set_Pr4231240917283394591_state,F4: state] :
      ( ! [A5: state,U3: state,T4: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
         => ( ( defined @ A5 @ U3 )
            & ( greater @ ( core @ A5 ) @ ( core @ U3 ) )
            & ( packag4817583915586052229_state @ plus @ unit @ T4 )
            & ( greater @ A5 @ ( core @ ( T4 @ F4 ) ) ) ) )
     => ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F4 ) ) ).

% PartialSA.valid_package_setI
thf(fact_536_option_Ocollapse,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( ( some_state @ ( the_state @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_537_option_Ocollapse,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( ( some_val @ ( the_val @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_538_option_Osel,axiom,
    ! [X22: state] :
      ( ( the_state @ ( some_state @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_539_option_Osel,axiom,
    ! [X22: val] :
      ( ( the_val @ ( some_val @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_540_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_541_option_Oexpand,axiom,
    ! [Option: option_val,Option2: option_val] :
      ( ( ( Option = none_val )
        = ( Option2 = none_val ) )
     => ( ( ( Option != none_val )
         => ( ( Option2 != none_val )
           => ( ( the_val @ Option )
              = ( the_val @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_542_option_Oexhaust__sel,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( Option
        = ( some_state @ ( the_state @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_543_option_Oexhaust__sel,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( Option
        = ( some_val @ ( the_val @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_544_in__multiply__refl,axiom,
    ! [X2: state,A3: set_state,P2: prat] :
      ( ( member_state @ X2 @ A3 )
     => ( member_state @ ( multiply @ P2 @ X2 ) @ ( multip8064567061438756306ertion @ P2 @ A3 ) ) ) ).

% in_multiply_refl
thf(fact_545_combinable__def,axiom,
    ( combinable
    = ( ^ [P6: set_state] :
        ! [Alpha: prat,Beta: prat] :
          ( ( ( ppos @ Alpha )
            & ( ppos @ Beta )
            & ( pgte @ pwrite @ ( padd @ Alpha @ Beta ) ) )
         => ( ord_le2494988322063910608_state @ ( add_set @ ( multip8064567061438756306ertion @ Alpha @ P6 ) @ ( multip8064567061438756306ertion @ Beta @ P6 ) ) @ ( multip8064567061438756306ertion @ ( padd @ Alpha @ Beta ) @ P6 ) ) ) ) ) ).

% combinable_def
thf(fact_546_PartialSA_Opackage__satI,axiom,
    ! [Pc: state > $o,A4: state,U5: state,U: state,A3: aassertion_state] :
      ( ( ( Pc @ ( core @ A4 ) )
       => ? [X6: state] :
            ( ( ( some_state @ X6 )
              = ( plus @ ( core @ A4 ) @ ( minus @ U5 @ U ) ) )
            & ( package_sat_state @ plus @ A3 @ X6 ) ) )
     => ( packag8070727096299822599_state @ plus @ core @ Pc @ A3 @ A4 @ U5 @ U ) ) ).

% PartialSA.package_satI
thf(fact_547_PartialSA_Opackage__sat__def,axiom,
    ! [Pc: state > $o,A3: aassertion_state,A4: state,U5: state,U: state] :
      ( ( packag8070727096299822599_state @ plus @ core @ Pc @ A3 @ A4 @ U5 @ U )
      = ( ( Pc @ ( core @ A4 ) )
       => ? [X: state] :
            ( ( ( some_state @ X )
              = ( plus @ ( core @ A4 ) @ ( minus @ U5 @ U ) ) )
            & ( package_sat_state @ plus @ A3 @ X ) ) ) ) ).

% PartialSA.package_sat_def
thf(fact_548_double__inclusion,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% double_inclusion
thf(fact_549_cwand__stronger,axiom,
    ! [A3: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ ( cwand @ A3 @ B3 ) @ ( wand @ A3 @ B3 ) ) ).

% cwand_stronger
thf(fact_550_asm0,axiom,
    ord_le2494988322063910608_state @ ( add_set @ a @ ( insert_state @ w @ bot_bot_set_state ) ) @ b ).

% asm0
thf(fact_551_package__logic_Ois__footprint__standard__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,W: val,A3: aassertion_val,B3: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag685230357167683994rd_val @ Plus @ W @ A3 @ B3 )
        = ( ! [A2: val,B2: val] :
              ( ( ( package_sat_val @ Plus @ A3 @ A2 )
                & ( ( some_val @ B2 )
                  = ( Plus @ A2 @ W ) ) )
             => ( package_sat_val @ Plus @ B3 @ B2 ) ) ) ) ) ).

% package_logic.is_footprint_standard_def
thf(fact_552_package__logic_Ois__footprint__standard__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,W: state,A3: aassertion_state,B3: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7096211993527611489_state @ Plus @ W @ A3 @ B3 )
        = ( ! [A2: state,B2: state] :
              ( ( ( package_sat_state @ Plus @ A3 @ A2 )
                & ( ( some_state @ B2 )
                  = ( Plus @ A2 @ W ) ) )
             => ( package_sat_state @ Plus @ B3 @ B2 ) ) ) ) ) ).

% package_logic.is_footprint_standard_def
thf(fact_553_package__logic_Ois__footprint__general__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,W: val,R2: val > val > val,A3: aassertion_val,B3: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag4897621287243763553al_val @ Plus @ W @ R2 @ A3 @ B3 )
        = ( ! [A2: val,B2: val] :
              ( ( ( package_sat_val @ Plus @ A3 @ A2 )
                & ( ( some_val @ B2 )
                  = ( Plus @ A2 @ ( R2 @ A2 @ W ) ) ) )
             => ( package_sat_val @ Plus @ B3 @ B2 ) ) ) ) ) ).

% package_logic.is_footprint_general_def
thf(fact_554_package__logic_Ois__footprint__general__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,W: state,R2: state > state > state,A3: aassertion_state,B3: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6192661653625646554_state @ Plus @ W @ R2 @ A3 @ B3 )
        = ( ! [A2: state,B2: state] :
              ( ( ( package_sat_state @ Plus @ A3 @ A2 )
                & ( ( some_state @ B2 )
                  = ( Plus @ A2 @ ( R2 @ A2 @ W ) ) ) )
             => ( package_sat_state @ Plus @ B3 @ B2 ) ) ) ) ) ).

% package_logic.is_footprint_general_def
thf(fact_555_package__logic_Ois__footprint__standardI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A3: aassertion_val,W: val,B3: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: val,B4: val] :
            ( ( ( package_sat_val @ Plus @ A3 @ A5 )
              & ( ( some_val @ B4 )
                = ( Plus @ A5 @ W ) ) )
           => ( package_sat_val @ Plus @ B3 @ B4 ) )
       => ( packag685230357167683994rd_val @ Plus @ W @ A3 @ B3 ) ) ) ).

% package_logic.is_footprint_standardI
thf(fact_556_package__logic_Ois__footprint__standardI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: aassertion_state,W: state,B3: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: state,B4: state] :
            ( ( ( package_sat_state @ Plus @ A3 @ A5 )
              & ( ( some_state @ B4 )
                = ( Plus @ A5 @ W ) ) )
           => ( package_sat_state @ Plus @ B3 @ B4 ) )
       => ( packag7096211993527611489_state @ Plus @ W @ A3 @ B3 ) ) ) ).

% package_logic.is_footprint_standardI
thf(fact_557_package__logic_Ois__footprint__generalI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A3: aassertion_val,R2: val > val > val,W: val,B3: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: val,B4: val] :
            ( ( ( package_sat_val @ Plus @ A3 @ A5 )
              & ( ( some_val @ B4 )
                = ( Plus @ A5 @ ( R2 @ A5 @ W ) ) ) )
           => ( package_sat_val @ Plus @ B3 @ B4 ) )
       => ( packag4897621287243763553al_val @ Plus @ W @ R2 @ A3 @ B3 ) ) ) ).

% package_logic.is_footprint_generalI
thf(fact_558_package__logic_Ois__footprint__generalI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: aassertion_state,R2: state > state > state,W: state,B3: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: state,B4: state] :
            ( ( ( package_sat_state @ Plus @ A3 @ A5 )
              & ( ( some_state @ B4 )
                = ( Plus @ A5 @ ( R2 @ A5 @ W ) ) ) )
           => ( package_sat_state @ Plus @ B3 @ B4 ) )
       => ( packag6192661653625646554_state @ Plus @ W @ R2 @ A3 @ B3 ) ) ) ).

% package_logic.is_footprint_generalI
thf(fact_559_ssubsetI,axiom,
    ! [A3: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ! [Pi4: state,H: produc3882902684156697774_state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi4 @ H ) @ A3 )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi4 @ H ) @ B3 ) )
     => ( ord_le4032880777312351359_state @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_560_ssubsetI,axiom,
    ! [A3: set_Pr2391701600042252046_state,B3: set_Pr2391701600042252046_state] :
      ( ! [Pi4: state,H: state > state] :
          ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi4 @ H ) @ A3 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi4 @ H ) @ B3 ) )
     => ( ord_le6483787752022044334_state @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_561_ssubsetI,axiom,
    ! [A3: set_Pr1688445902015331925_state,B3: set_Pr1688445902015331925_state] :
      ( ! [Pi4: option_state,H: option_state] :
          ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi4 @ H ) @ A3 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi4 @ H ) @ B3 ) )
     => ( ord_le6423325748750870005_state @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_562_ssubsetI,axiom,
    ! [A3: set_Pr6039890582625207499_state,B3: set_Pr6039890582625207499_state] :
      ( ! [Pi4: prat,H: state] :
          ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi4 @ H ) @ A3 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi4 @ H ) @ B3 ) )
     => ( ord_le262197048489888875_state @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_563_ssubsetI,axiom,
    ! [A3: set_Pr6847263425858009217_state,B3: set_Pr6847263425858009217_state] :
      ( ! [Pi4: prat,H: set_state] :
          ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi4 @ H ) @ A3 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi4 @ H ) @ B3 ) )
     => ( ord_le5761494056910127137_state @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_564_ssubsetI,axiom,
    ! [A3: set_Pr2507339680178222706t_char,B3: set_Pr2507339680178222706t_char] :
      ( ! [Pi4: nat,H: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi4 @ H ) @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi4 @ H ) @ B3 ) )
     => ( ord_le6315511645215477266t_char @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_565_ssubsetI,axiom,
    ! [A3: set_Pr7649655576376376530on_val,B3: set_Pr7649655576376376530on_val] :
      ( ! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi4 @ H ) @ A3 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi4 @ H ) @ B3 ) )
     => ( ord_le6900622908513426738on_val @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_566_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_567_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_568_sep__algebra_Osum__then__singleton,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ A )
          = ( Plus @ B @ C ) )
        = ( ( insert_val @ A @ bot_bot_set_val )
          = ( sep_add_set_val @ Plus @ ( insert_val @ B @ bot_bot_set_val ) @ ( insert_val @ C @ bot_bot_set_val ) ) ) ) ) ).

% sep_algebra.sum_then_singleton
thf(fact_569_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_570_PartialSA_Ois__footprint__standard__def,axiom,
    ! [W: state,A3: aassertion_state,B3: aassertion_state] :
      ( ( packag7096211993527611489_state @ plus @ W @ A3 @ B3 )
      = ( ! [A2: state,B2: state] :
            ( ( ( package_sat_state @ plus @ A3 @ A2 )
              & ( ( some_state @ B2 )
                = ( plus @ A2 @ W ) ) )
           => ( package_sat_state @ plus @ B3 @ B2 ) ) ) ) ).

% PartialSA.is_footprint_standard_def
thf(fact_571_PartialSA_Ois__footprint__general__def,axiom,
    ! [W: state,R2: state > state > state,A3: aassertion_state,B3: aassertion_state] :
      ( ( packag6192661653625646554_state @ plus @ W @ R2 @ A3 @ B3 )
      = ( ! [A2: state,B2: state] :
            ( ( ( package_sat_state @ plus @ A3 @ A2 )
              & ( ( some_state @ B2 )
                = ( plus @ A2 @ ( R2 @ A2 @ W ) ) ) )
           => ( package_sat_state @ plus @ B3 @ B2 ) ) ) ) ).

% PartialSA.is_footprint_general_def
thf(fact_572_PartialSA_Ois__footprint__standardI,axiom,
    ! [A3: aassertion_state,W: state,B3: aassertion_state] :
      ( ! [A5: state,B4: state] :
          ( ( ( package_sat_state @ plus @ A3 @ A5 )
            & ( ( some_state @ B4 )
              = ( plus @ A5 @ W ) ) )
         => ( package_sat_state @ plus @ B3 @ B4 ) )
     => ( packag7096211993527611489_state @ plus @ W @ A3 @ B3 ) ) ).

% PartialSA.is_footprint_standardI
thf(fact_573_PartialSA_Ois__footprint__generalI,axiom,
    ! [A3: aassertion_state,R2: state > state > state,W: state,B3: aassertion_state] :
      ( ! [A5: state,B4: state] :
          ( ( ( package_sat_state @ plus @ A3 @ A5 )
            & ( ( some_state @ B4 )
              = ( plus @ A5 @ ( R2 @ A5 @ W ) ) ) )
         => ( package_sat_state @ plus @ B3 @ B4 ) )
     => ( packag6192661653625646554_state @ plus @ W @ R2 @ A3 @ B3 ) ) ).

% PartialSA.is_footprint_generalI
thf(fact_574_PartialSA_Obigger__singleton,axiom,
    ! [Phi3: state,Phi2: state] :
      ( ( greater @ Phi3 @ Phi2 )
     => ( greater_set @ ( insert_state @ Phi3 @ bot_bot_set_state ) @ ( insert_state @ Phi2 @ bot_bot_set_state ) ) ) ).

% PartialSA.bigger_singleton
thf(fact_575_PartialSA_Osetify__sum,axiom,
    ! [P4: state > $o,A3: set_state,B3: set_state] :
      ( ( sep_setify_state @ P4 @ ( add_set @ A3 @ B3 ) )
      = ( ! [X: state] :
            ( ( member_state @ X @ A3 )
           => ( sep_setify_state @ P4 @ ( add_set @ ( insert_state @ X @ bot_bot_set_state ) @ B3 ) ) ) ) ) ).

% PartialSA.setify_sum
thf(fact_576_these__insert__Some,axiom,
    ! [X2: state,A3: set_option_state] :
      ( ( these_state @ ( insert_option_state @ ( some_state @ X2 ) @ A3 ) )
      = ( insert_state @ X2 @ ( these_state @ A3 ) ) ) ).

% these_insert_Some
thf(fact_577_these__insert__Some,axiom,
    ! [X2: val,A3: set_option_val] :
      ( ( these_val @ ( insert_option_val @ ( some_val @ X2 ) @ A3 ) )
      = ( insert_val @ X2 @ ( these_val @ A3 ) ) ) ).

% these_insert_Some
thf(fact_578_PartialSA_Owf__assertion__and__the,axiom,
    ! [A: state,B: state,A3: aassertion_state] :
      ( ( defined @ ( core @ A ) @ B )
     => ( ( package_sat_state @ plus @ A3 @ B )
       => ( ( packag7298478212905792770_state @ plus @ core @ A3 )
         => ( package_sat_state @ plus @ A3 @ ( the_state @ ( plus @ ( core @ A ) @ B ) ) ) ) ) ) ).

% PartialSA.wf_assertion_and_the
thf(fact_579_package__logic_Opackage__satI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Pc: val > $o,A4: val,U5: val,U: val,A3: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Pc @ ( Core @ A4 ) )
         => ? [X6: val] :
              ( ( ( some_val @ X6 )
                = ( Plus @ ( Core @ A4 ) @ ( sep_minus_val @ Plus @ Core @ U5 @ U ) ) )
              & ( package_sat_val @ Plus @ A3 @ X6 ) ) )
       => ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A3 @ A4 @ U5 @ U ) ) ) ).

% package_logic.package_satI
thf(fact_580_package__logic_Opackage__satI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,A4: state,U5: state,U: state,A3: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Pc @ ( Core @ A4 ) )
         => ? [X6: state] :
              ( ( ( some_state @ X6 )
                = ( Plus @ ( Core @ A4 ) @ ( sep_minus_state @ Plus @ Core @ U5 @ U ) ) )
              & ( package_sat_state @ Plus @ A3 @ X6 ) ) )
       => ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A3 @ A4 @ U5 @ U ) ) ) ).

% package_logic.package_satI
thf(fact_581_these__empty,axiom,
    ( ( these_state @ bot_bo710180891245420500_state )
    = bot_bot_set_state ) ).

% these_empty
thf(fact_582_these__insert__None,axiom,
    ! [A3: set_option_state] :
      ( ( these_state @ ( insert_option_state @ none_state @ A3 ) )
      = ( these_state @ A3 ) ) ).

% these_insert_None
thf(fact_583_these__insert__None,axiom,
    ! [A3: set_option_val] :
      ( ( these_val @ ( insert_option_val @ none_val @ A3 ) )
      = ( these_val @ A3 ) ) ).

% these_insert_None
thf(fact_584_PartialSA_Olarger__set__refl,axiom,
    ! [A3: set_state] : ( greater_set @ A3 @ A3 ) ).

% PartialSA.larger_set_refl
thf(fact_585_PartialSA_Osucc__set__trans,axiom,
    ! [A3: set_state,B3: set_state,C2: set_state] :
      ( ( greater_set @ A3 @ B3 )
     => ( ( greater_set @ B3 @ C2 )
       => ( greater_set @ A3 @ C2 ) ) ) ).

% PartialSA.succ_set_trans
thf(fact_586_PartialSA_Osetify__def,axiom,
    ( sep_setify_state
    = ( ^ [P6: state > $o,A6: set_state] :
        ! [X: state] :
          ( ( member_state @ X @ A6 )
         => ( P6 @ X ) ) ) ) ).

% PartialSA.setify_def
thf(fact_587_these__not__empty__eq,axiom,
    ! [B3: set_option_val] :
      ( ( ( these_val @ B3 )
       != bot_bot_set_val )
      = ( ( B3 != bot_bo5906790151302476563on_val )
        & ( B3
         != ( insert_option_val @ none_val @ bot_bo5906790151302476563on_val ) ) ) ) ).

% these_not_empty_eq
thf(fact_588_these__not__empty__eq,axiom,
    ! [B3: set_option_state] :
      ( ( ( these_state @ B3 )
       != bot_bot_set_state )
      = ( ( B3 != bot_bo710180891245420500_state )
        & ( B3
         != ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_not_empty_eq
thf(fact_589_these__empty__eq,axiom,
    ! [B3: set_option_val] :
      ( ( ( these_val @ B3 )
        = bot_bot_set_val )
      = ( ( B3 = bot_bo5906790151302476563on_val )
        | ( B3
          = ( insert_option_val @ none_val @ bot_bo5906790151302476563on_val ) ) ) ) ).

% these_empty_eq
thf(fact_590_these__empty__eq,axiom,
    ! [B3: set_option_state] :
      ( ( ( these_state @ B3 )
        = bot_bot_set_state )
      = ( ( B3 = bot_bo710180891245420500_state )
        | ( B3
          = ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_empty_eq
thf(fact_591_PartialSA_Ogreater__set__def,axiom,
    ( greater_set
    = ( ^ [A6: set_state,B6: set_state] :
        ! [X: state] :
          ( ( member_state @ X @ A6 )
         => ? [Y3: state] :
              ( ( member_state @ Y3 @ B6 )
              & ( greater @ X @ Y3 ) ) ) ) ) ).

% PartialSA.greater_set_def
thf(fact_592_PartialSA_Ogreater__setI,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ! [A5: state] :
          ( ( member_state @ A5 @ A3 )
         => ? [X6: state] :
              ( ( member_state @ X6 @ B3 )
              & ( greater @ A5 @ X6 ) ) )
     => ( greater_set @ A3 @ B3 ) ) ).

% PartialSA.greater_setI
thf(fact_593_PartialSA_Obigger__set,axiom,
    ! [A9: set_state,A3: set_state,B3: set_state] :
      ( ( greater_set @ A9 @ A3 )
     => ( greater_set @ ( add_set @ A9 @ B3 ) @ ( add_set @ A3 @ B3 ) ) ) ).

% PartialSA.bigger_set
thf(fact_594_in__these__eq,axiom,
    ! [X2: produc7272280185086996969_state,A3: set_op8023886964038656367_state] :
      ( ( member7461423308196221312_state @ X2 @ ( these_4347185934205712492_state @ A3 ) )
      = ( member3894984953261475024_state @ ( some_P6297086604321900740_state @ X2 ) @ A3 ) ) ).

% in_these_eq
thf(fact_595_in__these__eq,axiom,
    ! [X2: produc9003781588162623004on_val,A3: set_op6240181410663953058on_val] :
      ( ( member2879030624002546611on_val @ X2 @ ( these_3919539662624406175on_val @ A3 ) )
      = ( member4602090469187516291on_val @ ( some_P2581650809551911671on_val @ X2 ) @ A3 ) ) ).

% in_these_eq
thf(fact_596_in__these__eq,axiom,
    ! [X2: produc3882902684156697774_state,A3: set_op3387789762728285652_state] :
      ( ( member6715010751053357271_state @ X2 @ ( these_7452845017025203435_state @ A3 ) )
      = ( member5919342556599141533_state @ ( some_P5383513176685068947_state @ X2 ) @ A3 ) ) ).

% in_these_eq
thf(fact_597_in__these__eq,axiom,
    ! [X2: state,A3: set_option_state] :
      ( ( member_state @ X2 @ ( these_state @ A3 ) )
      = ( member_option_state @ ( some_state @ X2 ) @ A3 ) ) ).

% in_these_eq
thf(fact_598_in__these__eq,axiom,
    ! [X2: val,A3: set_option_val] :
      ( ( member_val @ X2 @ ( these_val @ A3 ) )
      = ( member_option_val @ ( some_val @ X2 ) @ A3 ) ) ).

% in_these_eq
thf(fact_599_PartialSA_Osub__bigger,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B3 )
     => ( greater_set @ A3 @ B3 ) ) ).

% PartialSA.sub_bigger
thf(fact_600_sep__algebra_Ominus__equiv__def__any__elem,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ X2 )
          = ( Plus @ A @ B ) )
       => ( ( some_val @ ( sep_minus_val @ Plus @ Core @ X2 @ A ) )
          = ( Plus @ B @ ( Core @ X2 ) ) ) ) ) ).

% sep_algebra.minus_equiv_def_any_elem
thf(fact_601_sep__algebra_Ominus__equiv__def__any__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X2 )
          = ( Plus @ A @ B ) )
       => ( ( some_state @ ( sep_minus_state @ Plus @ Core @ X2 @ A ) )
          = ( Plus @ B @ ( Core @ X2 ) ) ) ) ) ).

% sep_algebra.minus_equiv_def_any_elem
thf(fact_602_minus__def,axiom,
    ( minus
    = ( sep_minus_state @ plus @ core ) ) ).

% minus_def
thf(fact_603_package__logic_Opackage__sat__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Pc: val > $o,A3: aassertion_val,A4: val,U5: val,U: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A3 @ A4 @ U5 @ U )
        = ( ( Pc @ ( Core @ A4 ) )
         => ? [X: val] :
              ( ( ( some_val @ X )
                = ( Plus @ ( Core @ A4 ) @ ( sep_minus_val @ Plus @ Core @ U5 @ U ) ) )
              & ( package_sat_val @ Plus @ A3 @ X ) ) ) ) ) ).

% package_logic.package_sat_def
thf(fact_604_package__logic_Opackage__sat__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,A3: aassertion_state,A4: state,U5: state,U: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A3 @ A4 @ U5 @ U )
        = ( ( Pc @ ( Core @ A4 ) )
         => ? [X: state] :
              ( ( ( some_state @ X )
                = ( Plus @ ( Core @ A4 ) @ ( sep_minus_state @ Plus @ Core @ U5 @ U ) ) )
              & ( package_sat_state @ Plus @ A3 @ X ) ) ) ) ) ).

% package_logic.package_sat_def
thf(fact_605_subrelI,axiom,
    ! [R3: set_Pr4231240917283394591_state,S4: set_Pr4231240917283394591_state] :
      ( ! [X3: state,Y4: produc3882902684156697774_state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ Y4 ) @ R3 )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ Y4 ) @ S4 ) )
     => ( ord_le4032880777312351359_state @ R3 @ S4 ) ) ).

% subrelI
thf(fact_606_subrelI,axiom,
    ! [R3: set_Pr2391701600042252046_state,S4: set_Pr2391701600042252046_state] :
      ( ! [X3: state,Y4: state > state] :
          ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X3 @ Y4 ) @ R3 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X3 @ Y4 ) @ S4 ) )
     => ( ord_le6483787752022044334_state @ R3 @ S4 ) ) ).

% subrelI
thf(fact_607_subrelI,axiom,
    ! [R3: set_Pr1688445902015331925_state,S4: set_Pr1688445902015331925_state] :
      ( ! [X3: option_state,Y4: option_state] :
          ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ R3 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ S4 ) )
     => ( ord_le6423325748750870005_state @ R3 @ S4 ) ) ).

% subrelI
thf(fact_608_subrelI,axiom,
    ! [R3: set_Pr6039890582625207499_state,S4: set_Pr6039890582625207499_state] :
      ( ! [X3: prat,Y4: state] :
          ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X3 @ Y4 ) @ R3 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X3 @ Y4 ) @ S4 ) )
     => ( ord_le262197048489888875_state @ R3 @ S4 ) ) ).

% subrelI
thf(fact_609_subrelI,axiom,
    ! [R3: set_Pr6847263425858009217_state,S4: set_Pr6847263425858009217_state] :
      ( ! [X3: prat,Y4: set_state] :
          ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X3 @ Y4 ) @ R3 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X3 @ Y4 ) @ S4 ) )
     => ( ord_le5761494056910127137_state @ R3 @ S4 ) ) ).

% subrelI
thf(fact_610_subrelI,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S4: set_Pr2507339680178222706t_char] :
      ( ! [X3: nat,Y4: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y4 ) @ R3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y4 ) @ S4 ) )
     => ( ord_le6315511645215477266t_char @ R3 @ S4 ) ) ).

% subrelI
thf(fact_611_subrelI,axiom,
    ! [R3: set_Pr7649655576376376530on_val,S4: set_Pr7649655576376376530on_val] :
      ( ! [X3: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y4 ) @ R3 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y4 ) @ S4 ) )
     => ( ord_le6900622908513426738on_val @ R3 @ S4 ) ) ).

% subrelI
thf(fact_612_PartialSA_Owf__assertion__sat__larger__pure,axiom,
    ! [A3: aassertion_state,Phi2: state,Phi3: state,R3: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ A3 )
     => ( ( package_sat_state @ plus @ A3 @ Phi2 )
       => ( ( ( some_state @ Phi3 )
            = ( plus @ Phi2 @ R3 ) )
         => ( ( sep_pure_state @ plus @ R3 )
           => ( package_sat_state @ plus @ A3 @ Phi3 ) ) ) ) ) ).

% PartialSA.wf_assertion_sat_larger_pure
thf(fact_613_PartialSA_Omono__prop__set,axiom,
    ! [A3: set_state,B3: set_state,P4: state > $o] :
      ( ( greater_set @ A3 @ B3 )
     => ( ( sep_setify_state @ P4 @ B3 )
       => ( ( sep_mono_prop_state @ plus @ P4 )
         => ( sep_setify_state @ P4 @ A3 ) ) ) ) ).

% PartialSA.mono_prop_set
thf(fact_614_PartialSA_Osat__star__exists__bigger,axiom,
    ! [A3: aassertion_state,B3: aassertion_state,Phi2: state] :
      ( ( package_sat_state @ plus @ ( aStar_state @ A3 @ B3 ) @ Phi2 )
     => ( ( packag7298478212905792770_state @ plus @ core @ A3 )
       => ( ( packag7298478212905792770_state @ plus @ core @ B3 )
         => ? [A5: state,B4: state] :
              ( ( greater @ ( core @ A5 ) @ ( core @ Phi2 ) )
              & ( greater @ ( core @ B4 ) @ ( core @ Phi2 ) )
              & ( ( some_state @ Phi2 )
                = ( plus @ A5 @ B4 ) )
              & ( package_sat_state @ plus @ A3 @ A5 )
              & ( package_sat_state @ plus @ B3 @ B4 ) ) ) ) ) ).

% PartialSA.sat_star_exists_bigger
thf(fact_615_package__logic_Owf__assertion__sat__larger__pure,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A3: aassertion_val,Phi2: val,Phi3: val,R3: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ A3 )
       => ( ( package_sat_val @ Plus @ A3 @ Phi2 )
         => ( ( ( some_val @ Phi3 )
              = ( Plus @ Phi2 @ R3 ) )
           => ( ( sep_pure_val @ Plus @ R3 )
             => ( package_sat_val @ Plus @ A3 @ Phi3 ) ) ) ) ) ) ).

% package_logic.wf_assertion_sat_larger_pure
thf(fact_616_package__logic_Owf__assertion__sat__larger__pure,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: aassertion_state,Phi2: state,Phi3: state,R3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ A3 )
       => ( ( package_sat_state @ Plus @ A3 @ Phi2 )
         => ( ( ( some_state @ Phi3 )
              = ( Plus @ Phi2 @ R3 ) )
           => ( ( sep_pure_state @ Plus @ R3 )
             => ( package_sat_state @ Plus @ A3 @ Phi3 ) ) ) ) ) ) ).

% package_logic.wf_assertion_sat_larger_pure
thf(fact_617_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_618_PartialSA_Opure__def,axiom,
    ! [A: state] :
      ( ( sep_pure_state @ plus @ A )
      = ( ( some_state @ A )
        = ( plus @ A @ A ) ) ) ).

% PartialSA.pure_def
thf(fact_619_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_620_sep__algebra_Opure__stable,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_pure_val @ Plus @ A )
       => ( ( sep_pure_val @ Plus @ B )
         => ( ( ( some_val @ C )
              = ( Plus @ A @ B ) )
           => ( sep_pure_val @ Plus @ C ) ) ) ) ) ).

% sep_algebra.pure_stable
thf(fact_621_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_622_sep__algebra_Opure__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_pure_val @ Plus @ A )
        = ( ( some_val @ A )
          = ( Plus @ A @ A ) ) ) ) ).

% sep_algebra.pure_def
thf(fact_623_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_624_PartialSA_Osat_Osimps_I1_J,axiom,
    ! [Aa2: aassertion_state,B3: aassertion_state,Phi2: state] :
      ( ( package_sat_state @ plus @ ( aStar_state @ Aa2 @ B3 ) @ Phi2 )
      = ( ? [A2: state,B2: state] :
            ( ( ( some_state @ Phi2 )
              = ( plus @ A2 @ B2 ) )
            & ( package_sat_state @ plus @ Aa2 @ A2 )
            & ( package_sat_state @ plus @ B3 @ B2 ) ) ) ) ).

% PartialSA.sat.simps(1)
thf(fact_625_PartialSA_Owf__assertion_Osimps_I1_J,axiom,
    ! [Aa2: aassertion_state,B3: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aStar_state @ Aa2 @ B3 ) )
      = ( ( packag7298478212905792770_state @ plus @ core @ Aa2 )
        & ( packag7298478212905792770_state @ plus @ core @ B3 ) ) ) ).

% PartialSA.wf_assertion.simps(1)
thf(fact_626_package__logic_Osat_Osimps_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A3: aassertion_val,B3: aassertion_val,Phi2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ ( aStar_val @ A3 @ B3 ) @ Phi2 )
        = ( ? [A2: val,B2: val] :
              ( ( ( some_val @ Phi2 )
                = ( Plus @ A2 @ B2 ) )
              & ( package_sat_val @ Plus @ A3 @ A2 )
              & ( package_sat_val @ Plus @ B3 @ B2 ) ) ) ) ) ).

% package_logic.sat.simps(1)
thf(fact_627_package__logic_Osat_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: aassertion_state,B3: aassertion_state,Phi2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aStar_state @ A3 @ B3 ) @ Phi2 )
        = ( ? [A2: state,B2: state] :
              ( ( ( some_state @ Phi2 )
                = ( Plus @ A2 @ B2 ) )
              & ( package_sat_state @ Plus @ A3 @ A2 )
              & ( package_sat_state @ Plus @ B3 @ B2 ) ) ) ) ) ).

% package_logic.sat.simps(1)
thf(fact_628_PartialSA_Omono__propI,axiom,
    ! [P4: state > $o] :
      ( ! [X3: state,Y4: state] :
          ( ( ( greater @ Y4 @ X3 )
            & ( P4 @ X3 ) )
         => ( P4 @ Y4 ) )
     => ( sep_mono_prop_state @ plus @ P4 ) ) ).

% PartialSA.mono_propI
thf(fact_629_PartialSA_Omono__prop__def,axiom,
    ! [P4: state > $o] :
      ( ( sep_mono_prop_state @ plus @ P4 )
      = ( ! [X: state,Y3: state] :
            ( ( ( greater @ Y3 @ X )
              & ( P4 @ X ) )
           => ( P4 @ Y3 ) ) ) ) ).

% PartialSA.mono_prop_def
thf(fact_630_PartialSA_Osmaller__pure__sum__smaller,axiom,
    ! [Y: state,A: state,B: state,X2: state] :
      ( ( greater @ Y @ A )
     => ( ( greater @ Y @ B )
       => ( ( ( some_state @ X2 )
            = ( plus @ A @ B ) )
         => ( ( sep_pure_state @ plus @ B )
           => ( greater @ Y @ X2 ) ) ) ) ) ).

% PartialSA.smaller_pure_sum_smaller
thf(fact_631_PartialSA_Omono__prop__set__equiv,axiom,
    ! [P4: state > $o,A3: set_state,B3: set_state] :
      ( ( sep_mono_prop_state @ plus @ P4 )
     => ( ( sep_equiv_state @ plus @ A3 @ B3 )
       => ( ( sep_setify_state @ P4 @ A3 )
          = ( sep_setify_state @ P4 @ B3 ) ) ) ) ).

% PartialSA.mono_prop_set_equiv
thf(fact_632_package__logic_Omono__pruner__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,P2: val > $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8699728003424501129er_val @ Plus @ P2 )
        = ( ! [Phi5: val,Phi4: val,R: val] :
              ( ( ( sep_pure_val @ Plus @ R )
                & ( P2 @ Phi4 )
                & ( ( some_val @ Phi5 )
                  = ( Plus @ Phi4 @ R ) ) )
             => ( P2 @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pruner_def
thf(fact_633_package__logic_Omono__pruner__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,P2: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2456304381420842418_state @ Plus @ P2 )
        = ( ! [Phi5: state,Phi4: state,R: state] :
              ( ( ( sep_pure_state @ Plus @ R )
                & ( P2 @ Phi4 )
                & ( ( some_state @ Phi5 )
                  = ( Plus @ Phi4 @ R ) ) )
             => ( P2 @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pruner_def
thf(fact_634_PartialSA_Omono__pure__condI,axiom,
    ! [B: state > $o] :
      ( ! [Phi: state] :
          ( ( B @ Phi )
          = ( B @ ( core @ Phi ) ) )
     => ( ! [Phi: state,Phi6: state,R4: state] :
            ( ( ( sep_pure_state @ plus @ R4 )
              & ( ( some_state @ Phi6 )
                = ( plus @ Phi @ R4 ) )
              & ~ ( B @ Phi ) )
           => ~ ( B @ Phi6 ) )
       => ( packag6595153354952283300_state @ plus @ core @ B ) ) ) ).

% PartialSA.mono_pure_condI
thf(fact_635_PartialSA_Omono__pure__cond__def,axiom,
    ! [B: state > $o] :
      ( ( packag6595153354952283300_state @ plus @ core @ B )
      = ( ! [Phi4: state] :
            ( ( B @ Phi4 )
            = ( B @ ( core @ Phi4 ) ) )
        & ! [Phi5: state,Phi4: state,R: state] :
            ( ( ( sep_pure_state @ plus @ R )
              & ( ( some_state @ Phi5 )
                = ( plus @ Phi4 @ R ) )
              & ~ ( B @ Phi4 ) )
           => ~ ( B @ Phi5 ) ) ) ) ).

% PartialSA.mono_pure_cond_def
thf(fact_636_PartialSA_Osetify__sum__image,axiom,
    ! [P4: state > $o,F4: state > state,A3: set_state,B3: set_state] :
      ( ( sep_setify_state @ P4 @ ( add_set @ ( image_state_state @ F4 @ A3 ) @ B3 ) )
      = ( ! [X: state] :
            ( ( member_state @ X @ A3 )
           => ( sep_setify_state @ P4 @ ( add_set @ ( insert_state @ ( F4 @ X ) @ bot_bot_set_state ) @ B3 ) ) ) ) ) ).

% PartialSA.setify_sum_image
thf(fact_637_PartialSA_Oequiv__stable__sum,axiom,
    ! [A3: set_state,B3: set_state,C2: set_state] :
      ( ( sep_equiv_state @ plus @ A3 @ B3 )
     => ( sep_equiv_state @ plus @ ( add_set @ A3 @ C2 ) @ ( add_set @ B3 @ C2 ) ) ) ).

% PartialSA.equiv_stable_sum
thf(fact_638_PartialSA_Oequiv__def,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( sep_equiv_state @ plus @ A3 @ B3 )
      = ( ( greater_set @ A3 @ B3 )
        & ( greater_set @ B3 @ A3 ) ) ) ).

% PartialSA.equiv_def
thf(fact_639_PartialSA_OequivI,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( greater_set @ A3 @ B3 )
     => ( ( greater_set @ B3 @ A3 )
       => ( sep_equiv_state @ plus @ A3 @ B3 ) ) ) ).

% PartialSA.equivI
thf(fact_640_PartialSA_Omono__pruner__def,axiom,
    ! [P2: state > $o] :
      ( ( packag2456304381420842418_state @ plus @ P2 )
      = ( ! [Phi5: state,Phi4: state,R: state] :
            ( ( ( sep_pure_state @ plus @ R )
              & ( P2 @ Phi4 )
              & ( ( some_state @ Phi5 )
                = ( plus @ Phi4 @ R ) ) )
           => ( P2 @ Phi5 ) ) ) ) ).

% PartialSA.mono_pruner_def
thf(fact_641_package__logic_Omono__pure__cond__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B: val > $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6714868506247462167nd_val @ Plus @ Core @ B )
        = ( ! [Phi4: val] :
              ( ( B @ Phi4 )
              = ( B @ ( Core @ Phi4 ) ) )
          & ! [Phi5: val,Phi4: val,R: val] :
              ( ( ( sep_pure_val @ Plus @ R )
                & ( ( some_val @ Phi5 )
                  = ( Plus @ Phi4 @ R ) )
                & ~ ( B @ Phi4 ) )
             => ~ ( B @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pure_cond_def
thf(fact_642_package__logic_Omono__pure__cond__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6595153354952283300_state @ Plus @ Core @ B )
        = ( ! [Phi4: state] :
              ( ( B @ Phi4 )
              = ( B @ ( Core @ Phi4 ) ) )
          & ! [Phi5: state,Phi4: state,R: state] :
              ( ( ( sep_pure_state @ Plus @ R )
                & ( ( some_state @ Phi5 )
                  = ( Plus @ Phi4 @ R ) )
                & ~ ( B @ Phi4 ) )
             => ~ ( B @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pure_cond_def
thf(fact_643_package__logic_Omono__pure__condI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B: val > $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [Phi: val] :
            ( ( B @ Phi )
            = ( B @ ( Core @ Phi ) ) )
       => ( ! [Phi: val,Phi6: val,R4: val] :
              ( ( ( sep_pure_val @ Plus @ R4 )
                & ( ( some_val @ Phi6 )
                  = ( Plus @ Phi @ R4 ) )
                & ~ ( B @ Phi ) )
             => ~ ( B @ Phi6 ) )
         => ( packag6714868506247462167nd_val @ Plus @ Core @ B ) ) ) ) ).

% package_logic.mono_pure_condI
thf(fact_644_package__logic_Omono__pure__condI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [Phi: state] :
            ( ( B @ Phi )
            = ( B @ ( Core @ Phi ) ) )
       => ( ! [Phi: state,Phi6: state,R4: state] :
              ( ( ( sep_pure_state @ Plus @ R4 )
                & ( ( some_state @ Phi6 )
                  = ( Plus @ Phi @ R4 ) )
                & ~ ( B @ Phi ) )
             => ~ ( B @ Phi6 ) )
         => ( packag6595153354952283300_state @ Plus @ Core @ B ) ) ) ) ).

% package_logic.mono_pure_condI
thf(fact_645_PartialSA_Opackage__rhs__proof,axiom,
    ! [Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
     => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F4 )
       => ( ( packag7298478212905792770_state @ plus @ core @ A3 )
         => ( ( packag6595153354952283300_state @ plus @ core @ Pc )
           => ( ( stable @ F4 )
             => ( ( defined @ Phi2 @ F4 )
               => ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
                  & ( packag3877571396724326311_state @ plus @ core @ unit @ S3 @ F5 ) ) ) ) ) ) ) ) ).

% PartialSA.package_rhs_proof
thf(fact_646_multiply__sem__assertion_Osimps,axiom,
    ( multip8064567061438756306ertion
    = ( ^ [P3: prat,P6: set_state] : ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ P3 ) @ P6 ) ) ) ) ).

% multiply_sem_assertion.simps
thf(fact_647_multiply__sem__assertion_Oelims,axiom,
    ! [X2: prat,Xa: set_state,Y: set_state] :
      ( ( ( multip8064567061438756306ertion @ X2 @ Xa )
        = Y )
     => ( Y
        = ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X2 ) @ Xa ) ) ) ) ).

% multiply_sem_assertion.elims
thf(fact_648_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_649_PartialSA_Oequiv__up__closed__subset,axiom,
    ! [A3: set_state,B3: set_state,C2: set_state] :
      ( ( sep_up_closed_state @ plus @ A3 )
     => ( ( sep_equiv_state @ plus @ B3 @ C2 )
       => ( ( ord_le2494988322063910608_state @ B3 @ A3 )
          = ( ord_le2494988322063910608_state @ C2 @ A3 ) ) ) ) ).

% PartialSA.equiv_up_closed_subset
thf(fact_650_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_651_PartialSA_Obool__conj__def,axiom,
    ( packag1330488657391168505_state
    = ( ^ [A2: state > $o,B2: state > $o,X: state] :
          ( ( A2 @ X )
          & ( B2 @ X ) ) ) ) ).

% PartialSA.bool_conj_def
thf(fact_652_PartialSA_Oup__closedI,axiom,
    ! [A3: set_state] :
      ( ! [Phi6: state,Phi: state] :
          ( ( ( greater @ Phi6 @ Phi )
            & ( member_state @ Phi @ A3 ) )
         => ( member_state @ Phi6 @ A3 ) )
     => ( sep_up_closed_state @ plus @ A3 ) ) ).

% PartialSA.up_closedI
thf(fact_653_PartialSA_Oup__closed__def,axiom,
    ! [A3: set_state] :
      ( ( sep_up_closed_state @ plus @ A3 )
      = ( ! [Phi5: state] :
            ( ? [X: state] :
                ( ( member_state @ X @ A3 )
                & ( greater @ Phi5 @ X ) )
           => ( member_state @ Phi5 @ A3 ) ) ) ) ).

% PartialSA.up_closed_def
thf(fact_654_PartialSA_Oup__closed__sum,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( sep_up_closed_state @ plus @ A3 )
     => ( sep_up_closed_state @ plus @ ( add_set @ A3 @ B3 ) ) ) ).

% PartialSA.up_closed_sum
thf(fact_655_PartialSA_Oup__close__equiv,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( sep_up_closed_state @ plus @ A3 )
     => ( ( sep_up_closed_state @ plus @ B3 )
       => ( ( sep_equiv_state @ plus @ A3 @ B3 )
          = ( A3 = B3 ) ) ) ) ).

% PartialSA.up_close_equiv
thf(fact_656_PartialSA_OAStar,axiom,
    ! [Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state,B3: aassertion_state,Phi7: state,F6: state,S5: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F5 @ S3 @ Pc @ B3 @ Phi7 @ F6 @ S5 )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ ( aStar_state @ A3 @ B3 ) @ Phi7 @ F6 @ S5 ) ) ) ).

% PartialSA.AStar
thf(fact_657_PartialSA_Opackage__rhs_Ointros_I1_J,axiom,
    ! [Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state,Ba: aassertion_state,Phi7: state,F6: state,S5: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F5 @ S3 @ Pc @ Ba @ Phi7 @ F6 @ S5 )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ ( aStar_state @ A3 @ Ba ) @ Phi7 @ F6 @ S5 ) ) ) ).

% PartialSA.package_rhs.intros(1)
thf(fact_658_PartialSA_Oup__closed__bigger__subset,axiom,
    ! [B3: set_state,A3: set_state] :
      ( ( sep_up_closed_state @ plus @ B3 )
     => ( ( greater_set @ A3 @ B3 )
       => ( ord_le2494988322063910608_state @ A3 @ B3 ) ) ) ).

% PartialSA.up_closed_bigger_subset
thf(fact_659_multiply__sem__assertion_Opelims,axiom,
    ! [X2: prat,Xa: set_state,Y: set_state] :
      ( ( ( multip8064567061438756306ertion @ X2 @ Xa )
        = Y )
     => ( ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X2 @ Xa ) )
       => ~ ( ( Y
              = ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X2 ) @ Xa ) ) )
           => ~ ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X2 @ Xa ) ) ) ) ) ).

% multiply_sem_assertion.pelims
thf(fact_660_PartialSA_OAImp,axiom,
    ! [Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,B: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ ( packag1330488657391168505_state @ Pc @ B ) @ A3 @ Phi3 @ F5 @ S3 )
     => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ ( aImp_state @ B @ A3 ) @ Phi3 @ F5 @ S3 ) ) ).

% PartialSA.AImp
thf(fact_661_PartialSA_Ogeneral__completeness,axiom,
    ! [S2: set_Pr4231240917283394591_state,F4: state,A3: aassertion_state,Phi2: state,Phi3: state] :
      ( ! [A5: state,U3: state,T4: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
         => ! [X3: state] :
              ( ( ( some_state @ X3 )
                = ( plus @ A5 @ ( T4 @ F4 ) ) )
             => ( package_sat_state @ plus @ A3 @ X3 ) ) )
     => ( ( ( some_state @ Phi2 )
          = ( plus @ Phi3 @ F4 ) )
       => ( ( stable @ F4 )
         => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ unit )
           => ( ( packag7298478212905792770_state @ plus @ core @ A3 )
             => ? [X_1: set_Pr4231240917283394591_state] :
                  ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ unit @ S2
                  @ ^ [Uu3: state] : $true
                  @ A3
                  @ Phi3
                  @ F4
                  @ X_1 ) ) ) ) ) ) ).

% PartialSA.general_completeness
thf(fact_662_package__logic_Osat__star__exists__bigger,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A3: aassertion_val,B3: aassertion_val,Phi2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ ( aStar_val @ A3 @ B3 ) @ Phi2 )
       => ( ( packag5732365195153136185on_val @ Plus @ Core @ A3 )
         => ( ( packag5732365195153136185on_val @ Plus @ Core @ B3 )
           => ? [A5: val,B4: val] :
                ( ( sep_greater_val @ Plus @ ( Core @ A5 ) @ ( Core @ Phi2 ) )
                & ( sep_greater_val @ Plus @ ( Core @ B4 ) @ ( Core @ Phi2 ) )
                & ( ( some_val @ Phi2 )
                  = ( Plus @ A5 @ B4 ) )
                & ( package_sat_val @ Plus @ A3 @ A5 )
                & ( package_sat_val @ Plus @ B3 @ B4 ) ) ) ) ) ) ).

% package_logic.sat_star_exists_bigger
thf(fact_663_package__logic_Osat__star__exists__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: aassertion_state,B3: aassertion_state,Phi2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aStar_state @ A3 @ B3 ) @ Phi2 )
       => ( ( packag7298478212905792770_state @ Plus @ Core @ A3 )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ B3 )
           => ? [A5: state,B4: state] :
                ( ( sep_greater_state @ Plus @ ( Core @ A5 ) @ ( Core @ Phi2 ) )
                & ( sep_greater_state @ Plus @ ( Core @ B4 ) @ ( Core @ Phi2 ) )
                & ( ( some_state @ Phi2 )
                  = ( Plus @ A5 @ B4 ) )
                & ( package_sat_state @ Plus @ A3 @ A5 )
                & ( package_sat_state @ Plus @ B3 @ B4 ) ) ) ) ) ) ).

% package_logic.sat_star_exists_bigger
thf(fact_664_empty__map__add,axiom,
    ! [M: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val
        @ ^ [X: produc1457211279475724562t_char] : none_val
        @ M )
      = M ) ).

% empty_map_add
thf(fact_665_map__add__empty,axiom,
    ! [M: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ M
        @ ^ [X: produc1457211279475724562t_char] : none_val )
      = M ) ).

% map_add_empty
thf(fact_666_empty__eq__map__add__iff,axiom,
    ! [F4: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( ( ^ [X: produc1457211279475724562t_char] : none_val )
        = ( map_ad2908576761495157028ar_val @ F4 @ G ) )
      = ( ( F4
          = ( ^ [X: produc1457211279475724562t_char] : none_val ) )
        & ( G
          = ( ^ [X: produc1457211279475724562t_char] : none_val ) ) ) ) ).

% empty_eq_map_add_iff
thf(fact_667_pred__equals__eq2,axiom,
    ! [R2: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( ( ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_668_pred__equals__eq2,axiom,
    ! [R2: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( ( ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_669_pred__equals__eq2,axiom,
    ! [R2: set_Pr1688445902015331925_state,S2: set_Pr1688445902015331925_state] :
      ( ( ( ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_670_pred__equals__eq2,axiom,
    ! [R2: set_Pr6039890582625207499_state,S2: set_Pr6039890582625207499_state] :
      ( ( ( ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_671_pred__equals__eq2,axiom,
    ! [R2: set_Pr6847263425858009217_state,S2: set_Pr6847263425858009217_state] :
      ( ( ( ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_672_pred__equals__eq2,axiom,
    ! [R2: set_Pr2507339680178222706t_char,S2: set_Pr2507339680178222706t_char] :
      ( ( ( ^ [X: nat,Y3: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: nat,Y3: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_673_pred__equals__eq2,axiom,
    ! [R2: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( ( ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ R2 ) )
        = ( ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_674_multiply__sem__assertion_Ocases,axiom,
    ! [X2: produc6251377953716854561_state] :
      ~ ! [P: prat,P7: set_state] :
          ( X2
         != ( produc9080284787853255769_state @ P @ P7 ) ) ).

% multiply_sem_assertion.cases
thf(fact_675_map__add__def,axiom,
    ( map_ad2908576761495157028ar_val
    = ( ^ [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char] : ( case_o7794706883426356101al_val @ ( M1 @ X ) @ some_val @ ( M2 @ X ) ) ) ) ).

% map_add_def
thf(fact_676_case__optionE,axiom,
    ! [P4: $o,Q: state > $o,X2: option_state] :
      ( ( case_option_o_state @ P4 @ Q @ X2 )
     => ( ( ( X2 = none_state )
         => ~ P4 )
       => ~ ! [Y4: state] :
              ( ( X2
                = ( some_state @ Y4 ) )
             => ~ ( Q @ Y4 ) ) ) ) ).

% case_optionE
thf(fact_677_case__optionE,axiom,
    ! [P4: $o,Q: val > $o,X2: option_val] :
      ( ( case_option_o_val @ P4 @ Q @ X2 )
     => ( ( ( X2 = none_val )
         => ~ P4 )
       => ~ ! [Y4: val] :
              ( ( X2
                = ( some_val @ Y4 ) )
             => ~ ( Q @ Y4 ) ) ) ) ).

% case_optionE
thf(fact_678_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
      = ( case_option_o_state @ $false
        @ ^ [Uu3: state] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_679_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
      = ( case_option_o_val @ $false
        @ ^ [Uu3: val] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_680_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_state] :
      ( ( Option = none_state )
      = ( case_option_o_state @ $true
        @ ^ [Uu3: state] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_681_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_val] :
      ( ( Option = none_val )
      = ( case_option_o_val @ $true
        @ ^ [Uu3: val] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_682_sep__algebra_Ogreater__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ A @ B )
        = ( ? [C4: val] :
              ( ( some_val @ A )
              = ( Plus @ B @ C4 ) ) ) ) ) ).

% sep_algebra.greater_def
thf(fact_683_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 )
        = ( ? [C4: state] :
              ( ( some_state @ A )
              = ( Plus @ B @ C4 ) ) ) ) ) ).

% sep_algebra.greater_def
thf(fact_684_sep__algebra_Oextract__core,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val,X2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( some_val @ B )
            = ( Plus @ A @ X2 ) )
          & ( sep_greater_val @ Plus @ X2 @ ( Core @ B ) ) )
       => ( ( Core @ X2 )
          = ( Core @ B ) ) ) ) ).

% sep_algebra.extract_core
thf(fact_685_sep__algebra_Oextract__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( some_state @ B )
            = ( Plus @ A @ X2 ) )
          & ( sep_greater_state @ Plus @ X2 @ ( Core @ B ) ) )
       => ( ( Core @ X2 )
          = ( Core @ B ) ) ) ) ).

% sep_algebra.extract_core
thf(fact_686_sep__algebra_Ominus__exists,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ B @ A )
       => ? [X3: val] :
            ( ( ( some_val @ B )
              = ( Plus @ A @ X3 ) )
            & ( sep_greater_val @ Plus @ X3 @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.minus_exists
thf(fact_687_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 )
       => ? [X3: state] :
            ( ( ( some_state @ B )
              = ( Plus @ A @ X3 ) )
            & ( sep_greater_state @ Plus @ X3 @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.minus_exists
thf(fact_688_sep__algebra_Ominus__unique,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val,X2: val,Y: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( some_val @ B )
            = ( Plus @ A @ X2 ) )
          & ( sep_greater_val @ Plus @ X2 @ ( Core @ B ) ) )
       => ( ( ( ( some_val @ B )
              = ( Plus @ A @ Y ) )
            & ( sep_greater_val @ Plus @ Y @ ( Core @ B ) ) )
         => ( X2 = Y ) ) ) ) ).

% sep_algebra.minus_unique
thf(fact_689_sep__algebra_Ominus__unique,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X2: state,Y: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( some_state @ B )
            = ( Plus @ A @ X2 ) )
          & ( sep_greater_state @ Plus @ X2 @ ( Core @ B ) ) )
       => ( ( ( ( some_state @ B )
              = ( Plus @ A @ Y ) )
            & ( sep_greater_state @ Plus @ Y @ ( Core @ B ) ) )
         => ( X2 = Y ) ) ) ) ).

% sep_algebra.minus_unique
thf(fact_690_sep__algebra_Ogreater__equiv,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ A @ B )
        = ( ? [C4: val] :
              ( ( some_val @ A )
              = ( Plus @ C4 @ B ) ) ) ) ) ).

% sep_algebra.greater_equiv
thf(fact_691_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 )
        = ( ? [C4: state] :
              ( ( some_state @ A )
              = ( Plus @ C4 @ B ) ) ) ) ) ).

% sep_algebra.greater_equiv
thf(fact_692_sep__algebra_Oaddition__bigger,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A4: val,A: val,X5: val,B: val,X2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ A4 @ A )
       => ( ( ( some_val @ X5 )
            = ( Plus @ A4 @ B ) )
         => ( ( ( some_val @ X2 )
              = ( Plus @ A @ B ) )
           => ( sep_greater_val @ Plus @ X5 @ X2 ) ) ) ) ) ).

% sep_algebra.addition_bigger
thf(fact_693_sep__algebra_Oaddition__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: state,A: state,X5: state,B: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A4 @ A )
       => ( ( ( some_state @ X5 )
            = ( Plus @ A4 @ B ) )
         => ( ( ( some_state @ X2 )
              = ( Plus @ A @ B ) )
           => ( sep_greater_state @ Plus @ X5 @ X2 ) ) ) ) ) ).

% sep_algebra.addition_bigger
thf(fact_694_sep__algebra_Osmaller__than__core,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Y: val,X2: val,Z: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ Y @ X2 )
       => ( ( ( some_val @ Z )
            = ( Plus @ X2 @ ( Core @ Y ) ) )
         => ( ( Core @ Z )
            = ( Core @ Y ) ) ) ) ) ).

% sep_algebra.smaller_than_core
thf(fact_695_sep__algebra_Osmaller__than__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,X2: state,Z: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ X2 )
       => ( ( ( some_state @ Z )
            = ( Plus @ X2 @ ( Core @ Y ) ) )
         => ( ( Core @ Z )
            = ( Core @ Y ) ) ) ) ) ).

% sep_algebra.smaller_than_core
thf(fact_696_sep__algebra_Obigger__sum__smaller,axiom,
    ! [Plus: val > val > option_val,Core: val > val,C: val,A: val,B: val,A4: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ C )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_val @ Plus @ A @ A4 )
         => ? [B7: val] :
              ( ( sep_greater_val @ Plus @ B7 @ B )
              & ( ( some_val @ C )
                = ( Plus @ A4 @ B7 ) ) ) ) ) ) ).

% sep_algebra.bigger_sum_smaller
thf(fact_697_sep__algebra_Obigger__sum__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,C: state,A: state,B: state,A4: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ C )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_state @ Plus @ A @ A4 )
         => ? [B7: state] :
              ( ( sep_greater_state @ Plus @ B7 @ B )
              & ( ( some_state @ C )
                = ( Plus @ A4 @ B7 ) ) ) ) ) ) ).

% sep_algebra.bigger_sum_smaller
thf(fact_698_greater__def,axiom,
    ( greater
    = ( sep_greater_state @ plus ) ) ).

% greater_def
thf(fact_699_package__logic_Ogreater__than__sum__exists,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val,B1: val,B22: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ A @ B )
       => ( ( ( some_val @ B )
            = ( Plus @ B1 @ B22 ) )
         => ? [R4: val] :
              ( ( ( some_val @ A )
                = ( Plus @ R4 @ B22 ) )
              & ( sep_greater_val @ Plus @ ( Core @ R4 ) @ ( Core @ A ) )
              & ( sep_greater_val @ Plus @ R4 @ B1 ) ) ) ) ) ).

% package_logic.greater_than_sum_exists
thf(fact_700_package__logic_Ogreater__than__sum__exists,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state,B1: state,B22: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A @ B )
       => ( ( ( some_state @ B )
            = ( Plus @ B1 @ B22 ) )
         => ? [R4: state] :
              ( ( ( some_state @ A )
                = ( Plus @ R4 @ B22 ) )
              & ( sep_greater_state @ Plus @ ( Core @ R4 ) @ ( Core @ A ) )
              & ( sep_greater_state @ Plus @ R4 @ B1 ) ) ) ) ) ).

% package_logic.greater_than_sum_exists
thf(fact_701_package__logic_Obigger__core__sum__defined,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ ( Core @ A ) @ B )
       => ( ( some_val @ A )
          = ( Plus @ A @ B ) ) ) ) ).

% package_logic.bigger_core_sum_defined
thf(fact_702_package__logic_Obigger__core__sum__defined,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ ( Core @ A ) @ B )
       => ( ( some_state @ A )
          = ( Plus @ A @ B ) ) ) ) ).

% package_logic.bigger_core_sum_defined
thf(fact_703_package__logic_Obigger__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi2: val,A: val,B: val,Phi3: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_val @ Phi2 )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_val @ Plus @ Phi3 @ Phi2 )
         => ? [B7: val] :
              ( ( sep_greater_val @ Plus @ B7 @ B )
              & ( ( some_val @ Phi3 )
                = ( Plus @ A @ B7 ) ) ) ) ) ) ).

% package_logic.bigger_sum
thf(fact_704_package__logic_Obigger__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi2: state,A: state,B: state,Phi3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ Phi2 )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_state @ Plus @ Phi3 @ Phi2 )
         => ? [B7: state] :
              ( ( sep_greater_state @ Plus @ B7 @ B )
              & ( ( some_state @ Phi3 )
                = ( Plus @ A @ B7 ) ) ) ) ) ) ).

% package_logic.bigger_sum
thf(fact_705_PartialSA_Osat_Osimps_I2_J,axiom,
    ! [B: state > $o,Aa2: aassertion_state,Phi2: state] :
      ( ( package_sat_state @ plus @ ( aImp_state @ B @ Aa2 ) @ Phi2 )
      = ( ( B @ Phi2 )
       => ( package_sat_state @ plus @ Aa2 @ Phi2 ) ) ) ).

% PartialSA.sat.simps(2)
thf(fact_706_package__logic_Obigger__the,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,X5: val,Y: val,X2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_val @ A )
          = ( Plus @ X5 @ Y ) )
       => ( ( sep_greater_val @ Plus @ X5 @ X2 )
         => ( sep_greater_val @ Plus @ ( the_val @ ( Plus @ ( Core @ A ) @ X5 ) ) @ ( the_val @ ( Plus @ ( Core @ A ) @ X2 ) ) ) ) ) ) ).

% package_logic.bigger_the
thf(fact_707_package__logic_Obigger__the,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,X5: state,Y: state,X2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ A )
          = ( Plus @ X5 @ Y ) )
       => ( ( sep_greater_state @ Plus @ X5 @ X2 )
         => ( sep_greater_state @ Plus @ ( the_state @ ( Plus @ ( Core @ A ) @ X5 ) ) @ ( the_state @ ( Plus @ ( Core @ A ) @ X2 ) ) ) ) ) ) ).

% package_logic.bigger_the
thf(fact_708_sep__algebra_Ominus__equiv__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ B @ A )
       => ( ( ( some_val @ B )
            = ( Plus @ A @ ( sep_minus_val @ Plus @ Core @ B @ A ) ) )
          & ( sep_greater_val @ Plus @ ( sep_minus_val @ Plus @ Core @ B @ A ) @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.minus_equiv_def
thf(fact_709_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_710_sep__algebra_Ominus__and__plus,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Omega: val,Omega2: val,R3: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ Omega )
          = ( Plus @ Omega2 @ R3 ) )
       => ( ( sep_greater_val @ Plus @ Omega2 @ A )
         => ( ( some_val @ ( sep_minus_val @ Plus @ Core @ Omega @ A ) )
            = ( Plus @ ( sep_minus_val @ Plus @ Core @ Omega2 @ A ) @ R3 ) ) ) ) ) ).

% sep_algebra.minus_and_plus
thf(fact_711_sep__algebra_Ominus__and__plus,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Omega: state,Omega2: state,R3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ Omega )
          = ( Plus @ Omega2 @ R3 ) )
       => ( ( sep_greater_state @ Plus @ Omega2 @ A )
         => ( ( some_state @ ( sep_minus_state @ Plus @ Core @ Omega @ A ) )
            = ( Plus @ ( sep_minus_state @ Plus @ Core @ Omega2 @ A ) @ R3 ) ) ) ) ) ).

% sep_algebra.minus_and_plus
thf(fact_712_sep__algebra_Ominus__bigger,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X2: val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ X2 )
          = ( Plus @ A @ B ) )
       => ( sep_greater_val @ Plus @ ( sep_minus_val @ Plus @ Core @ X2 @ A ) @ B ) ) ) ).

% sep_algebra.minus_bigger
thf(fact_713_sep__algebra_Ominus__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X2 )
          = ( Plus @ A @ B ) )
       => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ X2 @ A ) @ B ) ) ) ).

% sep_algebra.minus_bigger
thf(fact_714_sep__algebra_Ominus__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val,X2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ A )
          = ( Plus @ B @ C ) )
       => ( ( sep_greater_val @ Plus @ X2 @ A )
         => ( ( sep_minus_val @ Plus @ Core @ X2 @ A )
            = ( sep_minus_val @ Plus @ Core @ ( sep_minus_val @ Plus @ Core @ X2 @ B ) @ C ) ) ) ) ) ).

% sep_algebra.minus_sum
thf(fact_715_sep__algebra_Ominus__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B @ C ) )
       => ( ( sep_greater_state @ Plus @ X2 @ A )
         => ( ( sep_minus_state @ Plus @ Core @ X2 @ A )
            = ( sep_minus_state @ Plus @ Core @ ( sep_minus_state @ Plus @ Core @ X2 @ B ) @ C ) ) ) ) ) ).

% sep_algebra.minus_sum
thf(fact_716_sep__algebra_OminusI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val,X2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ B )
          = ( Plus @ A @ X2 ) )
       => ( ( sep_greater_val @ Plus @ X2 @ ( Core @ B ) )
         => ( X2
            = ( sep_minus_val @ Plus @ Core @ B @ A ) ) ) ) ) ).

% sep_algebra.minusI
thf(fact_717_sep__algebra_OminusI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ B )
          = ( Plus @ A @ X2 ) )
       => ( ( sep_greater_state @ Plus @ X2 @ ( Core @ B ) )
         => ( X2
            = ( sep_minus_state @ Plus @ Core @ B @ A ) ) ) ) ) ).

% sep_algebra.minusI
thf(fact_718_package__logic_Oprove__last__completeness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A4: val,A: val,Nf1: val,F22: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ A4 @ A )
       => ( ( ( some_val @ A )
            = ( Plus @ Nf1 @ F22 ) )
         => ( sep_greater_val @ Plus @ ( sep_minus_val @ Plus @ Core @ A4 @ Nf1 ) @ F22 ) ) ) ) ).

% package_logic.prove_last_completeness
thf(fact_719_package__logic_Oprove__last__completeness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A4: state,A: state,Nf1: state,F22: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A4 @ A )
       => ( ( ( some_state @ A )
            = ( Plus @ Nf1 @ F22 ) )
         => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ A4 @ Nf1 ) @ F22 ) ) ) ) ).

% package_logic.prove_last_completeness
thf(fact_720_package__logic_Ominus__some,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ A @ B )
       => ( ( some_val @ A )
          = ( Plus @ B @ ( sep_minus_val @ Plus @ Core @ A @ B ) ) ) ) ) ).

% package_logic.minus_some
thf(fact_721_package__logic_Ominus__some,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A @ B )
       => ( ( some_state @ A )
          = ( Plus @ B @ ( sep_minus_state @ Plus @ Core @ A @ B ) ) ) ) ) ).

% package_logic.minus_some
thf(fact_722_sep__algebra_Osmaller__pure__sum__smaller,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Y: val,A: val,B: val,X2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ Y @ A )
       => ( ( sep_greater_val @ Plus @ Y @ B )
         => ( ( ( some_val @ X2 )
              = ( Plus @ A @ B ) )
           => ( ( sep_pure_val @ Plus @ B )
             => ( sep_greater_val @ Plus @ Y @ X2 ) ) ) ) ) ) ).

% sep_algebra.smaller_pure_sum_smaller
thf(fact_723_sep__algebra_Osmaller__pure__sum__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,A: state,B: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ A )
       => ( ( sep_greater_state @ Plus @ Y @ B )
         => ( ( ( some_state @ X2 )
              = ( Plus @ A @ B ) )
           => ( ( sep_pure_state @ Plus @ B )
             => ( sep_greater_state @ Plus @ Y @ X2 ) ) ) ) ) ) ).

% sep_algebra.smaller_pure_sum_smaller
thf(fact_724_package__logic_Ogeneral__completeness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,S2: set_Pr4723104092121316667al_val,F4: val,A3: aassertion_val,Phi2: val,Phi3: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: val,U3: val,T4: val > val] :
            ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A5 @ ( produc303288909735261391al_val @ U3 @ T4 ) ) @ S2 )
           => ! [X3: val] :
                ( ( ( some_val @ X3 )
                  = ( Plus @ A5 @ ( T4 @ F4 ) ) )
               => ( package_sat_val @ Plus @ A3 @ X3 ) ) )
       => ( ( ( some_val @ Phi2 )
            = ( Plus @ Phi3 @ F4 ) )
         => ( ( Stable @ F4 )
           => ( ( packag7517340229427718612et_val @ Plus @ Core @ Unit @ S2 @ Unit )
             => ( ( packag5732365195153136185on_val @ Plus @ Core @ A3 )
               => ? [X_1: set_Pr4723104092121316667al_val] :
                    ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi2 @ Unit @ S2
                    @ ^ [Uu3: val] : $true
                    @ A3
                    @ Phi3
                    @ F4
                    @ X_1 ) ) ) ) ) ) ) ).

% package_logic.general_completeness
thf(fact_725_package__logic_Ogeneral__completeness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F4: state,A3: aassertion_state,Phi2: state,Phi3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: state,U3: state,T4: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
           => ! [X3: state] :
                ( ( ( some_state @ X3 )
                  = ( Plus @ A5 @ ( T4 @ F4 ) ) )
               => ( package_sat_state @ Plus @ A3 @ X3 ) ) )
       => ( ( ( some_state @ Phi2 )
            = ( Plus @ Phi3 @ F4 ) )
         => ( ( Stable @ F4 )
           => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ Unit )
             => ( ( packag7298478212905792770_state @ Plus @ Core @ A3 )
               => ? [X_1: set_Pr4231240917283394591_state] :
                    ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ Unit @ S2
                    @ ^ [Uu3: state] : $true
                    @ A3
                    @ Phi3
                    @ F4
                    @ X_1 ) ) ) ) ) ) ) ).

% package_logic.general_completeness
thf(fact_726_PartialSA_Owf__assertion_Osimps_I2_J,axiom,
    ! [B: state > $o,Aa2: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aImp_state @ B @ Aa2 ) )
      = ( ( packag6595153354952283300_state @ plus @ core @ B )
        & ( packag7298478212905792770_state @ plus @ core @ Aa2 ) ) ) ).

% PartialSA.wf_assertion.simps(2)
thf(fact_727_PartialSA_Ocompleteness,axiom,
    ! [B3: aassertion_state,W: state,R2: state > state > state,A3: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B3 )
     => ( ( ( stable @ W )
          & ( packag6192661653625646554_state @ plus @ W @ R2 @ A3 @ B3 ) )
       => ( ( ( some_state @ Sigma )
            = ( plus @ Sigma2 @ W ) )
         => ( ! [A5: state] :
                ( ( package_sat_state @ plus @ A3 @ A5 )
               => ( packag4817583915586052229_state @ plus @ unit @ ( R2 @ A5 ) ) )
           => ? [X_1: set_Pr4231240917283394591_state] :
                ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ ( R2 @ A2 ) ) ) )
                      & ( package_sat_state @ plus @ A3 @ A2 ) ) )
                @ ^ [Uu3: state] : $true
                @ B3
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% PartialSA.completeness
thf(fact_728_package__logic_Opackage__rhs__connectionI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi2: val,F4: val,Phi3: val,F5: val,S2: set_Pr4723104092121316667al_val,S3: set_Pr4723104092121316667al_val,Pc: val > $o,A3: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Plus @ Phi2 @ F4 )
          = ( Plus @ Phi3 @ F5 ) )
       => ( ( Stable @ F5 )
         => ( ( sep_defined_val @ Plus @ Phi2 @ F4 )
           => ( ( sep_greater_val @ Plus @ F5 @ F4 )
             => ( ! [A5: val,U3: val,T4: val > val] :
                    ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A5 @ ( produc303288909735261391al_val @ U3 @ T4 ) ) @ S2 )
                   => ? [Au2: val] :
                        ( ( ( some_val @ Au2 )
                          = ( Plus @ A5 @ U3 ) )
                        & ( ( sep_defined_val @ Plus @ Au2 @ ( sep_minus_val @ Plus @ Core @ ( T4 @ F5 ) @ ( T4 @ F4 ) ) )
                         => ? [A8: val,U4: val] :
                              ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A8 @ ( produc303288909735261391al_val @ U4 @ T4 ) ) @ S3 )
                              & ( sep_greater_val @ Plus @ ( Core @ A8 ) @ ( Core @ A5 ) )
                              & ( ( Plus @ Au2 @ ( sep_minus_val @ Plus @ Core @ ( T4 @ F5 ) @ ( T4 @ F4 ) ) )
                                = ( Plus @ A8 @ U4 ) )
                              & ( sep_greater_val @ Plus @ U4 @ U3 )
                              & ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A3 @ A8 @ U4 @ U3 ) ) ) ) )
               => ( packag1944724486178758143on_val @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connectionI
thf(fact_729_package__logic_Opackage__rhs__connectionI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi2: state,F4: state,Phi3: state,F5: state,S2: set_Pr4231240917283394591_state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Plus @ Phi2 @ F4 )
          = ( Plus @ Phi3 @ F5 ) )
       => ( ( Stable @ F5 )
         => ( ( sep_defined_state @ Plus @ Phi2 @ F4 )
           => ( ( sep_greater_state @ Plus @ F5 @ F4 )
             => ( ! [A5: state,U3: state,T4: state > state] :
                    ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
                   => ? [Au2: state] :
                        ( ( ( some_state @ Au2 )
                          = ( Plus @ A5 @ U3 ) )
                        & ( ( sep_defined_state @ Plus @ Au2 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F5 ) @ ( T4 @ F4 ) ) )
                         => ? [A8: state,U4: state] :
                              ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A8 @ ( produc8632743443909064678_state @ U4 @ T4 ) ) @ S3 )
                              & ( sep_greater_state @ Plus @ ( Core @ A8 ) @ ( Core @ A5 ) )
                              & ( ( Plus @ Au2 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F5 ) @ ( T4 @ F4 ) ) )
                                = ( Plus @ A8 @ U4 ) )
                              & ( sep_greater_state @ Plus @ U4 @ U3 )
                              & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A3 @ A8 @ U4 @ U3 ) ) ) ) )
               => ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connectionI
thf(fact_730_package__logic_Opackage__rhs__connection__instantiate,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi2: val,F4: val,S2: set_Pr4723104092121316667al_val,Pc: val > $o,A3: aassertion_val,Phi3: val,F5: val,S3: set_Pr4723104092121316667al_val,A: val,U: val,T3: val > val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag1944724486178758143on_val @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
       => ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A @ ( produc303288909735261391al_val @ U @ T3 ) ) @ S2 )
         => ~ ! [Au: val] :
                ( ( ( some_val @ Au )
                  = ( Plus @ A @ U ) )
               => ~ ( ( sep_defined_val @ Plus @ Au @ ( sep_minus_val @ Plus @ Core @ ( T3 @ F5 ) @ ( T3 @ F4 ) ) )
                   => ? [A7: val,U2: val] :
                        ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A7 @ ( produc303288909735261391al_val @ U2 @ T3 ) ) @ S3 )
                        & ( sep_greater_val @ Plus @ ( Core @ A7 ) @ ( Core @ A ) )
                        & ( ( Plus @ Au @ ( sep_minus_val @ Plus @ Core @ ( T3 @ F5 ) @ ( T3 @ F4 ) ) )
                          = ( Plus @ A7 @ U2 ) )
                        & ( sep_greater_val @ Plus @ U2 @ U )
                        & ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A3 @ A7 @ U2 @ U ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connection_instantiate
thf(fact_731_package__logic_Opackage__rhs__connection__instantiate,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state,A: state,U: state,T3: state > state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
       => ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ~ ! [Au: state] :
                ( ( ( some_state @ Au )
                  = ( Plus @ A @ U ) )
               => ~ ( ( sep_defined_state @ Plus @ Au @ ( sep_minus_state @ Plus @ Core @ ( T3 @ F5 ) @ ( T3 @ F4 ) ) )
                   => ? [A7: state,U2: state] :
                        ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A7 @ ( produc8632743443909064678_state @ U2 @ T3 ) ) @ S3 )
                        & ( sep_greater_state @ Plus @ ( Core @ A7 ) @ ( Core @ A ) )
                        & ( ( Plus @ Au @ ( sep_minus_state @ Plus @ Core @ ( T3 @ F5 ) @ ( T3 @ F4 ) ) )
                          = ( Plus @ A7 @ U2 ) )
                        & ( sep_greater_state @ Plus @ U2 @ U )
                        & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A3 @ A7 @ U2 @ U ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connection_instantiate
thf(fact_732_package__logic_Ocompleteness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B3: aassertion_val,W: val,R2: val > val > val,A3: aassertion_val,Sigma: val,Sigma2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B3 )
       => ( ( ( Stable @ W )
            & ( packag4897621287243763553al_val @ Plus @ W @ R2 @ A3 @ B3 ) )
         => ( ( ( some_val @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ( ! [A5: val] :
                  ( ( package_sat_val @ Plus @ A3 @ A5 )
                 => ( packag4175576287423129206er_val @ Plus @ Unit @ ( R2 @ A5 ) ) )
             => ? [X_1: set_Pr4723104092121316667al_val] :
                  ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
                  @ ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A2: val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ Unit @ ( R2 @ A2 ) ) ) )
                        & ( package_sat_val @ Plus @ A3 @ A2 ) ) )
                  @ ^ [Uu3: val] : $true
                  @ B3
                  @ Sigma2
                  @ W
                  @ X_1 ) ) ) ) ) ) ).

% package_logic.completeness
thf(fact_733_package__logic_Ocompleteness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: aassertion_state,W: state,R2: state > state > state,A3: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B3 )
       => ( ( ( Stable @ W )
            & ( packag6192661653625646554_state @ Plus @ W @ R2 @ A3 @ B3 ) )
         => ( ( ( some_state @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ( ! [A5: state] :
                  ( ( package_sat_state @ Plus @ A3 @ A5 )
                 => ( packag4817583915586052229_state @ Plus @ Unit @ ( R2 @ A5 ) ) )
             => ? [X_1: set_Pr4231240917283394591_state] :
                  ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ ( R2 @ A2 ) ) ) )
                        & ( package_sat_state @ Plus @ A3 @ A2 ) ) )
                  @ ^ [Uu3: state] : $true
                  @ B3
                  @ Sigma2
                  @ W
                  @ X_1 ) ) ) ) ) ) ).

% package_logic.completeness
thf(fact_734_package__logic_OAddFromOutside,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi2: val,Phi3: val,M: val,F5: val,S3: set_Pr4723104092121316667al_val,Pc: val > $o,A3: aassertion_val,Phi7: val,F6: val,S5: set_Pr4723104092121316667al_val,F4: val,S2: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_val @ Phi2 )
          = ( Plus @ Phi3 @ M ) )
       => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi3 @ F5 @ S3 @ Pc @ A3 @ Phi7 @ F6 @ S5 )
         => ( ( Stable @ M )
           => ( ( ( some_val @ F5 )
                = ( Plus @ F4 @ M ) )
             => ( ( S3
                  = ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A2: val,U6: val,T5: val > val,R: val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ R @ ( produc303288909735261391al_val @ U6 @ T5 ) ) )
                        & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ S2 )
                        & ( ( some_val @ R )
                          = ( Plus @ A2 @ ( sep_minus_val @ Plus @ Core @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) ) )
                        & ( sep_defined_val @ Plus @ R @ U6 ) ) ) )
               => ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi7 @ F6 @ S5 ) ) ) ) ) ) ) ).

% package_logic.AddFromOutside
thf(fact_735_package__logic_OAddFromOutside,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi2: state,Phi3: state,M: state,F5: state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi7: state,F6: state,S5: set_Pr4231240917283394591_state,F4: state,S2: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ Phi2 )
          = ( Plus @ Phi3 @ M ) )
       => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi3 @ F5 @ S3 @ Pc @ A3 @ Phi7 @ F6 @ S5 )
         => ( ( Stable @ M )
           => ( ( ( some_state @ F5 )
                = ( Plus @ F4 @ M ) )
             => ( ( S3
                  = ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state,U6: state,T5: state > state,R: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ R @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                        & ( ( some_state @ R )
                          = ( Plus @ A2 @ ( sep_minus_state @ Plus @ Core @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) ) )
                        & ( sep_defined_state @ Plus @ R @ U6 ) ) ) )
               => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi7 @ F6 @ S5 ) ) ) ) ) ) ) ).

% package_logic.AddFromOutside
thf(fact_736_Rep__state__induct,axiom,
    ! [Y: produc9003781588162623004on_val,P4: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi4: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi4 )
              & ( valid_state @ Phi4 ) ) ) )
     => ( ! [X3: state] : ( P4 @ ( rep_state @ X3 ) )
       => ( P4 @ Y ) ) ) ).

% Rep_state_induct
thf(fact_737_Rep__state__cases,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y @ ( collec7475294502966882545on_val @ valid_state ) )
     => ~ ! [X3: state] :
            ( Y
           != ( rep_state @ X3 ) ) ) ).

% Rep_state_cases
thf(fact_738_Rep__state,axiom,
    ! [X2: state] :
      ( member2879030624002546611on_val @ ( rep_state @ X2 )
      @ ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [Phi4: produc9003781588162623004on_val] :
            ( ( Uu3 = Phi4 )
            & ( valid_state @ Phi4 ) ) ) ) ).

% Rep_state
thf(fact_739_Abs__state__cases,axiom,
    ! [X2: state] :
      ~ ! [Y4: produc9003781588162623004on_val] :
          ( ( X2
            = ( abs_state @ Y4 ) )
         => ~ ( member2879030624002546611on_val @ Y4 @ ( collec7475294502966882545on_val @ valid_state ) ) ) ).

% Abs_state_cases
thf(fact_740_Abs__state__induct,axiom,
    ! [P4: state > $o,X2: state] :
      ( ! [Y4: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ Y4
            @ ( collec7475294502966882545on_val
              @ ^ [Uu3: produc9003781588162623004on_val] :
                ? [Phi4: produc9003781588162623004on_val] :
                  ( ( Uu3 = Phi4 )
                  & ( valid_state @ Phi4 ) ) ) )
         => ( P4 @ ( abs_state @ Y4 ) ) )
     => ( P4 @ X2 ) ) ).

% Abs_state_induct
thf(fact_741_Abs__state__inject,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ X2
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi4: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi4 )
              & ( valid_state @ Phi4 ) ) ) )
     => ( ( member2879030624002546611on_val @ Y
          @ ( collec7475294502966882545on_val
            @ ^ [Uu3: produc9003781588162623004on_val] :
              ? [Phi4: produc9003781588162623004on_val] :
                ( ( Uu3 = Phi4 )
                & ( valid_state @ Phi4 ) ) ) )
       => ( ( ( abs_state @ X2 )
            = ( abs_state @ Y ) )
          = ( X2 = Y ) ) ) ) ).

% Abs_state_inject
thf(fact_742_sep__algebra_Odefined__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_defined_val @ Plus @ A @ B )
        = ( ( Plus @ A @ B )
         != none_val ) ) ) ).

% sep_algebra.defined_def
thf(fact_743_sep__algebra_Odefined__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_defined_state @ Plus @ A @ B )
        = ( ( Plus @ A @ B )
         != none_state ) ) ) ).

% sep_algebra.defined_def
thf(fact_744_package__logic_Odefined__sum__move,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val,X2: val,Y: val,A4: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_defined_val @ Plus @ A @ B )
       => ( ( ( some_val @ B )
            = ( Plus @ X2 @ Y ) )
         => ( ( ( some_val @ A4 )
              = ( Plus @ A @ X2 ) )
           => ( sep_defined_val @ Plus @ A4 @ Y ) ) ) ) ) ).

% package_logic.defined_sum_move
thf(fact_745_package__logic_Odefined__sum__move,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state,X2: state,Y: state,A4: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_defined_state @ Plus @ A @ B )
       => ( ( ( some_state @ B )
            = ( Plus @ X2 @ Y ) )
         => ( ( ( some_state @ A4 )
              = ( Plus @ A @ X2 ) )
           => ( sep_defined_state @ Plus @ A4 @ Y ) ) ) ) ) ).

% package_logic.defined_sum_move
thf(fact_746_defined__def,axiom,
    ( defined
    = ( sep_defined_state @ plus ) ) ).

% defined_def
thf(fact_747_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A3: set_Pr2391701600042252046_state,B3: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_ad8203089552910907580_state @ Plus @ A3 @ B3 )
        = ( collec394671971739345689_state
          @ ^ [Uu3: produc3882902684156697774_state] :
            ? [Phi4: produc3882902684156697774_state,A2: produc3882902684156697774_state,B2: produc3882902684156697774_state] :
              ( ( Uu3 = Phi4 )
              & ( member6715010751053357271_state @ A2 @ A3 )
              & ( member6715010751053357271_state @ B2 @ B3 )
              & ( ( some_P5383513176685068947_state @ Phi4 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_748_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A3: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_ad3908482799505143387_state @ Plus @ A3 @ B3 )
        = ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [Phi4: produc7272280185086996969_state,A2: produc7272280185086996969_state,B2: produc7272280185086996969_state] :
              ( ( Uu3 = Phi4 )
              & ( member7461423308196221312_state @ A2 @ A3 )
              & ( member7461423308196221312_state @ B2 @ B3 )
              & ( ( some_P6297086604321900740_state @ Phi4 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_749_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A3: set_Pr7649655576376376530on_val,B3: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_ad8320153009520438926on_val @ Plus @ A3 @ B3 )
        = ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi4: produc9003781588162623004on_val,A2: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi4 )
              & ( member2879030624002546611on_val @ A2 @ A3 )
              & ( member2879030624002546611on_val @ B2 @ B3 )
              & ( ( some_P2581650809551911671on_val @ Phi4 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_750_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A3: set_val,B3: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_add_set_val @ Plus @ A3 @ B3 )
        = ( collect_val
          @ ^ [Uu3: val] :
            ? [Phi4: val,A2: val,B2: val] :
              ( ( Uu3 = Phi4 )
              & ( member_val @ A2 @ A3 )
              & ( member_val @ B2 @ B3 )
              & ( ( some_val @ Phi4 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_751_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ A3 @ B3 )
        = ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi4: state,A2: state,B2: state] :
              ( ( Uu3 = Phi4 )
              & ( member_state @ A2 @ A3 )
              & ( member_state @ B2 @ B3 )
              & ( ( some_state @ Phi4 )
                = ( Plus @ A2 @ B2 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_752_PartialSA_Oadd__set__def,axiom,
    ( add_set
    = ( ^ [A6: set_state,B6: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi4: state,A2: state,B2: state] :
              ( ( Uu3 = Phi4 )
              & ( member_state @ A2 @ A6 )
              & ( member_state @ B2 @ B6 )
              & ( ( some_state @ Phi4 )
                = ( plus @ A2 @ B2 ) ) ) ) ) ) ).

% PartialSA.add_set_def
thf(fact_753_Abs__state__inverse,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi4: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi4 )
              & ( valid_state @ Phi4 ) ) ) )
     => ( ( rep_state @ ( abs_state @ Y ) )
        = Y ) ) ).

% Abs_state_inverse
thf(fact_754_PartialSA_Oupper__closure__def,axiom,
    ! [A3: set_state] :
      ( ( sep_up1246176804924251236_state @ plus @ A3 )
      = ( collect_state
        @ ^ [Uu3: state] :
          ? [Phi5: state,Phi4: state] :
            ( ( Uu3 = Phi5 )
            & ( greater @ Phi5 @ Phi4 )
            & ( member_state @ Phi4 @ A3 ) ) ) ) ).

% PartialSA.upper_closure_def
thf(fact_755_PartialSA_OAddFromOutside,axiom,
    ! [Phi2: state,Phi3: state,M: state,F5: state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi7: state,F6: state,S5: set_Pr4231240917283394591_state,F4: state,S2: set_Pr4231240917283394591_state] :
      ( ( ( some_state @ Phi2 )
        = ( plus @ Phi3 @ M ) )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F5 @ S3 @ Pc @ A3 @ Phi7 @ F6 @ S5 )
       => ( ( stable @ M )
         => ( ( ( some_state @ F5 )
              = ( plus @ F4 @ M ) )
           => ( ( S3
                = ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T5: state > state,R: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ R @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                      & ( ( some_state @ R )
                        = ( plus @ A2 @ ( minus @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) ) )
                      & ( defined @ R @ U6 ) ) ) )
             => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi7 @ F6 @ S5 ) ) ) ) ) ) ).

% PartialSA.AddFromOutside
thf(fact_756_scaled__def,axiom,
    ( scaled
    = ( ^ [Phi4: state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [P3: prat] :
              ( ( Uu3
                = ( multiply @ P3 @ Phi4 ) )
              & ( ppos @ P3 )
              & ( pgte @ pwrite @ P3 ) ) ) ) ) ).

% scaled_def
thf(fact_757_wand__equiv__def,axiom,
    ( wand
    = ( ^ [A6: set_state,B6: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi4: state] :
              ( ( Uu3 = Phi4 )
              & ( ord_le2494988322063910608_state @ ( add_set @ A6 @ ( insert_state @ Phi4 @ bot_bot_set_state ) ) @ B6 ) ) ) ) ) ).

% wand_equiv_def
thf(fact_758_package__logic_Ovalid__package__setI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F4: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: state,U3: state,T4: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
           => ( ( sep_defined_state @ Plus @ A5 @ U3 )
              & ( sep_greater_state @ Plus @ ( Core @ A5 ) @ ( Core @ U3 ) )
              & ( packag4817583915586052229_state @ Plus @ Unit @ T4 )
              & ( sep_greater_state @ Plus @ A5 @ ( Core @ ( T4 @ F4 ) ) ) ) )
       => ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F4 ) ) ) ).

% package_logic.valid_package_setI
thf(fact_759_PartialSA_Ocompleteness__paper,axiom,
    ! [B3: aassertion_state,W: state,A3: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B3 )
     => ( ( ( stable @ W )
          & ( packag7096211993527611489_state @ plus @ W @ A3 @ B3 ) )
       => ( ( ( some_state @ Sigma )
            = ( plus @ Sigma2 @ W ) )
         => ? [X_1: set_Pr4231240917283394591_state] :
              ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
                    & ( package_sat_state @ plus @ A3 @ A2 ) ) )
              @ ^ [Uu3: state] : $true
              @ B3
              @ Sigma2
              @ W
              @ X_1 ) ) ) ) ).

% PartialSA.completeness_paper
thf(fact_760_PartialSA_Osoundness,axiom,
    ! [B3: aassertion_state,A3: aassertion_state,S2: set_state,R2: state > state > state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B3 )
     => ( ! [A5: state] :
            ( ( package_sat_state @ plus @ A3 @ A5 )
           => ( member_state @ A5 @ S2 ) )
       => ( ! [A5: state] :
              ( ( member_state @ A5 @ S2 )
             => ( packag4817583915586052229_state @ plus @ unit @ ( R2 @ A5 ) ) )
         => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ ( R2 @ A2 ) ) ) )
                    & ( member_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: state] : $true
              @ B3
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ B3 ) )
                | ( packag4152378406607043437_state @ plus @ S3 ) )
             => ( ( stable @ W )
                & ( ( some_state @ Sigma )
                  = ( plus @ Sigma2 @ W ) )
                & ( packag6192661653625646554_state @ plus @ W @ R2 @ A3 @ B3 ) ) ) ) ) ) ) ).

% PartialSA.soundness
thf(fact_761_package__logic_Osoundness,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,Unit: produc7272280185086996969_state,Stable: produc7272280185086996969_state > $o,B3: aasser7019605857572776418_state,A3: aasser7019605857572776418_state,S2: set_Pr4231240917283394591_state,R2: produc7272280185086996969_state > produc7272280185086996969_state > produc7272280185086996969_state,Sigma: produc7272280185086996969_state,Sigma2: produc7272280185086996969_state,W: produc7272280185086996969_state,S3: set_Pr3289333603297600445_state] :
      ( ( packag2204431621656667814_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6254535061403973169_state @ Plus @ Core @ B3 )
       => ( ! [A5: produc7272280185086996969_state] :
              ( ( packag4865960244449834244_state @ Plus @ A3 @ A5 )
             => ( member7461423308196221312_state @ A5 @ S2 ) )
         => ( ! [A5: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ A5 @ S2 )
               => ( packag3884151059541177780_state @ Plus @ Unit @ ( R2 @ A5 ) ) )
           => ( ( packag3630742828623380589_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec7677102015826254172_state
                  @ ^ [Uu3: produc3057782296003698823_state] :
                    ? [A2: produc7272280185086996969_state] :
                      ( ( Uu3
                        = ( produc1904032489304825401_state @ A2 @ ( produc7648782313880683123_state @ Unit @ ( R2 @ A2 ) ) ) )
                      & ( member7461423308196221312_state @ A2 @ S2 ) ) )
                @ ^ [Uu3: produc7272280185086996969_state] : $true
                @ B3
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag1160585701684337059_state @ Plus @ ( packag4865960244449834244_state @ Plus @ B3 ) )
                  | ( packag3875795242785447337_state @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_P6297086604321900740_state @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag7261830501500456713_state @ Plus @ W @ R2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_762_package__logic_Osoundness,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,Unit: produc9003781588162623004on_val,Stable: produc9003781588162623004on_val > $o,B3: aasser1163550960519286933on_val,A3: aasser1163550960519286933on_val,S2: set_Pr7649655576376376530on_val,R2: produc9003781588162623004on_val > produc9003781588162623004on_val > produc9003781588162623004on_val,Sigma: produc9003781588162623004on_val,Sigma2: produc9003781588162623004on_val,W: produc9003781588162623004on_val,S3: set_Pr1565700067222966883on_val] :
      ( ( packag4549880205214723289on_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5541956234368469604on_val @ Plus @ Core @ B3 )
       => ( ! [A5: produc9003781588162623004on_val] :
              ( ( packag3677137384183703351on_val @ Plus @ A3 @ A5 )
             => ( member2879030624002546611on_val @ A5 @ S2 ) )
         => ( ! [A5: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ A5 @ S2 )
               => ( packag5955012701910386663on_val @ Plus @ Unit @ ( R2 @ A5 ) ) )
           => ( ( packag6843733491101478048on_val @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec3787697688312515330on_val
                  @ ^ [Uu3: produc217507841372899885on_val] :
                    ? [A2: produc9003781588162623004on_val] :
                      ( ( Uu3
                        = ( produc6985948203474072671on_val @ A2 @ ( produc6949276712527400972on_val @ Unit @ ( R2 @ A2 ) ) ) )
                      & ( member2879030624002546611on_val @ A2 @ S2 ) ) )
                @ ^ [Uu3: produc9003781588162623004on_val] : $true
                @ B3
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag1465513459401564630on_val @ Plus @ ( packag3677137384183703351on_val @ Plus @ B3 ) )
                  | ( packag161099568566171125on_val @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_P2581650809551911671on_val @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag8228529196479615804on_val @ Plus @ W @ R2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_763_package__logic_Osoundness,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,Unit: produc3882902684156697774_state,Stable: produc3882902684156697774_state > $o,B3: aasser3179708585289803787_state,A3: aasser3179708585289803787_state,S2: set_Pr2391701600042252046_state,R2: produc3882902684156697774_state > produc3882902684156697774_state > produc3882902684156697774_state,Sigma: produc3882902684156697774_state,Sigma2: produc3882902684156697774_state,W: produc3882902684156697774_state,S3: set_Pr5308428015838290133_state] :
      ( ( packag9124413901382861105_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8517968027831007846_state @ Plus @ Core @ B3 )
       => ( ! [A5: produc3882902684156697774_state] :
              ( ( packag2518615827549365843_state @ Plus @ A3 @ A5 )
             => ( member6715010751053357271_state @ A5 @ S2 ) )
         => ( ! [A5: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ A5 @ S2 )
               => ( packag5827594511758291363_state @ Plus @ Unit @ ( R2 @ A5 ) ) )
           => ( ( packag4061699384143886122_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec6806307004801468532_state
                  @ ^ [Uu3: produc6455357576889117855_state] :
                    ? [A2: produc3882902684156697774_state] :
                      ( ( Uu3
                        = ( produc5427679348311974809_state @ A2 @ ( produc3987245858496418492_state @ Unit @ ( R2 @ A2 ) ) ) )
                      & ( member6715010751053357271_state @ A2 @ S2 ) ) )
                @ ^ [Uu3: produc3882902684156697774_state] : $true
                @ B3
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag1930333982449825652_state @ Plus @ ( packag2518615827549365843_state @ Plus @ B3 ) )
                  | ( packag1386629450763126729_state @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_P5383513176685068947_state @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag4438072966337169294_state @ Plus @ W @ R2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_764_package__logic_Osoundness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B3: aassertion_val,A3: aassertion_val,S2: set_val,R2: val > val > val,Sigma: val,Sigma2: val,W: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B3 )
       => ( ! [A5: val] :
              ( ( package_sat_val @ Plus @ A3 @ A5 )
             => ( member_val @ A5 @ S2 ) )
         => ( ! [A5: val] :
                ( ( member_val @ A5 @ S2 )
               => ( packag4175576287423129206er_val @ Plus @ Unit @ ( R2 @ A5 ) ) )
           => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec8300311316619386842al_val
                  @ ^ [Uu3: produc2072440392815185925al_val] :
                    ? [A2: val] :
                      ( ( Uu3
                        = ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ Unit @ ( R2 @ A2 ) ) ) )
                      & ( member_val @ A2 @ S2 ) ) )
                @ ^ [Uu3: val] : $true
                @ B3
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag2085742556458808647ic_val @ Plus @ ( package_sat_val @ Plus @ B3 ) )
                  | ( packag3202507263771882441al_val @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_val @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag4897621287243763553al_val @ Plus @ W @ R2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_765_package__logic_Osoundness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: aassertion_state,A3: aassertion_state,S2: set_state,R2: state > state > state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B3 )
       => ( ! [A5: state] :
              ( ( package_sat_state @ Plus @ A3 @ A5 )
             => ( member_state @ A5 @ S2 ) )
         => ( ! [A5: state] :
                ( ( member_state @ A5 @ S2 )
               => ( packag4817583915586052229_state @ Plus @ Unit @ ( R2 @ A5 ) ) )
           => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ ( R2 @ A2 ) ) ) )
                      & ( member_state @ A2 @ S2 ) ) )
                @ ^ [Uu3: state] : $true
                @ B3
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag8361946002163212404_state @ Plus @ ( package_sat_state @ Plus @ B3 ) )
                  | ( packag4152378406607043437_state @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_state @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag6192661653625646554_state @ Plus @ W @ R2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_766_PartialSA_Ominus__def,axiom,
    ( minus
    = ( ^ [B2: state,A2: state] :
          ( fun_TH8221542053827138557_state @ B2
          @ ^ [X: state] :
              ( ( ( some_state @ B2 )
                = ( plus @ A2 @ X ) )
              & ( greater @ X @ ( core @ B2 ) ) ) ) ) ) ).

% PartialSA.minus_def
thf(fact_767_PartialSA_Ointuitionistic__def,axiom,
    ! [A3: state > $o] :
      ( ( packag8361946002163212404_state @ plus @ A3 )
      = ( ! [Phi5: state,Phi4: state] :
            ( ( ( greater @ Phi5 @ Phi4 )
              & ( A3 @ Phi4 ) )
           => ( A3 @ Phi5 ) ) ) ) ).

% PartialSA.intuitionistic_def
thf(fact_768_package__logic_Osoundness__paper,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,Unit: produc7272280185086996969_state,Stable: produc7272280185086996969_state > $o,B3: aasser7019605857572776418_state,A3: aasser7019605857572776418_state,S2: set_Pr4231240917283394591_state,Sigma: produc7272280185086996969_state,Sigma2: produc7272280185086996969_state,W: produc7272280185086996969_state,S3: set_Pr3289333603297600445_state] :
      ( ( packag2204431621656667814_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6254535061403973169_state @ Plus @ Core @ B3 )
       => ( ! [A5: produc7272280185086996969_state] :
              ( ( packag4865960244449834244_state @ Plus @ A3 @ A5 )
             => ( member7461423308196221312_state @ A5 @ S2 ) )
         => ( ( packag3630742828623380589_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec7677102015826254172_state
                @ ^ [Uu3: produc3057782296003698823_state] :
                  ? [A2: produc7272280185086996969_state] :
                    ( ( Uu3
                      = ( produc1904032489304825401_state @ A2 @ ( produc7648782313880683123_state @ Unit @ id_Pro2933125508726022968_state ) ) )
                    & ( member7461423308196221312_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: produc7272280185086996969_state] : $true
              @ B3
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag1160585701684337059_state @ Plus @ ( packag4865960244449834244_state @ Plus @ B3 ) )
                | ( packag3875795242785447337_state @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_P6297086604321900740_state @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag8077625386405451152_state @ Plus @ W @ A3 @ B3 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_769_package__logic_Osoundness__paper,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,Unit: produc9003781588162623004on_val,Stable: produc9003781588162623004on_val > $o,B3: aasser1163550960519286933on_val,A3: aasser1163550960519286933on_val,S2: set_Pr7649655576376376530on_val,Sigma: produc9003781588162623004on_val,Sigma2: produc9003781588162623004on_val,W: produc9003781588162623004on_val,S3: set_Pr1565700067222966883on_val] :
      ( ( packag4549880205214723289on_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5541956234368469604on_val @ Plus @ Core @ B3 )
       => ( ! [A5: produc9003781588162623004on_val] :
              ( ( packag3677137384183703351on_val @ Plus @ A3 @ A5 )
             => ( member2879030624002546611on_val @ A5 @ S2 ) )
         => ( ( packag6843733491101478048on_val @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec3787697688312515330on_val
                @ ^ [Uu3: produc217507841372899885on_val] :
                  ? [A2: produc9003781588162623004on_val] :
                    ( ( Uu3
                      = ( produc6985948203474072671on_val @ A2 @ ( produc6949276712527400972on_val @ Unit @ id_Pro1402184692240358763on_val ) ) )
                    & ( member2879030624002546611on_val @ A2 @ S2 ) ) )
              @ ^ [Uu3: produc9003781588162623004on_val] : $true
              @ B3
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag1465513459401564630on_val @ Plus @ ( packag3677137384183703351on_val @ Plus @ B3 ) )
                | ( packag161099568566171125on_val @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_P2581650809551911671on_val @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag6258220590826782659on_val @ Plus @ W @ A3 @ B3 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_770_package__logic_Osoundness__paper,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,Unit: produc3882902684156697774_state,Stable: produc3882902684156697774_state > $o,B3: aasser3179708585289803787_state,A3: aasser3179708585289803787_state,S2: set_Pr2391701600042252046_state,Sigma: produc3882902684156697774_state,Sigma2: produc3882902684156697774_state,W: produc3882902684156697774_state,S3: set_Pr5308428015838290133_state] :
      ( ( packag9124413901382861105_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8517968027831007846_state @ Plus @ Core @ B3 )
       => ( ! [A5: produc3882902684156697774_state] :
              ( ( packag2518615827549365843_state @ Plus @ A3 @ A5 )
             => ( member6715010751053357271_state @ A5 @ S2 ) )
         => ( ( packag4061699384143886122_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec6806307004801468532_state
                @ ^ [Uu3: produc6455357576889117855_state] :
                  ? [A2: produc3882902684156697774_state] :
                    ( ( Uu3
                      = ( produc5427679348311974809_state @ A2 @ ( produc3987245858496418492_state @ Unit @ id_Pro689249233197853983_state ) ) )
                    & ( member6715010751053357271_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: produc3882902684156697774_state] : $true
              @ B3
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag1930333982449825652_state @ Plus @ ( packag2518615827549365843_state @ Plus @ B3 ) )
                | ( packag1386629450763126729_state @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_P5383513176685068947_state @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag3130788337556013255_state @ Plus @ W @ A3 @ B3 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_771_package__logic_Osoundness__paper,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B3: aassertion_val,A3: aassertion_val,S2: set_val,Sigma: val,Sigma2: val,W: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B3 )
       => ( ! [A5: val] :
              ( ( package_sat_val @ Plus @ A3 @ A5 )
             => ( member_val @ A5 @ S2 ) )
         => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec8300311316619386842al_val
                @ ^ [Uu3: produc2072440392815185925al_val] :
                  ? [A2: val] :
                    ( ( Uu3
                      = ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ Unit @ id_val ) ) )
                    & ( member_val @ A2 @ S2 ) ) )
              @ ^ [Uu3: val] : $true
              @ B3
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag2085742556458808647ic_val @ Plus @ ( package_sat_val @ Plus @ B3 ) )
                | ( packag3202507263771882441al_val @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_val @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag685230357167683994rd_val @ Plus @ W @ A3 @ B3 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_772_package__logic_Osoundness__paper,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: aassertion_state,A3: aassertion_state,S2: set_state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B3 )
       => ( ! [A5: state] :
              ( ( package_sat_state @ Plus @ A3 @ A5 )
             => ( member_state @ A5 @ S2 ) )
         => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ id_state ) ) )
                    & ( member_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: state] : $true
              @ B3
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag8361946002163212404_state @ Plus @ ( package_sat_state @ Plus @ B3 ) )
                | ( packag4152378406607043437_state @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_state @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag7096211993527611489_state @ Plus @ W @ A3 @ B3 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_773_PartialSA_Osoundness__paper,axiom,
    ! [B3: aassertion_state,A3: aassertion_state,S2: set_state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B3 )
     => ( ! [A5: state] :
            ( ( package_sat_state @ plus @ A3 @ A5 )
           => ( member_state @ A5 @ S2 ) )
       => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A2: state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
                  & ( member_state @ A2 @ S2 ) ) )
            @ ^ [Uu3: state] : $true
            @ B3
            @ Sigma2
            @ W
            @ S3 )
         => ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ B3 ) )
              | ( packag4152378406607043437_state @ plus @ S3 ) )
           => ( ( stable @ W )
              & ( ( some_state @ Sigma )
                = ( plus @ Sigma2 @ W ) )
              & ( packag7096211993527611489_state @ plus @ W @ A3 @ B3 ) ) ) ) ) ) ).

% PartialSA.soundness_paper
thf(fact_774_sep__algebra_Ominus__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_minus_val @ Plus @ Core @ B @ A )
        = ( fun_THE_default_val @ B
          @ ^ [X: val] :
              ( ( ( some_val @ B )
                = ( Plus @ A @ X ) )
              & ( sep_greater_val @ Plus @ X @ ( Core @ B ) ) ) ) ) ) ).

% sep_algebra.minus_def
thf(fact_775_sep__algebra_Ominus__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_minus_state @ Plus @ Core @ B @ A )
        = ( fun_TH8221542053827138557_state @ B
          @ ^ [X: state] :
              ( ( ( some_state @ B )
                = ( Plus @ A @ X ) )
              & ( sep_greater_state @ Plus @ X @ ( Core @ B ) ) ) ) ) ) ).

% sep_algebra.minus_def
thf(fact_776_package__logic_Ocompleteness__paper,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B3: aassertion_val,W: val,A3: aassertion_val,Sigma: val,Sigma2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B3 )
       => ( ( ( Stable @ W )
            & ( packag685230357167683994rd_val @ Plus @ W @ A3 @ B3 ) )
         => ( ( ( some_val @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ? [X_1: set_Pr4723104092121316667al_val] :
                ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec8300311316619386842al_val
                  @ ^ [Uu3: produc2072440392815185925al_val] :
                    ? [A2: val] :
                      ( ( Uu3
                        = ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ Unit @ id_val ) ) )
                      & ( package_sat_val @ Plus @ A3 @ A2 ) ) )
                @ ^ [Uu3: val] : $true
                @ B3
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% package_logic.completeness_paper
thf(fact_777_package__logic_Ocompleteness__paper,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: aassertion_state,W: state,A3: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B3 )
       => ( ( ( Stable @ W )
            & ( packag7096211993527611489_state @ Plus @ W @ A3 @ B3 ) )
         => ( ( ( some_state @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ? [X_1: set_Pr4231240917283394591_state] :
                ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ id_state ) ) )
                      & ( package_sat_state @ Plus @ A3 @ A2 ) ) )
                @ ^ [Uu3: state] : $true
                @ B3
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% package_logic.completeness_paper
thf(fact_778_PartialSA_Ogeneral__soundness,axiom,
    ! [Phi2: state,S2: set_Pr2391701600042252046_state,A3: aassertion_state,Phi3: state,F4: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ unit
        @ ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [A2: state,T5: state > state] :
              ( ( Uu3
                = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ T5 ) ) )
              & ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A2 @ T5 ) @ S2 ) ) )
        @ ^ [Uu3: state] : $true
        @ A3
        @ Phi3
        @ F4
        @ S3 )
     => ( ! [A5: state,T4: state > state] :
            ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A5 @ T4 ) @ S2 )
           => ( packag4817583915586052229_state @ plus @ unit @ T4 ) )
       => ( ( packag7298478212905792770_state @ plus @ core @ A3 )
         => ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ A3 ) )
              | ( packag4152378406607043437_state @ plus @ S3 ) )
           => ( ( ( some_state @ Phi2 )
                = ( plus @ Phi3 @ F4 ) )
              & ( stable @ F4 )
              & ! [X6: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X6 @ S2 )
                 => ( produc3645877548210629341tate_o
                    @ ^ [A2: state,T5: state > state] :
                        ( ( defined @ A2 @ ( T5 @ F4 ) )
                       => ( package_sat_state @ plus @ A3 @ ( the_state @ ( plus @ A2 @ ( T5 @ F4 ) ) ) ) )
                    @ X6 ) ) ) ) ) ) ) ).

% PartialSA.general_soundness
thf(fact_779_PartialSA_Ounder__def,axiom,
    ! [A3: set_state,Omega2: state] :
      ( ( sep_under_state @ plus @ A3 @ Omega2 )
      = ( collect_state
        @ ^ [Uu3: state] :
          ? [Omega3: state] :
            ( ( Uu3 = Omega3 )
            & ( member_state @ Omega3 @ A3 )
            & ( greater @ Omega2 @ Omega3 ) ) ) ) ).

% PartialSA.under_def
thf(fact_780_PartialSA_Opackage__rhs_Ocases,axiom,
    ! [A12: state,A23: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ A12 @ A23 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
     => ( ! [A10: aassertion_state,Phi6: state,F7: state,S6: set_Pr4231240917283394591_state,B8: aassertion_state] :
            ( ( A52
              = ( aStar_state @ A10 @ B8 ) )
           => ( ( packag3171892606229212990_state @ plus @ core @ stable @ A12 @ A23 @ A32 @ A42 @ A10 @ Phi6 @ F7 @ S6 )
             => ~ ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F7 @ S6 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
       => ( ! [B4: state > $o,A10: aassertion_state] :
              ( ( A52
                = ( aImp_state @ B4 @ A10 ) )
             => ~ ( packag3171892606229212990_state @ plus @ core @ stable @ A12 @ A23 @ A32 @ ( packag1330488657391168505_state @ A42 @ B4 ) @ A10 @ A62 @ A72 @ A82 ) )
         => ( ! [Witness: produc7272280185086996969_state > state,B8: state > $o] :
                ( ( A52
                  = ( aSem_state @ B8 ) )
               => ( ( A62 = A12 )
                 => ( ( A72 = A23 )
                   => ( ! [A11: state,U7: state,T6: state > state] :
                          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A11 @ ( produc8632743443909064678_state @ U7 @ T6 ) ) @ A32 )
                         => ( ( A42 @ A11 )
                           => ( ( greater @ A11 @ ( Witness @ ( produc5886875603596089627_state @ A11 @ ( produc8632743443909064678_state @ U7 @ T6 ) ) ) )
                              & ( B8 @ ( Witness @ ( produc5886875603596089627_state @ A11 @ ( produc8632743443909064678_state @ U7 @ T6 ) ) ) ) ) ) )
                     => ( A82
                       != ( sup_su3837536599209510219_state
                          @ ( collec906845016635587262_state
                            @ ^ [Uu3: produc7272280185086996969_state] :
                              ? [A2: state,U6: state,T5: state > state] :
                                ( ( Uu3
                                  = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                                & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
                                & ~ ( A42 @ A2 ) ) )
                          @ ( collec906845016635587262_state
                            @ ^ [Uu3: produc7272280185086996969_state] :
                              ? [A2: state,U6: state,T5: state > state] :
                                ( ( Uu3
                                  = ( produc5886875603596089627_state @ ( minus @ A2 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) @ T5 ) ) )
                                & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
                                & ( A42 @ A2 ) ) ) ) ) ) ) ) )
           => ~ ! [Phi6: state,M3: state] :
                  ( ( ( some_state @ A12 )
                    = ( plus @ Phi6 @ M3 ) )
                 => ! [F7: state,S6: set_Pr4231240917283394591_state] :
                      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F7 @ S6 @ A42 @ A52 @ A62 @ A72 @ A82 )
                     => ( ( stable @ M3 )
                       => ( ( ( some_state @ F7 )
                            = ( plus @ A23 @ M3 ) )
                         => ( S6
                           != ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A2: state,U6: state,T5: state > state,R: state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ R @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
                                  & ( ( some_state @ R )
                                    = ( plus @ A2 @ ( minus @ ( T5 @ F7 ) @ ( T5 @ A23 ) ) ) )
                                  & ( defined @ R @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% PartialSA.package_rhs.cases
thf(fact_781_PartialSA_Opackage__rhs_Osimps,axiom,
    ! [A12: state,A23: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ A12 @ A23 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
      = ( ? [Phi4: state,F3: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,A6: aassertion_state,Phi5: state,F8: state,S8: set_Pr4231240917283394591_state,B6: aassertion_state,Phi8: state,F9: state,S9: set_Pr4231240917283394591_state] :
            ( ( A12 = Phi4 )
            & ( A23 = F3 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aStar_state @ A6 @ B6 ) )
            & ( A62 = Phi8 )
            & ( A72 = F9 )
            & ( A82 = S9 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi4 @ F3 @ S7 @ Pc2 @ A6 @ Phi5 @ F8 @ S8 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F8 @ S8 @ Pc2 @ B6 @ Phi8 @ F9 @ S9 ) )
        | ? [Phi4: state,F3: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,B2: state > $o,A6: aassertion_state,Phi5: state,F8: state,S8: set_Pr4231240917283394591_state] :
            ( ( A12 = Phi4 )
            & ( A23 = F3 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aImp_state @ B2 @ A6 ) )
            & ( A62 = Phi5 )
            & ( A72 = F8 )
            & ( A82 = S8 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi4 @ F3 @ S7 @ ( packag1330488657391168505_state @ Pc2 @ B2 ) @ A6 @ Phi5 @ F8 @ S8 ) )
        | ? [S7: set_Pr4231240917283394591_state,Pc2: state > $o,Witness2: produc7272280185086996969_state > state,B6: state > $o,S8: set_Pr4231240917283394591_state,Phi4: state,F3: state] :
            ( ( A12 = Phi4 )
            & ( A23 = F3 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aSem_state @ B6 ) )
            & ( A62 = Phi4 )
            & ( A72 = F3 )
            & ( A82 = S8 )
            & ! [X: state,Y3: state,Z3: state > state,Aa3: state] :
                ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z3 ) ) @ S7 )
               => ( ( Pc2 @ X )
                 => ( ( Aa3
                      = ( Witness2 @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z3 ) ) ) )
                   => ( ( greater @ X @ Aa3 )
                      & ( B6 @ Aa3 ) ) ) ) )
            & ( S8
              = ( sup_su3837536599209510219_state
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T5: state > state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S7 )
                      & ~ ( Pc2 @ A2 ) ) )
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T5: state > state,B2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ ( minus @ A2 @ B2 ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B2 ) ) @ T5 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S7 )
                      & ( Pc2 @ A2 )
                      & ( B2
                        = ( Witness2 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) ) ) ) )
        | ? [Phi4: state,Phi5: state,M4: state,F8: state,S8: set_Pr4231240917283394591_state,Pc2: state > $o,A6: aassertion_state,Phi8: state,F9: state,S9: set_Pr4231240917283394591_state,F3: state,S7: set_Pr4231240917283394591_state] :
            ( ( A12 = Phi4 )
            & ( A23 = F3 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52 = A6 )
            & ( A62 = Phi8 )
            & ( A72 = F9 )
            & ( A82 = S9 )
            & ( ( some_state @ Phi4 )
              = ( plus @ Phi5 @ M4 ) )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F8 @ S8 @ Pc2 @ A6 @ Phi8 @ F9 @ S9 )
            & ( stable @ M4 )
            & ( ( some_state @ F8 )
              = ( plus @ F3 @ M4 ) )
            & ( S8
              = ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state,U6: state,T5: state > state,R: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ R @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S7 )
                    & ( ( some_state @ R )
                      = ( plus @ A2 @ ( minus @ ( T5 @ F8 ) @ ( T5 @ F3 ) ) ) )
                    & ( defined @ R @ U6 ) ) ) ) ) ) ) ).

% PartialSA.package_rhs.simps
thf(fact_782_split__part,axiom,
    ! [P4: $o,Q: state > ( state > state ) > $o] :
      ( ( produc3645877548210629341tate_o
        @ ^ [A2: state,B2: state > state] :
            ( P4
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab3: produc3882902684156697774_state] :
            ( P4
            & ( produc3645877548210629341tate_o @ Q @ Ab3 ) ) ) ) ).

% split_part
thf(fact_783_split__part,axiom,
    ! [P4: $o,Q: state > produc3882902684156697774_state > $o] :
      ( ( produc1745419677418529872tate_o
        @ ^ [A2: state,B2: produc3882902684156697774_state] :
            ( P4
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab3: produc7272280185086996969_state] :
            ( P4
            & ( produc1745419677418529872tate_o @ Q @ Ab3 ) ) ) ) ).

% split_part
thf(fact_784_case__prod__conv,axiom,
    ! [F4: state > ( state > state ) > $o,A: state,B: state > state] :
      ( ( produc3645877548210629341tate_o @ F4 @ ( produc8632743443909064678_state @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_785_case__prod__conv,axiom,
    ! [F4: state > produc3882902684156697774_state > $o,A: state,B: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F4 @ ( produc5886875603596089627_state @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_786_case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,C: option_state > option_state > $o] :
      ( ! [A5: option_state,B4: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc6082990154846732246tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_787_case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,C: prat > state > $o] :
      ( ! [A5: prat,B4: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc1228703470279370976tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_788_case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,C: prat > set_state > $o] :
      ( ! [A5: prat,B4: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc7925652287191295402tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_789_case__prodI2,axiom,
    ! [P2: produc1457211279475724562t_char,C: nat > list_char > $o] :
      ( ! [A5: nat,B4: list_char] :
          ( ( P2
            = ( produc120197956887798346t_char @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc8526122873445342073char_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_790_case__prodI2,axiom,
    ! [P2: produc9003781588162623004on_val,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ! [A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
          ( ( P2
            = ( produc74620467203698198on_val @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc5248837549992124053_val_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_791_case__prodI2,axiom,
    ! [P2: produc3882902684156697774_state,C: state > ( state > state ) > $o] :
      ( ! [A5: state,B4: state > state] :
          ( ( P2
            = ( produc8632743443909064678_state @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc3645877548210629341tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_792_case__prodI2,axiom,
    ! [P2: produc7272280185086996969_state,C: state > produc3882902684156697774_state > $o] :
      ( ! [A5: state,B4: produc3882902684156697774_state] :
          ( ( P2
            = ( produc5886875603596089627_state @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc1745419677418529872tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_793_case__prodI,axiom,
    ! [F4: option_state > option_state > $o,A: option_state,B: option_state] :
      ( ( F4 @ A @ B )
     => ( produc6082990154846732246tate_o @ F4 @ ( produc9160152616361873709_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_794_case__prodI,axiom,
    ! [F4: prat > state > $o,A: prat,B: state] :
      ( ( F4 @ A @ B )
     => ( produc1228703470279370976tate_o @ F4 @ ( produc6205719116230455971_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_795_case__prodI,axiom,
    ! [F4: prat > set_state > $o,A: prat,B: set_state] :
      ( ( F4 @ A @ B )
     => ( produc7925652287191295402tate_o @ F4 @ ( produc9080284787853255769_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_796_case__prodI,axiom,
    ! [F4: nat > list_char > $o,A: nat,B: list_char] :
      ( ( F4 @ A @ B )
     => ( produc8526122873445342073char_o @ F4 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% case_prodI
thf(fact_797_case__prodI,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( F4 @ A @ B )
     => ( produc5248837549992124053_val_o @ F4 @ ( produc74620467203698198on_val @ A @ B ) ) ) ).

% case_prodI
thf(fact_798_case__prodI,axiom,
    ! [F4: state > ( state > state ) > $o,A: state,B: state > state] :
      ( ( F4 @ A @ B )
     => ( produc3645877548210629341tate_o @ F4 @ ( produc8632743443909064678_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_799_case__prodI,axiom,
    ! [F4: state > produc3882902684156697774_state > $o,A: state,B: produc3882902684156697774_state] :
      ( ( F4 @ A @ B )
     => ( produc1745419677418529872tate_o @ F4 @ ( produc5886875603596089627_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_800_case__swap,axiom,
    ! [F4: state > ( state > state ) > $o,P2: produc3882902684156697774_state] :
      ( ( produc7947965470134731869tate_o
        @ ^ [Y3: state > state,X: state] : ( F4 @ X @ Y3 )
        @ ( produc6438927260845192582_state @ P2 ) )
      = ( produc3645877548210629341tate_o @ F4 @ P2 ) ) ).

% case_swap
thf(fact_801_case__swap,axiom,
    ! [F4: state > produc3882902684156697774_state > $o,P2: produc7272280185086996969_state] :
      ( ( produc2011331576729698010tate_o
        @ ^ [Y3: produc3882902684156697774_state,X: state] : ( F4 @ X @ Y3 )
        @ ( produc6564544480275723131_state @ P2 ) )
      = ( produc1745419677418529872tate_o @ F4 @ P2 ) ) ).

% case_swap
thf(fact_802_case__swap,axiom,
    ! [F4: ( state > state ) > state > $o,P2: produc5876769945872133550_state] :
      ( ( produc3645877548210629341tate_o
        @ ^ [Y3: state,X: state > state] : ( F4 @ X @ Y3 )
        @ ( produc2543546466205187078_state @ P2 ) )
      = ( produc7947965470134731869tate_o @ F4 @ P2 ) ) ).

% case_swap
thf(fact_803_case__swap,axiom,
    ! [F4: produc3882902684156697774_state > state > $o,P2: produc1357540302793366807_state] :
      ( ( produc1745419677418529872tate_o
        @ ^ [Y3: state,X: produc3882902684156697774_state] : ( F4 @ X @ Y3 )
        @ ( produc8011752947996305265_state @ P2 ) )
      = ( produc2011331576729698010tate_o @ F4 @ P2 ) ) ).

% case_swap
thf(fact_804_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F4: state > ( state > state ) > $o,Prod: produc3882902684156697774_state] :
      ( ( H2 @ ( produc3645877548210629341tate_o @ F4 @ Prod ) )
      = ( produc3645877548210629341tate_o
        @ ^ [X12: state,X24: state > state] : ( H2 @ ( F4 @ X12 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_805_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F4: state > produc3882902684156697774_state > $o,Prod: produc7272280185086996969_state] :
      ( ( H2 @ ( produc1745419677418529872tate_o @ F4 @ Prod ) )
      = ( produc1745419677418529872tate_o
        @ ^ [X12: state,X24: produc3882902684156697774_state] : ( H2 @ ( F4 @ X12 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_806_prod_Odisc__eq__case,axiom,
    ! [Prod: produc3882902684156697774_state] :
      ( produc3645877548210629341tate_o
      @ ^ [Uu3: state,Uv3: state > state] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_807_prod_Odisc__eq__case,axiom,
    ! [Prod: produc7272280185086996969_state] :
      ( produc1745419677418529872tate_o
      @ ^ [Uu3: state,Uv3: produc3882902684156697774_state] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_808_case__prodE,axiom,
    ! [C: option_state > option_state > $o,P2: produc3142500478612311029_state] :
      ( ( produc6082990154846732246tate_o @ C @ P2 )
     => ~ ! [X3: option_state,Y4: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_809_case__prodE,axiom,
    ! [C: prat > state > $o,P2: produc4844402304972151915_state] :
      ( ( produc1228703470279370976tate_o @ C @ P2 )
     => ~ ! [X3: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_810_case__prodE,axiom,
    ! [C: prat > set_state > $o,P2: produc6251377953716854561_state] :
      ( ( produc7925652287191295402tate_o @ C @ P2 )
     => ~ ! [X3: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_811_case__prodE,axiom,
    ! [C: nat > list_char > $o,P2: produc1457211279475724562t_char] :
      ( ( produc8526122873445342073char_o @ C @ P2 )
     => ~ ! [X3: nat,Y4: list_char] :
            ( ( P2
              = ( produc120197956887798346t_char @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_812_case__prodE,axiom,
    ! [C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P2: produc9003781588162623004on_val] :
      ( ( produc5248837549992124053_val_o @ C @ P2 )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
            ( ( P2
              = ( produc74620467203698198on_val @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_813_case__prodE,axiom,
    ! [C: state > ( state > state ) > $o,P2: produc3882902684156697774_state] :
      ( ( produc3645877548210629341tate_o @ C @ P2 )
     => ~ ! [X3: state,Y4: state > state] :
            ( ( P2
              = ( produc8632743443909064678_state @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_814_case__prodE,axiom,
    ! [C: state > produc3882902684156697774_state > $o,P2: produc7272280185086996969_state] :
      ( ( produc1745419677418529872tate_o @ C @ P2 )
     => ~ ! [X3: state,Y4: produc3882902684156697774_state] :
            ( ( P2
              = ( produc5886875603596089627_state @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_815_case__prodD,axiom,
    ! [F4: option_state > option_state > $o,A: option_state,B: option_state] :
      ( ( produc6082990154846732246tate_o @ F4 @ ( produc9160152616361873709_state @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_816_case__prodD,axiom,
    ! [F4: prat > state > $o,A: prat,B: state] :
      ( ( produc1228703470279370976tate_o @ F4 @ ( produc6205719116230455971_state @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_817_case__prodD,axiom,
    ! [F4: prat > set_state > $o,A: prat,B: set_state] :
      ( ( produc7925652287191295402tate_o @ F4 @ ( produc9080284787853255769_state @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_818_case__prodD,axiom,
    ! [F4: nat > list_char > $o,A: nat,B: list_char] :
      ( ( produc8526122873445342073char_o @ F4 @ ( produc120197956887798346t_char @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_819_case__prodD,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc5248837549992124053_val_o @ F4 @ ( produc74620467203698198on_val @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_820_case__prodD,axiom,
    ! [F4: state > ( state > state ) > $o,A: state,B: state > state] :
      ( ( produc3645877548210629341tate_o @ F4 @ ( produc8632743443909064678_state @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_821_case__prodD,axiom,
    ! [F4: state > produc3882902684156697774_state > $o,A: state,B: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F4 @ ( produc5886875603596089627_state @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_822_old_Oprod_Ocase,axiom,
    ! [F4: state > ( state > state ) > $o,X1: state,X22: state > state] :
      ( ( produc3645877548210629341tate_o @ F4 @ ( produc8632743443909064678_state @ X1 @ X22 ) )
      = ( F4 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_823_old_Oprod_Ocase,axiom,
    ! [F4: state > produc3882902684156697774_state > $o,X1: state,X22: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F4 @ ( produc5886875603596089627_state @ X1 @ X22 ) )
      = ( F4 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_824_cond__case__prod__eta,axiom,
    ! [F4: state > ( state > state ) > $o,G: produc3882902684156697774_state > $o] :
      ( ! [X3: state,Y4: state > state] :
          ( ( F4 @ X3 @ Y4 )
          = ( G @ ( produc8632743443909064678_state @ X3 @ Y4 ) ) )
     => ( ( produc3645877548210629341tate_o @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_825_cond__case__prod__eta,axiom,
    ! [F4: state > produc3882902684156697774_state > $o,G: produc7272280185086996969_state > $o] :
      ( ! [X3: state,Y4: produc3882902684156697774_state] :
          ( ( F4 @ X3 @ Y4 )
          = ( G @ ( produc5886875603596089627_state @ X3 @ Y4 ) ) )
     => ( ( produc1745419677418529872tate_o @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_826_case__prod__eta,axiom,
    ! [F4: produc3882902684156697774_state > $o] :
      ( ( produc3645877548210629341tate_o
        @ ^ [X: state,Y3: state > state] : ( F4 @ ( produc8632743443909064678_state @ X @ Y3 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_827_case__prod__eta,axiom,
    ! [F4: produc7272280185086996969_state > $o] :
      ( ( produc1745419677418529872tate_o
        @ ^ [X: state,Y3: produc3882902684156697774_state] : ( F4 @ ( produc5886875603596089627_state @ X @ Y3 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_828_case__prodE2,axiom,
    ! [Q: $o > $o,P4: state > ( state > state ) > $o,Z: produc3882902684156697774_state] :
      ( ( Q @ ( produc3645877548210629341tate_o @ P4 @ Z ) )
     => ~ ! [X3: state,Y4: state > state] :
            ( ( Z
              = ( produc8632743443909064678_state @ X3 @ Y4 ) )
           => ~ ( Q @ ( P4 @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_829_case__prodE2,axiom,
    ! [Q: $o > $o,P4: state > produc3882902684156697774_state > $o,Z: produc7272280185086996969_state] :
      ( ( Q @ ( produc1745419677418529872tate_o @ P4 @ Z ) )
     => ~ ! [X3: state,Y4: produc3882902684156697774_state] :
            ( ( Z
              = ( produc5886875603596089627_state @ X3 @ Y4 ) )
           => ~ ( Q @ ( P4 @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_830_fst__def,axiom,
    ( produc6204653154488278070t_char
    = ( produc9163108120003956271ar_nat
      @ ^ [X12: nat,X24: list_char] : X12 ) ) ).

% fst_def
thf(fact_831_fst__def,axiom,
    ( produc5239619271231976746on_val
    = ( produc5787256515117077072r_prat
      @ ^ [X12: produc1457211279475724562t_char > prat,X24: produc1457211279475724562t_char > option_val] : X12 ) ) ).

% fst_def
thf(fact_832_snd__def,axiom,
    ( produc2319605628138516840on_val
    = ( produc985966260317727453on_val
      @ ^ [X12: produc1457211279475724562t_char > prat,X24: produc1457211279475724562t_char > option_val] : X24 ) ) ).

% snd_def
thf(fact_833_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc1457211279475724562t_char,A3: nat > list_char > $o] :
      ( ( member484235747325421115t_char @ X2 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ A3 ) ) )
     => ( A3 @ ( produc6204653154488278070t_char @ X2 ) @ ( produc1900778367047502200t_char @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_834_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc9003781588162623004on_val,A3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member2879030624002546611on_val @ X2 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ A3 ) ) )
     => ( A3 @ ( produc5239619271231976746on_val @ X2 ) @ ( produc2319605628138516840on_val @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_835_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc3882902684156697774_state,A3: state > ( state > state ) > $o] :
      ( ( member6715010751053357271_state @ X2 @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ A3 ) ) )
     => ( A3 @ ( produc7095885521192019410_state @ X2 ) @ ( produc1734700239402139924_state @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_836_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc7272280185086996969_state,A3: state > produc3882902684156697774_state > $o] :
      ( ( member7461423308196221312_state @ X2 @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ A3 ) ) )
     => ( A3 @ ( produc848024588167747631_state @ X2 ) @ ( produc6310399636379773037_state @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_837_PartialSA_Osat_Osimps_I3_J,axiom,
    ! [A3: state > $o,Phi2: state] :
      ( ( package_sat_state @ plus @ ( aSem_state @ A3 ) @ Phi2 )
      = ( A3 @ Phi2 ) ) ).

% PartialSA.sat.simps(3)
thf(fact_838_split__beta,axiom,
    ( produc3645877548210629341tate_o
    = ( ^ [F3: state > ( state > state ) > $o,Prod3: produc3882902684156697774_state] : ( F3 @ ( produc7095885521192019410_state @ Prod3 ) @ ( produc1734700239402139924_state @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_839_split__beta,axiom,
    ( produc1745419677418529872tate_o
    = ( ^ [F3: state > produc3882902684156697774_state > $o,Prod3: produc7272280185086996969_state] : ( F3 @ ( produc848024588167747631_state @ Prod3 ) @ ( produc6310399636379773037_state @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_840_case__prod__beta,axiom,
    ( produc3645877548210629341tate_o
    = ( ^ [F3: state > ( state > state ) > $o,P3: produc3882902684156697774_state] : ( F3 @ ( produc7095885521192019410_state @ P3 ) @ ( produc1734700239402139924_state @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_841_case__prod__beta,axiom,
    ( produc1745419677418529872tate_o
    = ( ^ [F3: state > produc3882902684156697774_state > $o,P3: produc7272280185086996969_state] : ( F3 @ ( produc848024588167747631_state @ P3 ) @ ( produc6310399636379773037_state @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_842_internal__case__prod__def,axiom,
    produc1225464857956169863tate_o = produc3645877548210629341tate_o ).

% internal_case_prod_def
thf(fact_843_internal__case__prod__def,axiom,
    produc3899216655821101094tate_o = produc1745419677418529872tate_o ).

% internal_case_prod_def
thf(fact_844_PartialSA_Owf__assertion_Ocases,axiom,
    ! [X2: aassertion_state] :
      ( ! [A10: aassertion_state,B8: aassertion_state] :
          ( X2
         != ( aStar_state @ A10 @ B8 ) )
     => ( ! [B4: state > $o,A10: aassertion_state] :
            ( X2
           != ( aImp_state @ B4 @ A10 ) )
       => ~ ! [A10: state > $o] :
              ( X2
             != ( aSem_state @ A10 ) ) ) ) ).

% PartialSA.wf_assertion.cases
thf(fact_845_PartialSA_Osat_Ocases,axiom,
    ! [X2: produc2756261550280135502_state] :
      ( ! [A10: aassertion_state,B8: aassertion_state,Phi: state] :
          ( X2
         != ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Phi ) )
     => ( ! [B4: state > $o,A10: aassertion_state,Phi: state] :
            ( X2
           != ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Phi ) )
       => ~ ! [A10: state > $o,Phi: state] :
              ( X2
             != ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Phi ) ) ) ) ).

% PartialSA.sat.cases
thf(fact_846_case__prod__beta_H,axiom,
    ( produc3645877548210629341tate_o
    = ( ^ [F3: state > ( state > state ) > $o,X: produc3882902684156697774_state] : ( F3 @ ( produc7095885521192019410_state @ X ) @ ( produc1734700239402139924_state @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_847_case__prod__beta_H,axiom,
    ( produc1745419677418529872tate_o
    = ( ^ [F3: state > produc3882902684156697774_state > $o,X: produc7272280185086996969_state] : ( F3 @ ( produc848024588167747631_state @ X ) @ ( produc6310399636379773037_state @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_848_case__prod__unfold,axiom,
    ( produc3645877548210629341tate_o
    = ( ^ [C4: state > ( state > state ) > $o,P3: produc3882902684156697774_state] : ( C4 @ ( produc7095885521192019410_state @ P3 ) @ ( produc1734700239402139924_state @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_849_case__prod__unfold,axiom,
    ( produc1745419677418529872tate_o
    = ( ^ [C4: state > produc3882902684156697774_state > $o,P3: produc7272280185086996969_state] : ( C4 @ ( produc848024588167747631_state @ P3 ) @ ( produc6310399636379773037_state @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_850_prod_Osplit__sel,axiom,
    ! [P4: $o > $o,F4: state > ( state > state ) > $o,Prod: produc3882902684156697774_state] :
      ( ( P4 @ ( produc3645877548210629341tate_o @ F4 @ Prod ) )
      = ( ( Prod
          = ( produc8632743443909064678_state @ ( produc7095885521192019410_state @ Prod ) @ ( produc1734700239402139924_state @ Prod ) ) )
       => ( P4 @ ( F4 @ ( produc7095885521192019410_state @ Prod ) @ ( produc1734700239402139924_state @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_851_prod_Osplit__sel,axiom,
    ! [P4: $o > $o,F4: state > produc3882902684156697774_state > $o,Prod: produc7272280185086996969_state] :
      ( ( P4 @ ( produc1745419677418529872tate_o @ F4 @ Prod ) )
      = ( ( Prod
          = ( produc5886875603596089627_state @ ( produc848024588167747631_state @ Prod ) @ ( produc6310399636379773037_state @ Prod ) ) )
       => ( P4 @ ( F4 @ ( produc848024588167747631_state @ Prod ) @ ( produc6310399636379773037_state @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_852_prod_Osplit__sel__asm,axiom,
    ! [P4: $o > $o,F4: state > ( state > state ) > $o,Prod: produc3882902684156697774_state] :
      ( ( P4 @ ( produc3645877548210629341tate_o @ F4 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc8632743443909064678_state @ ( produc7095885521192019410_state @ Prod ) @ ( produc1734700239402139924_state @ Prod ) ) )
            & ~ ( P4 @ ( F4 @ ( produc7095885521192019410_state @ Prod ) @ ( produc1734700239402139924_state @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_853_prod_Osplit__sel__asm,axiom,
    ! [P4: $o > $o,F4: state > produc3882902684156697774_state > $o,Prod: produc7272280185086996969_state] :
      ( ( P4 @ ( produc1745419677418529872tate_o @ F4 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc5886875603596089627_state @ ( produc848024588167747631_state @ Prod ) @ ( produc6310399636379773037_state @ Prod ) ) )
            & ~ ( P4 @ ( F4 @ ( produc848024588167747631_state @ Prod ) @ ( produc6310399636379773037_state @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_854_package__logic_Osat_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: produc2756261550280135502_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A10: aassertion_state,B8: aassertion_state,Phi: state] :
            ( X2
           != ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Phi ) )
       => ( ! [B4: state > $o,A10: aassertion_state,Phi: state] :
              ( X2
             != ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Phi ) )
         => ~ ! [A10: state > $o,Phi: state] :
                ( X2
               != ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Phi ) ) ) ) ) ).

% package_logic.sat.cases
thf(fact_855_PartialSA_Owf__assertion_Osimps_I3_J,axiom,
    ! [A3: state > $o] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aSem_state @ A3 ) )
      = ( packag2456304381420842418_state @ plus @ A3 ) ) ).

% PartialSA.wf_assertion.simps(3)
thf(fact_856_package__logic_OASem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,Pc: state > $o,Witness3: produc7272280185086996969_state > state,B3: state > $o,S3: set_Pr4231240917283394591_state,Phi2: state,F4: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: state,U3: state,T4: state > state,B4: state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
           => ( ( Pc @ A5 )
             => ( ( B4
                  = ( Witness3 @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) ) )
               => ( ( sep_greater_state @ Plus @ A5 @ B4 )
                  & ( B3 @ B4 ) ) ) ) )
       => ( ( S3
            = ( sup_su3837536599209510219_state
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state,U6: state,T5: state > state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                    & ~ ( Pc @ A2 ) ) )
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state,U6: state,T5: state > state,B2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A2 @ B2 ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ B2 ) ) @ T5 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                    & ( Pc @ A2 )
                    & ( B2
                      = ( Witness3 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) ) ) )
         => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ ( aSem_state @ B3 ) @ Phi2 @ F4 @ S3 ) ) ) ) ).

% package_logic.ASem
thf(fact_857_PartialSA_Osat_Oelims_I3_J,axiom,
    ! [X2: aassertion_state,Xa: state] :
      ( ~ ( package_sat_state @ plus @ X2 @ Xa )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A10 @ B8 ) )
           => ? [A11: state,B9: state] :
                ( ( ( some_state @ Xa )
                  = ( plus @ A11 @ B9 ) )
                & ( package_sat_state @ plus @ A10 @ A11 )
                & ( package_sat_state @ plus @ B8 @ B9 ) ) )
       => ( ! [B4: state > $o,A10: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B4 @ A10 ) )
             => ( ( B4 @ Xa )
               => ( package_sat_state @ plus @ A10 @ Xa ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X2
                  = ( aSem_state @ A10 ) )
               => ( A10 @ Xa ) ) ) ) ) ).

% PartialSA.sat.elims(3)
thf(fact_858_PartialSA_Osat_Oelims_I2_J,axiom,
    ! [X2: aassertion_state,Xa: state] :
      ( ( package_sat_state @ plus @ X2 @ Xa )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A10 @ B8 ) )
           => ~ ? [A5: state,B4: state] :
                  ( ( ( some_state @ Xa )
                    = ( plus @ A5 @ B4 ) )
                  & ( package_sat_state @ plus @ A10 @ A5 )
                  & ( package_sat_state @ plus @ B8 @ B4 ) ) )
       => ( ! [B4: state > $o,A10: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B4 @ A10 ) )
             => ~ ( ( B4 @ Xa )
                 => ( package_sat_state @ plus @ A10 @ Xa ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X2
                  = ( aSem_state @ A10 ) )
               => ~ ( A10 @ Xa ) ) ) ) ) ).

% PartialSA.sat.elims(2)
thf(fact_859_PartialSA_Osat_Oelims_I1_J,axiom,
    ! [X2: aassertion_state,Xa: state,Y: $o] :
      ( ( ( package_sat_state @ plus @ X2 @ Xa )
        = Y )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A10 @ B8 ) )
           => ( Y
              = ( ~ ? [A2: state,B2: state] :
                      ( ( ( some_state @ Xa )
                        = ( plus @ A2 @ B2 ) )
                      & ( package_sat_state @ plus @ A10 @ A2 )
                      & ( package_sat_state @ plus @ B8 @ B2 ) ) ) ) )
       => ( ! [B4: state > $o,A10: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B4 @ A10 ) )
             => ( Y
                = ( ~ ( ( B4 @ Xa )
                     => ( package_sat_state @ plus @ A10 @ Xa ) ) ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X2
                  = ( aSem_state @ A10 ) )
               => ( Y
                  = ( ~ ( A10 @ Xa ) ) ) ) ) ) ) ).

% PartialSA.sat.elims(1)
thf(fact_860_package__logic_Osat_Oelims_I3_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X2: aassertion_val,Xa: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_val @ Plus @ X2 @ Xa )
       => ( ! [A10: aassertion_val,B8: aassertion_val] :
              ( ( X2
                = ( aStar_val @ A10 @ B8 ) )
             => ? [A11: val,B9: val] :
                  ( ( ( some_val @ Xa )
                    = ( Plus @ A11 @ B9 ) )
                  & ( package_sat_val @ Plus @ A10 @ A11 )
                  & ( package_sat_val @ Plus @ B8 @ B9 ) ) )
         => ( ! [B4: val > $o,A10: aassertion_val] :
                ( ( X2
                  = ( aImp_val @ B4 @ A10 ) )
               => ( ( B4 @ Xa )
                 => ( package_sat_val @ Plus @ A10 @ Xa ) ) )
           => ~ ! [A10: val > $o] :
                  ( ( X2
                    = ( aSem_val @ A10 ) )
                 => ( A10 @ Xa ) ) ) ) ) ) ).

% package_logic.sat.elims(3)
thf(fact_861_package__logic_Osat_Oelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_state @ Plus @ X2 @ Xa )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ? [A11: state,B9: state] :
                  ( ( ( some_state @ Xa )
                    = ( Plus @ A11 @ B9 ) )
                  & ( package_sat_state @ Plus @ A10 @ A11 )
                  & ( package_sat_state @ Plus @ B8 @ B9 ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( ( B4 @ Xa )
                 => ( package_sat_state @ Plus @ A10 @ Xa ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( A10 @ Xa ) ) ) ) ) ) ).

% package_logic.sat.elims(3)
thf(fact_862_package__logic_Osat_Oelims_I2_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X2: aassertion_val,Xa: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ X2 @ Xa )
       => ( ! [A10: aassertion_val,B8: aassertion_val] :
              ( ( X2
                = ( aStar_val @ A10 @ B8 ) )
             => ~ ? [A5: val,B4: val] :
                    ( ( ( some_val @ Xa )
                      = ( Plus @ A5 @ B4 ) )
                    & ( package_sat_val @ Plus @ A10 @ A5 )
                    & ( package_sat_val @ Plus @ B8 @ B4 ) ) )
         => ( ! [B4: val > $o,A10: aassertion_val] :
                ( ( X2
                  = ( aImp_val @ B4 @ A10 ) )
               => ~ ( ( B4 @ Xa )
                   => ( package_sat_val @ Plus @ A10 @ Xa ) ) )
           => ~ ! [A10: val > $o] :
                  ( ( X2
                    = ( aSem_val @ A10 ) )
                 => ~ ( A10 @ Xa ) ) ) ) ) ) ).

% package_logic.sat.elims(2)
thf(fact_863_package__logic_Osat_Oelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ X2 @ Xa )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ~ ? [A5: state,B4: state] :
                    ( ( ( some_state @ Xa )
                      = ( Plus @ A5 @ B4 ) )
                    & ( package_sat_state @ Plus @ A10 @ A5 )
                    & ( package_sat_state @ Plus @ B8 @ B4 ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ~ ( ( B4 @ Xa )
                   => ( package_sat_state @ Plus @ A10 @ Xa ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ~ ( A10 @ Xa ) ) ) ) ) ) ).

% package_logic.sat.elims(2)
thf(fact_864_package__logic_Osat_Oelims_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X2: aassertion_val,Xa: val,Y: $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_val @ Plus @ X2 @ Xa )
          = Y )
       => ( ! [A10: aassertion_val,B8: aassertion_val] :
              ( ( X2
                = ( aStar_val @ A10 @ B8 ) )
             => ( Y
                = ( ~ ? [A2: val,B2: val] :
                        ( ( ( some_val @ Xa )
                          = ( Plus @ A2 @ B2 ) )
                        & ( package_sat_val @ Plus @ A10 @ A2 )
                        & ( package_sat_val @ Plus @ B8 @ B2 ) ) ) ) )
         => ( ! [B4: val > $o,A10: aassertion_val] :
                ( ( X2
                  = ( aImp_val @ B4 @ A10 ) )
               => ( Y
                  = ( ~ ( ( B4 @ Xa )
                       => ( package_sat_val @ Plus @ A10 @ Xa ) ) ) ) )
           => ~ ! [A10: val > $o] :
                  ( ( X2
                    = ( aSem_val @ A10 ) )
                 => ( Y
                    = ( ~ ( A10 @ Xa ) ) ) ) ) ) ) ) ).

% package_logic.sat.elims(1)
thf(fact_865_package__logic_Osat_Oelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa: state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_state @ Plus @ X2 @ Xa )
          = Y )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ( Y
                = ( ~ ? [A2: state,B2: state] :
                        ( ( ( some_state @ Xa )
                          = ( Plus @ A2 @ B2 ) )
                        & ( package_sat_state @ Plus @ A10 @ A2 )
                        & ( package_sat_state @ Plus @ B8 @ B2 ) ) ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( Y
                  = ( ~ ( ( B4 @ Xa )
                       => ( package_sat_state @ Plus @ A10 @ Xa ) ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( Y
                    = ( ~ ( A10 @ Xa ) ) ) ) ) ) ) ) ).

% package_logic.sat.elims(1)
thf(fact_866_PartialSA_OASem,axiom,
    ! [S2: set_Pr4231240917283394591_state,Pc: state > $o,Witness3: produc7272280185086996969_state > state,B3: state > $o,S3: set_Pr4231240917283394591_state,Phi2: state,F4: state] :
      ( ! [A5: state,U3: state,T4: state > state,B4: state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
         => ( ( Pc @ A5 )
           => ( ( B4
                = ( Witness3 @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) ) )
             => ( ( greater @ A5 @ B4 )
                & ( B3 @ B4 ) ) ) ) )
     => ( ( S3
          = ( sup_su3837536599209510219_state
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A2: state,U6: state,T5: state > state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                  & ~ ( Pc @ A2 ) ) )
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A2: state,U6: state,T5: state > state,B2: state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ ( minus @ A2 @ B2 ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B2 ) ) @ T5 ) ) )
                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                  & ( Pc @ A2 )
                  & ( B2
                    = ( Witness3 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) ) ) )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ ( aSem_state @ B3 ) @ Phi2 @ F4 @ S3 ) ) ) ).

% PartialSA.ASem
thf(fact_867_package__logic_Ogeneral__soundness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi2: val,S2: set_Pr6546617632949187467al_val,A3: aassertion_val,Phi3: val,F4: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi2 @ Unit
          @ ( collec8300311316619386842al_val
            @ ^ [Uu3: produc2072440392815185925al_val] :
              ? [A2: val,T5: val > val] :
                ( ( Uu3
                  = ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ Unit @ T5 ) ) )
                & ( member331055627401653996al_val @ ( produc303288909735261391al_val @ A2 @ T5 ) @ S2 ) ) )
          @ ^ [Uu3: val] : $true
          @ A3
          @ Phi3
          @ F4
          @ S3 )
       => ( ! [A5: val,T4: val > val] :
              ( ( member331055627401653996al_val @ ( produc303288909735261391al_val @ A5 @ T4 ) @ S2 )
             => ( packag4175576287423129206er_val @ Plus @ Unit @ T4 ) )
         => ( ( packag5732365195153136185on_val @ Plus @ Core @ A3 )
           => ( ( ( packag2085742556458808647ic_val @ Plus @ ( package_sat_val @ Plus @ A3 ) )
                | ( packag3202507263771882441al_val @ Plus @ S3 ) )
             => ( ( ( some_val @ Phi2 )
                  = ( Plus @ Phi3 @ F4 ) )
                & ( Stable @ F4 )
                & ! [X6: produc5503543764249810261al_val] :
                    ( ( member331055627401653996al_val @ X6 @ S2 )
                   => ( produc7666680587469814684_val_o
                      @ ^ [A2: val,T5: val > val] :
                          ( ( sep_defined_val @ Plus @ A2 @ ( T5 @ F4 ) )
                         => ( package_sat_val @ Plus @ A3 @ ( the_val @ ( Plus @ A2 @ ( T5 @ F4 ) ) ) ) )
                      @ X6 ) ) ) ) ) ) ) ) ).

% package_logic.general_soundness
thf(fact_868_package__logic_Ogeneral__soundness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi2: state,S2: set_Pr2391701600042252046_state,A3: aassertion_state,Phi3: state,F4: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ Unit
          @ ( collec906845016635587262_state
            @ ^ [Uu3: produc7272280185086996969_state] :
              ? [A2: state,T5: state > state] :
                ( ( Uu3
                  = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ T5 ) ) )
                & ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A2 @ T5 ) @ S2 ) ) )
          @ ^ [Uu3: state] : $true
          @ A3
          @ Phi3
          @ F4
          @ S3 )
       => ( ! [A5: state,T4: state > state] :
              ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A5 @ T4 ) @ S2 )
             => ( packag4817583915586052229_state @ Plus @ Unit @ T4 ) )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A3 )
           => ( ( ( packag8361946002163212404_state @ Plus @ ( package_sat_state @ Plus @ A3 ) )
                | ( packag4152378406607043437_state @ Plus @ S3 ) )
             => ( ( ( some_state @ Phi2 )
                  = ( Plus @ Phi3 @ F4 ) )
                & ( Stable @ F4 )
                & ! [X6: produc3882902684156697774_state] :
                    ( ( member6715010751053357271_state @ X6 @ S2 )
                   => ( produc3645877548210629341tate_o
                      @ ^ [A2: state,T5: state > state] :
                          ( ( sep_defined_state @ Plus @ A2 @ ( T5 @ F4 ) )
                         => ( package_sat_state @ Plus @ A3 @ ( the_state @ ( Plus @ A2 @ ( T5 @ F4 ) ) ) ) )
                      @ X6 ) ) ) ) ) ) ) ) ).

% package_logic.general_soundness
thf(fact_869_PartialSA_Owf__assertion_Oelims_I1_J,axiom,
    ! [X2: aassertion_state,Y: $o] :
      ( ( ( packag7298478212905792770_state @ plus @ core @ X2 )
        = Y )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A10 @ B8 ) )
           => ( Y
              = ( ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) ) )
       => ( ! [B4: state > $o,A10: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B4 @ A10 ) )
             => ( Y
                = ( ~ ( ( packag6595153354952283300_state @ plus @ core @ B4 )
                      & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X2
                  = ( aSem_state @ A10 ) )
               => ( Y
                  = ( ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(1)
thf(fact_870_PartialSA_Owf__assertion_Oelims_I2_J,axiom,
    ! [X2: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A10 @ B8 ) )
           => ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
       => ( ! [B4: state > $o,A10: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B4 @ A10 ) )
             => ~ ( ( packag6595153354952283300_state @ plus @ core @ B4 )
                  & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X2
                  = ( aSem_state @ A10 ) )
               => ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(2)
thf(fact_871_PartialSA_Owf__assertion_Oelims_I3_J,axiom,
    ! [X2: aassertion_state] :
      ( ~ ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A10 @ B8 ) )
           => ( ( packag7298478212905792770_state @ plus @ core @ A10 )
              & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
       => ( ! [B4: state > $o,A10: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B4 @ A10 ) )
             => ( ( packag6595153354952283300_state @ plus @ core @ B4 )
                & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X2
                  = ( aSem_state @ A10 ) )
               => ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(3)
thf(fact_872_package__logic_Opackage__rhs_Osimps,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A12: val,A23: val,A32: set_Pr4723104092121316667al_val,A42: val > $o,A52: aassertion_val,A62: val,A72: val,A82: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
        = ( ? [Phi4: val,F3: val,S7: set_Pr4723104092121316667al_val,Pc2: val > $o,A6: aassertion_val,Phi5: val,F8: val,S8: set_Pr4723104092121316667al_val,B6: aassertion_val,Phi8: val,F9: val,S9: set_Pr4723104092121316667al_val] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aStar_val @ A6 @ B6 ) )
              & ( A62 = Phi8 )
              & ( A72 = F9 )
              & ( A82 = S9 )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi4 @ F3 @ S7 @ Pc2 @ A6 @ Phi5 @ F8 @ S8 )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi5 @ F8 @ S8 @ Pc2 @ B6 @ Phi8 @ F9 @ S9 ) )
          | ? [Phi4: val,F3: val,S7: set_Pr4723104092121316667al_val,Pc2: val > $o,B2: val > $o,A6: aassertion_val,Phi5: val,F8: val,S8: set_Pr4723104092121316667al_val] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aImp_val @ B2 @ A6 ) )
              & ( A62 = Phi5 )
              & ( A72 = F8 )
              & ( A82 = S8 )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi4 @ F3 @ S7 @ ( packag6750194596558674946nj_val @ Pc2 @ B2 ) @ A6 @ Phi5 @ F8 @ S8 ) )
          | ? [S7: set_Pr4723104092121316667al_val,Pc2: val > $o,Witness2: produc2072440392815185925al_val > val,B6: val > $o,S8: set_Pr4723104092121316667al_val,Phi4: val,F3: val] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aSem_val @ B6 ) )
              & ( A62 = Phi4 )
              & ( A72 = F3 )
              & ( A82 = S8 )
              & ! [X: val,Y3: val,Z3: val > val,Aa3: val] :
                  ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ X @ ( produc303288909735261391al_val @ Y3 @ Z3 ) ) @ S7 )
                 => ( ( Pc2 @ X )
                   => ( ( Aa3
                        = ( Witness2 @ ( produc7187740677493149823al_val @ X @ ( produc303288909735261391al_val @ Y3 @ Z3 ) ) ) )
                     => ( ( sep_greater_val @ Plus @ X @ Aa3 )
                        & ( B6 @ Aa3 ) ) ) ) )
              & ( S8
                = ( sup_su4800132526891881063al_val
                  @ ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A2: val,U6: val,T5: val > val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) )
                        & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ S7 )
                        & ~ ( Pc2 @ A2 ) ) )
                  @ ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A2: val,U6: val,T5: val > val,B2: val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ ( sep_minus_val @ Plus @ Core @ A2 @ B2 ) @ ( produc303288909735261391al_val @ ( the_val @ ( Plus @ U6 @ B2 ) ) @ T5 ) ) )
                        & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ S7 )
                        & ( Pc2 @ A2 )
                        & ( B2
                          = ( Witness2 @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) ) ) ) ) ) ) )
          | ? [Phi4: val,Phi5: val,M4: val,F8: val,S8: set_Pr4723104092121316667al_val,Pc2: val > $o,A6: aassertion_val,Phi8: val,F9: val,S9: set_Pr4723104092121316667al_val,F3: val,S7: set_Pr4723104092121316667al_val] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52 = A6 )
              & ( A62 = Phi8 )
              & ( A72 = F9 )
              & ( A82 = S9 )
              & ( ( some_val @ Phi4 )
                = ( Plus @ Phi5 @ M4 ) )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi5 @ F8 @ S8 @ Pc2 @ A6 @ Phi8 @ F9 @ S9 )
              & ( Stable @ M4 )
              & ( ( some_val @ F8 )
                = ( Plus @ F3 @ M4 ) )
              & ( S8
                = ( collec8300311316619386842al_val
                  @ ^ [Uu3: produc2072440392815185925al_val] :
                    ? [A2: val,U6: val,T5: val > val,R: val] :
                      ( ( Uu3
                        = ( produc7187740677493149823al_val @ R @ ( produc303288909735261391al_val @ U6 @ T5 ) ) )
                      & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ S7 )
                      & ( ( some_val @ R )
                        = ( Plus @ A2 @ ( sep_minus_val @ Plus @ Core @ ( T5 @ F8 ) @ ( T5 @ F3 ) ) ) )
                      & ( sep_defined_val @ Plus @ R @ U6 ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.simps
thf(fact_873_package__logic_Opackage__rhs_Osimps,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A12: state,A23: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
        = ( ? [Phi4: state,F3: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,A6: aassertion_state,Phi5: state,F8: state,S8: set_Pr4231240917283394591_state,B6: aassertion_state,Phi8: state,F9: state,S9: set_Pr4231240917283394591_state] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aStar_state @ A6 @ B6 ) )
              & ( A62 = Phi8 )
              & ( A72 = F9 )
              & ( A82 = S9 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi4 @ F3 @ S7 @ Pc2 @ A6 @ Phi5 @ F8 @ S8 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi5 @ F8 @ S8 @ Pc2 @ B6 @ Phi8 @ F9 @ S9 ) )
          | ? [Phi4: state,F3: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,B2: state > $o,A6: aassertion_state,Phi5: state,F8: state,S8: set_Pr4231240917283394591_state] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aImp_state @ B2 @ A6 ) )
              & ( A62 = Phi5 )
              & ( A72 = F8 )
              & ( A82 = S8 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi4 @ F3 @ S7 @ ( packag1330488657391168505_state @ Pc2 @ B2 ) @ A6 @ Phi5 @ F8 @ S8 ) )
          | ? [S7: set_Pr4231240917283394591_state,Pc2: state > $o,Witness2: produc7272280185086996969_state > state,B6: state > $o,S8: set_Pr4231240917283394591_state,Phi4: state,F3: state] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aSem_state @ B6 ) )
              & ( A62 = Phi4 )
              & ( A72 = F3 )
              & ( A82 = S8 )
              & ! [X: state,Y3: state,Z3: state > state,Aa3: state] :
                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z3 ) ) @ S7 )
                 => ( ( Pc2 @ X )
                   => ( ( Aa3
                        = ( Witness2 @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z3 ) ) ) )
                     => ( ( sep_greater_state @ Plus @ X @ Aa3 )
                        & ( B6 @ Aa3 ) ) ) ) )
              & ( S8
                = ( sup_su3837536599209510219_state
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state,U6: state,T5: state > state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S7 )
                        & ~ ( Pc2 @ A2 ) ) )
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state,U6: state,T5: state > state,B2: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A2 @ B2 ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ B2 ) ) @ T5 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S7 )
                        & ( Pc2 @ A2 )
                        & ( B2
                          = ( Witness2 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) ) ) ) )
          | ? [Phi4: state,Phi5: state,M4: state,F8: state,S8: set_Pr4231240917283394591_state,Pc2: state > $o,A6: aassertion_state,Phi8: state,F9: state,S9: set_Pr4231240917283394591_state,F3: state,S7: set_Pr4231240917283394591_state] :
              ( ( A12 = Phi4 )
              & ( A23 = F3 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52 = A6 )
              & ( A62 = Phi8 )
              & ( A72 = F9 )
              & ( A82 = S9 )
              & ( ( some_state @ Phi4 )
                = ( Plus @ Phi5 @ M4 ) )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi5 @ F8 @ S8 @ Pc2 @ A6 @ Phi8 @ F9 @ S9 )
              & ( Stable @ M4 )
              & ( ( some_state @ F8 )
                = ( Plus @ F3 @ M4 ) )
              & ( S8
                = ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T5: state > state,R: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ R @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S7 )
                      & ( ( some_state @ R )
                        = ( Plus @ A2 @ ( sep_minus_state @ Plus @ Core @ ( T5 @ F8 ) @ ( T5 @ F3 ) ) ) )
                      & ( sep_defined_state @ Plus @ R @ U6 ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.simps
thf(fact_874_package__logic_Opackage__rhs_Ocases,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A12: val,A23: val,A32: set_Pr4723104092121316667al_val,A42: val > $o,A52: aassertion_val,A62: val,A72: val,A82: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
       => ( ! [A10: aassertion_val,Phi6: val,F7: val,S6: set_Pr4723104092121316667al_val,B8: aassertion_val] :
              ( ( A52
                = ( aStar_val @ A10 @ B8 ) )
             => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ A42 @ A10 @ Phi6 @ F7 @ S6 )
               => ~ ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi6 @ F7 @ S6 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
         => ( ! [B4: val > $o,A10: aassertion_val] :
                ( ( A52
                  = ( aImp_val @ B4 @ A10 ) )
               => ~ ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ ( packag6750194596558674946nj_val @ A42 @ B4 ) @ A10 @ A62 @ A72 @ A82 ) )
           => ( ! [Witness: produc2072440392815185925al_val > val,B8: val > $o] :
                  ( ( A52
                    = ( aSem_val @ B8 ) )
                 => ( ( A62 = A12 )
                   => ( ( A72 = A23 )
                     => ( ! [A11: val,U7: val,T6: val > val] :
                            ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A11 @ ( produc303288909735261391al_val @ U7 @ T6 ) ) @ A32 )
                           => ( ( A42 @ A11 )
                             => ( ( sep_greater_val @ Plus @ A11 @ ( Witness @ ( produc7187740677493149823al_val @ A11 @ ( produc303288909735261391al_val @ U7 @ T6 ) ) ) )
                                & ( B8 @ ( Witness @ ( produc7187740677493149823al_val @ A11 @ ( produc303288909735261391al_val @ U7 @ T6 ) ) ) ) ) ) )
                       => ( A82
                         != ( sup_su4800132526891881063al_val
                            @ ( collec8300311316619386842al_val
                              @ ^ [Uu3: produc2072440392815185925al_val] :
                                ? [A2: val,U6: val,T5: val > val] :
                                  ( ( Uu3
                                    = ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) )
                                  & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ A32 )
                                  & ~ ( A42 @ A2 ) ) )
                            @ ( collec8300311316619386842al_val
                              @ ^ [Uu3: produc2072440392815185925al_val] :
                                ? [A2: val,U6: val,T5: val > val] :
                                  ( ( Uu3
                                    = ( produc7187740677493149823al_val @ ( sep_minus_val @ Plus @ Core @ A2 @ ( Witness @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) ) ) @ ( produc303288909735261391al_val @ ( the_val @ ( Plus @ U6 @ ( Witness @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) ) ) ) @ T5 ) ) )
                                  & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ A32 )
                                  & ( A42 @ A2 ) ) ) ) ) ) ) ) )
             => ~ ! [Phi6: val,M3: val] :
                    ( ( ( some_val @ A12 )
                      = ( Plus @ Phi6 @ M3 ) )
                   => ! [F7: val,S6: set_Pr4723104092121316667al_val] :
                        ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi6 @ F7 @ S6 @ A42 @ A52 @ A62 @ A72 @ A82 )
                       => ( ( Stable @ M3 )
                         => ( ( ( some_val @ F7 )
                              = ( Plus @ A23 @ M3 ) )
                           => ( S6
                             != ( collec8300311316619386842al_val
                                @ ^ [Uu3: produc2072440392815185925al_val] :
                                  ? [A2: val,U6: val,T5: val > val,R: val] :
                                    ( ( Uu3
                                      = ( produc7187740677493149823al_val @ R @ ( produc303288909735261391al_val @ U6 @ T5 ) ) )
                                    & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ A32 )
                                    & ( ( some_val @ R )
                                      = ( Plus @ A2 @ ( sep_minus_val @ Plus @ Core @ ( T5 @ F7 ) @ ( T5 @ A23 ) ) ) )
                                    & ( sep_defined_val @ Plus @ R @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.cases
thf(fact_875_package__logic_Opackage__rhs_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A12: state,A23: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
       => ( ! [A10: aassertion_state,Phi6: state,F7: state,S6: set_Pr4231240917283394591_state,B8: aassertion_state] :
              ( ( A52
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ A42 @ A10 @ Phi6 @ F7 @ S6 )
               => ~ ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi6 @ F7 @ S6 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( A52
                  = ( aImp_state @ B4 @ A10 ) )
               => ~ ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A12 @ A23 @ A32 @ ( packag1330488657391168505_state @ A42 @ B4 ) @ A10 @ A62 @ A72 @ A82 ) )
           => ( ! [Witness: produc7272280185086996969_state > state,B8: state > $o] :
                  ( ( A52
                    = ( aSem_state @ B8 ) )
                 => ( ( A62 = A12 )
                   => ( ( A72 = A23 )
                     => ( ! [A11: state,U7: state,T6: state > state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A11 @ ( produc8632743443909064678_state @ U7 @ T6 ) ) @ A32 )
                           => ( ( A42 @ A11 )
                             => ( ( sep_greater_state @ Plus @ A11 @ ( Witness @ ( produc5886875603596089627_state @ A11 @ ( produc8632743443909064678_state @ U7 @ T6 ) ) ) )
                                & ( B8 @ ( Witness @ ( produc5886875603596089627_state @ A11 @ ( produc8632743443909064678_state @ U7 @ T6 ) ) ) ) ) ) )
                       => ( A82
                         != ( sup_su3837536599209510219_state
                            @ ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A2: state,U6: state,T5: state > state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
                                  & ~ ( A42 @ A2 ) ) )
                            @ ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A2: state,U6: state,T5: state > state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A2 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) @ T5 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
                                  & ( A42 @ A2 ) ) ) ) ) ) ) ) )
             => ~ ! [Phi6: state,M3: state] :
                    ( ( ( some_state @ A12 )
                      = ( Plus @ Phi6 @ M3 ) )
                   => ! [F7: state,S6: set_Pr4231240917283394591_state] :
                        ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi6 @ F7 @ S6 @ A42 @ A52 @ A62 @ A72 @ A82 )
                       => ( ( Stable @ M3 )
                         => ( ( ( some_state @ F7 )
                              = ( Plus @ A23 @ M3 ) )
                           => ( S6
                             != ( collec906845016635587262_state
                                @ ^ [Uu3: produc7272280185086996969_state] :
                                  ? [A2: state,U6: state,T5: state > state,R: state] :
                                    ( ( Uu3
                                      = ( produc5886875603596089627_state @ R @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
                                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
                                    & ( ( some_state @ R )
                                      = ( Plus @ A2 @ ( sep_minus_state @ Plus @ Core @ ( T5 @ F7 ) @ ( T5 @ A23 ) ) ) )
                                    & ( sep_defined_state @ Plus @ R @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.cases
thf(fact_876_PartialSA_Owf__assertion_Opelims_I1_J,axiom,
    ! [X2: aassertion_state,Y: $o] :
      ( ( ( packag7298478212905792770_state @ plus @ core @ X2 )
        = Y )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( Y
                  = ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
               => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A10 @ B8 ) ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( ( Y
                    = ( ( packag6595153354952283300_state @ plus @ core @ B4 )
                      & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
                 => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B4 @ A10 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( ( Y
                      = ( packag2456304381420842418_state @ plus @ A10 ) )
                   => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(1)
thf(fact_877_PartialSA_Owf__assertion_Opelims_I2_J,axiom,
    ! [X2: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A10 @ B8 ) )
               => ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B4 @ A10 ) )
                 => ~ ( ( packag6595153354952283300_state @ plus @ core @ B4 )
                      & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) )
                   => ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(2)
thf(fact_878_PartialSA_Owf__assertion_Opelims_I3_J,axiom,
    ! [X2: aassertion_state] :
      ( ~ ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A10 @ B8 ) )
               => ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                  & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B4 @ A10 ) )
                 => ( ( packag6595153354952283300_state @ plus @ core @ B4 )
                    & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) )
                   => ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(3)
thf(fact_879_package__logic_Osat_Opelims_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X2: aassertion_val,Xa: val,Y: $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_val @ Plus @ X2 @ Xa )
          = Y )
       => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ X2 @ Xa ) )
         => ( ! [A10: aassertion_val,B8: aassertion_val] :
                ( ( X2
                  = ( aStar_val @ A10 @ B8 ) )
               => ( ( Y
                    = ( ? [A2: val,B2: val] :
                          ( ( ( some_val @ Xa )
                            = ( Plus @ A2 @ B2 ) )
                          & ( package_sat_val @ Plus @ A10 @ A2 )
                          & ( package_sat_val @ Plus @ B8 @ B2 ) ) ) )
                 => ~ ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aStar_val @ A10 @ B8 ) @ Xa ) ) ) )
           => ( ! [B4: val > $o,A10: aassertion_val] :
                  ( ( X2
                    = ( aImp_val @ B4 @ A10 ) )
                 => ( ( Y
                      = ( ( B4 @ Xa )
                       => ( package_sat_val @ Plus @ A10 @ Xa ) ) )
                   => ~ ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aImp_val @ B4 @ A10 ) @ Xa ) ) ) )
             => ~ ! [A10: val > $o] :
                    ( ( X2
                      = ( aSem_val @ A10 ) )
                   => ( ( Y
                        = ( A10 @ Xa ) )
                     => ~ ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aSem_val @ A10 ) @ Xa ) ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(1)
thf(fact_880_package__logic_Osat_Opelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa: state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_state @ Plus @ X2 @ Xa )
          = Y )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa ) )
         => ( ! [A10: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A10 @ B8 ) )
               => ( ( Y
                    = ( ? [A2: state,B2: state] :
                          ( ( ( some_state @ Xa )
                            = ( Plus @ A2 @ B2 ) )
                          & ( package_sat_state @ Plus @ A10 @ A2 )
                          & ( package_sat_state @ Plus @ B8 @ B2 ) ) ) )
                 => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) ) ) )
           => ( ! [B4: state > $o,A10: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B4 @ A10 ) )
                 => ( ( Y
                      = ( ( B4 @ Xa )
                       => ( package_sat_state @ Plus @ A10 @ Xa ) ) )
                   => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Xa ) ) ) )
             => ~ ! [A10: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A10 ) )
                   => ( ( Y
                        = ( A10 @ Xa ) )
                     => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(1)
thf(fact_881_case__prod__Pair,axiom,
    ( ( produc803252138055838771_state @ produc9160152616361873709_state )
    = id_Pro8715877757508063718_state ) ).

% case_prod_Pair
thf(fact_882_case__prod__Pair,axiom,
    ( ( produc1922103714420728243_state @ produc6205719116230455971_state )
    = id_Pro5916511905602125660_state ) ).

% case_prod_Pair
thf(fact_883_case__prod__Pair,axiom,
    ( ( produc897481912382009139_state @ produc9080284787853255769_state )
    = id_Pro8860473219607035410_state ) ).

% case_prod_Pair
thf(fact_884_case__prod__Pair,axiom,
    ( ( produc4071494739252518515t_char @ produc120197956887798346t_char )
    = id_Pro6264179822262827651t_char ) ).

% case_prod_Pair
thf(fact_885_case__prod__Pair,axiom,
    ( ( produc4192255817509754207on_val @ produc74620467203698198on_val )
    = id_Pro1402184692240358763on_val ) ).

% case_prod_Pair
thf(fact_886_mem__case__prodI,axiom,
    ! [Z: state,C: prat > state > set_state,A: prat,B: state] :
      ( ( member_state @ Z @ ( C @ A @ B ) )
     => ( member_state @ Z @ ( produc2825319096956678952_state @ C @ ( produc6205719116230455971_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_887_mem__case__prodI,axiom,
    ! [Z: state,C: prat > set_state > set_state,A: prat,B: set_state] :
      ( ( member_state @ Z @ ( C @ A @ B ) )
     => ( member_state @ Z @ ( produc5034060725774254814_state @ C @ ( produc9080284787853255769_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_888_mem__case__prodI,axiom,
    ! [Z: state,C: nat > list_char > set_state,A: nat,B: list_char] :
      ( ( member_state @ Z @ ( C @ A @ B ) )
     => ( member_state @ Z @ ( produc7881066356751411151_state @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_889_mem__case__prodI,axiom,
    ! [Z: state,C: option_state > option_state > set_state,A: option_state,B: option_state] :
      ( ( member_state @ Z @ ( C @ A @ B ) )
     => ( member_state @ Z @ ( produc3424155258940669874_state @ C @ ( produc9160152616361873709_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_890_mem__case__prodI,axiom,
    ! [Z: produc3882902684156697774_state,C: prat > state > set_Pr2391701600042252046_state,A: prat,B: state] :
      ( ( member6715010751053357271_state @ Z @ ( C @ A @ B ) )
     => ( member6715010751053357271_state @ Z @ ( produc4369308447073708118_state @ C @ ( produc6205719116230455971_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_891_mem__case__prodI,axiom,
    ! [Z: produc3882902684156697774_state,C: prat > set_state > set_Pr2391701600042252046_state,A: prat,B: set_state] :
      ( ( member6715010751053357271_state @ Z @ ( C @ A @ B ) )
     => ( member6715010751053357271_state @ Z @ ( produc9043967962556017184_state @ C @ ( produc9080284787853255769_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_892_mem__case__prodI,axiom,
    ! [Z: produc3882902684156697774_state,C: nat > list_char > set_Pr2391701600042252046_state,A: nat,B: list_char] :
      ( ( member6715010751053357271_state @ Z @ ( C @ A @ B ) )
     => ( member6715010751053357271_state @ Z @ ( produc3312000583120074607_state @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_893_mem__case__prodI,axiom,
    ! [Z: produc3882902684156697774_state,C: option_state > option_state > set_Pr2391701600042252046_state,A: option_state,B: option_state] :
      ( ( member6715010751053357271_state @ Z @ ( C @ A @ B ) )
     => ( member6715010751053357271_state @ Z @ ( produc2209043727768083020_state @ C @ ( produc9160152616361873709_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_894_mem__case__prodI,axiom,
    ! [Z: produc7272280185086996969_state,C: prat > state > set_Pr4231240917283394591_state,A: prat,B: state] :
      ( ( member7461423308196221312_state @ Z @ ( C @ A @ B ) )
     => ( member7461423308196221312_state @ Z @ ( produc569869127943504087_state @ C @ ( produc6205719116230455971_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_895_mem__case__prodI,axiom,
    ! [Z: produc7272280185086996969_state,C: prat > set_state > set_Pr4231240917283394591_state,A: prat,B: set_state] :
      ( ( member7461423308196221312_state @ Z @ ( C @ A @ B ) )
     => ( member7461423308196221312_state @ Z @ ( produc7793558081207154829_state @ C @ ( produc9080284787853255769_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_896_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z: state,C: prat > state > set_state] :
      ( ! [A5: prat,B4: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A5 @ B4 ) )
         => ( member_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_state @ Z @ ( produc2825319096956678952_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_897_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z: state,C: prat > set_state > set_state] :
      ( ! [A5: prat,B4: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A5 @ B4 ) )
         => ( member_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_state @ Z @ ( produc5034060725774254814_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_898_mem__case__prodI2,axiom,
    ! [P2: produc1457211279475724562t_char,Z: state,C: nat > list_char > set_state] :
      ( ! [A5: nat,B4: list_char] :
          ( ( P2
            = ( produc120197956887798346t_char @ A5 @ B4 ) )
         => ( member_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_state @ Z @ ( produc7881066356751411151_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_899_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z: state,C: option_state > option_state > set_state] :
      ( ! [A5: option_state,B4: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A5 @ B4 ) )
         => ( member_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_state @ Z @ ( produc3424155258940669874_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_900_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z: produc3882902684156697774_state,C: prat > state > set_Pr2391701600042252046_state] :
      ( ! [A5: prat,B4: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A5 @ B4 ) )
         => ( member6715010751053357271_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member6715010751053357271_state @ Z @ ( produc4369308447073708118_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_901_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z: produc3882902684156697774_state,C: prat > set_state > set_Pr2391701600042252046_state] :
      ( ! [A5: prat,B4: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A5 @ B4 ) )
         => ( member6715010751053357271_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member6715010751053357271_state @ Z @ ( produc9043967962556017184_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_902_mem__case__prodI2,axiom,
    ! [P2: produc1457211279475724562t_char,Z: produc3882902684156697774_state,C: nat > list_char > set_Pr2391701600042252046_state] :
      ( ! [A5: nat,B4: list_char] :
          ( ( P2
            = ( produc120197956887798346t_char @ A5 @ B4 ) )
         => ( member6715010751053357271_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member6715010751053357271_state @ Z @ ( produc3312000583120074607_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_903_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z: produc3882902684156697774_state,C: option_state > option_state > set_Pr2391701600042252046_state] :
      ( ! [A5: option_state,B4: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A5 @ B4 ) )
         => ( member6715010751053357271_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member6715010751053357271_state @ Z @ ( produc2209043727768083020_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_904_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z: produc7272280185086996969_state,C: prat > state > set_Pr4231240917283394591_state] :
      ( ! [A5: prat,B4: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A5 @ B4 ) )
         => ( member7461423308196221312_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member7461423308196221312_state @ Z @ ( produc569869127943504087_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_905_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z: produc7272280185086996969_state,C: prat > set_state > set_Pr4231240917283394591_state] :
      ( ! [A5: prat,B4: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A5 @ B4 ) )
         => ( member7461423308196221312_state @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member7461423308196221312_state @ Z @ ( produc7793558081207154829_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_906_mem__case__prodE,axiom,
    ! [Z: state,C: prat > state > set_state,P2: produc4844402304972151915_state] :
      ( ( member_state @ Z @ ( produc2825319096956678952_state @ C @ P2 ) )
     => ~ ! [X3: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X3 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_907_mem__case__prodE,axiom,
    ! [Z: state,C: prat > set_state > set_state,P2: produc6251377953716854561_state] :
      ( ( member_state @ Z @ ( produc5034060725774254814_state @ C @ P2 ) )
     => ~ ! [X3: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X3 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_908_mem__case__prodE,axiom,
    ! [Z: state,C: nat > list_char > set_state,P2: produc1457211279475724562t_char] :
      ( ( member_state @ Z @ ( produc7881066356751411151_state @ C @ P2 ) )
     => ~ ! [X3: nat,Y4: list_char] :
            ( ( P2
              = ( produc120197956887798346t_char @ X3 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_909_mem__case__prodE,axiom,
    ! [Z: state,C: option_state > option_state > set_state,P2: produc3142500478612311029_state] :
      ( ( member_state @ Z @ ( produc3424155258940669874_state @ C @ P2 ) )
     => ~ ! [X3: option_state,Y4: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X3 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_910_mem__case__prodE,axiom,
    ! [Z: produc3882902684156697774_state,C: prat > state > set_Pr2391701600042252046_state,P2: produc4844402304972151915_state] :
      ( ( member6715010751053357271_state @ Z @ ( produc4369308447073708118_state @ C @ P2 ) )
     => ~ ! [X3: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X3 @ Y4 ) )
           => ~ ( member6715010751053357271_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_911_mem__case__prodE,axiom,
    ! [Z: produc3882902684156697774_state,C: prat > set_state > set_Pr2391701600042252046_state,P2: produc6251377953716854561_state] :
      ( ( member6715010751053357271_state @ Z @ ( produc9043967962556017184_state @ C @ P2 ) )
     => ~ ! [X3: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X3 @ Y4 ) )
           => ~ ( member6715010751053357271_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_912_mem__case__prodE,axiom,
    ! [Z: produc3882902684156697774_state,C: nat > list_char > set_Pr2391701600042252046_state,P2: produc1457211279475724562t_char] :
      ( ( member6715010751053357271_state @ Z @ ( produc3312000583120074607_state @ C @ P2 ) )
     => ~ ! [X3: nat,Y4: list_char] :
            ( ( P2
              = ( produc120197956887798346t_char @ X3 @ Y4 ) )
           => ~ ( member6715010751053357271_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_913_mem__case__prodE,axiom,
    ! [Z: produc3882902684156697774_state,C: option_state > option_state > set_Pr2391701600042252046_state,P2: produc3142500478612311029_state] :
      ( ( member6715010751053357271_state @ Z @ ( produc2209043727768083020_state @ C @ P2 ) )
     => ~ ! [X3: option_state,Y4: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X3 @ Y4 ) )
           => ~ ( member6715010751053357271_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_914_mem__case__prodE,axiom,
    ! [Z: produc7272280185086996969_state,C: prat > state > set_Pr4231240917283394591_state,P2: produc4844402304972151915_state] :
      ( ( member7461423308196221312_state @ Z @ ( produc569869127943504087_state @ C @ P2 ) )
     => ~ ! [X3: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X3 @ Y4 ) )
           => ~ ( member7461423308196221312_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_915_mem__case__prodE,axiom,
    ! [Z: produc7272280185086996969_state,C: prat > set_state > set_Pr4231240917283394591_state,P2: produc6251377953716854561_state] :
      ( ( member7461423308196221312_state @ Z @ ( produc7793558081207154829_state @ C @ P2 ) )
     => ~ ! [X3: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X3 @ Y4 ) )
           => ~ ( member7461423308196221312_state @ Z @ ( C @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_916_same__fst__def,axiom,
    ( same_f3145481655561538747_state
    = ( ^ [P6: state > $o,R6: state > set_Pr5527899070135025255_state] :
          ( collec2083760461196787346_state
          @ ( produc7167296152490707052tate_o
            @ ( produc8588859638187112394tate_o
              @ ^ [X7: state,Y7: state > state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [X: state,Y3: state > state] :
                      ( ( X7 = X )
                      & ( P6 @ X )
                      & ( member6978531283771894448_state @ ( produc5362444129537099455_state @ Y7 @ Y3 ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_917_same__fst__def,axiom,
    ( same_f8765775015391913222_state
    = ( ^ [P6: state > $o,R6: state > set_Pr4846723933414538759_state] :
          ( collec4625684881440051358_state
          @ ( produc3133855996611560728tate_o
            @ ( produc6466221716036377566tate_o
              @ ^ [X7: state,Y7: produc3882902684156697774_state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [X: state,Y3: produc3882902684156697774_state] :
                      ( ( X7 = X )
                      & ( P6 @ X )
                      & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ Y7 @ Y3 ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_918_PartialSA_Osat_Opelims_I3_J,axiom,
    ! [X2: aassertion_state,Xa: state] :
      ( ~ ( package_sat_state @ plus @ X2 @ Xa )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa ) )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) )
               => ? [A11: state,B9: state] :
                    ( ( ( some_state @ Xa )
                      = ( plus @ A11 @ B9 ) )
                    & ( package_sat_state @ plus @ A10 @ A11 )
                    & ( package_sat_state @ plus @ B8 @ B9 ) ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Xa ) )
                 => ( ( B4 @ Xa )
                   => ( package_sat_state @ plus @ A10 @ Xa ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) )
                   => ( A10 @ Xa ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(3)
thf(fact_919_PartialSA_Osat_Opelims_I2_J,axiom,
    ! [X2: aassertion_state,Xa: state] :
      ( ( package_sat_state @ plus @ X2 @ Xa )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa ) )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) )
               => ~ ? [A5: state,B4: state] :
                      ( ( ( some_state @ Xa )
                        = ( plus @ A5 @ B4 ) )
                      & ( package_sat_state @ plus @ A10 @ A5 )
                      & ( package_sat_state @ plus @ B8 @ B4 ) ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Xa ) )
                 => ~ ( ( B4 @ Xa )
                     => ( package_sat_state @ plus @ A10 @ Xa ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) )
                   => ~ ( A10 @ Xa ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(2)
thf(fact_920_PartialSA_Osat_Opelims_I1_J,axiom,
    ! [X2: aassertion_state,Xa: state,Y: $o] :
      ( ( ( package_sat_state @ plus @ X2 @ Xa )
        = Y )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa ) )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( Y
                  = ( ? [A2: state,B2: state] :
                        ( ( ( some_state @ Xa )
                          = ( plus @ A2 @ B2 ) )
                        & ( package_sat_state @ plus @ A10 @ A2 )
                        & ( package_sat_state @ plus @ B8 @ B2 ) ) ) )
               => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) ) ) )
         => ( ! [B4: state > $o,A10: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B4 @ A10 ) )
               => ( ( Y
                    = ( ( B4 @ Xa )
                     => ( package_sat_state @ plus @ A10 @ Xa ) ) )
                 => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Xa ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A10 ) )
                 => ( ( Y
                      = ( A10 @ Xa ) )
                   => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(1)
thf(fact_921_package__logic_Osat_Opelims_I3_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X2: aassertion_val,Xa: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_val @ Plus @ X2 @ Xa )
       => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ X2 @ Xa ) )
         => ( ! [A10: aassertion_val,B8: aassertion_val] :
                ( ( X2
                  = ( aStar_val @ A10 @ B8 ) )
               => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aStar_val @ A10 @ B8 ) @ Xa ) )
                 => ? [A11: val,B9: val] :
                      ( ( ( some_val @ Xa )
                        = ( Plus @ A11 @ B9 ) )
                      & ( package_sat_val @ Plus @ A10 @ A11 )
                      & ( package_sat_val @ Plus @ B8 @ B9 ) ) ) )
           => ( ! [B4: val > $o,A10: aassertion_val] :
                  ( ( X2
                    = ( aImp_val @ B4 @ A10 ) )
                 => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aImp_val @ B4 @ A10 ) @ Xa ) )
                   => ( ( B4 @ Xa )
                     => ( package_sat_val @ Plus @ A10 @ Xa ) ) ) )
             => ~ ! [A10: val > $o] :
                    ( ( X2
                      = ( aSem_val @ A10 ) )
                   => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aSem_val @ A10 ) @ Xa ) )
                     => ( A10 @ Xa ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(3)
thf(fact_922_package__logic_Osat_Opelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_state @ Plus @ X2 @ Xa )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa ) )
         => ( ! [A10: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A10 @ B8 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) )
                 => ? [A11: state,B9: state] :
                      ( ( ( some_state @ Xa )
                        = ( Plus @ A11 @ B9 ) )
                      & ( package_sat_state @ Plus @ A10 @ A11 )
                      & ( package_sat_state @ Plus @ B8 @ B9 ) ) ) )
           => ( ! [B4: state > $o,A10: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B4 @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Xa ) )
                   => ( ( B4 @ Xa )
                     => ( package_sat_state @ Plus @ A10 @ Xa ) ) ) )
             => ~ ! [A10: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A10 ) )
                   => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) )
                     => ( A10 @ Xa ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(3)
thf(fact_923_package__logic_Osat_Opelims_I2_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X2: aassertion_val,Xa: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ X2 @ Xa )
       => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ X2 @ Xa ) )
         => ( ! [A10: aassertion_val,B8: aassertion_val] :
                ( ( X2
                  = ( aStar_val @ A10 @ B8 ) )
               => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aStar_val @ A10 @ B8 ) @ Xa ) )
                 => ~ ? [A5: val,B4: val] :
                        ( ( ( some_val @ Xa )
                          = ( Plus @ A5 @ B4 ) )
                        & ( package_sat_val @ Plus @ A10 @ A5 )
                        & ( package_sat_val @ Plus @ B8 @ B4 ) ) ) )
           => ( ! [B4: val > $o,A10: aassertion_val] :
                  ( ( X2
                    = ( aImp_val @ B4 @ A10 ) )
                 => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aImp_val @ B4 @ A10 ) @ Xa ) )
                   => ~ ( ( B4 @ Xa )
                       => ( package_sat_val @ Plus @ A10 @ Xa ) ) ) )
             => ~ ! [A10: val > $o] :
                    ( ( X2
                      = ( aSem_val @ A10 ) )
                   => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aSem_val @ A10 ) @ Xa ) )
                     => ~ ( A10 @ Xa ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(2)
thf(fact_924_package__logic_Osat_Opelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ X2 @ Xa )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa ) )
         => ( ! [A10: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A10 @ B8 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) )
                 => ~ ? [A5: state,B4: state] :
                        ( ( ( some_state @ Xa )
                          = ( Plus @ A5 @ B4 ) )
                        & ( package_sat_state @ Plus @ A10 @ A5 )
                        & ( package_sat_state @ Plus @ B8 @ B4 ) ) ) )
           => ( ! [B4: state > $o,A10: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B4 @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Xa ) )
                   => ~ ( ( B4 @ Xa )
                       => ( package_sat_state @ Plus @ A10 @ Xa ) ) ) )
             => ~ ! [A10: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A10 ) )
                   => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) )
                     => ~ ( A10 @ Xa ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(2)
thf(fact_925_PartialSA_Ocompleteness__aux,axiom,
    ! [S2: set_Pr4231240917283394591_state,F1: state > state > ( state > state ) > state,F22: state > state > ( state > state ) > state,Pc: state > $o,A3: aassertion_state,F4: state,Phi2: state] :
      ( ! [A5: state,U3: state,T4: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
         => ( ( greater @ ( core @ ( F1 @ A5 @ U3 @ T4 ) ) @ ( core @ A5 ) )
            & ( ( some_state @ A5 )
              = ( plus @ ( F1 @ A5 @ U3 @ T4 ) @ ( F22 @ A5 @ U3 @ T4 ) ) )
            & ( ( Pc @ ( core @ A5 ) )
             => ( package_sat_state @ plus @ A3 @ ( the_state @ ( plus @ ( core @ A5 ) @ ( F1 @ A5 @ U3 @ T4 ) ) ) ) ) ) )
     => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F4 )
       => ( ( packag7298478212905792770_state @ plus @ core @ A3 )
         => ( ( packag6595153354952283300_state @ plus @ core @ Pc )
           => ( ( ( stable @ F4 )
                & ( defined @ Phi2 @ F4 ) )
             => ? [S6: set_Pr4231240917283394591_state] :
                  ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi2 @ F4 @ S6 )
                  & ! [X6: produc7272280185086996969_state] :
                      ( ( member7461423308196221312_state @ X6 @ S6 )
                     => ( produc1745419677418529872tate_o
                        @ ^ [A13: state] :
                            ( produc3645877548210629341tate_o
                            @ ^ [U8: state,T5: state > state] :
                              ? [A2: state,U6: state] :
                                ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                                & ( greater @ A13 @ ( F22 @ A2 @ U6 @ T5 ) )
                                & ( ( core @ A13 )
                                  = ( core @ A2 ) ) ) )
                        @ X6 ) ) ) ) ) ) ) ) ).

% PartialSA.completeness_aux
thf(fact_926_PartialSA_Opackage__rhs__connection__def,axiom,
    ! [Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
      = ( ( greater @ F5 @ F4 )
        & ( defined @ Phi2 @ F4 )
        & ( ( plus @ Phi2 @ F4 )
          = ( plus @ Phi3 @ F5 ) )
        & ( stable @ F5 )
        & ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A2: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,T5: state > state] :
                    ? [Au3: state] :
                      ( ( ( some_state @ Au3 )
                        = ( plus @ A2 @ U6 ) )
                      & ( ( defined @ Au3 @ ( minus @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) )
                       => ? [A13: state,U8: state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A13 @ ( produc8632743443909064678_state @ U8 @ T5 ) ) @ S3 )
                            & ( greater @ ( core @ A13 ) @ ( core @ A2 ) )
                            & ( ( plus @ Au3 @ ( minus @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) )
                              = ( plus @ A13 @ U8 ) )
                            & ( greater @ U8 @ U6 )
                            & ( packag8070727096299822599_state @ plus @ core @ Pc @ A3 @ A13 @ U8 @ U6 ) ) ) ) )
              @ X ) ) ) ) ).

% PartialSA.package_rhs_connection_def
thf(fact_927_package__logic_Ocompleteness__aux,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,S2: set_Pr4723104092121316667al_val,F1: val > val > ( val > val ) > val,F22: val > val > ( val > val ) > val,Pc: val > $o,A3: aassertion_val,F4: val,Phi2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: val,U3: val,T4: val > val] :
            ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A5 @ ( produc303288909735261391al_val @ U3 @ T4 ) ) @ S2 )
           => ( ( sep_greater_val @ Plus @ ( Core @ ( F1 @ A5 @ U3 @ T4 ) ) @ ( Core @ A5 ) )
              & ( ( some_val @ A5 )
                = ( Plus @ ( F1 @ A5 @ U3 @ T4 ) @ ( F22 @ A5 @ U3 @ T4 ) ) )
              & ( ( Pc @ ( Core @ A5 ) )
               => ( package_sat_val @ Plus @ A3 @ ( the_val @ ( Plus @ ( Core @ A5 ) @ ( F1 @ A5 @ U3 @ T4 ) ) ) ) ) ) )
       => ( ( packag7517340229427718612et_val @ Plus @ Core @ Unit @ S2 @ F4 )
         => ( ( packag5732365195153136185on_val @ Plus @ Core @ A3 )
           => ( ( packag6714868506247462167nd_val @ Plus @ Core @ Pc )
             => ( ( ( Stable @ F4 )
                  & ( sep_defined_val @ Plus @ Phi2 @ F4 ) )
               => ? [S6: set_Pr4723104092121316667al_val] :
                    ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi2 @ F4 @ S6 )
                    & ! [X6: produc2072440392815185925al_val] :
                        ( ( member5640714103519202460al_val @ X6 @ S6 )
                       => ( produc6995717860488419308_val_o
                          @ ^ [A13: val] :
                              ( produc7666680587469814684_val_o
                              @ ^ [U8: val,T5: val > val] :
                                ? [A2: val,U6: val] :
                                  ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A2 @ ( produc303288909735261391al_val @ U6 @ T5 ) ) @ S2 )
                                  & ( sep_greater_val @ Plus @ A13 @ ( F22 @ A2 @ U6 @ T5 ) )
                                  & ( ( Core @ A13 )
                                    = ( Core @ A2 ) ) ) )
                          @ X6 ) ) ) ) ) ) ) ) ) ).

% package_logic.completeness_aux
thf(fact_928_package__logic_Ocompleteness__aux,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F1: state > state > ( state > state ) > state,F22: state > state > ( state > state ) > state,Pc: state > $o,A3: aassertion_state,F4: state,Phi2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A5: state,U3: state,T4: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U3 @ T4 ) ) @ S2 )
           => ( ( sep_greater_state @ Plus @ ( Core @ ( F1 @ A5 @ U3 @ T4 ) ) @ ( Core @ A5 ) )
              & ( ( some_state @ A5 )
                = ( Plus @ ( F1 @ A5 @ U3 @ T4 ) @ ( F22 @ A5 @ U3 @ T4 ) ) )
              & ( ( Pc @ ( Core @ A5 ) )
               => ( package_sat_state @ Plus @ A3 @ ( the_state @ ( Plus @ ( Core @ A5 ) @ ( F1 @ A5 @ U3 @ T4 ) ) ) ) ) ) )
       => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F4 )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A3 )
           => ( ( packag6595153354952283300_state @ Plus @ Core @ Pc )
             => ( ( ( Stable @ F4 )
                  & ( sep_defined_state @ Plus @ Phi2 @ F4 ) )
               => ? [S6: set_Pr4231240917283394591_state] :
                    ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi2 @ F4 @ S6 )
                    & ! [X6: produc7272280185086996969_state] :
                        ( ( member7461423308196221312_state @ X6 @ S6 )
                       => ( produc1745419677418529872tate_o
                          @ ^ [A13: state] :
                              ( produc3645877548210629341tate_o
                              @ ^ [U8: state,T5: state > state] :
                                ? [A2: state,U6: state] :
                                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S2 )
                                  & ( sep_greater_state @ Plus @ A13 @ ( F22 @ A2 @ U6 @ T5 ) )
                                  & ( ( Core @ A13 )
                                    = ( Core @ A2 ) ) ) )
                          @ X6 ) ) ) ) ) ) ) ) ) ).

% package_logic.completeness_aux
thf(fact_929_PartialSA_Opure__remains__def,axiom,
    ! [S2: set_Pr523392378426063460_state] :
      ( ( packag9013619817833129428_state @ plus @ S2 )
      = ( ! [X: produc4357056439811976324_state] :
            ( ( member5752234768399382829_state @ X @ S2 )
           => ( produc8801812494870056711tate_o
              @ ^ [A2: state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [U6: state,P3: produc3882902684156697774_state] : ( sep_pure_state @ plus @ A2 ) )
              @ X ) ) ) ) ).

% PartialSA.pure_remains_def
thf(fact_930_PartialSA_Opure__remains__def,axiom,
    ! [S2: set_Pr4231240917283394591_state] :
      ( ( packag4152378406607043437_state @ plus @ S2 )
      = ( ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A2: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,P3: state > state] : ( sep_pure_state @ plus @ A2 ) )
              @ X ) ) ) ) ).

% PartialSA.pure_remains_def
thf(fact_931_PartialSA_Ovalid__package__set__def,axiom,
    ! [S2: set_Pr4231240917283394591_state,F4: state] :
      ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F4 )
      = ( ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A2: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,T5: state > state] :
                      ( ( defined @ A2 @ U6 )
                      & ( greater @ ( core @ A2 ) @ ( core @ U6 ) )
                      & ( packag4817583915586052229_state @ plus @ unit @ T5 )
                      & ( greater @ A2 @ ( core @ ( T5 @ F4 ) ) ) ) )
              @ X ) ) ) ) ).

% PartialSA.valid_package_set_def
thf(fact_932_package__logic_Opackage__rhs__connection__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi2: val,F4: val,S2: set_Pr4723104092121316667al_val,Pc: val > $o,A3: aassertion_val,Phi3: val,F5: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag1944724486178758143on_val @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
        = ( ( sep_greater_val @ Plus @ F5 @ F4 )
          & ( sep_defined_val @ Plus @ Phi2 @ F4 )
          & ( ( Plus @ Phi2 @ F4 )
            = ( Plus @ Phi3 @ F5 ) )
          & ( Stable @ F5 )
          & ! [X: produc2072440392815185925al_val] :
              ( ( member5640714103519202460al_val @ X @ S2 )
             => ( produc6995717860488419308_val_o
                @ ^ [A2: val] :
                    ( produc7666680587469814684_val_o
                    @ ^ [U6: val,T5: val > val] :
                      ? [Au3: val] :
                        ( ( ( some_val @ Au3 )
                          = ( Plus @ A2 @ U6 ) )
                        & ( ( sep_defined_val @ Plus @ Au3 @ ( sep_minus_val @ Plus @ Core @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) )
                         => ? [A13: val,U8: val] :
                              ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A13 @ ( produc303288909735261391al_val @ U8 @ T5 ) ) @ S3 )
                              & ( sep_greater_val @ Plus @ ( Core @ A13 ) @ ( Core @ A2 ) )
                              & ( ( Plus @ Au3 @ ( sep_minus_val @ Plus @ Core @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) )
                                = ( Plus @ A13 @ U8 ) )
                              & ( sep_greater_val @ Plus @ U8 @ U6 )
                              & ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A3 @ A13 @ U8 @ U6 ) ) ) ) )
                @ X ) ) ) ) ) ).

% package_logic.package_rhs_connection_def
thf(fact_933_package__logic_Opackage__rhs__connection__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi2: state,F4: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A3: aassertion_state,Phi3: state,F5: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi2 @ F4 @ S2 @ Pc @ A3 @ Phi3 @ F5 @ S3 )
        = ( ( sep_greater_state @ Plus @ F5 @ F4 )
          & ( sep_defined_state @ Plus @ Phi2 @ F4 )
          & ( ( Plus @ Phi2 @ F4 )
            = ( Plus @ Phi3 @ F5 ) )
          & ( Stable @ F5 )
          & ! [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ S2 )
             => ( produc1745419677418529872tate_o
                @ ^ [A2: state] :
                    ( produc3645877548210629341tate_o
                    @ ^ [U6: state,T5: state > state] :
                      ? [Au3: state] :
                        ( ( ( some_state @ Au3 )
                          = ( Plus @ A2 @ U6 ) )
                        & ( ( sep_defined_state @ Plus @ Au3 @ ( sep_minus_state @ Plus @ Core @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) )
                         => ? [A13: state,U8: state] :
                              ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A13 @ ( produc8632743443909064678_state @ U8 @ T5 ) ) @ S3 )
                              & ( sep_greater_state @ Plus @ ( Core @ A13 ) @ ( Core @ A2 ) )
                              & ( ( Plus @ Au3 @ ( sep_minus_state @ Plus @ Core @ ( T5 @ F5 ) @ ( T5 @ F4 ) ) )
                                = ( Plus @ A13 @ U8 ) )
                              & ( sep_greater_state @ Plus @ U8 @ U6 )
                              & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A3 @ A13 @ U8 @ U6 ) ) ) ) )
                @ X ) ) ) ) ) ).

% package_logic.package_rhs_connection_def
thf(fact_934_case__prod__Pair__iden,axiom,
    ! [P2: produc3142500478612311029_state] :
      ( ( produc803252138055838771_state @ produc9160152616361873709_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_935_case__prod__Pair__iden,axiom,
    ! [P2: produc4844402304972151915_state] :
      ( ( produc1922103714420728243_state @ produc6205719116230455971_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_936_case__prod__Pair__iden,axiom,
    ! [P2: produc6251377953716854561_state] :
      ( ( produc897481912382009139_state @ produc9080284787853255769_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_937_case__prod__Pair__iden,axiom,
    ! [P2: produc1457211279475724562t_char] :
      ( ( produc4071494739252518515t_char @ produc120197956887798346t_char @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_938_case__prod__Pair__iden,axiom,
    ! [P2: produc9003781588162623004on_val] :
      ( ( produc4192255817509754207on_val @ produc74620467203698198on_val @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_939_image2__def,axiom,
    ( bNF_Gr1645861816300964896_state
    = ( ^ [A6: set_state,F3: state > prat,G2: state > state] :
          ( collec9096564466166568534_state
          @ ^ [Uu3: produc4844402304972151915_state] :
            ? [A2: state] :
              ( ( Uu3
                = ( produc6205719116230455971_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_940_image2__def,axiom,
    ( bNF_Gr8363538844033681878_state
    = ( ^ [A6: set_state,F3: state > prat,G2: state > set_state] :
          ( collec5094083866753559820_state
          @ ^ [Uu3: produc6251377953716854561_state] :
            ? [A2: state] :
              ( ( Uu3
                = ( produc9080284787853255769_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_941_image2__def,axiom,
    ( bNF_Gr71701291235176519t_char
    = ( ^ [A6: set_state,F3: state > nat,G2: state > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu3: produc1457211279475724562t_char] :
            ? [A2: state] :
              ( ( Uu3
                = ( produc120197956887798346t_char @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_942_image2__def,axiom,
    ( bNF_Gr8144397171961131434_state
    = ( ^ [A6: set_state,F3: state > option_state,G2: state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A2: state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_943_image2__def,axiom,
    ( bNF_Gr947463461346416350_state
    = ( ^ [A6: set_state,F3: state > state,G2: state > produc3882902684156697774_state] :
          ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [A2: state] :
              ( ( Uu3
                = ( produc5886875603596089627_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_944_image2__def,axiom,
    ( bNF_Gr5432878666621363808_state
    = ( ^ [A6: set_Pr2391701600042252046_state,F3: produc3882902684156697774_state > prat,G2: produc3882902684156697774_state > state] :
          ( collec9096564466166568534_state
          @ ^ [Uu3: produc4844402304972151915_state] :
            ? [A2: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc6205719116230455971_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member6715010751053357271_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_945_image2__def,axiom,
    ( bNF_Gr6818451474134737942_state
    = ( ^ [A6: set_Pr2391701600042252046_state,F3: produc3882902684156697774_state > prat,G2: produc3882902684156697774_state > set_state] :
          ( collec5094083866753559820_state
          @ ^ [Uu3: produc6251377953716854561_state] :
            ? [A2: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc9080284787853255769_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member6715010751053357271_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_946_image2__def,axiom,
    ( bNF_Gr5554302884217825927t_char
    = ( ^ [A6: set_Pr2391701600042252046_state,F3: produc3882902684156697774_state > nat,G2: produc3882902684156697774_state > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu3: produc1457211279475724562t_char] :
            ? [A2: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc120197956887798346t_char @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member6715010751053357271_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_947_image2__def,axiom,
    ( bNF_Gr4913868530397957098_state
    = ( ^ [A6: set_Pr2391701600042252046_state,F3: produc3882902684156697774_state > option_state,G2: produc3882902684156697774_state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A2: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member6715010751053357271_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_948_image2__def,axiom,
    ( bNF_Gr2250442185837678927_state
    = ( ^ [A6: set_Pr4231240917283394591_state,F3: produc7272280185086996969_state > prat,G2: produc7272280185086996969_state > state] :
          ( collec9096564466166568534_state
          @ ^ [Uu3: produc4844402304972151915_state] :
            ? [A2: produc7272280185086996969_state] :
              ( ( Uu3
                = ( produc6205719116230455971_state @ ( F3 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member7461423308196221312_state @ A2 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_949_split__cong,axiom,
    ! [Q2: produc3882902684156697774_state,F4: state > ( state > state ) > $o,G: state > ( state > state ) > $o,P2: produc3882902684156697774_state] :
      ( ! [X3: state,Y4: state > state] :
          ( ( ( produc8632743443909064678_state @ X3 @ Y4 )
            = Q2 )
         => ( ( F4 @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc3645877548210629341tate_o @ F4 @ P2 )
          = ( produc3645877548210629341tate_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_950_split__cong,axiom,
    ! [Q2: produc7272280185086996969_state,F4: state > produc3882902684156697774_state > $o,G: state > produc3882902684156697774_state > $o,P2: produc7272280185086996969_state] :
      ( ! [X3: state,Y4: produc3882902684156697774_state] :
          ( ( ( produc5886875603596089627_state @ X3 @ Y4 )
            = Q2 )
         => ( ( F4 @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc1745419677418529872tate_o @ F4 @ P2 )
          = ( produc1745419677418529872tate_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_951_image2__eqI,axiom,
    ! [B: prat,F4: state > prat,X2: state,C: state,G: state > state,A3: set_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_state @ X2 @ A3 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ B @ C ) @ ( bNF_Gr1645861816300964896_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_952_image2__eqI,axiom,
    ! [B: prat,F4: state > prat,X2: state,C: set_state,G: state > set_state,A3: set_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_state @ X2 @ A3 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ B @ C ) @ ( bNF_Gr8363538844033681878_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_953_image2__eqI,axiom,
    ! [B: nat,F4: state > nat,X2: state,C: list_char,G: state > list_char,A3: set_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_state @ X2 @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr71701291235176519t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_954_image2__eqI,axiom,
    ! [B: state,F4: state > state,X2: state,C: state > state,G: state > state > state,A3: set_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_state @ X2 @ A3 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ B @ C ) @ ( bNF_Gr7364880804673765603_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_955_image2__eqI,axiom,
    ! [B: option_state,F4: state > option_state,X2: state,C: option_state,G: state > option_state,A3: set_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_state @ X2 @ A3 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ C ) @ ( bNF_Gr8144397171961131434_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_956_image2__eqI,axiom,
    ! [B: state,F4: state > state,X2: state,C: produc3882902684156697774_state,G: state > produc3882902684156697774_state,A3: set_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member_state @ X2 @ A3 )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ B @ C ) @ ( bNF_Gr947463461346416350_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_957_image2__eqI,axiom,
    ! [B: prat,F4: produc3882902684156697774_state > prat,X2: produc3882902684156697774_state,C: state,G: produc3882902684156697774_state > state,A3: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member6715010751053357271_state @ X2 @ A3 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ B @ C ) @ ( bNF_Gr5432878666621363808_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_958_image2__eqI,axiom,
    ! [B: prat,F4: produc3882902684156697774_state > prat,X2: produc3882902684156697774_state,C: set_state,G: produc3882902684156697774_state > set_state,A3: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member6715010751053357271_state @ X2 @ A3 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ B @ C ) @ ( bNF_Gr6818451474134737942_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_959_image2__eqI,axiom,
    ! [B: nat,F4: produc3882902684156697774_state > nat,X2: produc3882902684156697774_state,C: list_char,G: produc3882902684156697774_state > list_char,A3: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member6715010751053357271_state @ X2 @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr5554302884217825927t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_960_image2__eqI,axiom,
    ! [B: state,F4: produc3882902684156697774_state > state,X2: produc3882902684156697774_state,C: state > state,G: produc3882902684156697774_state > state > state,A3: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F4 @ X2 ) )
     => ( ( C
          = ( G @ X2 ) )
       => ( ( member6715010751053357271_state @ X2 @ A3 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ B @ C ) @ ( bNF_Gr5413576055295359779_state @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_961_cwand__def,axiom,
    ( cwand
    = ( ^ [A6: set_state,B6: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [W2: state] :
              ( ( Uu3 = W2 )
              & ! [A2: state,X: state] :
                  ( ( ( member_state @ A2 @ A6 )
                    & ( ( some_state @ X )
                      = ( plus @ ( r @ A2 @ W2 ) @ A2 ) ) )
                 => ( member_state @ X @ B6 ) ) ) ) ) ) ).

% cwand_def
thf(fact_962_lex__prod__def,axiom,
    ( lex_pr8336311929016760645_state
    = ( ^ [Ra: set_Pr1688445902015331925_state,Rb: set_Pr1688445902015331925_state] :
          ( collec6784228404298526930_state
          @ ( produc1935756327278076076tate_o
            @ ( produc5315882979514849802tate_o
              @ ^ [A2: option_state,B2: option_state] :
                  ( produc6082990154846732246tate_o
                  @ ^ [A13: option_state,B10: option_state] :
                      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B2 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_963_lex__prod__def,axiom,
    ( lex_pr4041356198452583422_state
    = ( ^ [Ra: set_Pr1785066336555260981_state,Rb: set_Pr5527899070135025255_state] :
          ( collec2083760461196787346_state
          @ ( produc7167296152490707052tate_o
            @ ( produc8588859638187112394tate_o
              @ ^ [A2: state,B2: state > state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [A13: state,B10: state > state] :
                      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member6978531283771894448_state @ ( produc5362444129537099455_state @ B2 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_964_lex__prod__def,axiom,
    ( lex_pr7268379037978345219_state
    = ( ^ [Ra: set_Pr1785066336555260981_state,Rb: set_Pr4846723933414538759_state] :
          ( collec4625684881440051358_state
          @ ( produc3133855996611560728tate_o
            @ ( produc6466221716036377566tate_o
              @ ^ [A2: state,B2: produc3882902684156697774_state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [A13: state,B10: produc3882902684156697774_state] :
                      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ B2 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_965_type__definition__state,axiom,
    ( type_d7385716227342696708on_val @ rep_state @ abs_state
    @ ( collec7475294502966882545on_val
      @ ^ [Uu3: produc9003781588162623004on_val] :
        ? [Phi4: produc9003781588162623004on_val] :
          ( ( Uu3 = Phi4 )
          & ( valid_state @ Phi4 ) ) ) ) ).

% type_definition_state
thf(fact_966_in__lex__prod,axiom,
    ! [A: option_state,B: option_state,A4: option_state,B5: option_state,R3: set_Pr1688445902015331925_state,S4: set_Pr1688445902015331925_state] :
      ( ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc9160152616361873709_state @ A4 @ B5 ) ) @ ( lex_pr8336311929016760645_state @ R3 @ S4 ) )
      = ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ B5 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_967_in__lex__prod,axiom,
    ! [A: prat,B: state,A4: prat,B5: state,R3: set_Pr3298805672376727137t_prat,S4: set_Pr1785066336555260981_state] :
      ( ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc6205719116230455971_state @ A4 @ B5 ) ) @ ( lex_prod_prat_state @ R3 @ S4 ) )
      = ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member753036827967488894_state @ ( produc544653723139640077_state @ B @ B5 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_968_in__lex__prod,axiom,
    ! [A: prat,B: set_state,A4: prat,B5: set_state,R3: set_Pr3298805672376727137t_prat,S4: set_Pr5618112136499017889_state] :
      ( ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc9080284787853255769_state @ A4 @ B5 ) ) @ ( lex_pr1787351029354341489_state @ R3 @ S4 ) )
      = ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member1976142906092959210_state @ ( produc1834047914073684089_state @ B @ B5 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_969_in__lex__prod,axiom,
    ! [A: nat,B: list_char,A4: nat,B5: list_char,R3: set_Pr1261947904930325089at_nat,S4: set_Pr4480099295215233283t_char] :
      ( ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc120197956887798346t_char @ A4 @ B5 ) ) @ ( lex_pr1681056561245511266t_char @ R3 @ S4 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ B @ B5 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_970_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > prat,B5: produc1457211279475724562t_char > option_val,R3: set_Pr1685704261205283783r_prat,S4: set_Pr4698884584343881589on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc74620467203698198on_val @ A4 @ B5 ) ) @ ( lex_pr8780192377291449342on_val @ R3 @ S4 ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A4 ) @ R3 )
        | ( ( A = A4 )
          & ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B @ B5 ) @ S4 ) ) ) ) ).

% in_lex_prod
thf(fact_971_wand__def,axiom,
    ( wand
    = ( ^ [A6: set_state,B6: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [W2: state] :
              ( ( Uu3 = W2 )
              & ! [A2: state,X: state] :
                  ( ( ( member_state @ A2 @ A6 )
                    & ( ( some_state @ X )
                      = ( plus @ W2 @ A2 ) ) )
                 => ( member_state @ X @ B6 ) ) ) ) ) ) ).

% wand_def
thf(fact_972_relImage__def,axiom,
    ( bNF_Gr63754628658600314_state
    = ( ^ [R6: set_Pr1688445902015331925_state,F3: option_state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A14: option_state,A24: option_state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ ( F3 @ A14 ) @ ( F3 @ A24 ) ) )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A14 @ A24 ) @ R6 ) ) ) ) ) ).

% relImage_def
thf(fact_973_prod__set__simps_I1_J,axiom,
    ! [X2: option_state,Y: option_state] :
      ( ( basic_4654892366950893085_state @ ( produc9160152616361873709_state @ X2 @ Y ) )
      = ( insert_option_state @ X2 @ bot_bo710180891245420500_state ) ) ).

% prod_set_simps(1)
thf(fact_974_prod__set__simps_I1_J,axiom,
    ! [X2: prat,Y: state] :
      ( ( basic_6605916601615013779_state @ ( produc6205719116230455971_state @ X2 @ Y ) )
      = ( insert_prat @ X2 @ bot_bot_set_prat ) ) ).

% prod_set_simps(1)
thf(fact_975_prod__set__simps_I1_J,axiom,
    ! [X2: prat,Y: set_state] :
      ( ( basic_1308176908494473545_state @ ( produc9080284787853255769_state @ X2 @ Y ) )
      = ( insert_prat @ X2 @ bot_bot_set_prat ) ) ).

% prod_set_simps(1)
thf(fact_976_prod__set__simps_I1_J,axiom,
    ! [X2: nat,Y: list_char] :
      ( ( basic_4688923273072336698t_char @ ( produc120197956887798346t_char @ X2 @ Y ) )
      = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).

% prod_set_simps(1)
thf(fact_977_prod__set__simps_I1_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( basic_7826031515394172198on_val @ ( produc74620467203698198on_val @ X2 @ Y ) )
      = ( insert8113562578550770211r_prat @ X2 @ bot_bo3164680191535668511r_prat ) ) ).

% prod_set_simps(1)
thf(fact_978_prod__set__simps_I2_J,axiom,
    ! [X2: option_state,Y: option_state] :
      ( ( basic_3472187447467737179_state @ ( produc9160152616361873709_state @ X2 @ Y ) )
      = ( insert_option_state @ Y @ bot_bo710180891245420500_state ) ) ).

% prod_set_simps(2)
thf(fact_979_prod__set__simps_I2_J,axiom,
    ! [X2: prat,Y: set_state] :
      ( ( basic_9212611206362231175_state @ ( produc9080284787853255769_state @ X2 @ Y ) )
      = ( insert_set_state @ Y @ bot_bo2271482359692755898_state ) ) ).

% prod_set_simps(2)
thf(fact_980_prod__set__simps_I2_J,axiom,
    ! [X2: nat,Y: list_char] :
      ( ( basic_2224790070303523448t_char @ ( produc120197956887798346t_char @ X2 @ Y ) )
      = ( insert_list_char @ Y @ bot_bo8503570744482963627t_char ) ) ).

% prod_set_simps(2)
thf(fact_981_prod__set__simps_I2_J,axiom,
    ! [X2: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( basic_8841519627501779048on_val @ ( produc74620467203698198on_val @ X2 @ Y ) )
      = ( insert216038912599248202on_val @ Y @ bot_bo5671083314777306084on_val ) ) ).

% prod_set_simps(2)
thf(fact_982_prod__set__simps_I2_J,axiom,
    ! [X2: prat,Y: state] :
      ( ( basic_6203404420667979729_state @ ( produc6205719116230455971_state @ X2 @ Y ) )
      = ( insert_state @ Y @ bot_bot_set_state ) ) ).

% prod_set_simps(2)
thf(fact_983_relInvImage__def,axiom,
    ( bNF_Gr8759033791985395207_state
    = ( ^ [A6: set_state,R6: set_Pr1688445902015331925_state,F3: state > option_state] :
          ( collec7320380983431419584_state
          @ ^ [Uu3: produc8023240190789890773_state] :
            ? [A14: state,A24: state] :
              ( ( Uu3
                = ( produc544653723139640077_state @ A14 @ A24 ) )
              & ( member_state @ A14 @ A6 )
              & ( member_state @ A24 @ A6 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F3 @ A14 ) @ ( F3 @ A24 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_984_relInvImage__def,axiom,
    ( bNF_Gr5630821380644201398_state
    = ( ^ [A6: set_Pr4231240917283394591_state,R6: set_Pr1688445902015331925_state,F3: produc7272280185086996969_state > option_state] :
          ( collec4625684881440051358_state
          @ ^ [Uu3: produc1362221940583236787_state] :
            ? [A14: produc7272280185086996969_state,A24: produc7272280185086996969_state] :
              ( ( Uu3
                = ( produc2330690084503141483_state @ A14 @ A24 ) )
              & ( member7461423308196221312_state @ A14 @ A6 )
              & ( member7461423308196221312_state @ A24 @ A6 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F3 @ A14 ) @ ( F3 @ A24 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_985_relInvImage__def,axiom,
    ( bNF_Gr4364826243083562601_state
    = ( ^ [A6: set_Pr7649655576376376530on_val,R6: set_Pr1688445902015331925_state,F3: produc9003781588162623004on_val > option_state] :
          ( collec5166231387257825412on_val
          @ ^ [Uu3: produc1317171644805857433on_val] :
            ? [A14: produc9003781588162623004on_val,A24: produc9003781588162623004on_val] :
              ( ( Uu3
                = ( produc4958730763592889809on_val @ A14 @ A24 ) )
              & ( member2879030624002546611on_val @ A14 @ A6 )
              & ( member2879030624002546611on_val @ A24 @ A6 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F3 @ A14 ) @ ( F3 @ A24 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_986_relInvImage__def,axiom,
    ( bNF_Gr919661598790814775_state
    = ( ^ [A6: set_Pr2391701600042252046_state,R6: set_Pr1688445902015331925_state,F3: produc3882902684156697774_state > option_state] :
          ( collec2083760461196787346_state
          @ ^ [Uu3: produc2030649568307348519_state] :
            ? [A14: produc3882902684156697774_state,A24: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc2343482328048119831_state @ A14 @ A24 ) )
              & ( member6715010751053357271_state @ A14 @ A6 )
              & ( member6715010751053357271_state @ A24 @ A6 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F3 @ A14 ) @ ( F3 @ A24 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_987_relInvImage__def,axiom,
    ( bNF_Gr621135371174781015_state
    = ( ^ [A6: set_option_state,R6: set_Pr1688445902015331925_state,F3: option_state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A14: option_state,A24: option_state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ A14 @ A24 ) )
              & ( member_option_state @ A14 @ A6 )
              & ( member_option_state @ A24 @ A6 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F3 @ A14 ) @ ( F3 @ A24 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_988_comp__one__def,axiom,
    ( comp_one
    = ( ^ [P3: prat] :
          ( fChoice_prat
          @ ^ [R: prat] :
              ( ( padd @ P3 @ R )
              = pwrite ) ) ) ) ).

% comp_one_def
thf(fact_989_Field__insert,axiom,
    ! [A: option_state,B: option_state,R3: set_Pr1688445902015331925_state] :
      ( ( field_option_state @ ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ A @ B ) @ R3 ) )
      = ( sup_su99422728725954156_state @ ( insert_option_state @ A @ ( insert_option_state @ B @ bot_bo710180891245420500_state ) ) @ ( field_option_state @ R3 ) ) ) ).

% Field_insert
thf(fact_990_Field__insert,axiom,
    ! [A: produc7272280185086996969_state,B: produc7272280185086996969_state,R3: set_Pr7975217055296730899_state] :
      ( ( field_4847752008948210780_state @ ( insert3455112352123193987_state @ ( produc2330690084503141483_state @ A @ B ) @ R3 ) )
      = ( sup_su3837536599209510219_state @ ( insert175804457004744729_state @ A @ ( insert175804457004744729_state @ B @ bot_bo6430881413863817907_state ) ) @ ( field_4847752008948210780_state @ R3 ) ) ) ).

% Field_insert
thf(fact_991_Field__insert,axiom,
    ! [A: state,B: state,R3: set_Pr1785066336555260981_state] :
      ( ( field_state @ ( insert7525286303735658661_state @ ( produc544653723139640077_state @ A @ B ) @ R3 ) )
      = ( sup_sup_set_state @ ( insert_state @ A @ ( insert_state @ B @ bot_bot_set_state ) ) @ ( field_state @ R3 ) ) ) ).

% Field_insert
thf(fact_992_FieldI1,axiom,
    ! [I: state,J: state,R2: set_Pr1785066336555260981_state] :
      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ I @ J ) @ R2 )
     => ( member_state @ I @ ( field_state @ R2 ) ) ) ).

% FieldI1
thf(fact_993_FieldI1,axiom,
    ! [I: produc7272280185086996969_state,J: produc7272280185086996969_state,R2: set_Pr7975217055296730899_state] :
      ( ( member1349250210377977180_state @ ( produc2330690084503141483_state @ I @ J ) @ R2 )
     => ( member7461423308196221312_state @ I @ ( field_4847752008948210780_state @ R2 ) ) ) ).

% FieldI1
thf(fact_994_FieldI1,axiom,
    ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 )
     => ( member2879030624002546611on_val @ I @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI1
thf(fact_995_FieldI1,axiom,
    ! [I: produc3882902684156697774_state,J: produc3882902684156697774_state,R2: set_Pr4846723933414538759_state] :
      ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ I @ J ) @ R2 )
     => ( member6715010751053357271_state @ I @ ( field_1980264525152535803_state @ R2 ) ) ) ).

% FieldI1
thf(fact_996_FieldI1,axiom,
    ! [I: option_state,J: option_state,R2: set_Pr1688445902015331925_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I @ J ) @ R2 )
     => ( member_option_state @ I @ ( field_option_state @ R2 ) ) ) ).

% FieldI1
thf(fact_997_FieldI2,axiom,
    ! [I: state,J: state,R2: set_Pr1785066336555260981_state] :
      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ I @ J ) @ R2 )
     => ( member_state @ J @ ( field_state @ R2 ) ) ) ).

% FieldI2
thf(fact_998_FieldI2,axiom,
    ! [I: produc7272280185086996969_state,J: produc7272280185086996969_state,R2: set_Pr7975217055296730899_state] :
      ( ( member1349250210377977180_state @ ( produc2330690084503141483_state @ I @ J ) @ R2 )
     => ( member7461423308196221312_state @ J @ ( field_4847752008948210780_state @ R2 ) ) ) ).

% FieldI2
thf(fact_999_FieldI2,axiom,
    ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 )
     => ( member2879030624002546611on_val @ J @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI2
thf(fact_1000_FieldI2,axiom,
    ! [I: produc3882902684156697774_state,J: produc3882902684156697774_state,R2: set_Pr4846723933414538759_state] :
      ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ I @ J ) @ R2 )
     => ( member6715010751053357271_state @ J @ ( field_1980264525152535803_state @ R2 ) ) ) ).

% FieldI2
thf(fact_1001_FieldI2,axiom,
    ! [I: option_state,J: option_state,R2: set_Pr1688445902015331925_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I @ J ) @ R2 )
     => ( member_option_state @ J @ ( field_option_state @ R2 ) ) ) ).

% FieldI2
thf(fact_1002_UnderS__def,axiom,
    ( order_5015692507921881172_state
    = ( ^ [R: set_Pr4846723933414538759_state,A6: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ^ [B2: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ B2 @ ( field_1980264525152535803_state @ R ) )
              & ! [X: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ B2 @ X ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1003_UnderS__def,axiom,
    ( order_6633786773044248771_state
    = ( ^ [R: set_Pr7975217055296730899_state,A6: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ^ [B2: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ B2 @ ( field_4847752008948210780_state @ R ) )
              & ! [X: produc7272280185086996969_state] :
                  ( ( member7461423308196221312_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member1349250210377977180_state @ ( produc2330690084503141483_state @ B2 @ X ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1004_UnderS__def,axiom,
    ( order_2655018729228486902on_val
    = ( ^ [R: set_Pr4305388979797279737on_val,A6: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ B2 @ ( field_9010718613293825679on_val @ R ) )
              & ! [X: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B2 @ X ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1005_UnderS__def,axiom,
    ( order_UnderS_state
    = ( ^ [R: set_Pr1785066336555260981_state,A6: set_state] :
          ( collect_state
          @ ^ [B2: state] :
              ( ( member_state @ B2 @ ( field_state @ R ) )
              & ! [X: state] :
                  ( ( member_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member753036827967488894_state @ ( produc544653723139640077_state @ B2 @ X ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1006_UnderS__def,axiom,
    ( order_7351702338818143716_state
    = ( ^ [R: set_Pr1688445902015331925_state,A6: set_option_state] :
          ( collect_option_state
          @ ^ [B2: option_state] :
              ( ( member_option_state @ B2 @ ( field_option_state @ R ) )
              & ! [X: option_state] :
                  ( ( member_option_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B2 @ X ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1007_Under__def,axiom,
    ( order_4361110382399168307_state
    = ( ^ [R: set_Pr4846723933414538759_state,A6: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ^ [B2: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ B2 @ ( field_1980264525152535803_state @ R ) )
              & ! [X: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X @ A6 )
                 => ( member3801449062345016400_state @ ( produc2343482328048119831_state @ B2 @ X ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1008_Under__def,axiom,
    ( order_4023999085709689892_state
    = ( ^ [R: set_Pr7975217055296730899_state,A6: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ^ [B2: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ B2 @ ( field_4847752008948210780_state @ R ) )
              & ! [X: produc7272280185086996969_state] :
                  ( ( member7461423308196221312_state @ X @ A6 )
                 => ( member1349250210377977180_state @ ( produc2330690084503141483_state @ B2 @ X ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1009_Under__def,axiom,
    ( order_7239719736367216215on_val
    = ( ^ [R: set_Pr4305388979797279737on_val,A6: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ B2 @ ( field_9010718613293825679on_val @ R ) )
              & ! [X: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ X @ A6 )
                 => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B2 @ X ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1010_Under__def,axiom,
    ( order_Under_state
    = ( ^ [R: set_Pr1785066336555260981_state,A6: set_state] :
          ( collect_state
          @ ^ [B2: state] :
              ( ( member_state @ B2 @ ( field_state @ R ) )
              & ! [X: state] :
                  ( ( member_state @ X @ A6 )
                 => ( member753036827967488894_state @ ( produc544653723139640077_state @ B2 @ X ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1011_Under__def,axiom,
    ( order_9213223657572546501_state
    = ( ^ [R: set_Pr1688445902015331925_state,A6: set_option_state] :
          ( collect_option_state
          @ ^ [B2: option_state] :
              ( ( member_option_state @ B2 @ ( field_option_state @ R ) )
              & ! [X: option_state] :
                  ( ( member_option_state @ X @ A6 )
                 => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B2 @ X ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1012_Above__def,axiom,
    ( order_2730342495953330126_state
    = ( ^ [R: set_Pr4846723933414538759_state,A6: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ^ [B2: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ B2 @ ( field_1980264525152535803_state @ R ) )
              & ! [X: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X @ A6 )
                 => ( member3801449062345016400_state @ ( produc2343482328048119831_state @ X @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1013_Above__def,axiom,
    ( order_8148989763630163657_state
    = ( ^ [R: set_Pr7975217055296730899_state,A6: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ^ [B2: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ B2 @ ( field_4847752008948210780_state @ R ) )
              & ! [X: produc7272280185086996969_state] :
                  ( ( member7461423308196221312_state @ X @ A6 )
                 => ( member1349250210377977180_state @ ( produc2330690084503141483_state @ X @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1014_Above__def,axiom,
    ( order_5053926465803840252on_val
    = ( ^ [R: set_Pr4305388979797279737on_val,A6: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ B2 @ ( field_9010718613293825679on_val @ R ) )
              & ! [X: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ X @ A6 )
                 => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1015_Above__def,axiom,
    ( order_Above_state
    = ( ^ [R: set_Pr1785066336555260981_state,A6: set_state] :
          ( collect_state
          @ ^ [B2: state] :
              ( ( member_state @ B2 @ ( field_state @ R ) )
              & ! [X: state] :
                  ( ( member_state @ X @ A6 )
                 => ( member753036827967488894_state @ ( produc544653723139640077_state @ X @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1016_Above__def,axiom,
    ( order_7772079012763333354_state
    = ( ^ [R: set_Pr1688445902015331925_state,A6: set_option_state] :
          ( collect_option_state
          @ ^ [B2: option_state] :
              ( ( member_option_state @ B2 @ ( field_option_state @ R ) )
              & ! [X: option_state] :
                  ( ( member_option_state @ X @ A6 )
                 => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1017_linear__order__on__singleton,axiom,
    ! [X2: option_state] : ( order_7502633587729487893_state @ ( insert_option_state @ X2 @ bot_bo710180891245420500_state ) @ ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ X2 @ X2 ) @ bot_bo1080640394036989633_state ) ) ).

% linear_order_on_singleton
thf(fact_1018_linear__order__on__singleton,axiom,
    ! [X2: state] : ( order_7174064491916216133_state @ ( insert_state @ X2 @ bot_bot_set_state ) @ ( insert7525286303735658661_state @ ( produc544653723139640077_state @ X2 @ X2 ) @ bot_bo9041262728264437921_state ) ) ).

% linear_order_on_singleton
thf(fact_1019_AboveS__def,axiom,
    ( order_941992989072866937_state
    = ( ^ [R: set_Pr4846723933414538759_state,A6: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ^ [B2: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ B2 @ ( field_1980264525152535803_state @ R ) )
              & ! [X: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ X @ B2 ) @ R ) ) ) ) ) ) ) ).

% AboveS_def
thf(fact_1020_AboveS__def,axiom,
    ( order_6608450432790103902_state
    = ( ^ [R: set_Pr7975217055296730899_state,A6: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ^ [B2: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ B2 @ ( field_4847752008948210780_state @ R ) )
              & ! [X: produc7272280185086996969_state] :
                  ( ( member7461423308196221312_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member1349250210377977180_state @ ( produc2330690084503141483_state @ X @ B2 ) @ R ) ) ) ) ) ) ) ).

% AboveS_def
thf(fact_1021_AboveS__def,axiom,
    ( order_3343947986673401233on_val
    = ( ^ [R: set_Pr4305388979797279737on_val,A6: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ B2 @ ( field_9010718613293825679on_val @ R ) )
              & ! [X: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ B2 ) @ R ) ) ) ) ) ) ) ).

% AboveS_def
thf(fact_1022_AboveS__def,axiom,
    ( order_AboveS_state
    = ( ^ [R: set_Pr1785066336555260981_state,A6: set_state] :
          ( collect_state
          @ ^ [B2: state] :
              ( ( member_state @ B2 @ ( field_state @ R ) )
              & ! [X: state] :
                  ( ( member_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member753036827967488894_state @ ( produc544653723139640077_state @ X @ B2 ) @ R ) ) ) ) ) ) ) ).

% AboveS_def
thf(fact_1023_AboveS__def,axiom,
    ( order_44153223842169855_state
    = ( ^ [R: set_Pr1688445902015331925_state,A6: set_option_state] :
          ( collect_option_state
          @ ^ [B2: option_state] :
              ( ( member_option_state @ B2 @ ( field_option_state @ R ) )
              & ! [X: option_state] :
                  ( ( member_option_state @ X @ A6 )
                 => ( ( B2 != X )
                    & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ B2 ) @ R ) ) ) ) ) ) ) ).

% AboveS_def
thf(fact_1024_insert__Times__insert,axiom,
    ! [A: state,A3: set_state,B: state,B3: set_state] :
      ( ( produc4572292869282958142_state @ ( insert_state @ A @ A3 )
        @ ^ [Uu3: state] : ( insert_state @ B @ B3 ) )
      = ( insert7525286303735658661_state @ ( produc544653723139640077_state @ A @ B )
        @ ( sup_su8786683420481640457_state
          @ ( produc4572292869282958142_state @ A3
            @ ^ [Uu3: state] : ( insert_state @ B @ B3 ) )
          @ ( produc4572292869282958142_state @ ( insert_state @ A @ A3 )
            @ ^ [Uu3: state] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_1025_insert__Times__insert,axiom,
    ! [A: option_state,A3: set_option_state,B: option_state,B3: set_option_state] :
      ( ( produc7708045757989049950_state @ ( insert_option_state @ A @ A3 )
        @ ^ [Uu3: option_state] : ( insert_option_state @ B @ B3 ) )
      = ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ A @ B )
        @ ( sup_su5006661989123481129_state
          @ ( produc7708045757989049950_state @ A3
            @ ^ [Uu3: option_state] : ( insert_option_state @ B @ B3 ) )
          @ ( produc7708045757989049950_state @ ( insert_option_state @ A @ A3 )
            @ ^ [Uu3: option_state] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_1026_insert__Times__insert,axiom,
    ! [A: prat,A3: set_prat,B: state,B3: set_state] :
      ( ( produc1658785944879207636_state @ ( insert_prat @ A @ A3 )
        @ ^ [Uu3: prat] : ( insert_state @ B @ B3 ) )
      = ( insert3559717418277143099_state @ ( produc6205719116230455971_state @ A @ B )
        @ ( sup_su1826719071296284831_state
          @ ( produc1658785944879207636_state @ A3
            @ ^ [Uu3: prat] : ( insert_state @ B @ B3 ) )
          @ ( produc1658785944879207636_state @ ( insert_prat @ A @ A3 )
            @ ^ [Uu3: prat] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_1027_insert__Times__insert,axiom,
    ! [A: prat,A3: set_prat,B: set_state,B3: set_set_state] :
      ( ( produc4653271056321233546_state @ ( insert_prat @ A @ A3 )
        @ ^ [Uu3: prat] : ( insert_set_state @ B @ B3 ) )
      = ( insert2241947522006009073_state @ ( produc9080284787853255769_state @ A @ B )
        @ ( sup_su5611131606495579221_state
          @ ( produc4653271056321233546_state @ A3
            @ ^ [Uu3: prat] : ( insert_set_state @ B @ B3 ) )
          @ ( produc4653271056321233546_state @ ( insert_prat @ A @ A3 )
            @ ^ [Uu3: prat] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_1028_insert__Times__insert,axiom,
    ! [A: nat,A3: set_nat,B: list_char,B3: set_list_char] :
      ( ( produc6030673636645808891t_char @ ( insert_nat @ A @ A3 )
        @ ^ [Uu3: nat] : ( insert_list_char @ B @ B3 ) )
      = ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B )
        @ ( sup_su7899656352754981958t_char
          @ ( produc6030673636645808891t_char @ A3
            @ ^ [Uu3: nat] : ( insert_list_char @ B @ B3 ) )
          @ ( produc6030673636645808891t_char @ ( insert_nat @ A @ A3 )
            @ ^ [Uu3: nat] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_1029_insert__Times__insert,axiom,
    ! [A: produc1457211279475724562t_char > prat,A3: set_Pr3433046674199179443r_prat,B: produc1457211279475724562t_char > option_val,B3: set_Pr1004501353197068112on_val] :
      ( ( produc1366239611153933861on_val @ ( insert8113562578550770211r_prat @ A @ A3 )
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( insert216038912599248202on_val @ B @ B3 ) )
      = ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ A @ B )
        @ ( sup_su7506060154676209150on_val
          @ ( produc1366239611153933861on_val @ A3
            @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( insert216038912599248202on_val @ B @ B3 ) )
          @ ( produc1366239611153933861on_val @ ( insert8113562578550770211r_prat @ A @ A3 )
            @ ^ [Uu3: produc1457211279475724562t_char > prat] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_1030_insert__Times__insert,axiom,
    ! [A: state,A3: set_state,B: produc3882902684156697774_state,B3: set_Pr2391701600042252046_state] :
      ( ( produc2831624448089660202_state @ ( insert_state @ A @ A3 )
        @ ^ [Uu3: state] : ( insert94804854012721790_state @ B @ B3 ) )
      = ( insert175804457004744729_state @ ( produc5886875603596089627_state @ A @ B )
        @ ( sup_su3837536599209510219_state
          @ ( produc2831624448089660202_state @ A3
            @ ^ [Uu3: state] : ( insert94804854012721790_state @ B @ B3 ) )
          @ ( produc2831624448089660202_state @ ( insert_state @ A @ A3 )
            @ ^ [Uu3: state] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_1031_mem__Sigma__iff,axiom,
    ! [A: state,B: state,A3: set_state,B3: state > set_state] :
      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ B ) @ ( produc4572292869282958142_state @ A3 @ B3 ) )
      = ( ( member_state @ A @ A3 )
        & ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1032_mem__Sigma__iff,axiom,
    ! [A: prat,B: state,A3: set_prat,B3: prat > set_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc1658785944879207636_state @ A3 @ B3 ) )
      = ( ( member_prat @ A @ A3 )
        & ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1033_mem__Sigma__iff,axiom,
    ! [A: prat,B: set_state,A3: set_prat,B3: prat > set_set_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc4653271056321233546_state @ A3 @ B3 ) )
      = ( ( member_prat @ A @ A3 )
        & ( member_set_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1034_mem__Sigma__iff,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B3: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
      = ( ( member_nat @ A @ A3 )
        & ( member_list_char @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1035_mem__Sigma__iff,axiom,
    ! [A: state,B: state > state,A3: set_state,B3: state > set_state_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ ( produc6646226935541082007_state @ A3 @ B3 ) )
      = ( ( member_state @ A @ A3 )
        & ( member_state_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1036_mem__Sigma__iff,axiom,
    ! [A: option_state,B: option_state,A3: set_option_state,B3: option_state > set_option_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc7708045757989049950_state @ A3 @ B3 ) )
      = ( ( member_option_state @ A @ A3 )
        & ( member_option_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1037_mem__Sigma__iff,axiom,
    ! [A: produc3882902684156697774_state,B: state,A3: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state > set_state] :
      ( ( member1546683425902591150_state @ ( produc7334084071316671761_state @ A @ B ) @ ( produc4278832915810242336_state @ A3 @ B3 ) )
      = ( ( member6715010751053357271_state @ A @ A3 )
        & ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1038_mem__Sigma__iff,axiom,
    ! [A: state,B: produc3882902684156697774_state,A3: set_state,B3: state > set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ ( produc2831624448089660202_state @ A3 @ B3 ) )
      = ( ( member_state @ A @ A3 )
        & ( member6715010751053357271_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1039_mem__Sigma__iff,axiom,
    ! [A: state,B: produc7272280185086996969_state,A3: set_state,B3: state > set_Pr4231240917283394591_state] :
      ( ( member5752234768399382829_state @ ( produc5194695453622026300_state @ A @ B ) @ ( produc8552965988182875245_state @ A3 @ B3 ) )
      = ( ( member_state @ A @ A3 )
        & ( member7461423308196221312_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1040_mem__Sigma__iff,axiom,
    ! [A: produc7272280185086996969_state,B: state,A3: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state > set_state] :
      ( ( member5722111062937351085_state @ ( produc7271836056140555068_state @ A @ B ) @ ( produc1406734553846628205_state @ A3 @ B3 ) )
      = ( ( member7461423308196221312_state @ A @ A3 )
        & ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1041_SigmaI,axiom,
    ! [A: state,A3: set_state,B: state,B3: state > set_state] :
      ( ( member_state @ A @ A3 )
     => ( ( member_state @ B @ ( B3 @ A ) )
       => ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ B ) @ ( produc4572292869282958142_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1042_SigmaI,axiom,
    ! [A: prat,A3: set_prat,B: state,B3: prat > set_state] :
      ( ( member_prat @ A @ A3 )
     => ( ( member_state @ B @ ( B3 @ A ) )
       => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc1658785944879207636_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1043_SigmaI,axiom,
    ! [A: prat,A3: set_prat,B: set_state,B3: prat > set_set_state] :
      ( ( member_prat @ A @ A3 )
     => ( ( member_set_state @ B @ ( B3 @ A ) )
       => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc4653271056321233546_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1044_SigmaI,axiom,
    ! [A: nat,A3: set_nat,B: list_char,B3: nat > set_list_char] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_list_char @ B @ ( B3 @ A ) )
       => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1045_SigmaI,axiom,
    ! [A: state,A3: set_state,B: state > state,B3: state > set_state_state] :
      ( ( member_state @ A @ A3 )
     => ( ( member_state_state @ B @ ( B3 @ A ) )
       => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ ( produc6646226935541082007_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1046_SigmaI,axiom,
    ! [A: option_state,A3: set_option_state,B: option_state,B3: option_state > set_option_state] :
      ( ( member_option_state @ A @ A3 )
     => ( ( member_option_state @ B @ ( B3 @ A ) )
       => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc7708045757989049950_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1047_SigmaI,axiom,
    ! [A: state,A3: set_state,B: produc3882902684156697774_state,B3: state > set_Pr2391701600042252046_state] :
      ( ( member_state @ A @ A3 )
     => ( ( member6715010751053357271_state @ B @ ( B3 @ A ) )
       => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ ( produc2831624448089660202_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1048_SigmaI,axiom,
    ! [A: produc3882902684156697774_state,A3: set_Pr2391701600042252046_state,B: state,B3: produc3882902684156697774_state > set_state] :
      ( ( member6715010751053357271_state @ A @ A3 )
     => ( ( member_state @ B @ ( B3 @ A ) )
       => ( member1546683425902591150_state @ ( produc7334084071316671761_state @ A @ B ) @ ( produc4278832915810242336_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1049_SigmaI,axiom,
    ! [A: state,A3: set_state,B: produc7272280185086996969_state,B3: state > set_Pr4231240917283394591_state] :
      ( ( member_state @ A @ A3 )
     => ( ( member7461423308196221312_state @ B @ ( B3 @ A ) )
       => ( member5752234768399382829_state @ ( produc5194695453622026300_state @ A @ B ) @ ( produc8552965988182875245_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1050_SigmaI,axiom,
    ! [A: produc7272280185086996969_state,A3: set_Pr4231240917283394591_state,B: state,B3: produc7272280185086996969_state > set_state] :
      ( ( member7461423308196221312_state @ A @ A3 )
     => ( ( member_state @ B @ ( B3 @ A ) )
       => ( member5722111062937351085_state @ ( produc7271836056140555068_state @ A @ B ) @ ( produc1406734553846628205_state @ A3 @ B3 ) ) ) ) ).

% SigmaI
thf(fact_1051_Collect__case__prod,axiom,
    ! [P4: state > $o,Q: state > $o] :
      ( ( collec7320380983431419584_state
        @ ( produc5376313899190161910tate_o
          @ ^ [A2: state,B2: state] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc4572292869282958142_state @ ( collect_state @ P4 )
        @ ^ [Uu3: state] : ( collect_state @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1052_Collect__case__prod,axiom,
    ! [P4: state > $o,Q: ( state > state ) > $o] :
      ( ( collec394671971739345689_state
        @ ( produc3645877548210629341tate_o
          @ ^ [A2: state,B2: state > state] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc6646226935541082007_state @ ( collect_state @ P4 )
        @ ^ [Uu3: state] : ( collect_state_state @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1053_Collect__case__prod,axiom,
    ! [P4: state > $o,Q: produc3882902684156697774_state > $o] :
      ( ( collec906845016635587262_state
        @ ( produc1745419677418529872tate_o
          @ ^ [A2: state,B2: produc3882902684156697774_state] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc2831624448089660202_state @ ( collect_state @ P4 )
        @ ^ [Uu3: state] : ( collec394671971739345689_state @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1054_Collect__case__prod,axiom,
    ! [P4: produc7272280185086996969_state > $o,Q: state > $o] :
      ( ( collec8329173533518380783_state
        @ ( produc3925133191046228999tate_o
          @ ^ [A2: produc7272280185086996969_state,B2: state] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc1406734553846628205_state @ ( collec906845016635587262_state @ P4 )
        @ ^ [Uu3: produc7272280185086996969_state] : ( collect_state @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1055_Collect__case__prod,axiom,
    ! [P4: state > $o,Q: produc7272280185086996969_state > $o] :
      ( ( collec8359297238980412527_state
        @ ( produc8801812494870056711tate_o
          @ ^ [A2: state,B2: produc7272280185086996969_state] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc8552965988182875245_state @ ( collect_state @ P4 )
        @ ^ [Uu3: state] : ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1056_Collect__case__prod,axiom,
    ! [P4: ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( collec7475294502966882545on_val
        @ ( produc5248837549992124053_val_o
          @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc1366239611153933861on_val @ ( collec7093475260167521854r_prat @ P4 )
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( collec8455114619259388271on_val @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1057_Collect__case__prod,axiom,
    ! [P4: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ( collec4625684881440051358_state
        @ ( produc3133855996611560728tate_o
          @ ^ [A2: produc7272280185086996969_state,B2: produc7272280185086996969_state] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc8058815403421562012_state @ ( collec906845016635587262_state @ P4 )
        @ ^ [Uu3: produc7272280185086996969_state] : ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1058_Collect__case__prod,axiom,
    ! [P4: produc9003781588162623004on_val > $o,Q: state > $o] :
      ( ( collec7689951075418334754_state
        @ ( produc8998596806100491796tate_o
          @ ^ [A2: produc9003781588162623004on_val,B2: state] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc556505947853469856_state @ ( collec7475294502966882545on_val @ P4 )
        @ ^ [Uu3: produc9003781588162623004on_val] : ( collect_state @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1059_Collect__case__prod,axiom,
    ! [P4: state > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( collec4277618271863934242on_val
        @ ( produc3402325802329673236_val_o
          @ ^ [A2: state,B2: produc9003781588162623004on_val] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc4123186969287248032on_val @ ( collect_state @ P4 )
        @ ^ [Uu3: state] : ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1060_Collect__case__prod,axiom,
    ! [P4: produc7272280185086996969_state > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( collec7485704936941981521on_val
        @ ( produc7822554240239717925_val_o
          @ ^ [A2: produc7272280185086996969_state,B2: produc9003781588162623004on_val] :
              ( ( P4 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc2280571527155128015on_val @ ( collec906845016635587262_state @ P4 )
        @ ^ [Uu3: produc7272280185086996969_state] : ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1061_Times__empty,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( ( produc4572292869282958142_state @ A3
          @ ^ [Uu3: state] : B3 )
        = bot_bo9041262728264437921_state )
      = ( ( A3 = bot_bot_set_state )
        | ( B3 = bot_bot_set_state ) ) ) ).

% Times_empty
thf(fact_1062_Collect__case__prod__Sigma,axiom,
    ! [P4: state > $o,Q: state > state > $o] :
      ( ( collec7320380983431419584_state
        @ ( produc5376313899190161910tate_o
          @ ^ [X: state,Y3: state] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc4572292869282958142_state @ ( collect_state @ P4 )
        @ ^ [X: state] : ( collect_state @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1063_Collect__case__prod__Sigma,axiom,
    ! [P4: state > $o,Q: state > ( state > state ) > $o] :
      ( ( collec394671971739345689_state
        @ ( produc3645877548210629341tate_o
          @ ^ [X: state,Y3: state > state] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc6646226935541082007_state @ ( collect_state @ P4 )
        @ ^ [X: state] : ( collect_state_state @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1064_Collect__case__prod__Sigma,axiom,
    ! [P4: state > $o,Q: state > produc3882902684156697774_state > $o] :
      ( ( collec906845016635587262_state
        @ ( produc1745419677418529872tate_o
          @ ^ [X: state,Y3: produc3882902684156697774_state] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc2831624448089660202_state @ ( collect_state @ P4 )
        @ ^ [X: state] : ( collec394671971739345689_state @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1065_Collect__case__prod__Sigma,axiom,
    ! [P4: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > state > $o] :
      ( ( collec8329173533518380783_state
        @ ( produc3925133191046228999tate_o
          @ ^ [X: produc7272280185086996969_state,Y3: state] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc1406734553846628205_state @ ( collec906845016635587262_state @ P4 )
        @ ^ [X: produc7272280185086996969_state] : ( collect_state @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1066_Collect__case__prod__Sigma,axiom,
    ! [P4: state > $o,Q: state > produc7272280185086996969_state > $o] :
      ( ( collec8359297238980412527_state
        @ ( produc8801812494870056711tate_o
          @ ^ [X: state,Y3: produc7272280185086996969_state] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc8552965988182875245_state @ ( collect_state @ P4 )
        @ ^ [X: state] : ( collec906845016635587262_state @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1067_Collect__case__prod__Sigma,axiom,
    ! [P4: ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( collec7475294502966882545on_val
        @ ( produc5248837549992124053_val_o
          @ ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc1366239611153933861on_val @ ( collec7093475260167521854r_prat @ P4 )
        @ ^ [X: produc1457211279475724562t_char > prat] : ( collec8455114619259388271on_val @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1068_Collect__case__prod__Sigma,axiom,
    ! [P4: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > produc7272280185086996969_state > $o] :
      ( ( collec4625684881440051358_state
        @ ( produc3133855996611560728tate_o
          @ ^ [X: produc7272280185086996969_state,Y3: produc7272280185086996969_state] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc8058815403421562012_state @ ( collec906845016635587262_state @ P4 )
        @ ^ [X: produc7272280185086996969_state] : ( collec906845016635587262_state @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1069_Collect__case__prod__Sigma,axiom,
    ! [P4: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > state > $o] :
      ( ( collec7689951075418334754_state
        @ ( produc8998596806100491796tate_o
          @ ^ [X: produc9003781588162623004on_val,Y3: state] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc556505947853469856_state @ ( collec7475294502966882545on_val @ P4 )
        @ ^ [X: produc9003781588162623004on_val] : ( collect_state @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1070_Collect__case__prod__Sigma,axiom,
    ! [P4: state > $o,Q: state > produc9003781588162623004on_val > $o] :
      ( ( collec4277618271863934242on_val
        @ ( produc3402325802329673236_val_o
          @ ^ [X: state,Y3: produc9003781588162623004on_val] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc4123186969287248032on_val @ ( collect_state @ P4 )
        @ ^ [X: state] : ( collec7475294502966882545on_val @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1071_Collect__case__prod__Sigma,axiom,
    ! [P4: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > produc9003781588162623004on_val > $o] :
      ( ( collec7485704936941981521on_val
        @ ( produc7822554240239717925_val_o
          @ ^ [X: produc7272280185086996969_state,Y3: produc9003781588162623004on_val] :
              ( ( P4 @ X )
              & ( Q @ X @ Y3 ) ) ) )
      = ( produc2280571527155128015on_val @ ( collec906845016635587262_state @ P4 )
        @ ^ [X: produc7272280185086996969_state] : ( collec7475294502966882545on_val @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_1072_SigmaE2,axiom,
    ! [A: state,B: state,A3: set_state,B3: state > set_state] :
      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ B ) @ ( produc4572292869282958142_state @ A3 @ B3 ) )
     => ~ ( ( member_state @ A @ A3 )
         => ~ ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1073_SigmaE2,axiom,
    ! [A: prat,B: state,A3: set_prat,B3: prat > set_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc1658785944879207636_state @ A3 @ B3 ) )
     => ~ ( ( member_prat @ A @ A3 )
         => ~ ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1074_SigmaE2,axiom,
    ! [A: prat,B: set_state,A3: set_prat,B3: prat > set_set_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc4653271056321233546_state @ A3 @ B3 ) )
     => ~ ( ( member_prat @ A @ A3 )
         => ~ ( member_set_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1075_SigmaE2,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B3: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
     => ~ ( ( member_nat @ A @ A3 )
         => ~ ( member_list_char @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1076_SigmaE2,axiom,
    ! [A: state,B: state > state,A3: set_state,B3: state > set_state_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ ( produc6646226935541082007_state @ A3 @ B3 ) )
     => ~ ( ( member_state @ A @ A3 )
         => ~ ( member_state_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1077_SigmaE2,axiom,
    ! [A: option_state,B: option_state,A3: set_option_state,B3: option_state > set_option_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc7708045757989049950_state @ A3 @ B3 ) )
     => ~ ( ( member_option_state @ A @ A3 )
         => ~ ( member_option_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1078_SigmaE2,axiom,
    ! [A: produc3882902684156697774_state,B: state,A3: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state > set_state] :
      ( ( member1546683425902591150_state @ ( produc7334084071316671761_state @ A @ B ) @ ( produc4278832915810242336_state @ A3 @ B3 ) )
     => ~ ( ( member6715010751053357271_state @ A @ A3 )
         => ~ ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1079_SigmaE2,axiom,
    ! [A: state,B: produc3882902684156697774_state,A3: set_state,B3: state > set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ ( produc2831624448089660202_state @ A3 @ B3 ) )
     => ~ ( ( member_state @ A @ A3 )
         => ~ ( member6715010751053357271_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1080_SigmaE2,axiom,
    ! [A: state,B: produc7272280185086996969_state,A3: set_state,B3: state > set_Pr4231240917283394591_state] :
      ( ( member5752234768399382829_state @ ( produc5194695453622026300_state @ A @ B ) @ ( produc8552965988182875245_state @ A3 @ B3 ) )
     => ~ ( ( member_state @ A @ A3 )
         => ~ ( member7461423308196221312_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1081_SigmaE2,axiom,
    ! [A: produc7272280185086996969_state,B: state,A3: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state > set_state] :
      ( ( member5722111062937351085_state @ ( produc7271836056140555068_state @ A @ B ) @ ( produc1406734553846628205_state @ A3 @ B3 ) )
     => ~ ( ( member7461423308196221312_state @ A @ A3 )
         => ~ ( member_state @ B @ ( B3 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1082_SigmaD2,axiom,
    ! [A: state,B: produc3882902684156697774_state,A3: set_state,B3: state > set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ ( produc2831624448089660202_state @ A3 @ B3 ) )
     => ( member6715010751053357271_state @ B @ ( B3 @ A ) ) ) ).

% SigmaD2
thf(fact_1083_SigmaD2,axiom,
    ! [A: state,B: state > state,A3: set_state,B3: state > set_state_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ ( produc6646226935541082007_state @ A3 @ B3 ) )
     => ( member_state_state @ B @ ( B3 @ A ) ) ) ).

% SigmaD2
thf(fact_1084_SigmaD2,axiom,
    ! [A: option_state,B: option_state,A3: set_option_state,B3: option_state > set_option_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc7708045757989049950_state @ A3 @ B3 ) )
     => ( member_option_state @ B @ ( B3 @ A ) ) ) ).

% SigmaD2
thf(fact_1085_SigmaD2,axiom,
    ! [A: prat,B: state,A3: set_prat,B3: prat > set_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc1658785944879207636_state @ A3 @ B3 ) )
     => ( member_state @ B @ ( B3 @ A ) ) ) ).

% SigmaD2
thf(fact_1086_SigmaD2,axiom,
    ! [A: prat,B: set_state,A3: set_prat,B3: prat > set_set_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc4653271056321233546_state @ A3 @ B3 ) )
     => ( member_set_state @ B @ ( B3 @ A ) ) ) ).

% SigmaD2
thf(fact_1087_SigmaD2,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B3: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
     => ( member_list_char @ B @ ( B3 @ A ) ) ) ).

% SigmaD2
thf(fact_1088_SigmaD2,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A3: set_Pr3433046674199179443r_prat,B3: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A3 @ B3 ) )
     => ( member1577473829074448177on_val @ B @ ( B3 @ A ) ) ) ).

% SigmaD2
thf(fact_1089_SigmaD1,axiom,
    ! [A: state,B: produc3882902684156697774_state,A3: set_state,B3: state > set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ ( produc2831624448089660202_state @ A3 @ B3 ) )
     => ( member_state @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1090_SigmaD1,axiom,
    ! [A: state,B: state > state,A3: set_state,B3: state > set_state_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ ( produc6646226935541082007_state @ A3 @ B3 ) )
     => ( member_state @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1091_SigmaD1,axiom,
    ! [A: option_state,B: option_state,A3: set_option_state,B3: option_state > set_option_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc7708045757989049950_state @ A3 @ B3 ) )
     => ( member_option_state @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1092_SigmaD1,axiom,
    ! [A: prat,B: state,A3: set_prat,B3: prat > set_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc1658785944879207636_state @ A3 @ B3 ) )
     => ( member_prat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1093_SigmaD1,axiom,
    ! [A: prat,B: set_state,A3: set_prat,B3: prat > set_set_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc4653271056321233546_state @ A3 @ B3 ) )
     => ( member_prat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1094_SigmaD1,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B3: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
     => ( member_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1095_SigmaD1,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A3: set_Pr3433046674199179443r_prat,B3: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A3 @ B3 ) )
     => ( member2336342582916103932r_prat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1096_SigmaE,axiom,
    ! [C: produc8023240190789890773_state,A3: set_state,B3: state > set_state] :
      ( ( member753036827967488894_state @ C @ ( produc4572292869282958142_state @ A3 @ B3 ) )
     => ~ ! [X3: state] :
            ( ( member_state @ X3 @ A3 )
           => ! [Y4: state] :
                ( ( member_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc544653723139640077_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1097_SigmaE,axiom,
    ! [C: produc4844402304972151915_state,A3: set_prat,B3: prat > set_state] :
      ( ( member6226390569306242836_state @ C @ ( produc1658785944879207636_state @ A3 @ B3 ) )
     => ~ ! [X3: prat] :
            ( ( member_prat @ X3 @ A3 )
           => ! [Y4: state] :
                ( ( member_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc6205719116230455971_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1098_SigmaE,axiom,
    ! [C: produc6251377953716854561_state,A3: set_prat,B3: prat > set_set_state] :
      ( ( member6708919011922255306_state @ C @ ( produc4653271056321233546_state @ A3 @ B3 ) )
     => ~ ! [X3: prat] :
            ( ( member_prat @ X3 @ A3 )
           => ! [Y4: set_state] :
                ( ( member_set_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc9080284787853255769_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1099_SigmaE,axiom,
    ! [C: produc1457211279475724562t_char,A3: set_nat,B3: nat > set_list_char] :
      ( ( member484235747325421115t_char @ C @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ! [Y4: list_char] :
                ( ( member_list_char @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc120197956887798346t_char @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1100_SigmaE,axiom,
    ! [C: produc3882902684156697774_state,A3: set_state,B3: state > set_state_state] :
      ( ( member6715010751053357271_state @ C @ ( produc6646226935541082007_state @ A3 @ B3 ) )
     => ~ ! [X3: state] :
            ( ( member_state @ X3 @ A3 )
           => ! [Y4: state > state] :
                ( ( member_state_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc8632743443909064678_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1101_SigmaE,axiom,
    ! [C: produc3142500478612311029_state,A3: set_option_state,B3: option_state > set_option_state] :
      ( ( member3029510603097127326_state @ C @ ( produc7708045757989049950_state @ A3 @ B3 ) )
     => ~ ! [X3: option_state] :
            ( ( member_option_state @ X3 @ A3 )
           => ! [Y4: option_state] :
                ( ( member_option_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc9160152616361873709_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1102_SigmaE,axiom,
    ! [C: produc1357540302793366807_state,A3: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state > set_state] :
      ( ( member1546683425902591150_state @ C @ ( produc4278832915810242336_state @ A3 @ B3 ) )
     => ~ ! [X3: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X3 @ A3 )
           => ! [Y4: state] :
                ( ( member_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc7334084071316671761_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1103_SigmaE,axiom,
    ! [C: produc7272280185086996969_state,A3: set_state,B3: state > set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ C @ ( produc2831624448089660202_state @ A3 @ B3 ) )
     => ~ ! [X3: state] :
            ( ( member_state @ X3 @ A3 )
           => ! [Y4: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc5886875603596089627_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1104_SigmaE,axiom,
    ! [C: produc4357056439811976324_state,A3: set_state,B3: state > set_Pr4231240917283394591_state] :
      ( ( member5752234768399382829_state @ C @ ( produc8552965988182875245_state @ A3 @ B3 ) )
     => ~ ! [X3: state] :
            ( ( member_state @ X3 @ A3 )
           => ! [Y4: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc5194695453622026300_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1105_SigmaE,axiom,
    ! [C: produc4326932734349944580_state,A3: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state > set_state] :
      ( ( member5722111062937351085_state @ C @ ( produc1406734553846628205_state @ A3 @ B3 ) )
     => ~ ! [X3: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X3 @ A3 )
           => ! [Y4: state] :
                ( ( member_state @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc7271836056140555068_state @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_1106_times__eq__iff,axiom,
    ! [A3: set_state,B3: set_state,C2: set_state,D2: set_state] :
      ( ( ( produc4572292869282958142_state @ A3
          @ ^ [Uu3: state] : B3 )
        = ( produc4572292869282958142_state @ C2
          @ ^ [Uu3: state] : D2 ) )
      = ( ( ( A3 = C2 )
          & ( B3 = D2 ) )
        | ( ( ( A3 = bot_bot_set_state )
            | ( B3 = bot_bot_set_state ) )
          & ( ( C2 = bot_bot_set_state )
            | ( D2 = bot_bot_set_state ) ) ) ) ) ).

% times_eq_iff
thf(fact_1107_Sigma__mono,axiom,
    ! [A3: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state > set_state,D2: produc7272280185086996969_state > set_state] :
      ( ( ord_le4032880777312351359_state @ A3 @ C2 )
     => ( ! [X3: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X3 @ A3 )
           => ( ord_le2494988322063910608_state @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le9131031735280434308_state @ ( produc1406734553846628205_state @ A3 @ B3 ) @ ( produc1406734553846628205_state @ C2 @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_1108_Sigma__mono,axiom,
    ! [A3: set_Pr7649655576376376530on_val,C2: set_Pr7649655576376376530on_val,B3: produc9003781588162623004on_val > set_state,D2: produc9003781588162623004on_val > set_state] :
      ( ( ord_le6900622908513426738on_val @ A3 @ C2 )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ A3 )
           => ( ord_le2494988322063910608_state @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le801108382804549175_state @ ( produc556505947853469856_state @ A3 @ B3 ) @ ( produc556505947853469856_state @ C2 @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_1109_Sigma__mono,axiom,
    ! [A3: set_Pr2391701600042252046_state,C2: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state > set_state,D2: produc3882902684156697774_state > set_state] :
      ( ( ord_le6483787752022044334_state @ A3 @ C2 )
     => ( ! [X3: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X3 @ A3 )
           => ( ord_le2494988322063910608_state @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le4167756422255405229_state @ ( produc4278832915810242336_state @ A3 @ B3 ) @ ( produc4278832915810242336_state @ C2 @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_1110_Sigma__mono,axiom,
    ! [A3: set_state,C2: set_state,B3: state > set_state,D2: state > set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ C2 )
     => ( ! [X3: state] :
            ( ( member_state @ X3 @ A3 )
           => ( ord_le2494988322063910608_state @ ( B3 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le2777189432094499797_state @ ( produc4572292869282958142_state @ A3 @ B3 ) @ ( produc4572292869282958142_state @ C2 @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_1111_Sigma__Un__distrib1,axiom,
    ! [I2: set_state,J2: set_state,C2: state > set_Pr2391701600042252046_state] :
      ( ( produc2831624448089660202_state @ ( sup_sup_set_state @ I2 @ J2 ) @ C2 )
      = ( sup_su3837536599209510219_state @ ( produc2831624448089660202_state @ I2 @ C2 ) @ ( produc2831624448089660202_state @ J2 @ C2 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1112_Times__subset__cancel2,axiom,
    ! [X2: state,C2: set_state,A3: set_state,B3: set_state] :
      ( ( member_state @ X2 @ C2 )
     => ( ( ord_le2777189432094499797_state
          @ ( produc4572292869282958142_state @ A3
            @ ^ [Uu3: state] : C2 )
          @ ( produc4572292869282958142_state @ B3
            @ ^ [Uu3: state] : C2 ) )
        = ( ord_le2494988322063910608_state @ A3 @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_1113_Times__subset__cancel2,axiom,
    ! [X2: produc7272280185086996969_state,C2: set_Pr4231240917283394591_state,A3: set_state,B3: set_state] :
      ( ( member7461423308196221312_state @ X2 @ C2 )
     => ( ( ord_le6305643028697367044_state
          @ ( produc8552965988182875245_state @ A3
            @ ^ [Uu3: state] : C2 )
          @ ( produc8552965988182875245_state @ B3
            @ ^ [Uu3: state] : C2 ) )
        = ( ord_le2494988322063910608_state @ A3 @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_1114_Times__subset__cancel2,axiom,
    ! [X2: produc9003781588162623004on_val,C2: set_Pr7649655576376376530on_val,A3: set_state,B3: set_state] :
      ( ( member2879030624002546611on_val @ X2 @ C2 )
     => ( ( ord_le2093392236560368951on_val
          @ ( produc4123186969287248032on_val @ A3
            @ ^ [Uu3: state] : C2 )
          @ ( produc4123186969287248032on_val @ B3
            @ ^ [Uu3: state] : C2 ) )
        = ( ord_le2494988322063910608_state @ A3 @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_1115_Times__subset__cancel2,axiom,
    ! [X2: produc3882902684156697774_state,C2: set_Pr2391701600042252046_state,A3: set_state,B3: set_state] :
      ( ( member6715010751053357271_state @ X2 @ C2 )
     => ( ( ord_le4032880777312351359_state
          @ ( produc2831624448089660202_state @ A3
            @ ^ [Uu3: state] : C2 )
          @ ( produc2831624448089660202_state @ B3
            @ ^ [Uu3: state] : C2 ) )
        = ( ord_le2494988322063910608_state @ A3 @ B3 ) ) ) ).

% Times_subset_cancel2
thf(fact_1116_mem__Times__iff,axiom,
    ! [X2: produc8023240190789890773_state,A3: set_state,B3: set_state] :
      ( ( member753036827967488894_state @ X2
        @ ( produc4572292869282958142_state @ A3
          @ ^ [Uu3: state] : B3 ) )
      = ( ( member_state @ ( produc6071883272770572025_state @ X2 ) @ A3 )
        & ( member_state @ ( produc992860721336220475_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1117_mem__Times__iff,axiom,
    ! [X2: produc1457211279475724562t_char,A3: set_nat,B3: set_list_char] :
      ( ( member484235747325421115t_char @ X2
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu3: nat] : B3 ) )
      = ( ( member_nat @ ( produc6204653154488278070t_char @ X2 ) @ A3 )
        & ( member_list_char @ ( produc1900778367047502200t_char @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1118_mem__Times__iff,axiom,
    ! [X2: produc3882902684156697774_state,A3: set_state,B3: set_state_state] :
      ( ( member6715010751053357271_state @ X2
        @ ( produc6646226935541082007_state @ A3
          @ ^ [Uu3: state] : B3 ) )
      = ( ( member_state @ ( produc7095885521192019410_state @ X2 ) @ A3 )
        & ( member_state_state @ ( produc1734700239402139924_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1119_mem__Times__iff,axiom,
    ! [X2: produc1357540302793366807_state,A3: set_Pr2391701600042252046_state,B3: set_state] :
      ( ( member1546683425902591150_state @ X2
        @ ( produc4278832915810242336_state @ A3
          @ ^ [Uu3: produc3882902684156697774_state] : B3 ) )
      = ( ( member6715010751053357271_state @ ( produc2295233055888329765_state @ X2 ) @ A3 )
        & ( member_state @ ( produc7757608104100355171_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1120_mem__Times__iff,axiom,
    ! [X2: produc7272280185086996969_state,A3: set_state,B3: set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ X2
        @ ( produc2831624448089660202_state @ A3
          @ ^ [Uu3: state] : B3 ) )
      = ( ( member_state @ ( produc848024588167747631_state @ X2 ) @ A3 )
        & ( member6715010751053357271_state @ ( produc6310399636379773037_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1121_mem__Times__iff,axiom,
    ! [X2: produc4357056439811976324_state,A3: set_state,B3: set_Pr4231240917283394591_state] :
      ( ( member5752234768399382829_state @ X2
        @ ( produc8552965988182875245_state @ A3
          @ ^ [Uu3: state] : B3 ) )
      = ( ( member_state @ ( produc9028042817728042024_state @ X2 ) @ A3 )
        & ( member7461423308196221312_state @ ( produc8002322135157487722_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1122_mem__Times__iff,axiom,
    ! [X2: produc4326932734349944580_state,A3: set_Pr4231240917283394591_state,B3: set_state] :
      ( ( member5722111062937351085_state @ X2
        @ ( produc1406734553846628205_state @ A3
          @ ^ [Uu3: produc7272280185086996969_state] : B3 ) )
      = ( ( member7461423308196221312_state @ ( produc1881811383391794984_state @ X2 ) @ A3 )
        & ( member_state @ ( produc856090700821240682_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1123_mem__Times__iff,axiom,
    ! [X2: produc2030649568307348519_state,A3: set_Pr2391701600042252046_state,B3: set_Pr2391701600042252046_state] :
      ( ( member3801449062345016400_state @ X2
        @ ( produc5372175771586804680_state @ A3
          @ ^ [Uu3: produc3882902684156697774_state] : B3 ) )
      = ( ( member6715010751053357271_state @ ( produc528253918903283203_state @ X2 ) @ A3 )
        & ( member6715010751053357271_state @ ( produc1543742031010890053_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1124_mem__Times__iff,axiom,
    ! [X2: produc2932130718683550586_state,A3: set_Pr4231240917283394591_state,B3: set_Pr2391701600042252046_state] :
      ( ( member3497469645994473233_state @ X2
        @ ( produc7282391691264310459_state @ A3
          @ ^ [Uu3: produc7272280185086996969_state] : B3 ) )
      = ( ( member7461423308196221312_state @ ( produc3617983543337321920_state @ X2 ) @ A3 )
        & ( member6715010751053357271_state @ ( produc3196213569639636478_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1125_mem__Times__iff,axiom,
    ! [X2: produc3798933545658027718_state,A3: set_Pr2391701600042252046_state,B3: set_Pr4231240917283394591_state] :
      ( ( member4364272472968950365_state @ X2
        @ ( produc4180506523417663567_state @ A3
          @ ^ [Uu3: produc3882902684156697774_state] : B3 ) )
      = ( ( member6715010751053357271_state @ ( produc516098375490675028_state @ X2 ) @ A3 )
        & ( member7461423308196221312_state @ ( produc94328401792989586_state @ X2 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1126_Sigma__Un__distrib2,axiom,
    ! [I2: set_state,A3: state > set_Pr2391701600042252046_state,B3: state > set_Pr2391701600042252046_state] :
      ( ( produc2831624448089660202_state @ I2
        @ ^ [I3: state] : ( sup_su1984960714304006370_state @ ( A3 @ I3 ) @ ( B3 @ I3 ) ) )
      = ( sup_su3837536599209510219_state @ ( produc2831624448089660202_state @ I2 @ A3 ) @ ( produc2831624448089660202_state @ I2 @ B3 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1127_Times__Un__distrib1,axiom,
    ! [A3: set_state,B3: set_state,C2: set_Pr2391701600042252046_state] :
      ( ( produc2831624448089660202_state @ ( sup_sup_set_state @ A3 @ B3 )
        @ ^ [Uu3: state] : C2 )
      = ( sup_su3837536599209510219_state
        @ ( produc2831624448089660202_state @ A3
          @ ^ [Uu3: state] : C2 )
        @ ( produc2831624448089660202_state @ B3
          @ ^ [Uu3: state] : C2 ) ) ) ).

% Times_Un_distrib1
thf(fact_1128_times__subset__iff,axiom,
    ! [A3: set_state,C2: set_state,B3: set_state,D2: set_state] :
      ( ( ord_le2777189432094499797_state
        @ ( produc4572292869282958142_state @ A3
          @ ^ [Uu3: state] : C2 )
        @ ( produc4572292869282958142_state @ B3
          @ ^ [Uu3: state] : D2 ) )
      = ( ( A3 = bot_bot_set_state )
        | ( C2 = bot_bot_set_state )
        | ( ( ord_le2494988322063910608_state @ A3 @ B3 )
          & ( ord_le2494988322063910608_state @ C2 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_1129_image__paired__Times,axiom,
    ! [F4: state > state,G: state > state,A3: set_state,B3: set_state] :
      ( ( image_1391693815708851269_state
        @ ( produc4988896184150421299_state
          @ ^ [X: state,Y3: state] : ( produc544653723139640077_state @ ( F4 @ X ) @ ( G @ Y3 ) ) )
        @ ( produc4572292869282958142_state @ A3
          @ ^ [Uu3: state] : B3 ) )
      = ( produc4572292869282958142_state @ ( image_state_state @ F4 @ A3 )
        @ ^ [Uu3: state] : ( image_state_state @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_1130_relation__of__def,axiom,
    ( order_7651614119453947708_state
    = ( ^ [P6: option_state > option_state > $o,A6: set_option_state] :
          ( collec8144523193623705312_state
          @ ( produc6082990154846732246tate_o
            @ ^ [A2: option_state,B2: option_state] :
                ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A2 @ B2 )
                  @ ( produc7708045757989049950_state @ A6
                    @ ^ [Uu3: option_state] : A6 ) )
                & ( P6 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_1131_refl__on__singleton,axiom,
    ! [X2: option_state] : ( refl_on_option_state @ ( insert_option_state @ X2 @ bot_bo710180891245420500_state ) @ ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ X2 @ X2 ) @ bot_bo1080640394036989633_state ) ) ).

% refl_on_singleton
thf(fact_1132_refl__on__singleton,axiom,
    ! [X2: state] : ( refl_on_state @ ( insert_state @ X2 @ bot_bot_set_state ) @ ( insert7525286303735658661_state @ ( produc544653723139640077_state @ X2 @ X2 ) @ bot_bo9041262728264437921_state ) ) ).

% refl_on_singleton
thf(fact_1133_these__image__Some__eq,axiom,
    ! [A3: set_state] :
      ( ( these_state @ ( image_6076465424260689483_state @ some_state @ A3 ) )
      = A3 ) ).

% these_image_Some_eq
thf(fact_1134_these__image__Some__eq,axiom,
    ! [A3: set_val] :
      ( ( these_val @ ( image_val_option_val @ some_val @ A3 ) )
      = A3 ) ).

% these_image_Some_eq
thf(fact_1135_pair__imageI,axiom,
    ! [A: prat,B: state,A3: set_Pr6039890582625207499_state,F4: prat > state > state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ A3 )
     => ( member_state @ ( F4 @ A @ B ) @ ( image_6883929567913436896_state @ ( produc5191658985856023922_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1136_pair__imageI,axiom,
    ! [A: prat,B: set_state,A3: set_Pr6847263425858009217_state,F4: prat > set_state > state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ A3 )
     => ( member_state @ ( F4 @ A @ B ) @ ( image_1091725855630477206_state @ ( produc2626907587099862312_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1137_pair__imageI,axiom,
    ! [A: nat,B: list_char,A3: set_Pr2507339680178222706t_char,F4: nat > list_char > state] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A3 )
     => ( member_state @ ( F4 @ A @ B ) @ ( image_7385172899168281479_state @ ( produc217683206301814297_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1138_pair__imageI,axiom,
    ! [A: state,B: state > state,A3: set_Pr2391701600042252046_state,F4: state > ( state > state ) > state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ A3 )
     => ( member_state @ ( F4 @ A @ B ) @ ( image_6759111139246981411_state @ ( produc8575939086779714485_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1139_pair__imageI,axiom,
    ! [A: option_state,B: option_state,A3: set_Pr1688445902015331925_state,F4: option_state > option_state > state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ A3 )
     => ( member_state @ ( F4 @ A @ B ) @ ( image_393491784373993066_state @ ( produc7211634496914305532_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1140_pair__imageI,axiom,
    ! [A: state,B: state > state,A3: set_Pr2391701600042252046_state,F4: state > ( state > state ) > $o] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ A3 )
     => ( member_o @ ( F4 @ A @ B ) @ ( image_528283217102427119tate_o @ ( produc3645877548210629341tate_o @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1141_pair__imageI,axiom,
    ! [A: state,B: produc3882902684156697774_state,A3: set_Pr4231240917283394591_state,F4: state > produc3882902684156697774_state > state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ A3 )
     => ( member_state @ ( F4 @ A @ B ) @ ( image_2710133271811360426_state @ ( produc8258182183023037954_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1142_pair__imageI,axiom,
    ! [A: prat,B: state,A3: set_Pr6039890582625207499_state,F4: prat > state > produc3882902684156697774_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ A3 )
     => ( member6715010751053357271_state @ ( F4 @ A @ B ) @ ( image_1979208616201753608_state @ ( produc6727336724318121462_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1143_pair__imageI,axiom,
    ! [A: state,B: produc3882902684156697774_state,A3: set_Pr4231240917283394591_state,F4: state > produc3882902684156697774_state > $o] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ A3 )
     => ( member_o @ ( F4 @ A @ B ) @ ( image_6262870063963634088tate_o @ ( produc1745419677418529872tate_o @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1144_pair__imageI,axiom,
    ! [A: prat,B: set_state,A3: set_Pr6847263425858009217_state,F4: prat > set_state > produc3882902684156697774_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ A3 )
     => ( member6715010751053357271_state @ ( F4 @ A @ B ) @ ( image_7417905060788143314_state @ ( produc8936804542704588736_state @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1145_pair__in__swap__image,axiom,
    ! [Y: state,X2: prat,A3: set_Pr6039890582625207499_state] :
      ( ( member8716344949261458452e_prat @ ( produc5862297150877942435e_prat @ Y @ X2 ) @ ( image_2830361380300980677e_prat @ produc6686753061519163459_state @ A3 ) )
      = ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1146_pair__in__swap__image,axiom,
    ! [Y: prat,X2: state,A3: set_Pr8996442262779224523e_prat] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Y @ X2 ) @ ( image_8107462086123161541_state @ produc6343331096166649923e_prat @ A3 ) )
      = ( member8716344949261458452e_prat @ ( produc5862297150877942435e_prat @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1147_pair__in__swap__image,axiom,
    ! [Y: set_state,X2: prat,A3: set_Pr6847263425858009217_state] :
      ( ( member3111753566487124938e_prat @ ( produc5311113300630973017e_prat @ Y @ X2 ) @ ( image_8313154324233580357e_prat @ produc1562565776713478649_state @ A3 ) )
      = ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1148_pair__in__swap__image,axiom,
    ! [Y: list_char,X2: nat,A3: set_Pr2507339680178222706t_char] :
      ( ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ Y @ X2 ) @ ( image_652198164289537669ar_nat @ produc5051419219066781674t_char @ A3 ) )
      = ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1149_pair__in__swap__image,axiom,
    ! [Y: prat,X2: set_state,A3: set_Pr6566877984550221441e_prat] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Y @ X2 ) @ ( image_8063195947988251461_state @ produc7016766326345971705e_prat @ A3 ) )
      = ( member3111753566487124938e_prat @ ( produc5311113300630973017e_prat @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1150_pair__in__swap__image,axiom,
    ! [Y: nat,X2: list_char,A3: set_Pr8803067691650843250ar_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Y @ X2 ) @ ( image_688197443715608197t_char @ produc4410195085741832554ar_nat @ A3 ) )
      = ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1151_pair__in__swap__image,axiom,
    ! [Y: state > state,X2: state,A3: set_Pr2391701600042252046_state] :
      ( ( member8708878012768793047_state @ ( produc4737362649269059174_state @ Y @ X2 ) @ ( image_2502765500263079301_state @ produc6438927260845192582_state @ A3 ) )
      = ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1152_pair__in__swap__image,axiom,
    ! [Y: state,X2: state > state,A3: set_Pr2063381651806504974_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Y @ X2 ) @ ( image_5540908381507763077_state @ produc2543546466205187078_state @ A3 ) )
      = ( member8708878012768793047_state @ ( produc4737362649269059174_state @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1153_pair__in__swap__image,axiom,
    ! [Y: option_state,X2: option_state,A3: set_Pr1688445902015331925_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Y @ X2 ) @ ( image_3543735668343923269_state @ produc8425362190318354125_state @ A3 ) )
      = ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1154_pair__in__swap__image,axiom,
    ! [Y: produc3882902684156697774_state,X2: state,A3: set_Pr4231240917283394591_state] :
      ( ( member1546683425902591150_state @ ( produc7334084071316671761_state @ Y @ X2 ) @ ( image_6372718444620452615_state @ produc6564544480275723131_state @ A3 ) )
      = ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X2 @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_1155_fst__image__times,axiom,
    ! [B3: set_list_char,A3: set_nat] :
      ( ( ( B3 = bot_bo8503570744482963627t_char )
       => ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char
            @ ( produc6030673636645808891t_char @ A3
              @ ^ [Uu3: nat] : B3 ) )
          = bot_bot_set_nat ) )
      & ( ( B3 != bot_bo8503570744482963627t_char )
       => ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char
            @ ( produc6030673636645808891t_char @ A3
              @ ^ [Uu3: nat] : B3 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_1156_fst__image__times,axiom,
    ! [B3: set_Pr1004501353197068112on_val,A3: set_Pr3433046674199179443r_prat] :
      ( ( ( B3 = bot_bo5671083314777306084on_val )
       => ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val
            @ ( produc1366239611153933861on_val @ A3
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B3 ) )
          = bot_bo3164680191535668511r_prat ) )
      & ( ( B3 != bot_bo5671083314777306084on_val )
       => ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val
            @ ( produc1366239611153933861on_val @ A3
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B3 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_1157_fst__image__times,axiom,
    ! [B3: set_state,A3: set_state] :
      ( ( ( B3 = bot_bot_set_state )
       => ( ( image_5145414166468158282_state @ produc6071883272770572025_state
            @ ( produc4572292869282958142_state @ A3
              @ ^ [Uu3: state] : B3 ) )
          = bot_bot_set_state ) )
      & ( ( B3 != bot_bot_set_state )
       => ( ( image_5145414166468158282_state @ produc6071883272770572025_state
            @ ( produc4572292869282958142_state @ A3
              @ ^ [Uu3: state] : B3 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_1158_snd__image__times,axiom,
    ! [A3: set_Pr3433046674199179443r_prat,B3: set_Pr1004501353197068112on_val] :
      ( ( ( A3 = bot_bo3164680191535668511r_prat )
       => ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val
            @ ( produc1366239611153933861on_val @ A3
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B3 ) )
          = bot_bo5671083314777306084on_val ) )
      & ( ( A3 != bot_bo3164680191535668511r_prat )
       => ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val
            @ ( produc1366239611153933861on_val @ A3
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B3 ) )
          = B3 ) ) ) ).

% snd_image_times
thf(fact_1159_snd__image__times,axiom,
    ! [A3: set_state,B3: set_state] :
      ( ( ( A3 = bot_bot_set_state )
       => ( ( image_5145414166468158282_state @ produc992860721336220475_state
            @ ( produc4572292869282958142_state @ A3
              @ ^ [Uu3: state] : B3 ) )
          = bot_bot_set_state ) )
      & ( ( A3 != bot_bot_set_state )
       => ( ( image_5145414166468158282_state @ produc992860721336220475_state
            @ ( produc4572292869282958142_state @ A3
              @ ^ [Uu3: state] : B3 ) )
          = B3 ) ) ) ).

% snd_image_times
thf(fact_1160_refl__onD2,axiom,
    ! [A3: set_state,R3: set_Pr1785066336555260981_state,X2: state,Y: state] :
      ( ( refl_on_state @ A3 @ R3 )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ X2 @ Y ) @ R3 )
       => ( member_state @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_1161_refl__onD2,axiom,
    ! [A3: set_Pr4231240917283394591_state,R3: set_Pr7975217055296730899_state,X2: produc7272280185086996969_state,Y: produc7272280185086996969_state] :
      ( ( refl_o8189482246475458346_state @ A3 @ R3 )
     => ( ( member1349250210377977180_state @ ( produc2330690084503141483_state @ X2 @ Y ) @ R3 )
       => ( member7461423308196221312_state @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_1162_refl__onD2,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val,X2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R3 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y ) @ R3 )
       => ( member2879030624002546611on_val @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_1163_refl__onD2,axiom,
    ! [A3: set_Pr2391701600042252046_state,R3: set_Pr4846723933414538759_state,X2: produc3882902684156697774_state,Y: produc3882902684156697774_state] :
      ( ( refl_o3383612479072233901_state @ A3 @ R3 )
     => ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ X2 @ Y ) @ R3 )
       => ( member6715010751053357271_state @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_1164_refl__onD2,axiom,
    ! [A3: set_option_state,R3: set_Pr1688445902015331925_state,X2: option_state,Y: option_state] :
      ( ( refl_on_option_state @ A3 @ R3 )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X2 @ Y ) @ R3 )
       => ( member_option_state @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_1165_refl__onD1,axiom,
    ! [A3: set_state,R3: set_Pr1785066336555260981_state,X2: state,Y: state] :
      ( ( refl_on_state @ A3 @ R3 )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ X2 @ Y ) @ R3 )
       => ( member_state @ X2 @ A3 ) ) ) ).

% refl_onD1
thf(fact_1166_refl__onD1,axiom,
    ! [A3: set_Pr4231240917283394591_state,R3: set_Pr7975217055296730899_state,X2: produc7272280185086996969_state,Y: produc7272280185086996969_state] :
      ( ( refl_o8189482246475458346_state @ A3 @ R3 )
     => ( ( member1349250210377977180_state @ ( produc2330690084503141483_state @ X2 @ Y ) @ R3 )
       => ( member7461423308196221312_state @ X2 @ A3 ) ) ) ).

% refl_onD1
thf(fact_1167_refl__onD1,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val,X2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R3 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y ) @ R3 )
       => ( member2879030624002546611on_val @ X2 @ A3 ) ) ) ).

% refl_onD1
thf(fact_1168_refl__onD1,axiom,
    ! [A3: set_Pr2391701600042252046_state,R3: set_Pr4846723933414538759_state,X2: produc3882902684156697774_state,Y: produc3882902684156697774_state] :
      ( ( refl_o3383612479072233901_state @ A3 @ R3 )
     => ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ X2 @ Y ) @ R3 )
       => ( member6715010751053357271_state @ X2 @ A3 ) ) ) ).

% refl_onD1
thf(fact_1169_refl__onD1,axiom,
    ! [A3: set_option_state,R3: set_Pr1688445902015331925_state,X2: option_state,Y: option_state] :
      ( ( refl_on_option_state @ A3 @ R3 )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X2 @ Y ) @ R3 )
       => ( member_option_state @ X2 @ A3 ) ) ) ).

% refl_onD1
thf(fact_1170_refl__onD,axiom,
    ! [A3: set_state,R3: set_Pr1785066336555260981_state,A: state] :
      ( ( refl_on_state @ A3 @ R3 )
     => ( ( member_state @ A @ A3 )
       => ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_1171_refl__onD,axiom,
    ! [A3: set_Pr4231240917283394591_state,R3: set_Pr7975217055296730899_state,A: produc7272280185086996969_state] :
      ( ( refl_o8189482246475458346_state @ A3 @ R3 )
     => ( ( member7461423308196221312_state @ A @ A3 )
       => ( member1349250210377977180_state @ ( produc2330690084503141483_state @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_1172_refl__onD,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R3 )
     => ( ( member2879030624002546611on_val @ A @ A3 )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_1173_refl__onD,axiom,
    ! [A3: set_Pr2391701600042252046_state,R3: set_Pr4846723933414538759_state,A: produc3882902684156697774_state] :
      ( ( refl_o3383612479072233901_state @ A3 @ R3 )
     => ( ( member6715010751053357271_state @ A @ A3 )
       => ( member3801449062345016400_state @ ( produc2343482328048119831_state @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_1174_refl__onD,axiom,
    ! [A3: set_option_state,R3: set_Pr1688445902015331925_state,A: option_state] :
      ( ( refl_on_option_state @ A3 @ R3 )
     => ( ( member_option_state @ A @ A3 )
       => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A ) @ R3 ) ) ) ).

% refl_onD
thf(fact_1175_refl__on__domain,axiom,
    ! [A3: set_state,R3: set_Pr1785066336555260981_state,A: state,B: state] :
      ( ( refl_on_state @ A3 @ R3 )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ B ) @ R3 )
       => ( ( member_state @ A @ A3 )
          & ( member_state @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_1176_refl__on__domain,axiom,
    ! [A3: set_Pr4231240917283394591_state,R3: set_Pr7975217055296730899_state,A: produc7272280185086996969_state,B: produc7272280185086996969_state] :
      ( ( refl_o8189482246475458346_state @ A3 @ R3 )
     => ( ( member1349250210377977180_state @ ( produc2330690084503141483_state @ A @ B ) @ R3 )
       => ( ( member7461423308196221312_state @ A @ A3 )
          & ( member7461423308196221312_state @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_1177_refl__on__domain,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A3 @ R3 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R3 )
       => ( ( member2879030624002546611on_val @ A @ A3 )
          & ( member2879030624002546611on_val @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_1178_refl__on__domain,axiom,
    ! [A3: set_Pr2391701600042252046_state,R3: set_Pr4846723933414538759_state,A: produc3882902684156697774_state,B: produc3882902684156697774_state] :
      ( ( refl_o3383612479072233901_state @ A3 @ R3 )
     => ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ A @ B ) @ R3 )
       => ( ( member6715010751053357271_state @ A @ A3 )
          & ( member6715010751053357271_state @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_1179_refl__on__domain,axiom,
    ! [A3: set_option_state,R3: set_Pr1688445902015331925_state,A: option_state,B: option_state] :
      ( ( refl_on_option_state @ A3 @ R3 )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ R3 )
       => ( ( member_option_state @ A @ A3 )
          & ( member_option_state @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_1180_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_1181_None__notin__image__Some,axiom,
    ! [A3: set_val] :
      ~ ( member_option_val @ none_val @ ( image_val_option_val @ some_val @ A3 ) ) ).

% None_notin_image_Some
thf(fact_1182_fst__image__Sigma,axiom,
    ! [A3: set_nat,B3: nat > set_list_char] :
      ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
      = ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bo8503570744482963627t_char ) ) ) ) ).

% fst_image_Sigma
thf(fact_1183_fst__image__Sigma,axiom,
    ! [A3: set_Pr3433046674199179443r_prat,B3: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ ( produc1366239611153933861on_val @ A3 @ B3 ) )
      = ( collec7093475260167521854r_prat
        @ ^ [X: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bo5671083314777306084on_val ) ) ) ) ).

% fst_image_Sigma
thf(fact_1184_fst__image__Sigma,axiom,
    ! [A3: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state > set_state] :
      ( ( image_3025028170732149008_state @ produc2295233055888329765_state @ ( produc4278832915810242336_state @ A3 @ B3 ) )
      = ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bot_set_state ) ) ) ) ).

% fst_image_Sigma
thf(fact_1185_fst__image__Sigma,axiom,
    ! [A3: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state > set_state] :
      ( ( image_8377117390685227944_state @ produc1881811383391794984_state @ ( produc1406734553846628205_state @ A3 @ B3 ) )
      = ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bot_set_state ) ) ) ) ).

% fst_image_Sigma
thf(fact_1186_fst__image__Sigma,axiom,
    ! [A3: set_Pr7649655576376376530on_val,B3: produc9003781588162623004on_val > set_state] :
      ( ( image_3733899853514731534on_val @ produc6734235122797862491_state @ ( produc556505947853469856_state @ A3 @ B3 ) )
      = ( collec7475294502966882545on_val
        @ ^ [X: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bot_set_state ) ) ) ) ).

% fst_image_Sigma
thf(fact_1187_fst__image__Sigma,axiom,
    ! [A3: set_state,B3: state > set_state] :
      ( ( image_5145414166468158282_state @ produc6071883272770572025_state @ ( produc4572292869282958142_state @ A3 @ B3 ) )
      = ( collect_state
        @ ^ [X: state] :
            ( ( member_state @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bot_set_state ) ) ) ) ).

% fst_image_Sigma
thf(fact_1188_swap__product,axiom,
    ! [A3: set_option_state,B3: set_option_state] :
      ( ( image_3543735668343923269_state
        @ ( produc803252138055838771_state
          @ ^ [I3: option_state,J3: option_state] : ( produc9160152616361873709_state @ J3 @ I3 ) )
        @ ( produc7708045757989049950_state @ A3
          @ ^ [Uu3: option_state] : B3 ) )
      = ( produc7708045757989049950_state @ B3
        @ ^ [Uu3: option_state] : A3 ) ) ).

% swap_product
thf(fact_1189_swap__product,axiom,
    ! [A3: set_state,B3: set_prat] :
      ( ( image_8107462086123161541_state
        @ ( produc772268595108696499_state
          @ ^ [I3: state,J3: prat] : ( produc6205719116230455971_state @ J3 @ I3 ) )
        @ ( produc1315363979526694100e_prat @ A3
          @ ^ [Uu3: state] : B3 ) )
      = ( produc1658785944879207636_state @ B3
        @ ^ [Uu3: prat] : A3 ) ) ).

% swap_product
thf(fact_1190_swap__product,axiom,
    ! [A3: set_set_state,B3: set_prat] :
      ( ( image_8063195947988251461_state
        @ ( produc4505194488364329267_state
          @ ^ [I3: set_state,J3: prat] : ( produc9080284787853255769_state @ J3 @ I3 ) )
        @ ( produc884099569098950794e_prat @ A3
          @ ^ [Uu3: set_state] : B3 ) )
      = ( produc4653271056321233546_state @ B3
        @ ^ [Uu3: prat] : A3 ) ) ).

% swap_product
thf(fact_1191_swap__product,axiom,
    ! [A3: set_list_char,B3: set_nat] :
      ( ( image_688197443715608197t_char
        @ ( produc6166083679720228083t_char
          @ ^ [I3: list_char,J3: nat] : ( produc120197956887798346t_char @ J3 @ I3 ) )
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu3: list_char] : B3 ) )
      = ( produc6030673636645808891t_char @ B3
        @ ^ [Uu3: nat] : A3 ) ) ).

% swap_product
thf(fact_1192_swap__product,axiom,
    ! [A3: set_Pr1004501353197068112on_val,B3: set_Pr3433046674199179443r_prat] :
      ( ( image_1698057272686278225on_val
        @ ( produc4110721796449090601on_val
          @ ^ [I3: produc1457211279475724562t_char > option_val,J3: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ J3 @ I3 ) )
        @ ( produc3189049046222722799r_prat @ A3
          @ ^ [Uu3: produc1457211279475724562t_char > option_val] : B3 ) )
      = ( produc1366239611153933861on_val @ B3
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : A3 ) ) ).

% swap_product
thf(fact_1193_Some__image__these__eq,axiom,
    ! [A3: set_option_state] :
      ( ( image_6076465424260689483_state @ some_state @ ( these_state @ A3 ) )
      = ( collect_option_state
        @ ^ [X: option_state] :
            ( ( member_option_state @ X @ A3 )
            & ( X != none_state ) ) ) ) ).

% Some_image_these_eq
thf(fact_1194_Some__image__these__eq,axiom,
    ! [A3: set_option_val] :
      ( ( image_val_option_val @ some_val @ ( these_val @ A3 ) )
      = ( collect_option_val
        @ ^ [X: option_val] :
            ( ( member_option_val @ X @ A3 )
            & ( X != none_val ) ) ) ) ).

% Some_image_these_eq
thf(fact_1195_refl__on__def,axiom,
    ( refl_on_option_state
    = ( ^ [A6: set_option_state,R: set_Pr1688445902015331925_state] :
          ( ( ord_le6423325748750870005_state @ R
            @ ( produc7708045757989049950_state @ A6
              @ ^ [Uu3: option_state] : A6 ) )
          & ! [X: option_state] :
              ( ( member_option_state @ X @ A6 )
             => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def
thf(fact_1196_refl__onI,axiom,
    ! [R3: set_Pr1785066336555260981_state,A3: set_state] :
      ( ( ord_le2777189432094499797_state @ R3
        @ ( produc4572292869282958142_state @ A3
          @ ^ [Uu3: state] : A3 ) )
     => ( ! [X3: state] :
            ( ( member_state @ X3 @ A3 )
           => ( member753036827967488894_state @ ( produc544653723139640077_state @ X3 @ X3 ) @ R3 ) )
       => ( refl_on_state @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_1197_refl__onI,axiom,
    ! [R3: set_Pr7975217055296730899_state,A3: set_Pr4231240917283394591_state] :
      ( ( ord_le77907785264135859_state @ R3
        @ ( produc8058815403421562012_state @ A3
          @ ^ [Uu3: produc7272280185086996969_state] : A3 ) )
     => ( ! [X3: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X3 @ A3 )
           => ( member1349250210377977180_state @ ( produc2330690084503141483_state @ X3 @ X3 ) @ R3 ) )
       => ( refl_o8189482246475458346_state @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_1198_refl__onI,axiom,
    ! [R3: set_Pr4305388979797279737on_val,A3: set_Pr7649655576376376530on_val] :
      ( ( ord_le7224449410216889241on_val @ R3
        @ ( produc4067394217145614850on_val @ A3
          @ ^ [Uu3: produc9003781588162623004on_val] : A3 ) )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ A3 )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ X3 ) @ R3 ) )
       => ( refl_o6679066329051764061on_val @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_1199_refl__onI,axiom,
    ! [R3: set_Pr4846723933414538759_state,A3: set_Pr2391701600042252046_state] :
      ( ( ord_le1955585117161862567_state @ R3
        @ ( produc5372175771586804680_state @ A3
          @ ^ [Uu3: produc3882902684156697774_state] : A3 ) )
     => ( ! [X3: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X3 @ A3 )
           => ( member3801449062345016400_state @ ( produc2343482328048119831_state @ X3 @ X3 ) @ R3 ) )
       => ( refl_o3383612479072233901_state @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_1200_refl__onI,axiom,
    ! [R3: set_Pr1688445902015331925_state,A3: set_option_state] :
      ( ( ord_le6423325748750870005_state @ R3
        @ ( produc7708045757989049950_state @ A3
          @ ^ [Uu3: option_state] : A3 ) )
     => ( ! [X3: option_state] :
            ( ( member_option_state @ X3 @ A3 )
           => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ X3 ) @ R3 ) )
       => ( refl_on_option_state @ A3 @ R3 ) ) ) ).

% refl_onI
thf(fact_1201_Option_Othese__def,axiom,
    ( these_state
    = ( ^ [A6: set_option_state] :
          ( image_3532137647693456075_state @ the_state
          @ ( collect_option_state
            @ ^ [X: option_state] :
                ( ( member_option_state @ X @ A6 )
                & ( X != none_state ) ) ) ) ) ) ).

% Option.these_def
thf(fact_1202_Option_Othese__def,axiom,
    ( these_val
    = ( ^ [A6: set_option_val] :
          ( image_option_val_val @ the_val
          @ ( collect_option_val
            @ ^ [X: option_val] :
                ( ( member_option_val @ X @ A6 )
                & ( X != none_val ) ) ) ) ) ) ).

% Option.these_def
thf(fact_1203_refl__on__def_H,axiom,
    ( refl_on_state
    = ( ^ [A6: set_state,R: set_Pr1785066336555260981_state] :
          ( ! [X: produc8023240190789890773_state] :
              ( ( member753036827967488894_state @ X @ R )
             => ( produc5376313899190161910tate_o
                @ ^ [Y3: state,Z3: state] :
                    ( ( member_state @ Y3 @ A6 )
                    & ( member_state @ Z3 @ A6 ) )
                @ X ) )
          & ! [X: state] :
              ( ( member_state @ X @ A6 )
             => ( member753036827967488894_state @ ( produc544653723139640077_state @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1204_refl__on__def_H,axiom,
    ( refl_o8189482246475458346_state
    = ( ^ [A6: set_Pr4231240917283394591_state,R: set_Pr7975217055296730899_state] :
          ( ! [X: produc1362221940583236787_state] :
              ( ( member1349250210377977180_state @ X @ R )
             => ( produc3133855996611560728tate_o
                @ ^ [Y3: produc7272280185086996969_state,Z3: produc7272280185086996969_state] :
                    ( ( member7461423308196221312_state @ Y3 @ A6 )
                    & ( member7461423308196221312_state @ Z3 @ A6 ) )
                @ X ) )
          & ! [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ A6 )
             => ( member1349250210377977180_state @ ( produc2330690084503141483_state @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1205_refl__on__def_H,axiom,
    ( refl_o6679066329051764061on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
          ( ! [X: produc1317171644805857433on_val] :
              ( ( member6505152944228695362on_val @ X @ R )
             => ( produc1845727980445322802_val_o
                @ ^ [Y3: produc9003781588162623004on_val,Z3: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ Y3 @ A6 )
                    & ( member2879030624002546611on_val @ Z3 @ A6 ) )
                @ X ) )
          & ! [X: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X @ A6 )
             => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1206_refl__on__def_H,axiom,
    ( refl_o3383612479072233901_state
    = ( ^ [A6: set_Pr2391701600042252046_state,R: set_Pr4846723933414538759_state] :
          ( ! [X: produc2030649568307348519_state] :
              ( ( member3801449062345016400_state @ X @ R )
             => ( produc7167296152490707052tate_o
                @ ^ [Y3: produc3882902684156697774_state,Z3: produc3882902684156697774_state] :
                    ( ( member6715010751053357271_state @ Y3 @ A6 )
                    & ( member6715010751053357271_state @ Z3 @ A6 ) )
                @ X ) )
          & ! [X: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X @ A6 )
             => ( member3801449062345016400_state @ ( produc2343482328048119831_state @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1207_refl__on__def_H,axiom,
    ( refl_on_option_state
    = ( ^ [A6: set_option_state,R: set_Pr1688445902015331925_state] :
          ( ! [X: produc3142500478612311029_state] :
              ( ( member3029510603097127326_state @ X @ R )
             => ( produc6082990154846732246tate_o
                @ ^ [Y3: option_state,Z3: option_state] :
                    ( ( member_option_state @ Y3 @ A6 )
                    & ( member_option_state @ Z3 @ A6 ) )
                @ X ) )
          & ! [X: option_state] :
              ( ( member_option_state @ X @ A6 )
             => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1208_Collect__split__mono__strong,axiom,
    ! [X8: set_nat,A3: set_Pr2507339680178222706t_char,Y8: set_list_char,P4: nat > list_char > $o,Q: nat > list_char > $o] :
      ( ( X8
        = ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ A3 ) )
     => ( ( Y8
          = ( image_3034139768025083822t_char @ produc1900778367047502200t_char @ A3 ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ X8 )
             => ! [Xa2: list_char] :
                  ( ( member_list_char @ Xa2 @ Y8 )
                 => ( ( P4 @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ P4 ) ) )
           => ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1209_Collect__split__mono__strong,axiom,
    ! [X8: set_Pr3433046674199179443r_prat,A3: set_Pr7649655576376376530on_val,Y8: set_Pr1004501353197068112on_val,P4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( X8
        = ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ A3 ) )
     => ( ( Y8
          = ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ A3 ) )
       => ( ! [X3: produc1457211279475724562t_char > prat] :
              ( ( member2336342582916103932r_prat @ X3 @ X8 )
             => ! [Xa2: produc1457211279475724562t_char > option_val] :
                  ( ( member1577473829074448177on_val @ Xa2 @ Y8 )
                 => ( ( P4 @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le6900622908513426738on_val @ A3 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ P4 ) ) )
           => ( ord_le6900622908513426738on_val @ A3 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1210_Collect__split__mono__strong,axiom,
    ! [X8: set_state,A3: set_Pr2391701600042252046_state,Y8: set_state_state,P4: state > ( state > state ) > $o,Q: state > ( state > state ) > $o] :
      ( ( X8
        = ( image_6759111139246981411_state @ produc7095885521192019410_state @ A3 ) )
     => ( ( Y8
          = ( image_6089932313839440636_state @ produc1734700239402139924_state @ A3 ) )
       => ( ! [X3: state] :
              ( ( member_state @ X3 @ X8 )
             => ! [Xa2: state > state] :
                  ( ( member_state_state @ Xa2 @ Y8 )
                 => ( ( P4 @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le6483787752022044334_state @ A3 @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ P4 ) ) )
           => ( ord_le6483787752022044334_state @ A3 @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1211_Collect__split__mono__strong,axiom,
    ! [X8: set_state,A3: set_Pr4231240917283394591_state,Y8: set_Pr2391701600042252046_state,P4: state > produc3882902684156697774_state > $o,Q: state > produc3882902684156697774_state > $o] :
      ( ( X8
        = ( image_2710133271811360426_state @ produc848024588167747631_state @ A3 ) )
     => ( ( Y8
          = ( image_7466395622266389182_state @ produc6310399636379773037_state @ A3 ) )
       => ( ! [X3: state] :
              ( ( member_state @ X3 @ X8 )
             => ! [Xa2: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ Xa2 @ Y8 )
                 => ( ( P4 @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le4032880777312351359_state @ A3 @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ P4 ) ) )
           => ( ord_le4032880777312351359_state @ A3 @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1212_subset__fst__snd,axiom,
    ! [A3: set_Pr2507339680178222706t_char] :
      ( ord_le6315511645215477266t_char @ A3
      @ ( produc6030673636645808891t_char @ ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ A3 )
        @ ^ [Uu3: nat] : ( image_3034139768025083822t_char @ produc1900778367047502200t_char @ A3 ) ) ) ).

% subset_fst_snd
thf(fact_1213_subset__fst__snd,axiom,
    ! [A3: set_Pr7649655576376376530on_val] :
      ( ord_le6900622908513426738on_val @ A3
      @ ( produc1366239611153933861on_val @ ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ A3 )
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ A3 ) ) ) ).

% subset_fst_snd
thf(fact_1214_member__product,axiom,
    ! [X2: produc7272280185086996969_state,A3: set_state,B3: set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ X2 @ ( produc3518108405486217632_state @ A3 @ B3 ) )
      = ( member7461423308196221312_state @ X2
        @ ( produc2831624448089660202_state @ A3
          @ ^ [Uu3: state] : B3 ) ) ) ).

% member_product
thf(fact_1215_member__product,axiom,
    ! [X2: produc9003781588162623004on_val,A3: set_Pr3433046674199179443r_prat,B3: set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ X2 @ ( produc747609993485717659on_val @ A3 @ B3 ) )
      = ( member2879030624002546611on_val @ X2
        @ ( produc1366239611153933861on_val @ A3
          @ ^ [Uu3: produc1457211279475724562t_char > prat] : B3 ) ) ) ).

% member_product
thf(fact_1216_member__product,axiom,
    ! [X2: produc3882902684156697774_state,A3: set_state,B3: set_state_state] :
      ( ( member6715010751053357271_state @ X2 @ ( produc7466003549325918881_state @ A3 @ B3 ) )
      = ( member6715010751053357271_state @ X2
        @ ( produc6646226935541082007_state @ A3
          @ ^ [Uu3: state] : B3 ) ) ) ).

% member_product
thf(fact_1217_flip__pred,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R2: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6900622908513426738on_val @ A3 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( conver4900003399366114050r_prat @ R2 ) ) ) )
     => ( ord_le2122960673345045316r_prat
        @ ( image_2943372182239103057r_prat
          @ ( produc2964738055033037937r_prat
            @ ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( produc1897429902272487136r_prat @ Y3 @ X ) )
          @ A3 )
        @ ( collec6247776740490166275r_prat @ ( produc5226666187743293259prat_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1218_flip__pred,axiom,
    ! [A3: set_Pr1688445902015331925_state,R2: option_state > option_state > $o] :
      ( ( ord_le6423325748750870005_state @ A3 @ ( collec8144523193623705312_state @ ( produc6082990154846732246tate_o @ ( conver5693912221178358411_state @ R2 ) ) ) )
     => ( ord_le6423325748750870005_state
        @ ( image_3543735668343923269_state
          @ ( produc803252138055838771_state
            @ ^ [X: option_state,Y3: option_state] : ( produc9160152616361873709_state @ Y3 @ X ) )
          @ A3 )
        @ ( collec8144523193623705312_state @ ( produc6082990154846732246tate_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1219_flip__pred,axiom,
    ! [A3: set_Pr8996442262779224523e_prat,R2: prat > state > $o] :
      ( ( ord_le3218748728643905899e_prat @ A3 @ ( collec2363146809267008342e_prat @ ( produc5248533930912372960prat_o @ ( conversep_prat_state @ R2 ) ) ) )
     => ( ord_le262197048489888875_state
        @ ( image_8107462086123161541_state
          @ ( produc772268595108696499_state
            @ ^ [X: state,Y3: prat] : ( produc6205719116230455971_state @ Y3 @ X ) )
          @ A3 )
        @ ( collec9096564466166568534_state @ ( produc1228703470279370976tate_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1220_flip__pred,axiom,
    ! [A3: set_Pr6566877984550221441e_prat,R2: prat > set_state > $o] :
      ( ( ord_le5481108615602339361e_prat @ A3 @ ( collec1496918421318429452e_prat @ ( produc6803115863956348842prat_o @ ( conver7791356076405787575_state @ R2 ) ) ) )
     => ( ord_le5761494056910127137_state
        @ ( image_8063195947988251461_state
          @ ( produc4505194488364329267_state
            @ ^ [X: set_state,Y3: prat] : ( produc9080284787853255769_state @ Y3 @ X ) )
          @ A3 )
        @ ( collec5094083866753559820_state @ ( produc7925652287191295402tate_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1221_flip__pred,axiom,
    ! [A3: set_Pr8803067691650843250ar_nat,R2: nat > list_char > $o] :
      ( ( ord_le3387867619833322002ar_nat @ A3 @ ( collec8608152508653127293ar_nat @ ( produc2492928673396900345_nat_o @ ( conver4209408378539545768t_char @ R2 ) ) ) )
     => ( ord_le6315511645215477266t_char
        @ ( image_688197443715608197t_char
          @ ( produc6166083679720228083t_char
            @ ^ [X: list_char,Y3: nat] : ( produc120197956887798346t_char @ Y3 @ X ) )
          @ A3 )
        @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1222_flip__pred,axiom,
    ! [A3: set_Pr2871993341207995108r_prat,R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( ord_le2122960673345045316r_prat @ A3 @ ( collec6247776740490166275r_prat @ ( produc5226666187743293259prat_o @ ( conver3077193964297325112on_val @ R2 ) ) ) )
     => ( ord_le6900622908513426738on_val
        @ ( image_1698057272686278225on_val
          @ ( produc4110721796449090601on_val
            @ ^ [X: produc1457211279475724562t_char > option_val,Y3: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ Y3 @ X ) )
          @ A3 )
        @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1223_flip__pred,axiom,
    ! [A3: set_Pr2063381651806504974_state,R2: state > ( state > state ) > $o] :
      ( ( ord_le6155467803786297262_state @ A3 @ ( collec2388539233454781465_state @ ( produc7947965470134731869tate_o @ ( conver5193656574937193540_state @ R2 ) ) ) )
     => ( ord_le6483787752022044334_state
        @ ( image_5540908381507763077_state
          @ ( produc3696886681956174835_state
            @ ^ [X: state > state,Y3: state] : ( produc8632743443909064678_state @ Y3 @ X ) )
          @ A3 )
        @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1224_flip__pred,axiom,
    ! [A3: set_Pr4366116562226448461_state,R2: state > produc3882902684156697774_state > $o] :
      ( ( ord_le4167756422255405229_state @ A3 @ ( collec4215477171196732908_state @ ( produc2011331576729698010tate_o @ ( conver7967167495199203645_state @ R2 ) ) ) )
     => ( ord_le4032880777312351359_state
        @ ( image_726267498729412871_state
          @ ( produc5156651381015392551_state
            @ ^ [X: produc3882902684156697774_state,Y3: state] : ( produc5886875603596089627_state @ Y3 @ X ) )
          @ A3 )
        @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1225_flip__pred,axiom,
    ! [A3: set_Pr2391701600042252046_state,R2: ( state > state ) > state > $o] :
      ( ( ord_le6483787752022044334_state @ A3 @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ ( conver1298275780297188036_state @ R2 ) ) ) )
     => ( ord_le6155467803786297262_state
        @ ( image_2502765500263079301_state
          @ ( produc7521540169313256307_state
            @ ^ [X: state,Y3: state > state] : ( produc4737362649269059174_state @ Y3 @ X ) )
          @ A3 )
        @ ( collec2388539233454781465_state @ ( produc7947965470134731869tate_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1226_flip__pred,axiom,
    ! [A3: set_Pr4231240917283394591_state,R2: produc3882902684156697774_state > state > $o] :
      ( ( ord_le4032880777312351359_state @ A3 @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ ( conver191003926065009971_state @ R2 ) ) ) )
     => ( ord_le4167756422255405229_state
        @ ( image_6372718444620452615_state
          @ ( produc6605482068524530271_state
            @ ^ [X: state,Y3: produc3882902684156697774_state] : ( produc7334084071316671761_state @ Y3 @ X ) )
          @ A3 )
        @ ( collec4215477171196732908_state @ ( produc2011331576729698010tate_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_1227_aboveS__def,axiom,
    ( order_2272932805307567934_state
    = ( ^ [R: set_Pr7975217055296730899_state,A2: produc7272280185086996969_state] :
          ( collec906845016635587262_state
          @ ^ [B2: produc7272280185086996969_state] :
              ( ( B2 != A2 )
              & ( member1349250210377977180_state @ ( produc2330690084503141483_state @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_1228_aboveS__def,axiom,
    ( order_86566288019506545on_val
    = ( ^ [R: set_Pr4305388979797279737on_val,A2: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B2: produc9003781588162623004on_val] :
              ( ( B2 != A2 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_1229_aboveS__def,axiom,
    ( order_aboveS_state
    = ( ^ [R: set_Pr1785066336555260981_state,A2: state] :
          ( collect_state
          @ ^ [B2: state] :
              ( ( B2 != A2 )
              & ( member753036827967488894_state @ ( produc544653723139640077_state @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_1230_aboveS__def,axiom,
    ( order_7110021593729121247_state
    = ( ^ [R: set_Pr1688445902015331925_state,A2: option_state] :
          ( collect_option_state
          @ ^ [B2: option_state] :
              ( ( B2 != A2 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_1231_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr1688445902015331925_state] :
      ( ( order_7502633587729487893_state @ ( field_option_state @ R3 ) @ R3 )
     => ( ( order_4461428600198971116_state @ ( field_option_state @ R3 ) @ R3 )
        = ( ! [A6: set_option_state] :
              ( ( ord_le7116032884704190368_state @ A6 @ ( field_option_state @ R3 ) )
             => ( ( A6 != bot_bo710180891245420500_state )
               => ? [X: option_state] :
                    ( ( member_option_state @ X @ A6 )
                    & ! [Y3: option_state] :
                        ( ( member_option_state @ Y3 @ A6 )
                       => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1232_Linear__order__Well__order__iff,axiom,
    ! [R3: set_Pr1785066336555260981_state] :
      ( ( order_7174064491916216133_state @ ( field_state @ R3 ) @ R3 )
     => ( ( order_3972865417222356124_state @ ( field_state @ R3 ) @ R3 )
        = ( ! [A6: set_state] :
              ( ( ord_le2494988322063910608_state @ A6 @ ( field_state @ R3 ) )
             => ( ( A6 != bot_bot_set_state )
               => ? [X: state] :
                    ( ( member_state @ X @ A6 )
                    & ! [Y3: state] :
                        ( ( member_state @ Y3 @ A6 )
                       => ( member753036827967488894_state @ ( produc544653723139640077_state @ X @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1233_Sigma__def,axiom,
    ( produc7708045757989049950_state
    = ( ^ [A6: set_option_state,B6: option_state > set_option_state] :
          ( comple3552586563226307650_state
          @ ( image_5951889428630037764_state
            @ ^ [X: option_state] :
                ( comple3552586563226307650_state
                @ ( image_5951889428630037764_state
                  @ ^ [Y3: option_state] : ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ bot_bo1080640394036989633_state )
                  @ ( B6 @ X ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1234_Sigma__def,axiom,
    ( produc1658785944879207636_state
    = ( ^ [A6: set_prat,B6: prat > set_state] :
          ( comple2806200606391696312_state
          @ ( image_4777334102424180148_state
            @ ^ [X: prat] :
                ( comple2806200606391696312_state
                @ ( image_366767958430777034_state
                  @ ^ [Y3: state] : ( insert3559717418277143099_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ bot_bo1267616847652589879_state )
                  @ ( B6 @ X ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1235_Sigma__def,axiom,
    ( produc4653271056321233546_state
    = ( ^ [A6: set_prat,B6: prat > set_set_state] :
          ( comple4239529300265302894_state
          @ ( image_4183996178746729834_state
            @ ^ [X: prat] :
                ( comple4239529300265302894_state
                @ ( image_8880686358524681546_state
                  @ ^ [Y3: set_state] : ( insert2241947522006009073_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ bot_bo6311991824895087853_state )
                  @ ( B6 @ X ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1236_Sigma__def,axiom,
    ( produc6030673636645808891t_char
    = ( ^ [A6: set_nat,B6: nat > set_list_char] :
          ( comple9178978156834306527t_char
          @ ( image_7918911184113263195t_char
            @ ^ [X: nat] :
                ( comple9178978156834306527t_char
                @ ( image_366428816823531018t_char
                  @ ^ [Y3: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X @ Y3 ) @ bot_bo1414094081888755934t_char )
                  @ ( B6 @ X ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1237_Sigma__def,axiom,
    ( produc1366239611153933861on_val
    = ( ^ [A6: set_Pr3433046674199179443r_prat,B6: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
          ( comple5258588689616361253on_val
          @ ( image_3306202642189553376on_val
            @ ^ [X: produc1457211279475724562t_char > prat] :
                ( comple5258588689616361253on_val
                @ ( image_8136294080383932403on_val
                  @ ^ [Y3: produc1457211279475724562t_char > option_val] : ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ bot_bo6722054719108133734on_val )
                  @ ( B6 @ X ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1238_underS__incl__iff,axiom,
    ! [R3: set_Pr7975217055296730899_state,A: produc7272280185086996969_state,B: produc7272280185086996969_state] :
      ( ( order_5206819535706528884_state @ ( field_4847752008948210780_state @ R3 ) @ R3 )
     => ( ( member7461423308196221312_state @ A @ ( field_4847752008948210780_state @ R3 ) )
       => ( ( member7461423308196221312_state @ B @ ( field_4847752008948210780_state @ R3 ) )
         => ( ( ord_le4032880777312351359_state @ ( order_2298269145561712803_state @ R3 @ A ) @ ( order_2298269145561712803_state @ R3 @ B ) )
            = ( member1349250210377977180_state @ ( produc2330690084503141483_state @ A @ B ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1239_underS__incl__iff,axiom,
    ! [R3: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R3 ) @ R3 )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R3 ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R3 ) )
         => ( ( ord_le6900622908513426738on_val @ ( order_8621009067429368022on_val @ R3 @ A ) @ ( order_8621009067429368022on_val @ R3 @ B ) )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1240_underS__incl__iff,axiom,
    ! [R3: set_Pr4846723933414538759_state,A: produc3882902684156697774_state,B: produc3882902684156697774_state] :
      ( ( order_95835485040556771_state @ ( field_1980264525152535803_state @ R3 ) @ R3 )
     => ( ( member6715010751053357271_state @ A @ ( field_1980264525152535803_state @ R3 ) )
       => ( ( member6715010751053357271_state @ B @ ( field_1980264525152535803_state @ R3 ) )
         => ( ( ord_le6483787752022044334_state @ ( order_3619705406259710068_state @ R3 @ A ) @ ( order_3619705406259710068_state @ R3 @ B ) )
            = ( member3801449062345016400_state @ ( produc2343482328048119831_state @ A @ B ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1241_underS__incl__iff,axiom,
    ! [R3: set_Pr1688445902015331925_state,A: option_state,B: option_state] :
      ( ( order_7502633587729487893_state @ ( field_option_state @ R3 ) @ R3 )
     => ( ( member_option_state @ A @ ( field_option_state @ R3 ) )
       => ( ( member_option_state @ B @ ( field_option_state @ R3 ) )
         => ( ( ord_le7116032884704190368_state @ ( order_5194198671850319300_state @ R3 @ A ) @ ( order_5194198671850319300_state @ R3 @ B ) )
            = ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1242_underS__incl__iff,axiom,
    ! [R3: set_Pr1785066336555260981_state,A: state,B: state] :
      ( ( order_7174064491916216133_state @ ( field_state @ R3 ) @ R3 )
     => ( ( member_state @ A @ ( field_state @ R3 ) )
       => ( ( member_state @ B @ ( field_state @ R3 ) )
         => ( ( ord_le2494988322063910608_state @ ( order_underS_state @ R3 @ A ) @ ( order_underS_state @ R3 @ B ) )
            = ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ B ) @ R3 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1243_underS__def,axiom,
    ( order_2298269145561712803_state
    = ( ^ [R: set_Pr7975217055296730899_state,A2: produc7272280185086996969_state] :
          ( collec906845016635587262_state
          @ ^ [B2: produc7272280185086996969_state] :
              ( ( B2 != A2 )
              & ( member1349250210377977180_state @ ( produc2330690084503141483_state @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1244_underS__def,axiom,
    ( order_8621009067429368022on_val
    = ( ^ [R: set_Pr4305388979797279737on_val,A2: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B2: produc9003781588162623004on_val] :
              ( ( B2 != A2 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1245_underS__def,axiom,
    ( order_underS_state
    = ( ^ [R: set_Pr1785066336555260981_state,A2: state] :
          ( collect_state
          @ ^ [B2: state] :
              ( ( B2 != A2 )
              & ( member753036827967488894_state @ ( produc544653723139640077_state @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1246_underS__def,axiom,
    ( order_5194198671850319300_state
    = ( ^ [R: set_Pr1688445902015331925_state,A2: option_state] :
          ( collect_option_state
          @ ^ [B2: option_state] :
              ( ( B2 != A2 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1247_well__order__on__domain,axiom,
    ! [A3: set_state,R3: set_Pr1785066336555260981_state,A: state,B: state] :
      ( ( order_3972865417222356124_state @ A3 @ R3 )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ B ) @ R3 )
       => ( ( member_state @ A @ A3 )
          & ( member_state @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1248_well__order__on__domain,axiom,
    ! [A3: set_Pr4231240917283394591_state,R3: set_Pr7975217055296730899_state,A: produc7272280185086996969_state,B: produc7272280185086996969_state] :
      ( ( order_7673620040736478667_state @ A3 @ R3 )
     => ( ( member1349250210377977180_state @ ( produc2330690084503141483_state @ A @ B ) @ R3 )
       => ( ( member7461423308196221312_state @ A @ A3 )
          & ( member7461423308196221312_state @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1249_well__order__on__domain,axiom,
    ! [A3: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_2579646547231075838on_val @ A3 @ R3 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R3 )
       => ( ( member2879030624002546611on_val @ A @ A3 )
          & ( member2879030624002546611on_val @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1250_well__order__on__domain,axiom,
    ! [A3: set_Pr2391701600042252046_state,R3: set_Pr4846723933414538759_state,A: produc3882902684156697774_state,B: produc3882902684156697774_state] :
      ( ( order_6707500972528932172_state @ A3 @ R3 )
     => ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ A @ B ) @ R3 )
       => ( ( member6715010751053357271_state @ A @ A3 )
          & ( member6715010751053357271_state @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1251_well__order__on__domain,axiom,
    ! [A3: set_option_state,R3: set_Pr1688445902015331925_state,A: option_state,B: option_state] :
      ( ( order_4461428600198971116_state @ A3 @ R3 )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ R3 )
       => ( ( member_option_state @ A @ A3 )
          & ( member_option_state @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1252_snd__image__Sigma,axiom,
    ! [A3: set_Pr3433046674199179443r_prat,B3: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ ( produc1366239611153933861on_val @ A3 @ B3 ) )
      = ( comple6221860311599351331on_val @ ( image_4386470036779017822on_val @ B3 @ A3 ) ) ) ).

% snd_image_Sigma
thf(fact_1253_Id__on__def,axiom,
    ( id_on_option_state
    = ( ^ [A6: set_option_state] :
          ( comple3552586563226307650_state
          @ ( image_5951889428630037764_state
            @ ^ [X: option_state] : ( insert4171857611248116165_state @ ( produc9160152616361873709_state @ X @ X ) @ bot_bo1080640394036989633_state )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_1254_underS__I,axiom,
    ! [I: state,J: state,R2: set_Pr1785066336555260981_state] :
      ( ( I != J )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ I @ J ) @ R2 )
       => ( member_state @ I @ ( order_underS_state @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1255_underS__I,axiom,
    ! [I: produc7272280185086996969_state,J: produc7272280185086996969_state,R2: set_Pr7975217055296730899_state] :
      ( ( I != J )
     => ( ( member1349250210377977180_state @ ( produc2330690084503141483_state @ I @ J ) @ R2 )
       => ( member7461423308196221312_state @ I @ ( order_2298269145561712803_state @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1256_underS__I,axiom,
    ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( I != J )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 )
       => ( member2879030624002546611on_val @ I @ ( order_8621009067429368022on_val @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1257_underS__I,axiom,
    ! [I: produc3882902684156697774_state,J: produc3882902684156697774_state,R2: set_Pr4846723933414538759_state] :
      ( ( I != J )
     => ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ I @ J ) @ R2 )
       => ( member6715010751053357271_state @ I @ ( order_3619705406259710068_state @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1258_underS__I,axiom,
    ! [I: option_state,J: option_state,R2: set_Pr1688445902015331925_state] :
      ( ( I != J )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I @ J ) @ R2 )
       => ( member_option_state @ I @ ( order_5194198671850319300_state @ R2 @ J ) ) ) ) ).

% underS_I
thf(fact_1259_underS__E,axiom,
    ! [I: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,J: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ I @ ( order_8621009067429368022on_val @ R2 @ J ) )
     => ( ( I != J )
        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1260_underS__E,axiom,
    ! [I: produc3882902684156697774_state,R2: set_Pr4846723933414538759_state,J: produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ I @ ( order_3619705406259710068_state @ R2 @ J ) )
     => ( ( I != J )
        & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1261_underS__E,axiom,
    ! [I: option_state,R2: set_Pr1688445902015331925_state,J: option_state] :
      ( ( member_option_state @ I @ ( order_5194198671850319300_state @ R2 @ J ) )
     => ( ( I != J )
        & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I @ J ) @ R2 ) ) ) ).

% underS_E
thf(fact_1262_PartialSA_Omax__projection__prop__pure__core,axiom,
    sep_ma8214210560313151521_state @ plus @ ( sep_pure_state @ plus ) @ core ).

% PartialSA.max_projection_prop_pure_core
thf(fact_1263_PartialSA_Ompp__prop,axiom,
    ! [P4: state > $o,F4: state > state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
     => ( P4 @ ( F4 @ X2 ) ) ) ).

% PartialSA.mpp_prop
thf(fact_1264_PartialSA_Ompp__invo,axiom,
    ! [P4: state > $o,F4: state > state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
     => ( ( F4 @ ( F4 @ X2 ) )
        = ( F4 @ X2 ) ) ) ).

% PartialSA.mpp_invo
thf(fact_1265_PartialSA_Omax__projection__prop__def,axiom,
    ! [P4: state > $o,F4: state > state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
      = ( ! [X: state] :
            ( ( greater @ X @ ( F4 @ X ) )
            & ( P4 @ ( F4 @ X ) )
            & ! [P3: state] :
                ( ( ( P4 @ P3 )
                  & ( greater @ X @ P3 ) )
               => ( greater @ ( F4 @ X ) @ P3 ) ) ) ) ) ).

% PartialSA.max_projection_prop_def
thf(fact_1266_PartialSA_Omax__projection__propI,axiom,
    ! [F4: state > state,P4: state > $o] :
      ( ! [X3: state] : ( greater @ X3 @ ( F4 @ X3 ) )
     => ( ! [X3: state] : ( P4 @ ( F4 @ X3 ) )
       => ( ! [X3: state,P: state] :
              ( ( ( P4 @ P )
                & ( greater @ X3 @ P ) )
             => ( greater @ ( F4 @ X3 ) @ P ) )
         => ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 ) ) ) ) ).

% PartialSA.max_projection_propI
thf(fact_1267_PartialSA_Ompp__smaller,axiom,
    ! [P4: state > $o,F4: state > state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
     => ( greater @ X2 @ ( F4 @ X2 ) ) ) ).

% PartialSA.mpp_smaller
thf(fact_1268_PartialSA_Ompp__mono,axiom,
    ! [P4: state > $o,F4: state > state,A: state,B: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
     => ( ( greater @ A @ B )
       => ( greater @ ( F4 @ A ) @ ( F4 @ B ) ) ) ) ).

% PartialSA.mpp_mono
thf(fact_1269_PartialSA_OmppI,axiom,
    ! [P4: state > $o,F4: state > state,A: state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
     => ( ( greater @ A @ X2 )
       => ( ( P4 @ X2 )
         => ( ( greater @ X2 @ ( F4 @ A ) )
           => ( X2
              = ( F4 @ A ) ) ) ) ) ) ).

% PartialSA.mppI
thf(fact_1270_PartialSA_Omax__projection__propE_I3_J,axiom,
    ! [P4: state > $o,F4: state > state,P2: state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
     => ( ( ( P4 @ P2 )
          & ( greater @ X2 @ P2 ) )
       => ( greater @ ( F4 @ X2 ) @ P2 ) ) ) ).

% PartialSA.max_projection_propE(3)
thf(fact_1271_PartialSA_Ompp__compatible,axiom,
    ! [P4: state > $o,F4: state > state,A: state,B: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P4 @ F4 )
     => ( ( defined @ A @ B )
       => ( defined @ ( F4 @ A ) @ ( F4 @ B ) ) ) ) ).

% PartialSA.mpp_compatible
thf(fact_1272_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

% Helper facts (8)
thf(help_If_2_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X2: prat,Y: prat] :
      ( ( if_prat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X2: prat,Y: prat] :
      ( ( if_prat @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_fChoice_1_1_fChoice_001t__PosRat__Oprat_T,axiom,
    ! [P4: prat > $o] :
      ( ( P4 @ ( fChoice_prat @ P4 ) )
      = ( ? [X9: prat] : ( P4 @ X9 ) ) ) ).

thf(help_If_2_1_If_001t__PartialHeapSA__Ostate_T,axiom,
    ! [X2: state,Y: state] :
      ( ( if_state @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__PartialHeapSA__Ostate_T,axiom,
    ! [X2: state,Y: state] :
      ( ( if_state @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [P4: $o] :
      ( ( P4 = $true )
      | ( P4 = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X2: option_state,Y: option_state] :
      ( ( if_option_state @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X2: option_state,Y: option_state] :
      ( ( if_option_state @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ~ ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ ( multiply @ p @ w ) ) @ ( get_m @ a2 ) ) ) ).

%------------------------------------------------------------------------------