TPTP Problem File: SLH0573^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Combinable_Wands/0003_CombinableWands/prob_00214_007695__7643690_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1790 ( 588 unt; 506 typ; 0 def)
% Number of atoms : 3780 (1633 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 11872 ( 416 ~; 23 |; 400 &;9478 @)
% ( 0 <=>;1555 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 7 avg)
% Number of types : 78 ( 77 usr)
% Number of type conns : 2348 (2348 >; 0 *; 0 +; 0 <<)
% Number of symbols : 432 ( 429 usr; 37 con; 0-11 aty)
% Number of variables : 4192 ( 447 ^;3516 !; 229 ?;4192 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:09:01.442
%------------------------------------------------------------------------------
% Could-be-implicit typings (77)
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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,
intuitionistic: set_state > $o ).
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,
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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,
wand: set_state > set_state > set_state ).
thf(sy_c_Finite__Set_OFpow_001t__PartialHeapSA__Ostate,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,
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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_Mask_Omultiply__mask,type,
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thf(sy_c_Mask_Onull_001t__Nat__Onat,type,
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thf(sy_c_Mask_Opre__get__m_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_Mask_Opre__get__m_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Mask_Opre__get__m_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Mask_Opre__get__m__rel_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Mask_Opre__get__m__rel_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Mask_Oupper__bounded,type,
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thf(sy_c_Mask_Ovalid__mask,type,
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thf(sy_c_Mask_Ovalid__mask__rel,type,
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thf(sy_c_Mask_Ovalid__null,type,
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thf(sy_c_Option_Ooption_ONone_001t__Mask__Oval,type,
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thf(sy_c_Option_Ooption_ONone_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Option_Ooption_OSome_001t__Mask__Oval,type,
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thf(sy_c_Option_Ooption_OSome_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Option_Ooption_Othe_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
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packag1330488657391168505_state: ( state > $o ) > ( state > $o ) > state > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Ointuitionistic_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_PackageLogic_Opackage__logic_Ois__footprint__general_001t__PartialHeapSA__Ostate,type,
packag6192661653625646554_state: ( state > state > option_state ) > state > ( state > state > state ) > aassertion_state > aassertion_state > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Ois__footprint__standard_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_PackageLogic_Opackage__logic_Omono__pruner_001t__PartialHeapSA__Ostate,type,
packag2456304381420842418_state: ( state > state > option_state ) > ( state > $o ) > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Omono__pure__cond_001t__PartialHeapSA__Ostate,type,
packag6595153354952283300_state: ( state > state > option_state ) > ( state > state ) > ( state > $o ) > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Omono__transformer_001t__PartialHeapSA__Ostate,type,
packag4817583915586052229_state: ( state > state > option_state ) > state > ( state > state ) > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Opackage__rhs_001t__PartialHeapSA__Ostate,type,
packag3171892606229212990_state: ( state > state > option_state ) > ( state > state ) > ( state > $o ) > state > state > set_Pr4231240917283394591_state > ( state > $o ) > aassertion_state > state > state > set_Pr4231240917283394591_state > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Opackage__rhs__connection_001t__PartialHeapSA__Ostate,type,
packag2054414653763826108_state: ( state > state > option_state ) > ( state > state ) > ( state > $o ) > state > state > set_Pr4231240917283394591_state > ( state > $o ) > aassertion_state > state > state > set_Pr4231240917283394591_state > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Opackage__sat_001t__PartialHeapSA__Ostate,type,
packag8070727096299822599_state: ( state > state > option_state ) > ( state > state ) > ( state > $o ) > aassertion_state > state > state > state > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Opure__remains_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
packag4152378406607043437_state: ( state > state > option_state ) > set_Pr4231240917283394591_state > $o ).
thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__PartialHeapSA__Ostate,type,
package_sat_state: ( state > state > option_state ) > aassertion_state > state > $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__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__PartialHeapSA__Ostate,type,
packag7298478212905792770_state: ( state > state > option_state ) > ( state > state ) > aassertion_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_001t__PartialHeapSA__Ostate,type,
produc4737362649269059174_state: ( state > state ) > state > produc5876769945872133550_state ).
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produc8008623296994167373on_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > produc1757238238866670357on_val ).
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produc1897429902272487136r_prat: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > produc7776263825685906734r_prat ).
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produc74620467203698198on_val: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ).
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produc1164665353665699927r_prat: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > produc3336119888008106215r_prat ).
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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,
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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__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__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 ).
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thf(sy_c_Product__Type_OPair_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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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,
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produc4958730763592889809on_val: produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ).
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produc516693425924439767_state: produc6251377953716854561_state > produc6251377953716854561_state > produc3646891683153674343_state ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__PartialHeapSA__Ostate_J_001t__PosRat__Oprat,type,
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__PartialHeapSA__Ostate_J_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001_Eo,type,
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thf(sy_c_Product__Type_Ocurry_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001_Eo,type,
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thf(sy_c_Product__Type_Ocurry_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate_001_Eo,type,
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thf(sy_c_Product__Type_Ocurry_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J_001_Eo,type,
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accp_P6233459008282737652_state: ( produc4844402304972151915_state > produc4844402304972151915_state > $o ) > produc4844402304972151915_state > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
accp_P6554740477754186410_state: ( produc6251377953716854561_state > produc6251377953716854561_state > $o ) > produc6251377953716854561_state > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_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,
accp_P6797916352740768290on_val: ( produc1317171644805857433on_val > produc1317171644805857433on_val > $o ) > produc1317171644805857433on_val > $o ).
thf(sy_c_Wellfounded_Olex__prod_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,
lex_pr8780192377291449342on_val: set_Pr1685704261205283783r_prat > set_Pr4698884584343881589on_val > set_Pr4305388979797279737on_val ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
lex_pr1681056561245511266t_char: set_Pr1261947904930325089at_nat > set_Pr4480099295215233283t_char > set_Pr3996677194857090567t_char ).
thf(sy_c_Wellfounded_Olex__prod_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
lex_pr8336311929016760645_state: set_Pr1688445902015331925_state > set_Pr1688445902015331925_state > set_Pr2946632083460398663_state ).
thf(sy_c_Wellfounded_Olex__prod_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
lex_prod_prat_state: set_Pr3298805672376727137t_prat > set_Pr1785066336555260981_state > set_Pr5323702258453113287_state ).
thf(sy_c_Wellfounded_Olex__prod_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
lex_pr1787351029354341489_state: set_Pr3298805672376727137t_prat > set_Pr5618112136499017889_state > set_Pr401513967981718343_state ).
thf(sy_c_Wfrec_Osame__fst_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,
same_f8138205353876899841on_val: ( ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val ) > set_Pr4305388979797279737on_val ).
thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
same_f652031872680891423t_char: ( nat > $o ) > ( nat > set_Pr4480099295215233283t_char ) > set_Pr3996677194857090567t_char ).
thf(sy_c_Wfrec_Osame__fst_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
same_f2704544105346032002_state: ( option_state > $o ) > ( option_state > set_Pr1688445902015331925_state ) > set_Pr2946632083460398663_state ).
thf(sy_c_Wfrec_Osame__fst_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
same_fst_prat_state: ( prat > $o ) > ( prat > set_Pr1785066336555260981_state ) > set_Pr5323702258453113287_state ).
thf(sy_c_Wfrec_Osame__fst_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
same_f5333510722715586990_state: ( prat > $o ) > ( prat > set_Pr5618112136499017889_state ) > set_Pr401513967981718343_state ).
thf(sy_c_fChoice_001t__PosRat__Oprat,type,
fChoice_prat: ( prat > $o ) > prat ).
thf(sy_c_member_001t__Option__Ooption_It__Mask__Oval_J,type,
member_option_val: option_val > set_option_val > $o ).
thf(sy_c_member_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
member_option_state: option_state > set_option_state > $o ).
thf(sy_c_member_001t__PartialHeapSA__Ostate,type,
member_state: state > set_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_Mt__PartialHeapSA__Ostate_J,type,
member8708878012768793047_state: produc5876769945872133550_state > set_Pr2063381651806504974_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_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_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,
member5502714215428086462on_val: produc1757238238866670357on_val > set_Pr4698884584343881589on_val > $o ).
thf(sy_c_member_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,
member2879030624002546611on_val: produc9003781588162623004on_val > set_Pr7649655576376376530on_val > $o ).
thf(sy_c_member_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__PosRat__Oprat_J_J,type,
member1381939458143960592r_prat: produc3336119888008106215r_prat > set_Pr1685704261205283783r_prat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
member6023471406134104908t_char: produc6859378861088059299t_char > set_Pr4480099295215233283t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J,type,
member6633493567087309371ar_nat: produc7606469099237612818ar_nat > set_Pr8803067691650843250ar_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
member484235747325421115t_char: produc1457211279475724562t_char > set_Pr2507339680178222706t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
member3029510603097127326_state: produc3142500478612311029_state > set_Pr1688445902015331925_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
member6715010751053357271_state: produc3882902684156697774_state > set_Pr2391701600042252046_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
member753036827967488894_state: produc8023240190789890773_state > set_Pr1785066336555260981_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PosRat__Oprat_J,type,
member8716344949261458452e_prat: produc7334356684927367531e_prat > set_Pr8996442262779224523e_prat > $o ).
thf(sy_c_member_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,
member7461423308196221312_state: produc7272280185086996969_state > set_Pr4231240917283394591_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J,type,
member6226390569306242836_state: produc4844402304972151915_state > set_Pr6039890582625207499_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__PosRat__Oprat_J,type,
member5208769068039259050t_prat: produc2235545006120619777t_prat > set_Pr3298805672376727137t_prat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
member6708919011922255306_state: produc6251377953716854561_state > set_Pr6847263425858009217_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_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,
member6505152944228695362on_val: produc1317171644805857433on_val > set_Pr4305388979797279737on_val > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
member151853725258021968t_char: produc6071853487833700903t_char > set_Pr3996677194857090567t_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
member2361837818702807184_state: produc4946314520779599207_state > set_Pr2946632083460398663_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_Mt__PartialHeapSA__Ostate_J,type,
member1546683425902591150_state: produc1357540302793366807_state > set_Pr4366116562226448461_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_Mt__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_J,type,
member14790466581074960_state: produc4324712645871441639_state > set_Pr5323702258453113287_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_Mt__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
member647386464819295632_state: produc3646891683153674343_state > set_Pr401513967981718343_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__PartialHeapSA__Ostate_J_Mt__PosRat__Oprat_J,type,
member3111753566487124938e_prat: produc2654212508281724193e_prat > set_Pr6566877984550221441e_prat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__PartialHeapSA__Ostate_J_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
member1976142906092959210_state: produc597591266584407361_state > set_Pr5618112136499017889_state > $o ).
thf(sy_c_member_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
member_set_state: set_state > set_set_state > $o ).
thf(sy_v_A,type,
a: set_state ).
thf(sy_v_p,type,
p: prat ).
thf(sy_v_x,type,
x: state ).
% Relevant facts (1275)
thf(fact_0_assms,axiom,
member_state @ x @ a ).
% assms
thf(fact_1_in__multiply__sem,axiom,
! [X: state,P: prat,A: set_state] :
( ( member_state @ X @ ( multip8064567061438756306ertion @ P @ A ) )
=> ? [X2: state] :
( ( member_state @ X2 @ A )
& ( greater @ X @ ( multiply @ P @ X2 ) ) ) ) ).
% in_multiply_sem
thf(fact_2_multiply__sem__assertion_Oelims,axiom,
! [X: prat,Xa: set_state,Y: set_state] :
( ( ( multip8064567061438756306ertion @ X @ Xa )
= Y )
=> ( Y
= ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X ) @ Xa ) ) ) ) ).
% multiply_sem_assertion.elims
thf(fact_3_multiply__sem__assertion_Osimps,axiom,
( multip8064567061438756306ertion
= ( ^ [P2: prat,P3: set_state] : ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ P2 ) @ P3 ) ) ) ) ).
% multiply_sem_assertion.simps
thf(fact_4_w__in__scaled,axiom,
! [W: state] : ( member_state @ W @ ( scaled @ W ) ) ).
% w_in_scaled
thf(fact_5_multiply__sem__assertion_Opelims,axiom,
! [X: prat,Xa: set_state,Y: set_state] :
( ( ( multip8064567061438756306ertion @ X @ Xa )
= Y )
=> ( ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X @ Xa ) )
=> ~ ( ( Y
= ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X ) @ Xa ) ) )
=> ~ ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X @ Xa ) ) ) ) ) ).
% multiply_sem_assertion.pelims
thf(fact_6_get__h__multiply,axiom,
! [P: prat,X: state] :
( ( pgte @ pwrite @ P )
=> ( ( get_h @ ( multiply @ P @ X ) )
= ( get_h @ X ) ) ) ).
% get_h_multiply
thf(fact_7_multiply__sem__assertion_Ocases,axiom,
! [X: produc6251377953716854561_state] :
~ ! [P4: prat,P5: set_state] :
( X
!= ( produc9080284787853255769_state @ P4 @ P5 ) ) ).
% multiply_sem_assertion.cases
thf(fact_8_intuitionistic__def,axiom,
( intuitionistic
= ( ^ [A2: set_state] :
! [A3: state,B: state] :
( ( ( greater @ A3 @ B )
& ( member_state @ B @ A2 ) )
=> ( member_state @ A3 @ A2 ) ) ) ) ).
% intuitionistic_def
thf(fact_9_image__eqI,axiom,
! [B2: state,F: state > state,X: state,A: set_state] :
( ( B2
= ( F @ X ) )
=> ( ( member_state @ X @ A )
=> ( member_state @ B2 @ ( image_state_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_10_image__eqI,axiom,
! [B2: produc3882902684156697774_state,F: state > produc3882902684156697774_state,X: state,A: set_state] :
( ( B2
= ( F @ X ) )
=> ( ( member_state @ X @ A )
=> ( member6715010751053357271_state @ B2 @ ( image_5311902671526399277_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_11_image__eqI,axiom,
! [B2: state,F: produc3882902684156697774_state > state,X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( B2
= ( F @ X ) )
=> ( ( member6715010751053357271_state @ X @ A )
=> ( member_state @ B2 @ ( image_6759111139246981411_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_12_image__eqI,axiom,
! [B2: produc7272280185086996969_state,F: state > produc7272280185086996969_state,X: state,A: set_state] :
( ( B2
= ( F @ X ) )
=> ( ( member_state @ X @ A )
=> ( member7461423308196221312_state @ B2 @ ( image_632992669292831658_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_13_image__eqI,axiom,
! [B2: state,F: produc7272280185086996969_state > state,X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( B2
= ( F @ X ) )
=> ( ( member7461423308196221312_state @ X @ A )
=> ( member_state @ B2 @ ( image_2710133271811360426_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_image__eqI,axiom,
! [B2: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state,X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( B2
= ( F @ X ) )
=> ( ( member6715010751053357271_state @ X @ A )
=> ( member6715010751053357271_state @ B2 @ ( image_508898238547643525_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_15_image__eqI,axiom,
! [B2: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state,X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( B2
= ( F @ X ) )
=> ( ( member7461423308196221312_state @ X @ A )
=> ( member6715010751053357271_state @ B2 @ ( image_7466395622266389182_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_16_image__eqI,axiom,
! [B2: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state,X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( B2
= ( F @ X ) )
=> ( ( member6715010751053357271_state @ X @ A )
=> ( member7461423308196221312_state @ B2 @ ( image_4364510454419742290_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_17_image__eqI,axiom,
! [B2: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state,X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( B2
= ( F @ X ) )
=> ( ( member7461423308196221312_state @ X @ A )
=> ( member7461423308196221312_state @ B2 @ ( image_3064086290059306969_state @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_18_image__eqI,axiom,
! [B2: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val,X: state,A: set_state] :
( ( B2
= ( F @ X ) )
=> ( ( member_state @ X @ A )
=> ( member2879030624002546611on_val @ B2 @ ( image_4772210114854293981on_val @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_19_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_20_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_21_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_22_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_23_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_24_old_Oprod_Oinject,axiom,
! [A4: option_state,B2: option_state,A5: option_state,B3: option_state] :
( ( ( produc9160152616361873709_state @ A4 @ B2 )
= ( produc9160152616361873709_state @ A5 @ B3 ) )
= ( ( A4 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_25_old_Oprod_Oinject,axiom,
! [A4: prat,B2: state,A5: prat,B3: state] :
( ( ( produc6205719116230455971_state @ A4 @ B2 )
= ( produc6205719116230455971_state @ A5 @ B3 ) )
= ( ( A4 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_26_old_Oprod_Oinject,axiom,
! [A4: prat,B2: set_state,A5: prat,B3: set_state] :
( ( ( produc9080284787853255769_state @ A4 @ B2 )
= ( produc9080284787853255769_state @ A5 @ B3 ) )
= ( ( A4 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_27_old_Oprod_Oinject,axiom,
! [A4: nat,B2: list_char,A5: nat,B3: list_char] :
( ( ( produc120197956887798346t_char @ A4 @ B2 )
= ( produc120197956887798346t_char @ A5 @ B3 ) )
= ( ( A4 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val,A5: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
( ( ( produc74620467203698198on_val @ A4 @ B2 )
= ( produc74620467203698198on_val @ A5 @ B3 ) )
= ( ( A4 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_29_sep__algebra_Oupper__closure_Ocong,axiom,
sep_up1246176804924251236_state = sep_up1246176804924251236_state ).
% sep_algebra.upper_closure.cong
thf(fact_30_pgte__antisym,axiom,
! [A4: prat,B2: prat] :
( ( pgte @ A4 @ B2 )
=> ( ( pgte @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% pgte_antisym
thf(fact_31_PartialSA_Osucc__refl,axiom,
! [A4: state] : ( greater @ A4 @ A4 ) ).
% PartialSA.succ_refl
thf(fact_32_PartialSA_Osucc__trans,axiom,
! [A4: state,B2: state,C: state] :
( ( greater @ A4 @ B2 )
=> ( ( greater @ B2 @ C )
=> ( greater @ A4 @ C ) ) ) ).
% PartialSA.succ_trans
thf(fact_33_PartialSA_Osucc__antisym,axiom,
! [A4: state,B2: state] :
( ( greater @ A4 @ B2 )
=> ( ( greater @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% PartialSA.succ_antisym
thf(fact_34_accp_Ocases,axiom,
! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A4: produc1317171644805857433on_val] :
( ( accp_P6797916352740768290on_val @ R @ A4 )
=> ! [Y3: produc1317171644805857433on_val] :
( ( R @ Y3 @ A4 )
=> ( accp_P6797916352740768290on_val @ R @ Y3 ) ) ) ).
% accp.cases
thf(fact_35_accp_Ocases,axiom,
! [R: produc4844402304972151915_state > produc4844402304972151915_state > $o,A4: produc4844402304972151915_state] :
( ( accp_P6233459008282737652_state @ R @ A4 )
=> ! [Y3: produc4844402304972151915_state] :
( ( R @ Y3 @ A4 )
=> ( accp_P6233459008282737652_state @ R @ Y3 ) ) ) ).
% accp.cases
thf(fact_36_accp_Ocases,axiom,
! [R: produc6251377953716854561_state > produc6251377953716854561_state > $o,A4: produc6251377953716854561_state] :
( ( accp_P6554740477754186410_state @ R @ A4 )
=> ! [Y3: produc6251377953716854561_state] :
( ( R @ Y3 @ A4 )
=> ( accp_P6554740477754186410_state @ R @ Y3 ) ) ) ).
% accp.cases
thf(fact_37_accp_Ocases,axiom,
! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A4: produc9003781588162623004on_val] :
( ( accp_P4405556307444930259on_val @ R @ A4 )
=> ! [Y3: produc9003781588162623004on_val] :
( ( R @ Y3 @ A4 )
=> ( accp_P4405556307444930259on_val @ R @ Y3 ) ) ) ).
% accp.cases
thf(fact_38_accp_Ocases,axiom,
! [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A4: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R @ A4 )
=> ! [Y3: produc1457211279475724562t_char > prat] :
( ( R @ Y3 @ A4 )
=> ( accp_P4765339447491148764r_prat @ R @ Y3 ) ) ) ).
% accp.cases
thf(fact_39_accp_Osimps,axiom,
( accp_P6797916352740768290on_val
= ( ^ [R2: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A3: produc1317171644805857433on_val] :
? [X3: produc1317171644805857433on_val] :
( ( A3 = X3 )
& ! [Y4: produc1317171644805857433on_val] :
( ( R2 @ Y4 @ X3 )
=> ( accp_P6797916352740768290on_val @ R2 @ Y4 ) ) ) ) ) ).
% accp.simps
thf(fact_40_accp_Osimps,axiom,
( accp_P6233459008282737652_state
= ( ^ [R2: produc4844402304972151915_state > produc4844402304972151915_state > $o,A3: produc4844402304972151915_state] :
? [X3: produc4844402304972151915_state] :
( ( A3 = X3 )
& ! [Y4: produc4844402304972151915_state] :
( ( R2 @ Y4 @ X3 )
=> ( accp_P6233459008282737652_state @ R2 @ Y4 ) ) ) ) ) ).
% accp.simps
thf(fact_41_accp_Osimps,axiom,
( accp_P6554740477754186410_state
= ( ^ [R2: produc6251377953716854561_state > produc6251377953716854561_state > $o,A3: produc6251377953716854561_state] :
? [X3: produc6251377953716854561_state] :
( ( A3 = X3 )
& ! [Y4: produc6251377953716854561_state] :
( ( R2 @ Y4 @ X3 )
=> ( accp_P6554740477754186410_state @ R2 @ Y4 ) ) ) ) ) ).
% accp.simps
thf(fact_42_accp_Osimps,axiom,
( accp_P4405556307444930259on_val
= ( ^ [R2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A3: produc9003781588162623004on_val] :
? [X3: produc9003781588162623004on_val] :
( ( A3 = X3 )
& ! [Y4: produc9003781588162623004on_val] :
( ( R2 @ Y4 @ X3 )
=> ( accp_P4405556307444930259on_val @ R2 @ Y4 ) ) ) ) ) ).
% accp.simps
thf(fact_43_accp_Osimps,axiom,
( accp_P4765339447491148764r_prat
= ( ^ [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A3: produc1457211279475724562t_char > prat] :
? [X3: produc1457211279475724562t_char > prat] :
( ( A3 = X3 )
& ! [Y4: produc1457211279475724562t_char > prat] :
( ( R2 @ Y4 @ X3 )
=> ( accp_P4765339447491148764r_prat @ R2 @ Y4 ) ) ) ) ) ).
% accp.simps
thf(fact_44_Pair__inject,axiom,
! [A4: option_state,B2: option_state,A5: option_state,B3: option_state] :
( ( ( produc9160152616361873709_state @ A4 @ B2 )
= ( produc9160152616361873709_state @ A5 @ B3 ) )
=> ~ ( ( A4 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_45_Pair__inject,axiom,
! [A4: prat,B2: state,A5: prat,B3: state] :
( ( ( produc6205719116230455971_state @ A4 @ B2 )
= ( produc6205719116230455971_state @ A5 @ B3 ) )
=> ~ ( ( A4 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_46_Pair__inject,axiom,
! [A4: prat,B2: set_state,A5: prat,B3: set_state] :
( ( ( produc9080284787853255769_state @ A4 @ B2 )
= ( produc9080284787853255769_state @ A5 @ B3 ) )
=> ~ ( ( A4 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_47_Pair__inject,axiom,
! [A4: nat,B2: list_char,A5: nat,B3: list_char] :
( ( ( produc120197956887798346t_char @ A4 @ B2 )
= ( produc120197956887798346t_char @ A5 @ B3 ) )
=> ~ ( ( A4 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_48_Pair__inject,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val,A5: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
( ( ( produc74620467203698198on_val @ A4 @ B2 )
= ( produc74620467203698198on_val @ A5 @ B3 ) )
=> ~ ( ( A4 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_49_prod__cases,axiom,
! [P6: produc3142500478612311029_state > $o,P: produc3142500478612311029_state] :
( ! [A6: option_state,B4: option_state] : ( P6 @ ( produc9160152616361873709_state @ A6 @ B4 ) )
=> ( P6 @ P ) ) ).
% prod_cases
thf(fact_50_prod__cases,axiom,
! [P6: produc4844402304972151915_state > $o,P: produc4844402304972151915_state] :
( ! [A6: prat,B4: state] : ( P6 @ ( produc6205719116230455971_state @ A6 @ B4 ) )
=> ( P6 @ P ) ) ).
% prod_cases
thf(fact_51_prod__cases,axiom,
! [P6: produc6251377953716854561_state > $o,P: produc6251377953716854561_state] :
( ! [A6: prat,B4: set_state] : ( P6 @ ( produc9080284787853255769_state @ A6 @ B4 ) )
=> ( P6 @ P ) ) ).
% prod_cases
thf(fact_52_prod__cases,axiom,
! [P6: produc1457211279475724562t_char > $o,P: produc1457211279475724562t_char] :
( ! [A6: nat,B4: list_char] : ( P6 @ ( produc120197956887798346t_char @ A6 @ B4 ) )
=> ( P6 @ P ) ) ).
% prod_cases
thf(fact_53_prod__cases,axiom,
! [P6: produc9003781588162623004on_val > $o,P: produc9003781588162623004on_val] :
( ! [A6: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] : ( P6 @ ( produc74620467203698198on_val @ A6 @ B4 ) )
=> ( P6 @ P ) ) ).
% prod_cases
thf(fact_54_surj__pair,axiom,
! [P: produc3142500478612311029_state] :
? [X2: option_state,Y5: option_state] :
( P
= ( produc9160152616361873709_state @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_55_surj__pair,axiom,
! [P: produc4844402304972151915_state] :
? [X2: prat,Y5: state] :
( P
= ( produc6205719116230455971_state @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_56_surj__pair,axiom,
! [P: produc6251377953716854561_state] :
? [X2: prat,Y5: set_state] :
( P
= ( produc9080284787853255769_state @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_57_surj__pair,axiom,
! [P: produc1457211279475724562t_char] :
? [X2: nat,Y5: list_char] :
( P
= ( produc120197956887798346t_char @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_58_surj__pair,axiom,
! [P: produc9003781588162623004on_val] :
? [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] :
( P
= ( produc74620467203698198on_val @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_59_old_Oprod_Oexhaust,axiom,
! [Y: produc3142500478612311029_state] :
~ ! [A6: option_state,B4: option_state] :
( Y
!= ( produc9160152616361873709_state @ A6 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_60_old_Oprod_Oexhaust,axiom,
! [Y: produc4844402304972151915_state] :
~ ! [A6: prat,B4: state] :
( Y
!= ( produc6205719116230455971_state @ A6 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_61_old_Oprod_Oexhaust,axiom,
! [Y: produc6251377953716854561_state] :
~ ! [A6: prat,B4: set_state] :
( Y
!= ( produc9080284787853255769_state @ A6 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_62_old_Oprod_Oexhaust,axiom,
! [Y: produc1457211279475724562t_char] :
~ ! [A6: nat,B4: list_char] :
( Y
!= ( produc120197956887798346t_char @ A6 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_63_old_Oprod_Oexhaust,axiom,
! [Y: produc9003781588162623004on_val] :
~ ! [A6: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
( Y
!= ( produc74620467203698198on_val @ A6 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_64_rev__image__eqI,axiom,
! [X: state,A: set_state,B2: state,F: state > state] :
( ( member_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_state @ B2 @ ( image_state_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_65_rev__image__eqI,axiom,
! [X: state,A: set_state,B2: produc3882902684156697774_state,F: state > produc3882902684156697774_state] :
( ( member_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member6715010751053357271_state @ B2 @ ( image_5311902671526399277_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_66_rev__image__eqI,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,B2: state,F: produc3882902684156697774_state > state] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_state @ B2 @ ( image_6759111139246981411_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_67_rev__image__eqI,axiom,
! [X: state,A: set_state,B2: produc7272280185086996969_state,F: state > produc7272280185086996969_state] :
( ( member_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member7461423308196221312_state @ B2 @ ( image_632992669292831658_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_68_rev__image__eqI,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,B2: state,F: produc7272280185086996969_state > state] :
( ( member7461423308196221312_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_state @ B2 @ ( image_2710133271811360426_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_69_rev__image__eqI,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,B2: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member6715010751053357271_state @ B2 @ ( image_508898238547643525_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_70_rev__image__eqI,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,B2: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state] :
( ( member7461423308196221312_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member6715010751053357271_state @ B2 @ ( image_7466395622266389182_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_71_rev__image__eqI,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,B2: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member7461423308196221312_state @ B2 @ ( image_4364510454419742290_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_72_rev__image__eqI,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,B2: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member7461423308196221312_state @ B2 @ ( image_3064086290059306969_state @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_73_rev__image__eqI,axiom,
! [X: state,A: set_state,B2: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val] :
( ( member_state @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member2879030624002546611on_val @ B2 @ ( image_4772210114854293981on_val @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_74_ball__imageD,axiom,
! [F: state > state,A: set_state,P6: state > $o] :
( ! [X2: state] :
( ( member_state @ X2 @ ( image_state_state @ F @ A ) )
=> ( P6 @ X2 ) )
=> ! [X4: state] :
( ( member_state @ X4 @ A )
=> ( P6 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_75_ball__imageD,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state,P6: produc9003781588162623004on_val > $o] :
( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ ( image_4690842878458256517on_val @ F @ A ) )
=> ( P6 @ X2 ) )
=> ! [X4: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X4 @ A )
=> ( P6 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_76_ball__imageD,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,P6: state > $o] :
( ! [X2: state] :
( ( member_state @ X2 @ ( image_1205529093420515805_state @ F @ A ) )
=> ( P6 @ X2 ) )
=> ! [X4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X4 @ A )
=> ( P6 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_77_ball__imageD,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val,P6: produc7272280185086996969_state > $o] :
( ! [X2: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X2 @ ( image_3204238854701872908_state @ F @ A ) )
=> ( P6 @ X2 ) )
=> ! [X4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X4 @ A )
=> ( P6 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_78_ball__imageD,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val,P6: produc3882902684156697774_state > $o] :
( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ ( image_4494208899119068747_state @ F @ A ) )
=> ( P6 @ X2 ) )
=> ! [X4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X4 @ A )
=> ( P6 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_79_ball__imageD,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] :
( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ ( image_4170889944715819327on_val @ F @ A ) )
=> ( P6 @ X2 ) )
=> ! [X4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X4 @ A )
=> ( P6 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_80_image__cong,axiom,
! [M: set_state,N: set_state,F: state > state,G: state > state] :
( ( M = N )
=> ( ! [X2: state] :
( ( member_state @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_state_state @ F @ M )
= ( image_state_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_81_image__cong,axiom,
! [M: set_Pr7649655576376376530on_val,N: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > state,G: produc9003781588162623004on_val > state] :
( ( M = N )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_1205529093420515805_state @ F @ M )
= ( image_1205529093420515805_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_82_image__cong,axiom,
! [M: set_Pr7649655576376376530on_val,N: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc7272280185086996969_state,G: produc9003781588162623004on_val > produc7272280185086996969_state] :
( ( M = N )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_3204238854701872908_state @ F @ M )
= ( image_3204238854701872908_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_83_image__cong,axiom,
! [M: set_Pr7649655576376376530on_val,N: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc3882902684156697774_state,G: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( M = N )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_4494208899119068747_state @ F @ M )
= ( image_4494208899119068747_state @ G @ N ) ) ) ) ).
% image_cong
thf(fact_84_image__cong,axiom,
! [M: set_Pr7649655576376376530on_val,N: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,G: produc9003781588162623004on_val > produc9003781588162623004on_val] :
( ( M = N )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_4170889944715819327on_val @ F @ M )
= ( image_4170889944715819327on_val @ G @ N ) ) ) ) ).
% image_cong
thf(fact_85_image__cong,axiom,
! [M: set_Pr2391701600042252046_state,N: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc9003781588162623004on_val,G: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( M = N )
=> ( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_4690842878458256517on_val @ F @ M )
= ( image_4690842878458256517on_val @ G @ N ) ) ) ) ).
% image_cong
thf(fact_86_bex__imageD,axiom,
! [F: state > state,A: set_state,P6: state > $o] :
( ? [X4: state] :
( ( member_state @ X4 @ ( image_state_state @ F @ A ) )
& ( P6 @ X4 ) )
=> ? [X2: state] :
( ( member_state @ X2 @ A )
& ( P6 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_87_bex__imageD,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state,P6: produc9003781588162623004on_val > $o] :
( ? [X4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X4 @ ( image_4690842878458256517on_val @ F @ A ) )
& ( P6 @ X4 ) )
=> ? [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ A )
& ( P6 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_88_bex__imageD,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,P6: state > $o] :
( ? [X4: state] :
( ( member_state @ X4 @ ( image_1205529093420515805_state @ F @ A ) )
& ( P6 @ X4 ) )
=> ? [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ A )
& ( P6 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_89_bex__imageD,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val,P6: produc7272280185086996969_state > $o] :
( ? [X4: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X4 @ ( image_3204238854701872908_state @ F @ A ) )
& ( P6 @ X4 ) )
=> ? [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ A )
& ( P6 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_90_bex__imageD,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val,P6: produc3882902684156697774_state > $o] :
( ? [X4: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X4 @ ( image_4494208899119068747_state @ F @ A ) )
& ( P6 @ X4 ) )
=> ? [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ A )
& ( P6 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_91_bex__imageD,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] :
( ? [X4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X4 @ ( image_4170889944715819327on_val @ F @ A ) )
& ( P6 @ X4 ) )
=> ? [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ A )
& ( P6 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_92_image__iff,axiom,
! [Z: state,F: state > state,A: set_state] :
( ( member_state @ Z @ ( image_state_state @ F @ A ) )
= ( ? [X3: state] :
( ( member_state @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_93_image__iff,axiom,
! [Z: state,F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( member_state @ Z @ ( image_1205529093420515805_state @ F @ A ) )
= ( ? [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_94_image__iff,axiom,
! [Z: produc7272280185086996969_state,F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val] :
( ( member7461423308196221312_state @ Z @ ( image_3204238854701872908_state @ F @ A ) )
= ( ? [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_95_image__iff,axiom,
! [Z: produc9003781588162623004on_val,F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state] :
( ( member2879030624002546611on_val @ Z @ ( image_4690842878458256517on_val @ F @ A ) )
= ( ? [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_96_image__iff,axiom,
! [Z: produc9003781588162623004on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( member2879030624002546611on_val @ Z @ ( image_4170889944715819327on_val @ F @ A ) )
= ( ? [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_97_image__iff,axiom,
! [Z: produc3882902684156697774_state,F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val] :
( ( member6715010751053357271_state @ Z @ ( image_4494208899119068747_state @ F @ A ) )
= ( ? [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_98_imageI,axiom,
! [X: state,A: set_state,F: state > state] :
( ( member_state @ X @ A )
=> ( member_state @ ( F @ X ) @ ( image_state_state @ F @ A ) ) ) ).
% imageI
thf(fact_99_imageI,axiom,
! [X: state,A: set_state,F: state > produc3882902684156697774_state] :
( ( member_state @ X @ A )
=> ( member6715010751053357271_state @ ( F @ X ) @ ( image_5311902671526399277_state @ F @ A ) ) ) ).
% imageI
thf(fact_100_imageI,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state] :
( ( member6715010751053357271_state @ X @ A )
=> ( member_state @ ( F @ X ) @ ( image_6759111139246981411_state @ F @ A ) ) ) ).
% imageI
thf(fact_101_imageI,axiom,
! [X: state,A: set_state,F: state > produc7272280185086996969_state] :
( ( member_state @ X @ A )
=> ( member7461423308196221312_state @ ( F @ X ) @ ( image_632992669292831658_state @ F @ A ) ) ) ).
% imageI
thf(fact_102_imageI,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state] :
( ( member7461423308196221312_state @ X @ A )
=> ( member_state @ ( F @ X ) @ ( image_2710133271811360426_state @ F @ A ) ) ) ).
% imageI
thf(fact_103_imageI,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X @ A )
=> ( member6715010751053357271_state @ ( F @ X ) @ ( image_508898238547643525_state @ F @ A ) ) ) ).
% imageI
thf(fact_104_imageI,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc3882902684156697774_state] :
( ( member7461423308196221312_state @ X @ A )
=> ( member6715010751053357271_state @ ( F @ X ) @ ( image_7466395622266389182_state @ F @ A ) ) ) ).
% imageI
thf(fact_105_imageI,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc7272280185086996969_state] :
( ( member6715010751053357271_state @ X @ A )
=> ( member7461423308196221312_state @ ( F @ X ) @ ( image_4364510454419742290_state @ F @ A ) ) ) ).
% imageI
thf(fact_106_imageI,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X @ A )
=> ( member7461423308196221312_state @ ( F @ X ) @ ( image_3064086290059306969_state @ F @ A ) ) ) ).
% imageI
thf(fact_107_imageI,axiom,
! [X: state,A: set_state,F: state > produc9003781588162623004on_val] :
( ( member_state @ X @ A )
=> ( member2879030624002546611on_val @ ( F @ X ) @ ( image_4772210114854293981on_val @ F @ A ) ) ) ).
% imageI
thf(fact_108_commutative,axiom,
( plus
= ( ^ [A3: state,B: state] : ( plus @ B @ A3 ) ) ) ).
% commutative
thf(fact_109_accp__induct__rule,axiom,
! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A4: produc1317171644805857433on_val,P6: produc1317171644805857433on_val > $o] :
( ( accp_P6797916352740768290on_val @ R @ A4 )
=> ( ! [X2: produc1317171644805857433on_val] :
( ( accp_P6797916352740768290on_val @ R @ X2 )
=> ( ! [Y3: produc1317171644805857433on_val] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct_rule
thf(fact_110_accp__induct__rule,axiom,
! [R: produc4844402304972151915_state > produc4844402304972151915_state > $o,A4: produc4844402304972151915_state,P6: produc4844402304972151915_state > $o] :
( ( accp_P6233459008282737652_state @ R @ A4 )
=> ( ! [X2: produc4844402304972151915_state] :
( ( accp_P6233459008282737652_state @ R @ X2 )
=> ( ! [Y3: produc4844402304972151915_state] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct_rule
thf(fact_111_accp__induct__rule,axiom,
! [R: produc6251377953716854561_state > produc6251377953716854561_state > $o,A4: produc6251377953716854561_state,P6: produc6251377953716854561_state > $o] :
( ( accp_P6554740477754186410_state @ R @ A4 )
=> ( ! [X2: produc6251377953716854561_state] :
( ( accp_P6554740477754186410_state @ R @ X2 )
=> ( ! [Y3: produc6251377953716854561_state] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct_rule
thf(fact_112_accp__induct__rule,axiom,
! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A4: produc9003781588162623004on_val,P6: produc9003781588162623004on_val > $o] :
( ( accp_P4405556307444930259on_val @ R @ A4 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( accp_P4405556307444930259on_val @ R @ X2 )
=> ( ! [Y3: produc9003781588162623004on_val] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct_rule
thf(fact_113_accp__induct__rule,axiom,
! [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A4: produc1457211279475724562t_char > prat,P6: ( produc1457211279475724562t_char > prat ) > $o] :
( ( accp_P4765339447491148764r_prat @ R @ A4 )
=> ( ! [X2: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R @ X2 )
=> ( ! [Y3: produc1457211279475724562t_char > prat] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct_rule
thf(fact_114_not__accp__down,axiom,
! [R3: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X: produc1317171644805857433on_val] :
( ~ ( accp_P6797916352740768290on_val @ R3 @ X )
=> ~ ! [Z2: produc1317171644805857433on_val] :
( ( R3 @ Z2 @ X )
=> ( accp_P6797916352740768290on_val @ R3 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_115_not__accp__down,axiom,
! [R3: produc4844402304972151915_state > produc4844402304972151915_state > $o,X: produc4844402304972151915_state] :
( ~ ( accp_P6233459008282737652_state @ R3 @ X )
=> ~ ! [Z2: produc4844402304972151915_state] :
( ( R3 @ Z2 @ X )
=> ( accp_P6233459008282737652_state @ R3 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_116_not__accp__down,axiom,
! [R3: produc6251377953716854561_state > produc6251377953716854561_state > $o,X: produc6251377953716854561_state] :
( ~ ( accp_P6554740477754186410_state @ R3 @ X )
=> ~ ! [Z2: produc6251377953716854561_state] :
( ( R3 @ Z2 @ X )
=> ( accp_P6554740477754186410_state @ R3 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_117_not__accp__down,axiom,
! [R3: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val] :
( ~ ( accp_P4405556307444930259on_val @ R3 @ X )
=> ~ ! [Z2: produc9003781588162623004on_val] :
( ( R3 @ Z2 @ X )
=> ( accp_P4405556307444930259on_val @ R3 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_118_not__accp__down,axiom,
! [R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
( ~ ( accp_P4765339447491148764r_prat @ R3 @ X )
=> ~ ! [Z2: produc1457211279475724562t_char > prat] :
( ( R3 @ Z2 @ X )
=> ( accp_P4765339447491148764r_prat @ R3 @ Z2 ) ) ) ).
% not_accp_down
thf(fact_119_accp__downward,axiom,
! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,B2: produc1317171644805857433on_val,A4: produc1317171644805857433on_val] :
( ( accp_P6797916352740768290on_val @ R @ B2 )
=> ( ( R @ A4 @ B2 )
=> ( accp_P6797916352740768290on_val @ R @ A4 ) ) ) ).
% accp_downward
thf(fact_120_accp__downward,axiom,
! [R: produc4844402304972151915_state > produc4844402304972151915_state > $o,B2: produc4844402304972151915_state,A4: produc4844402304972151915_state] :
( ( accp_P6233459008282737652_state @ R @ B2 )
=> ( ( R @ A4 @ B2 )
=> ( accp_P6233459008282737652_state @ R @ A4 ) ) ) ).
% accp_downward
thf(fact_121_accp__downward,axiom,
! [R: produc6251377953716854561_state > produc6251377953716854561_state > $o,B2: produc6251377953716854561_state,A4: produc6251377953716854561_state] :
( ( accp_P6554740477754186410_state @ R @ B2 )
=> ( ( R @ A4 @ B2 )
=> ( accp_P6554740477754186410_state @ R @ A4 ) ) ) ).
% accp_downward
thf(fact_122_accp__downward,axiom,
! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,B2: produc9003781588162623004on_val,A4: produc9003781588162623004on_val] :
( ( accp_P4405556307444930259on_val @ R @ B2 )
=> ( ( R @ A4 @ B2 )
=> ( accp_P4405556307444930259on_val @ R @ A4 ) ) ) ).
% accp_downward
thf(fact_123_accp__downward,axiom,
! [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,B2: produc1457211279475724562t_char > prat,A4: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R @ B2 )
=> ( ( R @ A4 @ B2 )
=> ( accp_P4765339447491148764r_prat @ R @ A4 ) ) ) ).
% accp_downward
thf(fact_124_mem__Collect__eq,axiom,
! [A4: produc3882902684156697774_state,P6: produc3882902684156697774_state > $o] :
( ( member6715010751053357271_state @ A4 @ ( collec394671971739345689_state @ P6 ) )
= ( P6 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A4: produc7272280185086996969_state,P6: produc7272280185086996969_state > $o] :
( ( member7461423308196221312_state @ A4 @ ( collec906845016635587262_state @ P6 ) )
= ( P6 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A4: state,P6: state > $o] :
( ( member_state @ A4 @ ( collect_state @ P6 ) )
= ( P6 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A4: produc9003781588162623004on_val,P6: produc9003781588162623004on_val > $o] :
( ( member2879030624002546611on_val @ A4 @ ( collec7475294502966882545on_val @ P6 ) )
= ( P6 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A: set_Pr2391701600042252046_state] :
( ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_Pr4231240917283394591_state] :
( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_state] :
( ( collect_state
@ ^ [X3: state] : ( member_state @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A: set_Pr7649655576376376530on_val] :
( ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_132_Collect__cong,axiom,
! [P6: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
( ! [X2: produc7272280185086996969_state] :
( ( P6 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec906845016635587262_state @ P6 )
= ( collec906845016635587262_state @ Q ) ) ) ).
% Collect_cong
thf(fact_133_Collect__cong,axiom,
! [P6: state > $o,Q: state > $o] :
( ! [X2: state] :
( ( P6 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_state @ P6 )
= ( collect_state @ Q ) ) ) ).
% Collect_cong
thf(fact_134_Collect__cong,axiom,
! [P6: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
( ! [X2: produc9003781588162623004on_val] :
( ( P6 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec7475294502966882545on_val @ P6 )
= ( collec7475294502966882545on_val @ Q ) ) ) ).
% Collect_cong
thf(fact_135_accp__induct,axiom,
! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A4: produc1317171644805857433on_val,P6: produc1317171644805857433on_val > $o] :
( ( accp_P6797916352740768290on_val @ R @ A4 )
=> ( ! [X2: produc1317171644805857433on_val] :
( ( accp_P6797916352740768290on_val @ R @ X2 )
=> ( ! [Y3: produc1317171644805857433on_val] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct
thf(fact_136_accp__induct,axiom,
! [R: produc4844402304972151915_state > produc4844402304972151915_state > $o,A4: produc4844402304972151915_state,P6: produc4844402304972151915_state > $o] :
( ( accp_P6233459008282737652_state @ R @ A4 )
=> ( ! [X2: produc4844402304972151915_state] :
( ( accp_P6233459008282737652_state @ R @ X2 )
=> ( ! [Y3: produc4844402304972151915_state] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct
thf(fact_137_accp__induct,axiom,
! [R: produc6251377953716854561_state > produc6251377953716854561_state > $o,A4: produc6251377953716854561_state,P6: produc6251377953716854561_state > $o] :
( ( accp_P6554740477754186410_state @ R @ A4 )
=> ( ! [X2: produc6251377953716854561_state] :
( ( accp_P6554740477754186410_state @ R @ X2 )
=> ( ! [Y3: produc6251377953716854561_state] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct
thf(fact_138_accp__induct,axiom,
! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A4: produc9003781588162623004on_val,P6: produc9003781588162623004on_val > $o] :
( ( accp_P4405556307444930259on_val @ R @ A4 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( accp_P4405556307444930259on_val @ R @ X2 )
=> ( ! [Y3: produc9003781588162623004on_val] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct
thf(fact_139_accp__induct,axiom,
! [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A4: produc1457211279475724562t_char > prat,P6: ( produc1457211279475724562t_char > prat ) > $o] :
( ( accp_P4765339447491148764r_prat @ R @ A4 )
=> ( ! [X2: produc1457211279475724562t_char > prat] :
( ( accp_P4765339447491148764r_prat @ R @ X2 )
=> ( ! [Y3: produc1457211279475724562t_char > prat] :
( ( R @ Y3 @ X2 )
=> ( P6 @ Y3 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ A4 ) ) ) ).
% accp_induct
thf(fact_140_accpI,axiom,
! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X: produc1317171644805857433on_val] :
( ! [Y5: produc1317171644805857433on_val] :
( ( R @ Y5 @ X )
=> ( accp_P6797916352740768290on_val @ R @ Y5 ) )
=> ( accp_P6797916352740768290on_val @ R @ X ) ) ).
% accpI
thf(fact_141_accpI,axiom,
! [R: produc4844402304972151915_state > produc4844402304972151915_state > $o,X: produc4844402304972151915_state] :
( ! [Y5: produc4844402304972151915_state] :
( ( R @ Y5 @ X )
=> ( accp_P6233459008282737652_state @ R @ Y5 ) )
=> ( accp_P6233459008282737652_state @ R @ X ) ) ).
% accpI
thf(fact_142_accpI,axiom,
! [R: produc6251377953716854561_state > produc6251377953716854561_state > $o,X: produc6251377953716854561_state] :
( ! [Y5: produc6251377953716854561_state] :
( ( R @ Y5 @ X )
=> ( accp_P6554740477754186410_state @ R @ Y5 ) )
=> ( accp_P6554740477754186410_state @ R @ X ) ) ).
% accpI
thf(fact_143_accpI,axiom,
! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val] :
( ! [Y5: produc9003781588162623004on_val] :
( ( R @ Y5 @ X )
=> ( accp_P4405556307444930259on_val @ R @ Y5 ) )
=> ( accp_P4405556307444930259on_val @ R @ X ) ) ).
% accpI
thf(fact_144_accpI,axiom,
! [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
( ! [Y5: produc1457211279475724562t_char > prat] :
( ( R @ Y5 @ X )
=> ( accp_P4765339447491148764r_prat @ R @ Y5 ) )
=> ( accp_P4765339447491148764r_prat @ R @ X ) ) ).
% accpI
thf(fact_145_PartialSA_Ointuitionistic__def,axiom,
! [A: state > $o] :
( ( packag8361946002163212404_state @ plus @ A )
= ( ! [Phi: state,Phi2: state] :
( ( ( greater @ Phi @ Phi2 )
& ( A @ Phi2 ) )
=> ( A @ Phi ) ) ) ) ).
% PartialSA.intuitionistic_def
thf(fact_146_PartialSA_Omono__prop__def,axiom,
! [P6: state > $o] :
( ( sep_mono_prop_state @ plus @ P6 )
= ( ! [X3: state,Y4: state] :
( ( ( greater @ Y4 @ X3 )
& ( P6 @ X3 ) )
=> ( P6 @ Y4 ) ) ) ) ).
% PartialSA.mono_prop_def
thf(fact_147_PartialSA_Omono__propI,axiom,
! [P6: state > $o] :
( ! [X2: state,Y5: state] :
( ( ( greater @ Y5 @ X2 )
& ( P6 @ X2 ) )
=> ( P6 @ Y5 ) )
=> ( sep_mono_prop_state @ plus @ P6 ) ) ).
% PartialSA.mono_propI
thf(fact_148_PartialSA_Oupper__closure__up__closed,axiom,
! [A: set_state] : ( sep_up_closed_state @ plus @ ( sep_up1246176804924251236_state @ plus @ A ) ) ).
% PartialSA.upper_closure_up_closed
thf(fact_149_Sup_OSUP__cong,axiom,
! [A: set_state,B5: set_state,C2: state > state,D: state > state,Sup: set_state > state] :
( ( A = B5 )
=> ( ! [X2: state] :
( ( member_state @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_state_state @ C2 @ A ) )
= ( Sup @ ( image_state_state @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_150_Sup_OSUP__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > state,D: produc9003781588162623004on_val > state,Sup: set_state > state] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_1205529093420515805_state @ C2 @ A ) )
= ( Sup @ ( image_1205529093420515805_state @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_151_Sup_OSUP__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > produc7272280185086996969_state,D: produc9003781588162623004on_val > produc7272280185086996969_state,Sup: set_Pr4231240917283394591_state > produc7272280185086996969_state] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_3204238854701872908_state @ C2 @ A ) )
= ( Sup @ ( image_3204238854701872908_state @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_152_Sup_OSUP__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > produc3882902684156697774_state,D: produc9003781588162623004on_val > produc3882902684156697774_state,Sup: set_Pr2391701600042252046_state > produc3882902684156697774_state] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_4494208899119068747_state @ C2 @ A ) )
= ( Sup @ ( image_4494208899119068747_state @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_153_Sup_OSUP__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > produc9003781588162623004on_val,D: produc9003781588162623004on_val > produc9003781588162623004on_val,Sup: set_Pr7649655576376376530on_val > produc9003781588162623004on_val] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_4170889944715819327on_val @ C2 @ A ) )
= ( Sup @ ( image_4170889944715819327on_val @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_154_Sup_OSUP__cong,axiom,
! [A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state,C2: produc3882902684156697774_state > produc9003781588162623004on_val,D: produc3882902684156697774_state > produc9003781588162623004on_val,Sup: set_Pr7649655576376376530on_val > produc9003781588162623004on_val] :
( ( A = B5 )
=> ( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_4690842878458256517on_val @ C2 @ A ) )
= ( Sup @ ( image_4690842878458256517on_val @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_155_Inf_OINF__cong,axiom,
! [A: set_state,B5: set_state,C2: state > state,D: state > state,Inf: set_state > state] :
( ( A = B5 )
=> ( ! [X2: state] :
( ( member_state @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_state_state @ C2 @ A ) )
= ( Inf @ ( image_state_state @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_156_Inf_OINF__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > state,D: produc9003781588162623004on_val > state,Inf: set_state > state] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_1205529093420515805_state @ C2 @ A ) )
= ( Inf @ ( image_1205529093420515805_state @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_157_Inf_OINF__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > produc7272280185086996969_state,D: produc9003781588162623004on_val > produc7272280185086996969_state,Inf: set_Pr4231240917283394591_state > produc7272280185086996969_state] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_3204238854701872908_state @ C2 @ A ) )
= ( Inf @ ( image_3204238854701872908_state @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_158_Inf_OINF__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > produc3882902684156697774_state,D: produc9003781588162623004on_val > produc3882902684156697774_state,Inf: set_Pr2391701600042252046_state > produc3882902684156697774_state] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_4494208899119068747_state @ C2 @ A ) )
= ( Inf @ ( image_4494208899119068747_state @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_159_Inf_OINF__cong,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val > produc9003781588162623004on_val,D: produc9003781588162623004on_val > produc9003781588162623004on_val,Inf: set_Pr7649655576376376530on_val > produc9003781588162623004on_val] :
( ( A = B5 )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_4170889944715819327on_val @ C2 @ A ) )
= ( Inf @ ( image_4170889944715819327on_val @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_160_Inf_OINF__cong,axiom,
! [A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state,C2: produc3882902684156697774_state > produc9003781588162623004on_val,D: produc3882902684156697774_state > produc9003781588162623004on_val,Inf: set_Pr7649655576376376530on_val > produc9003781588162623004on_val] :
( ( A = B5 )
=> ( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ B5 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_4690842878458256517on_val @ C2 @ A ) )
= ( Inf @ ( image_4690842878458256517on_val @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_161_larger__implies__larger__heap,axiom,
! [A4: state,B2: state] :
( ( greater @ A4 @ B2 )
=> ( larger1532547405891354081ar_val @ ( get_h @ A4 ) @ ( get_h @ B2 ) ) ) ).
% larger_implies_larger_heap
thf(fact_162_ssubst__Pair__rhs,axiom,
! [R: state,S: produc3882902684156697774_state,R3: set_Pr4231240917283394591_state,S2: produc3882902684156697774_state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member7461423308196221312_state @ ( produc5886875603596089627_state @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_163_ssubst__Pair__rhs,axiom,
! [R: state,S: state > state,R3: set_Pr2391701600042252046_state,S2: state > state] :
( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member6715010751053357271_state @ ( produc8632743443909064678_state @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_164_ssubst__Pair__rhs,axiom,
! [R: option_state,S: option_state,R3: set_Pr1688445902015331925_state,S2: option_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_165_ssubst__Pair__rhs,axiom,
! [R: prat,S: state,R3: set_Pr6039890582625207499_state,S2: state] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_166_ssubst__Pair__rhs,axiom,
! [R: prat,S: set_state,R3: set_Pr6847263425858009217_state,S2: set_state] :
( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member6708919011922255306_state @ ( produc9080284787853255769_state @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_167_ssubst__Pair__rhs,axiom,
! [R: nat,S: list_char,R3: set_Pr2507339680178222706t_char,S2: list_char] :
( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member484235747325421115t_char @ ( produc120197956887798346t_char @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_168_ssubst__Pair__rhs,axiom,
! [R: produc1457211279475724562t_char > prat,S: produc1457211279475724562t_char > option_val,R3: set_Pr7649655576376376530on_val,S2: produc1457211279475724562t_char > option_val] :
( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_169_curryI,axiom,
! [F: produc3142500478612311029_state > $o,A4: option_state,B2: option_state] :
( ( F @ ( produc9160152616361873709_state @ A4 @ B2 ) )
=> ( produc4777492810592459639tate_o @ F @ A4 @ B2 ) ) ).
% curryI
thf(fact_170_curryI,axiom,
! [F: produc4844402304972151915_state > $o,A4: prat,B2: state] :
( ( F @ ( produc6205719116230455971_state @ A4 @ B2 ) )
=> ( produc6680356150136934785tate_o @ F @ A4 @ B2 ) ) ).
% curryI
thf(fact_171_curryI,axiom,
! [F: produc6251377953716854561_state > $o,A4: prat,B2: set_state] :
( ( F @ ( produc9080284787853255769_state @ A4 @ B2 ) )
=> ( produc936324473496865355tate_o @ F @ A4 @ B2 ) ) ).
% curryI
thf(fact_172_curryI,axiom,
! [F: produc1457211279475724562t_char > $o,A4: nat,B2: list_char] :
( ( F @ ( produc120197956887798346t_char @ A4 @ B2 ) )
=> ( produc80901228256969370char_o @ F @ A4 @ B2 ) ) ).
% curryI
thf(fact_173_curryI,axiom,
! [F: produc9003781588162623004on_val > $o,A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
( ( F @ ( produc74620467203698198on_val @ A4 @ B2 ) )
=> ( produc9184331568460336180_val_o @ F @ A4 @ B2 ) ) ).
% curryI
thf(fact_174_sep__algebra_Oup__closed_Ocong,axiom,
sep_up_closed_state = sep_up_closed_state ).
% sep_algebra.up_closed.cong
thf(fact_175_larger__heap__anti,axiom,
! [A4: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A4 @ B2 )
=> ( ( larger1532547405891354081ar_val @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% larger_heap_anti
thf(fact_176_larger__heap__refl,axiom,
! [H: produc1457211279475724562t_char > option_val] : ( larger1532547405891354081ar_val @ H @ H ) ).
% larger_heap_refl
thf(fact_177_larger__heap__trans,axiom,
! [A4: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A4 @ B2 )
=> ( ( larger1532547405891354081ar_val @ B2 @ C )
=> ( larger1532547405891354081ar_val @ A4 @ C ) ) ) ).
% larger_heap_trans
thf(fact_178_sep__algebra_Omono__prop_Ocong,axiom,
sep_mono_prop_state = sep_mono_prop_state ).
% sep_algebra.mono_prop.cong
thf(fact_179_curryD,axiom,
! [F: produc3142500478612311029_state > $o,A4: option_state,B2: option_state] :
( ( produc4777492810592459639tate_o @ F @ A4 @ B2 )
=> ( F @ ( produc9160152616361873709_state @ A4 @ B2 ) ) ) ).
% curryD
thf(fact_180_curryD,axiom,
! [F: produc4844402304972151915_state > $o,A4: prat,B2: state] :
( ( produc6680356150136934785tate_o @ F @ A4 @ B2 )
=> ( F @ ( produc6205719116230455971_state @ A4 @ B2 ) ) ) ).
% curryD
thf(fact_181_curryD,axiom,
! [F: produc6251377953716854561_state > $o,A4: prat,B2: set_state] :
( ( produc936324473496865355tate_o @ F @ A4 @ B2 )
=> ( F @ ( produc9080284787853255769_state @ A4 @ B2 ) ) ) ).
% curryD
thf(fact_182_curryD,axiom,
! [F: produc1457211279475724562t_char > $o,A4: nat,B2: list_char] :
( ( produc80901228256969370char_o @ F @ A4 @ B2 )
=> ( F @ ( produc120197956887798346t_char @ A4 @ B2 ) ) ) ).
% curryD
thf(fact_183_curryD,axiom,
! [F: produc9003781588162623004on_val > $o,A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
( ( produc9184331568460336180_val_o @ F @ A4 @ B2 )
=> ( F @ ( produc74620467203698198on_val @ A4 @ B2 ) ) ) ).
% curryD
thf(fact_184_curryE,axiom,
! [F: produc3142500478612311029_state > $o,A4: option_state,B2: option_state] :
( ( produc4777492810592459639tate_o @ F @ A4 @ B2 )
=> ( F @ ( produc9160152616361873709_state @ A4 @ B2 ) ) ) ).
% curryE
thf(fact_185_curryE,axiom,
! [F: produc4844402304972151915_state > $o,A4: prat,B2: state] :
( ( produc6680356150136934785tate_o @ F @ A4 @ B2 )
=> ( F @ ( produc6205719116230455971_state @ A4 @ B2 ) ) ) ).
% curryE
thf(fact_186_curryE,axiom,
! [F: produc6251377953716854561_state > $o,A4: prat,B2: set_state] :
( ( produc936324473496865355tate_o @ F @ A4 @ B2 )
=> ( F @ ( produc9080284787853255769_state @ A4 @ B2 ) ) ) ).
% curryE
thf(fact_187_curryE,axiom,
! [F: produc1457211279475724562t_char > $o,A4: nat,B2: list_char] :
( ( produc80901228256969370char_o @ F @ A4 @ B2 )
=> ( F @ ( produc120197956887798346t_char @ A4 @ B2 ) ) ) ).
% curryE
thf(fact_188_curryE,axiom,
! [F: produc9003781588162623004on_val > $o,A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
( ( produc9184331568460336180_val_o @ F @ A4 @ B2 )
=> ( F @ ( produc74620467203698198on_val @ A4 @ B2 ) ) ) ).
% curryE
thf(fact_189_PartialSA_Oup__closedI,axiom,
! [A: set_state] :
( ! [Phi3: state,Phi4: state] :
( ( ( greater @ Phi3 @ Phi4 )
& ( member_state @ Phi4 @ A ) )
=> ( member_state @ Phi3 @ A ) )
=> ( sep_up_closed_state @ plus @ A ) ) ).
% PartialSA.up_closedI
thf(fact_190_PartialSA_Oup__closed__def,axiom,
! [A: set_state] :
( ( sep_up_closed_state @ plus @ A )
= ( ! [Phi: state] :
( ? [X3: state] :
( ( member_state @ X3 @ A )
& ( greater @ Phi @ X3 ) )
=> ( member_state @ Phi @ A ) ) ) ) ).
% PartialSA.up_closed_def
thf(fact_191_package__logic_Ointuitionistic_Ocong,axiom,
packag8361946002163212404_state = packag8361946002163212404_state ).
% package_logic.intuitionistic.cong
thf(fact_192_PartialSA_Oup__close__equiv,axiom,
! [A: set_state,B5: set_state] :
( ( sep_up_closed_state @ plus @ A )
=> ( ( sep_up_closed_state @ plus @ B5 )
=> ( ( sep_equiv_state @ plus @ A @ B5 )
= ( A = B5 ) ) ) ) ).
% PartialSA.up_close_equiv
thf(fact_193_PartialSA_Oup__closed__sum,axiom,
! [A: set_state,B5: set_state] :
( ( sep_up_closed_state @ plus @ A )
=> ( sep_up_closed_state @ plus @ ( add_set @ A @ B5 ) ) ) ).
% PartialSA.up_closed_sum
thf(fact_194_multiply__smaller__pwrite,axiom,
! [A4: prat,B2: prat] :
( ( pgte @ pwrite @ A4 )
=> ( ( pgte @ pwrite @ B2 )
=> ( pgte @ pwrite @ ( pmult @ A4 @ B2 ) ) ) ) ).
% multiply_smaller_pwrite
thf(fact_195_PartialSA_Omax__projection__propE_I3_J,axiom,
! [P6: state > $o,F: state > state,P: state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
=> ( ( ( P6 @ P )
& ( greater @ X @ P ) )
=> ( greater @ ( F @ X ) @ P ) ) ) ).
% PartialSA.max_projection_propE(3)
thf(fact_196_PartialSA_OmppI,axiom,
! [P6: state > $o,F: state > state,A4: state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
=> ( ( greater @ A4 @ X )
=> ( ( P6 @ X )
=> ( ( greater @ X @ ( F @ A4 ) )
=> ( X
= ( F @ A4 ) ) ) ) ) ) ).
% PartialSA.mppI
thf(fact_197_PartialSA_Ompp__mono,axiom,
! [P6: state > $o,F: state > state,A4: state,B2: state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
=> ( ( greater @ A4 @ B2 )
=> ( greater @ ( F @ A4 ) @ ( F @ B2 ) ) ) ) ).
% PartialSA.mpp_mono
thf(fact_198_PartialSA_Ompp__smaller,axiom,
! [P6: state > $o,F: state > state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
=> ( greater @ X @ ( F @ X ) ) ) ).
% PartialSA.mpp_smaller
thf(fact_199_PartialSA_Omax__projection__propI,axiom,
! [F: state > state,P6: state > $o] :
( ! [X2: state] : ( greater @ X2 @ ( F @ X2 ) )
=> ( ! [X2: state] : ( P6 @ ( F @ X2 ) )
=> ( ! [X2: state,P4: state] :
( ( ( P6 @ P4 )
& ( greater @ X2 @ P4 ) )
=> ( greater @ ( F @ X2 ) @ P4 ) )
=> ( sep_ma8214210560313151521_state @ plus @ P6 @ F ) ) ) ) ).
% PartialSA.max_projection_propI
thf(fact_200_PartialSA_Omax__projection__prop__def,axiom,
! [P6: state > $o,F: state > state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
= ( ! [X3: state] :
( ( greater @ X3 @ ( F @ X3 ) )
& ( P6 @ ( F @ X3 ) )
& ! [P2: state] :
( ( ( P6 @ P2 )
& ( greater @ X3 @ P2 ) )
=> ( greater @ ( F @ X3 ) @ P2 ) ) ) ) ) ).
% PartialSA.max_projection_prop_def
thf(fact_201_larger__implies__greater__mask__hl,axiom,
! [A4: state,B2: state,Hl: produc1457211279475724562t_char] :
( ( greater @ A4 @ B2 )
=> ( pgte @ ( get_m @ A4 @ Hl ) @ ( get_m @ B2 @ Hl ) ) ) ).
% larger_implies_greater_mask_hl
thf(fact_202_sep__algebra_Oequiv_Ocong,axiom,
sep_equiv_state = sep_equiv_state ).
% sep_algebra.equiv.cong
thf(fact_203_PartialSA_Oadd__set__asso,axiom,
! [A: set_state,B5: set_state,C2: set_state] :
( ( add_set @ ( add_set @ A @ B5 ) @ C2 )
= ( add_set @ A @ ( add_set @ B5 @ C2 ) ) ) ).
% PartialSA.add_set_asso
thf(fact_204_PartialSA_Oadd__set__commm,axiom,
( add_set
= ( ^ [A2: set_state,B6: set_state] : ( add_set @ B6 @ A2 ) ) ) ).
% PartialSA.add_set_commm
thf(fact_205_sep__algebra_Omax__projection__prop_Ocong,axiom,
sep_ma8214210560313151521_state = sep_ma8214210560313151521_state ).
% sep_algebra.max_projection_prop.cong
thf(fact_206_pmult__comm,axiom,
( pmult
= ( ^ [A3: prat,B: prat] : ( pmult @ B @ A3 ) ) ) ).
% pmult_comm
thf(fact_207_PartialSA_Oequiv__stable__sum,axiom,
! [A: set_state,B5: set_state,C2: set_state] :
( ( sep_equiv_state @ plus @ A @ B5 )
=> ( sep_equiv_state @ plus @ ( add_set @ A @ C2 ) @ ( add_set @ B5 @ C2 ) ) ) ).
% PartialSA.equiv_stable_sum
thf(fact_208_PartialSA_Ompp__prop,axiom,
! [P6: state > $o,F: state > state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
=> ( P6 @ ( F @ X ) ) ) ).
% PartialSA.mpp_prop
thf(fact_209_PartialSA_Ompp__invo,axiom,
! [P6: state > $o,F: state > state,X: state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
=> ( ( F @ ( F @ X ) )
= ( F @ X ) ) ) ).
% PartialSA.mpp_invo
thf(fact_210_pmult__order,axiom,
! [A4: prat,B2: prat,P: prat] :
( ( pgte @ A4 @ B2 )
=> ( pgte @ ( pmult @ P @ A4 ) @ ( pmult @ B2 @ P ) ) ) ).
% pmult_order
thf(fact_211_state__ext,axiom,
! [A4: state,B2: state] :
( ( ( get_h @ A4 )
= ( get_h @ B2 ) )
=> ( ( ( get_m @ A4 )
= ( get_m @ B2 ) )
=> ( A4 = B2 ) ) ) ).
% state_ext
thf(fact_212_pmult__special_I1_J,axiom,
! [X: prat] :
( ( pmult @ pwrite @ X )
= X ) ).
% pmult_special(1)
thf(fact_213_greaterI,axiom,
! [A4: state,B2: state] :
( ( larger1532547405891354081ar_val @ ( get_h @ A4 ) @ ( get_h @ B2 ) )
=> ( ( greate7680731506090375811t_char @ ( get_m @ A4 ) @ ( get_m @ B2 ) )
=> ( greater @ A4 @ B2 ) ) ) ).
% greaterI
thf(fact_214_PartialSA_Omono__prop__set__equiv,axiom,
! [P6: state > $o,A: set_state,B5: set_state] :
( ( sep_mono_prop_state @ plus @ P6 )
=> ( ( sep_equiv_state @ plus @ A @ B5 )
=> ( ( sep_setify_state @ P6 @ A )
= ( sep_setify_state @ P6 @ B5 ) ) ) ) ).
% PartialSA.mono_prop_set_equiv
thf(fact_215_PartialSA_Oequiv__up__closed__subset,axiom,
! [A: set_state,B5: set_state,C2: set_state] :
( ( sep_up_closed_state @ plus @ A )
=> ( ( sep_equiv_state @ plus @ B5 @ C2 )
=> ( ( ord_le2494988322063910608_state @ B5 @ A )
= ( ord_le2494988322063910608_state @ C2 @ A ) ) ) ) ).
% PartialSA.equiv_up_closed_subset
thf(fact_216_PartialSA_Oup__closed__plus__UNIV,axiom,
! [A: set_state] : ( sep_up_closed_state @ plus @ ( add_set @ A @ top_top_set_state ) ) ).
% PartialSA.up_closed_plus_UNIV
thf(fact_217_padd__one__ineq__sum,axiom,
! [A4: prat,B2: prat,X: prat,Aa: prat,Bb: prat] :
( ( ( padd @ A4 @ B2 )
= pwrite )
=> ( ( pgte @ X @ Aa )
=> ( ( pgte @ X @ Bb )
=> ( pgte @ X @ ( padd @ ( pmult @ A4 @ Aa ) @ ( pmult @ B2 @ Bb ) ) ) ) ) ) ).
% padd_one_ineq_sum
thf(fact_218_PartialSA_Oequiv__def,axiom,
! [A: set_state,B5: set_state] :
( ( sep_equiv_state @ plus @ A @ B5 )
= ( ( greater_set @ A @ B5 )
& ( greater_set @ B5 @ A ) ) ) ).
% PartialSA.equiv_def
thf(fact_219_PartialSA_OequivI,axiom,
! [A: set_state,B5: set_state] :
( ( greater_set @ A @ B5 )
=> ( ( greater_set @ B5 @ A )
=> ( sep_equiv_state @ plus @ A @ B5 ) ) ) ).
% PartialSA.equivI
thf(fact_220_get__h__m,axiom,
( rep_state
= ( ^ [X3: state] : ( produc74620467203698198on_val @ ( get_m @ X3 ) @ ( get_h @ X3 ) ) ) ) ).
% get_h_m
thf(fact_221_PartialSA_Ompp__compatible,axiom,
! [P6: state > $o,F: state > state,A4: state,B2: state] :
( ( sep_ma8214210560313151521_state @ plus @ P6 @ F )
=> ( ( defined @ A4 @ B2 )
=> ( defined @ ( F @ A4 ) @ ( F @ B2 ) ) ) ) ).
% PartialSA.mpp_compatible
thf(fact_222_multiply__valid,axiom,
! [P: prat,Phi5: state] :
( ( pgte @ pwrite @ P )
=> ( valid_state @ ( produc74620467203698198on_val @ ( multiply_mask @ P @ ( get_m @ Phi5 ) ) @ ( get_h @ Phi5 ) ) ) ) ).
% multiply_valid
thf(fact_223_UNIV__I,axiom,
! [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ top_to174962084833059535_state ) ).
% UNIV_I
thf(fact_224_UNIV__I,axiom,
! [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ top_to1561438921946393474on_val ) ).
% UNIV_I
thf(fact_225_UNIV__I,axiom,
! [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ top_to1744471583079920734_state ) ).
% UNIV_I
thf(fact_226_UNIV__I,axiom,
! [X: state] : ( member_state @ X @ top_top_set_state ) ).
% UNIV_I
thf(fact_227_subsetI,axiom,
! [A: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
( ! [X2: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X2 @ A )
=> ( member7461423308196221312_state @ X2 @ B5 ) )
=> ( ord_le4032880777312351359_state @ A @ B5 ) ) ).
% subsetI
thf(fact_228_subsetI,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ A )
=> ( member2879030624002546611on_val @ X2 @ B5 ) )
=> ( ord_le6900622908513426738on_val @ A @ B5 ) ) ).
% subsetI
thf(fact_229_subsetI,axiom,
! [A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ A )
=> ( member6715010751053357271_state @ X2 @ B5 ) )
=> ( ord_le6483787752022044334_state @ A @ B5 ) ) ).
% subsetI
thf(fact_230_subsetI,axiom,
! [A: set_state,B5: set_state] :
( ! [X2: state] :
( ( member_state @ X2 @ A )
=> ( member_state @ X2 @ B5 ) )
=> ( ord_le2494988322063910608_state @ A @ B5 ) ) ).
% subsetI
thf(fact_231_padd__cancellative,axiom,
! [A4: prat,X: prat,B2: prat,Y: prat] :
( ( A4
= ( padd @ X @ B2 ) )
=> ( ( A4
= ( padd @ Y @ B2 ) )
=> ( X = Y ) ) ) ).
% padd_cancellative
thf(fact_232_Collect__mono__iff,axiom,
! [P6: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
( ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ P6 ) @ ( collec906845016635587262_state @ Q ) )
= ( ! [X3: produc7272280185086996969_state] :
( ( P6 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_233_Collect__mono__iff,axiom,
! [P6: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
( ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ P6 ) @ ( collec7475294502966882545on_val @ Q ) )
= ( ! [X3: produc9003781588162623004on_val] :
( ( P6 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_234_Collect__mono__iff,axiom,
! [P6: state > $o,Q: state > $o] :
( ( ord_le2494988322063910608_state @ ( collect_state @ P6 ) @ ( collect_state @ Q ) )
= ( ! [X3: state] :
( ( P6 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_235_set__eq__subset,axiom,
( ( ^ [Y6: set_state,Z3: set_state] : ( Y6 = Z3 ) )
= ( ^ [A2: set_state,B6: set_state] :
( ( ord_le2494988322063910608_state @ A2 @ B6 )
& ( ord_le2494988322063910608_state @ B6 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_236_subset__trans,axiom,
! [A: set_state,B5: set_state,C2: set_state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ( ord_le2494988322063910608_state @ B5 @ C2 )
=> ( ord_le2494988322063910608_state @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_237_UNIV__witness,axiom,
? [X2: produc7272280185086996969_state] : ( member7461423308196221312_state @ X2 @ top_to174962084833059535_state ) ).
% UNIV_witness
thf(fact_238_UNIV__witness,axiom,
? [X2: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X2 @ top_to1561438921946393474on_val ) ).
% UNIV_witness
thf(fact_239_UNIV__witness,axiom,
? [X2: produc3882902684156697774_state] : ( member6715010751053357271_state @ X2 @ top_to1744471583079920734_state ) ).
% UNIV_witness
thf(fact_240_UNIV__witness,axiom,
? [X2: state] : ( member_state @ X2 @ top_top_set_state ) ).
% UNIV_witness
thf(fact_241_Collect__mono,axiom,
! [P6: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
( ! [X2: produc7272280185086996969_state] :
( ( P6 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ P6 ) @ ( collec906845016635587262_state @ Q ) ) ) ).
% Collect_mono
thf(fact_242_Collect__mono,axiom,
! [P6: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
( ! [X2: produc9003781588162623004on_val] :
( ( P6 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ P6 ) @ ( collec7475294502966882545on_val @ Q ) ) ) ).
% Collect_mono
thf(fact_243_Collect__mono,axiom,
! [P6: state > $o,Q: state > $o] :
( ! [X2: state] :
( ( P6 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le2494988322063910608_state @ ( collect_state @ P6 ) @ ( collect_state @ Q ) ) ) ).
% Collect_mono
thf(fact_244_padd__comm,axiom,
( padd
= ( ^ [A3: prat,B: prat] : ( padd @ B @ A3 ) ) ) ).
% padd_comm
thf(fact_245_padd__asso,axiom,
! [A4: prat,B2: prat,C: prat] :
( ( padd @ ( padd @ A4 @ B2 ) @ C )
= ( padd @ A4 @ ( padd @ B2 @ C ) ) ) ).
% padd_asso
thf(fact_246_subset__refl,axiom,
! [A: set_state] : ( ord_le2494988322063910608_state @ A @ A ) ).
% subset_refl
thf(fact_247_subset__UNIV,axiom,
! [A: set_state] : ( ord_le2494988322063910608_state @ A @ top_top_set_state ) ).
% subset_UNIV
thf(fact_248_subset__iff,axiom,
( ord_le4032880777312351359_state
= ( ^ [A2: set_Pr4231240917283394591_state,B6: set_Pr4231240917283394591_state] :
! [T: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ T @ A2 )
=> ( member7461423308196221312_state @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_249_subset__iff,axiom,
( ord_le6900622908513426738on_val
= ( ^ [A2: set_Pr7649655576376376530on_val,B6: set_Pr7649655576376376530on_val] :
! [T: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ T @ A2 )
=> ( member2879030624002546611on_val @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_250_subset__iff,axiom,
( ord_le6483787752022044334_state
= ( ^ [A2: set_Pr2391701600042252046_state,B6: set_Pr2391701600042252046_state] :
! [T: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ T @ A2 )
=> ( member6715010751053357271_state @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_251_subset__iff,axiom,
( ord_le2494988322063910608_state
= ( ^ [A2: set_state,B6: set_state] :
! [T: state] :
( ( member_state @ T @ A2 )
=> ( member_state @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_252_equalityD2,axiom,
! [A: set_state,B5: set_state] :
( ( A = B5 )
=> ( ord_le2494988322063910608_state @ B5 @ A ) ) ).
% equalityD2
thf(fact_253_equalityD1,axiom,
! [A: set_state,B5: set_state] :
( ( A = B5 )
=> ( ord_le2494988322063910608_state @ A @ B5 ) ) ).
% equalityD1
thf(fact_254_subset__eq,axiom,
( ord_le4032880777312351359_state
= ( ^ [A2: set_Pr4231240917283394591_state,B6: set_Pr4231240917283394591_state] :
! [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ A2 )
=> ( member7461423308196221312_state @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_255_subset__eq,axiom,
( ord_le6900622908513426738on_val
= ( ^ [A2: set_Pr7649655576376376530on_val,B6: set_Pr7649655576376376530on_val] :
! [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A2 )
=> ( member2879030624002546611on_val @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_256_subset__eq,axiom,
( ord_le6483787752022044334_state
= ( ^ [A2: set_Pr2391701600042252046_state,B6: set_Pr2391701600042252046_state] :
! [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A2 )
=> ( member6715010751053357271_state @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_257_subset__eq,axiom,
( ord_le2494988322063910608_state
= ( ^ [A2: set_state,B6: set_state] :
! [X3: state] :
( ( member_state @ X3 @ A2 )
=> ( member_state @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_258_equalityE,axiom,
! [A: set_state,B5: set_state] :
( ( A = B5 )
=> ~ ( ( ord_le2494988322063910608_state @ A @ B5 )
=> ~ ( ord_le2494988322063910608_state @ B5 @ A ) ) ) ).
% equalityE
thf(fact_259_UNIV__eq__I,axiom,
! [A: set_Pr4231240917283394591_state] :
( ! [X2: produc7272280185086996969_state] : ( member7461423308196221312_state @ X2 @ A )
=> ( top_to174962084833059535_state = A ) ) ).
% UNIV_eq_I
thf(fact_260_UNIV__eq__I,axiom,
! [A: set_Pr7649655576376376530on_val] :
( ! [X2: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X2 @ A )
=> ( top_to1561438921946393474on_val = A ) ) ).
% UNIV_eq_I
thf(fact_261_UNIV__eq__I,axiom,
! [A: set_Pr2391701600042252046_state] :
( ! [X2: produc3882902684156697774_state] : ( member6715010751053357271_state @ X2 @ A )
=> ( top_to1744471583079920734_state = A ) ) ).
% UNIV_eq_I
thf(fact_262_UNIV__eq__I,axiom,
! [A: set_state] :
( ! [X2: state] : ( member_state @ X2 @ A )
=> ( top_top_set_state = A ) ) ).
% UNIV_eq_I
thf(fact_263_subsetD,axiom,
! [A: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state,C: produc7272280185086996969_state] :
( ( ord_le4032880777312351359_state @ A @ B5 )
=> ( ( member7461423308196221312_state @ C @ A )
=> ( member7461423308196221312_state @ C @ B5 ) ) ) ).
% subsetD
thf(fact_264_subsetD,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,C: produc9003781588162623004on_val] :
( ( ord_le6900622908513426738on_val @ A @ B5 )
=> ( ( member2879030624002546611on_val @ C @ A )
=> ( member2879030624002546611on_val @ C @ B5 ) ) ) ).
% subsetD
thf(fact_265_subsetD,axiom,
! [A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state,C: produc3882902684156697774_state] :
( ( ord_le6483787752022044334_state @ A @ B5 )
=> ( ( member6715010751053357271_state @ C @ A )
=> ( member6715010751053357271_state @ C @ B5 ) ) ) ).
% subsetD
thf(fact_266_subsetD,axiom,
! [A: set_state,B5: set_state,C: state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ( member_state @ C @ A )
=> ( member_state @ C @ B5 ) ) ) ).
% subsetD
thf(fact_267_in__mono,axiom,
! [A: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state,X: produc7272280185086996969_state] :
( ( ord_le4032880777312351359_state @ A @ B5 )
=> ( ( member7461423308196221312_state @ X @ A )
=> ( member7461423308196221312_state @ X @ B5 ) ) ) ).
% in_mono
thf(fact_268_in__mono,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,X: produc9003781588162623004on_val] :
( ( ord_le6900622908513426738on_val @ A @ B5 )
=> ( ( member2879030624002546611on_val @ X @ A )
=> ( member2879030624002546611on_val @ X @ B5 ) ) ) ).
% in_mono
thf(fact_269_in__mono,axiom,
! [A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state,X: produc3882902684156697774_state] :
( ( ord_le6483787752022044334_state @ A @ B5 )
=> ( ( member6715010751053357271_state @ X @ A )
=> ( member6715010751053357271_state @ X @ B5 ) ) ) ).
% in_mono
thf(fact_270_in__mono,axiom,
! [A: set_state,B5: set_state,X: state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ( member_state @ X @ A )
=> ( member_state @ X @ B5 ) ) ) ).
% in_mono
thf(fact_271_PartialSA_Olarger__set__refl,axiom,
! [A: set_state] : ( greater_set @ A @ A ) ).
% PartialSA.larger_set_refl
thf(fact_272_PartialSA_Osucc__set__trans,axiom,
! [A: set_state,B5: set_state,C2: set_state] :
( ( greater_set @ A @ B5 )
=> ( ( greater_set @ B5 @ C2 )
=> ( greater_set @ A @ C2 ) ) ) ).
% PartialSA.succ_set_trans
thf(fact_273_Rep__state__inject,axiom,
! [X: state,Y: state] :
( ( ( rep_state @ X )
= ( rep_state @ Y ) )
= ( X = Y ) ) ).
% Rep_state_inject
thf(fact_274_PartialSA_Osub__bigger,axiom,
! [A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( greater_set @ A @ B5 ) ) ).
% PartialSA.sub_bigger
thf(fact_275_PartialSA_Osetify__def,axiom,
( sep_setify_state
= ( ^ [P3: state > $o,A2: set_state] :
! [X3: state] :
( ( member_state @ X3 @ A2 )
=> ( P3 @ X3 ) ) ) ) ).
% PartialSA.setify_def
thf(fact_276_valid__state_Ocases,axiom,
! [X: produc9003781588162623004on_val] :
~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( X
!= ( produc74620467203698198on_val @ Pi @ H2 ) ) ).
% valid_state.cases
thf(fact_277_add__states_Ocases,axiom,
! [X: produc1317171644805857433on_val] :
~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val,Pi2: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
( X
!= ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H2 ) @ ( produc74620467203698198on_val @ Pi2 @ H3 ) ) ) ).
% add_states.cases
thf(fact_278_range__subsetD,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,B5: set_Pr4231240917283394591_state,I: produc9003781588162623004on_val] :
( ( ord_le4032880777312351359_state @ ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val ) @ B5 )
=> ( member7461423308196221312_state @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_279_range__subsetD,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val,I: produc3882902684156697774_state] :
( ( ord_le6900622908513426738on_val @ ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state ) @ B5 )
=> ( member2879030624002546611on_val @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_280_range__subsetD,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val,I: produc9003781588162623004on_val] :
( ( ord_le6900622908513426738on_val @ ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val ) @ B5 )
=> ( member2879030624002546611on_val @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_281_range__subsetD,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state,I: produc9003781588162623004on_val] :
( ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val ) @ B5 )
=> ( member6715010751053357271_state @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_282_range__subsetD,axiom,
! [F: state > produc7272280185086996969_state,B5: set_Pr4231240917283394591_state,I: state] :
( ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ top_top_set_state ) @ B5 )
=> ( member7461423308196221312_state @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_283_range__subsetD,axiom,
! [F: state > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val,I: state] :
( ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) @ B5 )
=> ( member2879030624002546611on_val @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_284_range__subsetD,axiom,
! [F: state > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state,I: state] :
( ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ top_top_set_state ) @ B5 )
=> ( member6715010751053357271_state @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_285_range__subsetD,axiom,
! [F: produc9003781588162623004on_val > state,B5: set_state,I: produc9003781588162623004on_val] :
( ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val ) @ B5 )
=> ( member_state @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_286_range__subsetD,axiom,
! [F: state > state,B5: set_state,I: state] :
( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ top_top_set_state ) @ B5 )
=> ( member_state @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_287_image__mono,axiom,
! [A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( ord_le6483787752022044334_state @ A @ B5 )
=> ( ord_le6900622908513426738on_val @ ( image_4690842878458256517on_val @ F @ A ) @ ( image_4690842878458256517on_val @ F @ B5 ) ) ) ).
% image_mono
thf(fact_288_image__mono,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc7272280185086996969_state] :
( ( ord_le6900622908513426738on_val @ A @ B5 )
=> ( ord_le4032880777312351359_state @ ( image_3204238854701872908_state @ F @ A ) @ ( image_3204238854701872908_state @ F @ B5 ) ) ) ).
% image_mono
thf(fact_289_image__mono,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( ord_le6900622908513426738on_val @ A @ B5 )
=> ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ A ) @ ( image_4494208899119068747_state @ F @ B5 ) ) ) ).
% image_mono
thf(fact_290_image__mono,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
( ( ord_le6900622908513426738on_val @ A @ B5 )
=> ( ord_le6900622908513426738on_val @ ( image_4170889944715819327on_val @ F @ A ) @ ( image_4170889944715819327on_val @ F @ B5 ) ) ) ).
% image_mono
thf(fact_291_image__mono,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > state] :
( ( ord_le6900622908513426738on_val @ A @ B5 )
=> ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ A ) @ ( image_1205529093420515805_state @ F @ B5 ) ) ) ).
% image_mono
thf(fact_292_image__mono,axiom,
! [A: set_state,B5: set_state,F: state > state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A ) @ ( image_state_state @ F @ B5 ) ) ) ).
% image_mono
thf(fact_293_image__subsetI,axiom,
! [A: set_state,F: state > state,B5: set_state] :
( ! [X2: state] :
( ( member_state @ X2 @ A )
=> ( member_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_294_image__subsetI,axiom,
! [A: set_state,F: state > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ! [X2: state] :
( ( member_state @ X2 @ A )
=> ( member6715010751053357271_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_295_image__subsetI,axiom,
! [A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state,B5: set_state] :
( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ A )
=> ( member_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le2494988322063910608_state @ ( image_6759111139246981411_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_296_image__subsetI,axiom,
! [A: set_state,F: state > produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] :
( ! [X2: state] :
( ( member_state @ X2 @ A )
=> ( member7461423308196221312_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_297_image__subsetI,axiom,
! [A: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state,B5: set_state] :
( ! [X2: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X2 @ A )
=> ( member_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le2494988322063910608_state @ ( image_2710133271811360426_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_298_image__subsetI,axiom,
! [A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ A )
=> ( member6715010751053357271_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6483787752022044334_state @ ( image_508898238547643525_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_299_image__subsetI,axiom,
! [A: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ! [X2: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X2 @ A )
=> ( member6715010751053357271_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6483787752022044334_state @ ( image_7466395622266389182_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_300_image__subsetI,axiom,
! [A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] :
( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ A )
=> ( member7461423308196221312_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le4032880777312351359_state @ ( image_4364510454419742290_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_301_image__subsetI,axiom,
! [A: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] :
( ! [X2: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X2 @ A )
=> ( member7461423308196221312_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le4032880777312351359_state @ ( image_3064086290059306969_state @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_302_image__subsetI,axiom,
! [A: set_state,F: state > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ! [X2: state] :
( ( member_state @ X2 @ A )
=> ( member2879030624002546611on_val @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_303_subset__imageE,axiom,
! [B5: set_Pr7649655576376376530on_val,F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state] :
( ( ord_le6900622908513426738on_val @ B5 @ ( image_4690842878458256517on_val @ F @ A ) )
=> ~ ! [C3: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ C3 @ A )
=> ( B5
!= ( image_4690842878458256517on_val @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_304_subset__imageE,axiom,
! [B5: set_Pr4231240917283394591_state,F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val] :
( ( ord_le4032880777312351359_state @ B5 @ ( image_3204238854701872908_state @ F @ A ) )
=> ~ ! [C3: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ C3 @ A )
=> ( B5
!= ( image_3204238854701872908_state @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_305_subset__imageE,axiom,
! [B5: set_Pr2391701600042252046_state,F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val] :
( ( ord_le6483787752022044334_state @ B5 @ ( image_4494208899119068747_state @ F @ A ) )
=> ~ ! [C3: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ C3 @ A )
=> ( B5
!= ( image_4494208899119068747_state @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_306_subset__imageE,axiom,
! [B5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B5 @ ( image_4170889944715819327on_val @ F @ A ) )
=> ~ ! [C3: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ C3 @ A )
=> ( B5
!= ( image_4170889944715819327on_val @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_307_subset__imageE,axiom,
! [B5: set_state,F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( ord_le2494988322063910608_state @ B5 @ ( image_1205529093420515805_state @ F @ A ) )
=> ~ ! [C3: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ C3 @ A )
=> ( B5
!= ( image_1205529093420515805_state @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_308_subset__imageE,axiom,
! [B5: set_state,F: state > state,A: set_state] :
( ( ord_le2494988322063910608_state @ B5 @ ( image_state_state @ F @ A ) )
=> ~ ! [C3: set_state] :
( ( ord_le2494988322063910608_state @ C3 @ A )
=> ( B5
!= ( image_state_state @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_309_image__subset__iff,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val,B5: set_Pr4231240917283394591_state] :
( ( ord_le4032880777312351359_state @ ( image_3204238854701872908_state @ F @ A ) @ B5 )
= ( ! [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
=> ( member7461423308196221312_state @ ( F @ X3 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_310_image__subset__iff,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state,B5: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ ( image_4690842878458256517on_val @ F @ A ) @ B5 )
= ( ! [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
=> ( member2879030624002546611on_val @ ( F @ X3 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_311_image__subset__iff,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ ( image_4170889944715819327on_val @ F @ A ) @ B5 )
= ( ! [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
=> ( member2879030624002546611on_val @ ( F @ X3 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_312_image__subset__iff,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val,B5: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ A ) @ B5 )
= ( ! [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
=> ( member6715010751053357271_state @ ( F @ X3 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_313_image__subset__iff,axiom,
! [F: state > state,A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A ) @ B5 )
= ( ! [X3: state] :
( ( member_state @ X3 @ A )
=> ( member_state @ ( F @ X3 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_314_image__subset__iff,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,B5: set_state] :
( ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ A ) @ B5 )
= ( ! [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
=> ( member_state @ ( F @ X3 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_315_subset__image__iff,axiom,
! [B5: set_Pr7649655576376376530on_val,F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state] :
( ( ord_le6900622908513426738on_val @ B5 @ ( image_4690842878458256517on_val @ F @ A ) )
= ( ? [AA: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ AA @ A )
& ( B5
= ( image_4690842878458256517on_val @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_316_subset__image__iff,axiom,
! [B5: set_Pr4231240917283394591_state,F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val] :
( ( ord_le4032880777312351359_state @ B5 @ ( image_3204238854701872908_state @ F @ A ) )
= ( ? [AA: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ AA @ A )
& ( B5
= ( image_3204238854701872908_state @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_317_subset__image__iff,axiom,
! [B5: set_Pr2391701600042252046_state,F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val] :
( ( ord_le6483787752022044334_state @ B5 @ ( image_4494208899119068747_state @ F @ A ) )
= ( ? [AA: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ AA @ A )
& ( B5
= ( image_4494208899119068747_state @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_318_subset__image__iff,axiom,
! [B5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B5 @ ( image_4170889944715819327on_val @ F @ A ) )
= ( ? [AA: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ AA @ A )
& ( B5
= ( image_4170889944715819327on_val @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_319_subset__image__iff,axiom,
! [B5: set_state,F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( ord_le2494988322063910608_state @ B5 @ ( image_1205529093420515805_state @ F @ A ) )
= ( ? [AA: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ AA @ A )
& ( B5
= ( image_1205529093420515805_state @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_320_subset__image__iff,axiom,
! [B5: set_state,F: state > state,A: set_state] :
( ( ord_le2494988322063910608_state @ B5 @ ( image_state_state @ F @ A ) )
= ( ? [AA: set_state] :
( ( ord_le2494988322063910608_state @ AA @ A )
& ( B5
= ( image_state_state @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_321_rangeI,axiom,
! [F: produc9003781588162623004on_val > state,X: produc9003781588162623004on_val] : ( member_state @ ( F @ X ) @ ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val ) ) ).
% rangeI
thf(fact_322_rangeI,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,X: produc9003781588162623004on_val] : ( member7461423308196221312_state @ ( F @ X ) @ ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val ) ) ).
% rangeI
thf(fact_323_rangeI,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,X: produc3882902684156697774_state] : ( member2879030624002546611on_val @ ( F @ X ) @ ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state ) ) ).
% rangeI
thf(fact_324_rangeI,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ ( F @ X ) @ ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val ) ) ).
% rangeI
thf(fact_325_rangeI,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,X: produc9003781588162623004on_val] : ( member6715010751053357271_state @ ( F @ X ) @ ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val ) ) ).
% rangeI
thf(fact_326_rangeI,axiom,
! [F: state > state,X: state] : ( member_state @ ( F @ X ) @ ( image_state_state @ F @ top_top_set_state ) ) ).
% rangeI
thf(fact_327_rangeI,axiom,
! [F: state > produc7272280185086996969_state,X: state] : ( member7461423308196221312_state @ ( F @ X ) @ ( image_632992669292831658_state @ F @ top_top_set_state ) ) ).
% rangeI
thf(fact_328_rangeI,axiom,
! [F: state > produc9003781588162623004on_val,X: state] : ( member2879030624002546611on_val @ ( F @ X ) @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) ) ).
% rangeI
thf(fact_329_rangeI,axiom,
! [F: state > produc3882902684156697774_state,X: state] : ( member6715010751053357271_state @ ( F @ X ) @ ( image_5311902671526399277_state @ F @ top_top_set_state ) ) ).
% rangeI
thf(fact_330_range__eqI,axiom,
! [B2: state,F: produc9003781588162623004on_val > state,X: produc9003781588162623004on_val] :
( ( B2
= ( F @ X ) )
=> ( member_state @ B2 @ ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val ) ) ) ).
% range_eqI
thf(fact_331_range__eqI,axiom,
! [B2: produc7272280185086996969_state,F: produc9003781588162623004on_val > produc7272280185086996969_state,X: produc9003781588162623004on_val] :
( ( B2
= ( F @ X ) )
=> ( member7461423308196221312_state @ B2 @ ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val ) ) ) ).
% range_eqI
thf(fact_332_range__eqI,axiom,
! [B2: produc9003781588162623004on_val,F: produc3882902684156697774_state > produc9003781588162623004on_val,X: produc3882902684156697774_state] :
( ( B2
= ( F @ X ) )
=> ( member2879030624002546611on_val @ B2 @ ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state ) ) ) ).
% range_eqI
thf(fact_333_range__eqI,axiom,
! [B2: produc9003781588162623004on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,X: produc9003781588162623004on_val] :
( ( B2
= ( F @ X ) )
=> ( member2879030624002546611on_val @ B2 @ ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val ) ) ) ).
% range_eqI
thf(fact_334_range__eqI,axiom,
! [B2: produc3882902684156697774_state,F: produc9003781588162623004on_val > produc3882902684156697774_state,X: produc9003781588162623004on_val] :
( ( B2
= ( F @ X ) )
=> ( member6715010751053357271_state @ B2 @ ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val ) ) ) ).
% range_eqI
thf(fact_335_range__eqI,axiom,
! [B2: state,F: state > state,X: state] :
( ( B2
= ( F @ X ) )
=> ( member_state @ B2 @ ( image_state_state @ F @ top_top_set_state ) ) ) ).
% range_eqI
thf(fact_336_range__eqI,axiom,
! [B2: produc7272280185086996969_state,F: state > produc7272280185086996969_state,X: state] :
( ( B2
= ( F @ X ) )
=> ( member7461423308196221312_state @ B2 @ ( image_632992669292831658_state @ F @ top_top_set_state ) ) ) ).
% range_eqI
thf(fact_337_range__eqI,axiom,
! [B2: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val,X: state] :
( ( B2
= ( F @ X ) )
=> ( member2879030624002546611on_val @ B2 @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) ) ) ).
% range_eqI
thf(fact_338_range__eqI,axiom,
! [B2: produc3882902684156697774_state,F: state > produc3882902684156697774_state,X: state] :
( ( B2
= ( F @ X ) )
=> ( member6715010751053357271_state @ B2 @ ( image_5311902671526399277_state @ F @ top_top_set_state ) ) ) ).
% range_eqI
thf(fact_339_greater__sum__both,axiom,
! [A4: prat,B2: prat,C: prat] :
( ( pgte @ A4 @ ( padd @ B2 @ C ) )
=> ? [A1: prat,A22: prat] :
( ( A4
= ( padd @ A1 @ A22 ) )
& ( pgte @ A1 @ B2 )
& ( pgte @ A22 @ C ) ) ) ).
% greater_sum_both
thf(fact_340_p__greater__exists,axiom,
( pgte
= ( ^ [A3: prat,B: prat] :
? [R2: prat] :
( A3
= ( padd @ B @ R2 ) ) ) ) ).
% p_greater_exists
thf(fact_341_PartialSA_Oup__closed__bigger__subset,axiom,
! [B5: set_state,A: set_state] :
( ( sep_up_closed_state @ plus @ B5 )
=> ( ( greater_set @ A @ B5 )
=> ( ord_le2494988322063910608_state @ A @ B5 ) ) ) ).
% PartialSA.up_closed_bigger_subset
thf(fact_342_pmult__distr,axiom,
! [A4: prat,B2: prat,C: prat] :
( ( pmult @ A4 @ ( padd @ B2 @ C ) )
= ( padd @ ( pmult @ A4 @ B2 ) @ ( pmult @ A4 @ C ) ) ) ).
% pmult_distr
thf(fact_343_pmult__padd,axiom,
! [A4: prat,B2: prat,X: prat,C: prat,Y: prat] :
( ( pmult @ A4 @ ( padd @ ( pmult @ B2 @ X ) @ ( pmult @ C @ Y ) ) )
= ( padd @ ( pmult @ ( pmult @ A4 @ B2 ) @ X ) @ ( pmult @ ( pmult @ A4 @ C ) @ Y ) ) ) ).
% pmult_padd
thf(fact_344_PartialSA_Osmaller__compatible,axiom,
! [A5: state,B2: state,A4: state] :
( ( defined @ A5 @ B2 )
=> ( ( greater @ A5 @ A4 )
=> ( defined @ A4 @ B2 ) ) ) ).
% PartialSA.smaller_compatible
thf(fact_345_PartialSA_Ogreater__setI,axiom,
! [A: set_state,B5: set_state] :
( ! [A6: state] :
( ( member_state @ A6 @ A )
=> ? [X4: state] :
( ( member_state @ X4 @ B5 )
& ( greater @ A6 @ X4 ) ) )
=> ( greater_set @ A @ B5 ) ) ).
% PartialSA.greater_setI
thf(fact_346_PartialSA_Ogreater__set__def,axiom,
( greater_set
= ( ^ [A2: set_state,B6: set_state] :
! [X3: state] :
( ( member_state @ X3 @ A2 )
=> ? [Y4: state] :
( ( member_state @ Y4 @ B6 )
& ( greater @ X3 @ Y4 ) ) ) ) ) ).
% PartialSA.greater_set_def
thf(fact_347_PartialSA_Omono__prop__set,axiom,
! [A: set_state,B5: set_state,P6: state > $o] :
( ( greater_set @ A @ B5 )
=> ( ( sep_setify_state @ P6 @ B5 )
=> ( ( sep_mono_prop_state @ plus @ P6 )
=> ( sep_setify_state @ P6 @ A ) ) ) ) ).
% PartialSA.mono_prop_set
thf(fact_348_PartialSA_Obigger__set,axiom,
! [A7: set_state,A: set_state,B5: set_state] :
( ( greater_set @ A7 @ A )
=> ( greater_set @ ( add_set @ A7 @ B5 ) @ ( add_set @ A @ B5 ) ) ) ).
% PartialSA.bigger_set
thf(fact_349_valid__bin,axiom,
! [A4: state] : ( valid_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ A4 ) ) @ ( get_h @ A4 ) ) ) ).
% valid_bin
thf(fact_350_padd__comp__one,axiom,
! [X: prat] :
( ( pgte @ pwrite @ X )
=> ( ( padd @ X @ ( comp_one @ X ) )
= pwrite ) ) ).
% padd_comp_one
thf(fact_351_mult__greater,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat,P: prat] :
( ( greate7680731506090375811t_char @ A4 @ B2 )
=> ( greate7680731506090375811t_char @ ( multiply_mask @ P @ A4 ) @ ( multiply_mask @ P @ B2 ) ) ) ).
% mult_greater
thf(fact_352_multiply_Oelims,axiom,
! [X: prat,Xa: state,Y: state] :
( ( ( multiply @ X @ Xa )
= Y )
=> ( Y
= ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X @ ( get_m @ Xa ) ) @ ( get_h @ Xa ) ) ) ) ) ).
% multiply.elims
thf(fact_353_multiply_Osimps,axiom,
( multiply
= ( ^ [P2: prat,Phi2: state] : ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ P2 @ ( get_m @ Phi2 ) ) @ ( get_h @ Phi2 ) ) ) ) ) ).
% multiply.simps
thf(fact_354_multiply__mask__def,axiom,
( multiply_mask
= ( ^ [P2: prat,Pi3: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( pmult @ P2 @ ( Pi3 @ Hl2 ) ) ) ) ).
% multiply_mask_def
thf(fact_355_scalable__def,axiom,
( scalable
= ( ^ [W2: state,A3: state] :
! [X3: state] :
( ( member_state @ X3 @ ( scaled @ W2 ) )
=> ~ ( defined @ A3 @ X3 ) ) ) ) ).
% scalable_def
thf(fact_356_double__inclusion,axiom,
! [A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ( ord_le2494988322063910608_state @ B5 @ A )
=> ( A = B5 ) ) ) ).
% double_inclusion
thf(fact_357_mult__write__mask,axiom,
! [Pi4: produc1457211279475724562t_char > prat] :
( ( multiply_mask @ pwrite @ Pi4 )
= Pi4 ) ).
% mult_write_mask
thf(fact_358_iso__tuple__UNIV__I,axiom,
! [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ top_to174962084833059535_state ) ).
% iso_tuple_UNIV_I
thf(fact_359_iso__tuple__UNIV__I,axiom,
! [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ top_to1561438921946393474on_val ) ).
% iso_tuple_UNIV_I
thf(fact_360_iso__tuple__UNIV__I,axiom,
! [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ top_to1744471583079920734_state ) ).
% iso_tuple_UNIV_I
thf(fact_361_iso__tuple__UNIV__I,axiom,
! [X: state] : ( member_state @ X @ top_top_set_state ) ).
% iso_tuple_UNIV_I
thf(fact_362_ssubsetI,axiom,
! [A: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
( ! [Pi: state,H2: produc3882902684156697774_state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi @ H2 ) @ A )
=> ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi @ H2 ) @ B5 ) )
=> ( ord_le4032880777312351359_state @ A @ B5 ) ) ).
% ssubsetI
thf(fact_363_ssubsetI,axiom,
! [A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
( ! [Pi: state,H2: state > state] :
( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi @ H2 ) @ A )
=> ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi @ H2 ) @ B5 ) )
=> ( ord_le6483787752022044334_state @ A @ B5 ) ) ).
% ssubsetI
thf(fact_364_ssubsetI,axiom,
! [A: set_Pr1688445902015331925_state,B5: set_Pr1688445902015331925_state] :
( ! [Pi: option_state,H2: option_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi @ H2 ) @ A )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi @ H2 ) @ B5 ) )
=> ( ord_le6423325748750870005_state @ A @ B5 ) ) ).
% ssubsetI
thf(fact_365_ssubsetI,axiom,
! [A: set_Pr6039890582625207499_state,B5: set_Pr6039890582625207499_state] :
( ! [Pi: prat,H2: state] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi @ H2 ) @ A )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi @ H2 ) @ B5 ) )
=> ( ord_le262197048489888875_state @ A @ B5 ) ) ).
% ssubsetI
thf(fact_366_ssubsetI,axiom,
! [A: set_Pr6847263425858009217_state,B5: set_Pr6847263425858009217_state] :
( ! [Pi: prat,H2: set_state] :
( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi @ H2 ) @ A )
=> ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi @ H2 ) @ B5 ) )
=> ( ord_le5761494056910127137_state @ A @ B5 ) ) ).
% ssubsetI
thf(fact_367_ssubsetI,axiom,
! [A: set_Pr2507339680178222706t_char,B5: set_Pr2507339680178222706t_char] :
( ! [Pi: nat,H2: list_char] :
( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi @ H2 ) @ A )
=> ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi @ H2 ) @ B5 ) )
=> ( ord_le6315511645215477266t_char @ A @ B5 ) ) ).
% ssubsetI
thf(fact_368_ssubsetI,axiom,
! [A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H2 ) @ A )
=> ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H2 ) @ B5 ) )
=> ( ord_le6900622908513426738on_val @ A @ B5 ) ) ).
% ssubsetI
thf(fact_369_Rep__state__inverse,axiom,
! [X: state] :
( ( abs_state @ ( rep_state @ X ) )
= X ) ).
% Rep_state_inverse
thf(fact_370_greater__mask__properties_I1_J,axiom,
! [Pi4: produc1457211279475724562t_char > prat] : ( greate7680731506090375811t_char @ Pi4 @ Pi4 ) ).
% greater_mask_properties(1)
thf(fact_371_greater__mask__properties_I2_J,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( ( greate7680731506090375811t_char @ A4 @ B2 )
& ( greate7680731506090375811t_char @ B2 @ C ) )
=> ( greate7680731506090375811t_char @ A4 @ C ) ) ).
% greater_mask_properties(2)
thf(fact_372_greater__mask__properties_I3_J,axiom,
! [Pi5: produc1457211279475724562t_char > prat,Pi4: produc1457211279475724562t_char > prat] :
( ( ( greate7680731506090375811t_char @ Pi5 @ Pi4 )
& ( greate7680731506090375811t_char @ Pi4 @ Pi5 ) )
=> ( Pi4 = Pi5 ) ) ).
% greater_mask_properties(3)
thf(fact_373_greater__mask__equiv__def,axiom,
( greate7680731506090375811t_char
= ( ^ [Pi6: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat] :
! [Hl2: produc1457211279475724562t_char] : ( pgte @ ( Pi6 @ Hl2 ) @ ( Pi3 @ Hl2 ) ) ) ) ).
% greater_mask_equiv_def
thf(fact_374_greater__maskI,axiom,
! [Pi5: produc1457211279475724562t_char > prat,Pi4: produc1457211279475724562t_char > prat] :
( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ ( Pi5 @ Hl3 ) @ ( Pi4 @ Hl3 ) )
=> ( greate7680731506090375811t_char @ Pi5 @ Pi4 ) ) ).
% greater_maskI
thf(fact_375_binary__def,axiom,
( binary
= ( ^ [A2: set_state] :
! [X3: state] :
( ( member_state @ X3 @ A2 )
=> ( member_state @ ( abs_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ X3 ) ) @ ( get_h @ X3 ) ) ) @ A2 ) ) ) ) ).
% binary_def
thf(fact_376_multiply_Opelims,axiom,
! [X: prat,Xa: state,Y: state] :
( ( ( multiply @ X @ Xa )
= Y )
=> ( ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X @ Xa ) )
=> ~ ( ( Y
= ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X @ ( get_m @ Xa ) ) @ ( get_h @ Xa ) ) ) )
=> ~ ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X @ Xa ) ) ) ) ) ).
% multiply.pelims
thf(fact_377_dual__order_Orefl,axiom,
! [A4: set_state] : ( ord_le2494988322063910608_state @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_378_order__refl,axiom,
! [X: set_state] : ( ord_le2494988322063910608_state @ X @ X ) ).
% order_refl
thf(fact_379_non__scalable__instantiate,axiom,
! [W: state,A4: state] :
( ~ ( scalable @ W @ A4 )
=> ? [P4: prat] :
( ( ppos @ P4 )
& ( pgte @ pwrite @ P4 )
& ( defined @ A4 @ ( multiply @ P4 @ W ) ) ) ) ).
% non_scalable_instantiate
thf(fact_380_combinable__def,axiom,
( combinable
= ( ^ [P3: set_state] :
! [Alpha: prat,Beta: prat] :
( ( ( ppos @ Alpha )
& ( ppos @ Beta )
& ( pgte @ pwrite @ ( padd @ Alpha @ Beta ) ) )
=> ( ord_le2494988322063910608_state @ ( add_set @ ( multip8064567061438756306ertion @ Alpha @ P3 ) @ ( multip8064567061438756306ertion @ Beta @ P3 ) ) @ ( multip8064567061438756306ertion @ ( padd @ Alpha @ Beta ) @ P3 ) ) ) ) ) ).
% combinable_def
thf(fact_381_R__def,axiom,
( r
= ( ^ [A3: state,W2: state] : ( if_state @ ( scalable @ W2 @ A3 ) @ W2 @ ( abs_state @ ( produc74620467203698198on_val @ ( comp_min_mask @ ( get_m @ A3 ) @ ( get_m @ W2 ) ) @ ( get_h @ W2 ) ) ) ) ) ) ).
% R_def
thf(fact_382_accp__subset,axiom,
! [R1: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,R22: produc1317171644805857433on_val > produc1317171644805857433on_val > $o] :
( ( ord_le2319172193063290574_val_o @ R1 @ R22 )
=> ( ord_le8528861841627966596_val_o @ ( accp_P6797916352740768290on_val @ R22 ) @ ( accp_P6797916352740768290on_val @ R1 ) ) ) ).
% accp_subset
thf(fact_383_accp__subset,axiom,
! [R1: produc4844402304972151915_state > produc4844402304972151915_state > $o,R22: produc4844402304972151915_state > produc4844402304972151915_state > $o] :
( ( ord_le2599682012455951694tate_o @ R1 @ R22 )
=> ( ord_le5681876185599785522tate_o @ ( accp_P6233459008282737652_state @ R22 ) @ ( accp_P6233459008282737652_state @ R1 ) ) ) ).
% accp_subset
thf(fact_384_accp__subset,axiom,
! [R1: produc6251377953716854561_state > produc6251377953716854561_state > $o,R22: produc6251377953716854561_state > produc6251377953716854561_state > $o] :
( ( ord_le249087669307848398tate_o @ R1 @ R22 )
=> ( ord_le8699832056682887676tate_o @ ( accp_P6554740477754186410_state @ R22 ) @ ( accp_P6554740477754186410_state @ R1 ) ) ) ).
% accp_subset
thf(fact_385_accp__subset,axiom,
! [R1: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,R22: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
( ( ord_le616650860478780142_val_o @ R1 @ R22 )
=> ( ord_le6140313152057869779_val_o @ ( accp_P4405556307444930259on_val @ R22 ) @ ( accp_P4405556307444930259on_val @ R1 ) ) ) ).
% accp_subset
thf(fact_386_accp__subset,axiom,
! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R22: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
( ( ord_le6723339807950303054prat_o @ R1 @ R22 )
=> ( ord_le2830794348966088778prat_o @ ( accp_P4765339447491148764r_prat @ R22 ) @ ( accp_P4765339447491148764r_prat @ R1 ) ) ) ).
% accp_subset
thf(fact_387_multiply_Ocases,axiom,
! [X: produc4844402304972151915_state] :
~ ! [P4: prat,Phi4: state] :
( X
!= ( produc6205719116230455971_state @ P4 @ Phi4 ) ) ).
% multiply.cases
thf(fact_388_top__set__def,axiom,
( top_to174962084833059535_state
= ( collec906845016635587262_state @ top_to3660658859742706806tate_o ) ) ).
% top_set_def
thf(fact_389_top__set__def,axiom,
( top_to1561438921946393474on_val
= ( collec7475294502966882545on_val @ top_to9203468014822947715_val_o ) ) ).
% top_set_def
thf(fact_390_top__set__def,axiom,
( top_top_set_state
= ( collect_state @ top_top_state_o ) ) ).
% top_set_def
thf(fact_391_accp__subset__induct,axiom,
! [D: produc1317171644805857433on_val > $o,R3: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X: produc1317171644805857433on_val,P6: produc1317171644805857433on_val > $o] :
( ( ord_le8528861841627966596_val_o @ D @ ( accp_P6797916352740768290on_val @ R3 ) )
=> ( ! [X2: produc1317171644805857433on_val,Z2: produc1317171644805857433on_val] :
( ( D @ X2 )
=> ( ( R3 @ Z2 @ X2 )
=> ( D @ Z2 ) ) )
=> ( ( D @ X )
=> ( ! [X2: produc1317171644805857433on_val] :
( ( D @ X2 )
=> ( ! [Z4: produc1317171644805857433on_val] :
( ( R3 @ Z4 @ X2 )
=> ( P6 @ Z4 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_392_accp__subset__induct,axiom,
! [D: produc4844402304972151915_state > $o,R3: produc4844402304972151915_state > produc4844402304972151915_state > $o,X: produc4844402304972151915_state,P6: produc4844402304972151915_state > $o] :
( ( ord_le5681876185599785522tate_o @ D @ ( accp_P6233459008282737652_state @ R3 ) )
=> ( ! [X2: produc4844402304972151915_state,Z2: produc4844402304972151915_state] :
( ( D @ X2 )
=> ( ( R3 @ Z2 @ X2 )
=> ( D @ Z2 ) ) )
=> ( ( D @ X )
=> ( ! [X2: produc4844402304972151915_state] :
( ( D @ X2 )
=> ( ! [Z4: produc4844402304972151915_state] :
( ( R3 @ Z4 @ X2 )
=> ( P6 @ Z4 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_393_accp__subset__induct,axiom,
! [D: produc6251377953716854561_state > $o,R3: produc6251377953716854561_state > produc6251377953716854561_state > $o,X: produc6251377953716854561_state,P6: produc6251377953716854561_state > $o] :
( ( ord_le8699832056682887676tate_o @ D @ ( accp_P6554740477754186410_state @ R3 ) )
=> ( ! [X2: produc6251377953716854561_state,Z2: produc6251377953716854561_state] :
( ( D @ X2 )
=> ( ( R3 @ Z2 @ X2 )
=> ( D @ Z2 ) ) )
=> ( ( D @ X )
=> ( ! [X2: produc6251377953716854561_state] :
( ( D @ X2 )
=> ( ! [Z4: produc6251377953716854561_state] :
( ( R3 @ Z4 @ X2 )
=> ( P6 @ Z4 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_394_accp__subset__induct,axiom,
! [D: produc9003781588162623004on_val > $o,R3: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val,P6: produc9003781588162623004on_val > $o] :
( ( ord_le6140313152057869779_val_o @ D @ ( accp_P4405556307444930259on_val @ R3 ) )
=> ( ! [X2: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] :
( ( D @ X2 )
=> ( ( R3 @ Z2 @ X2 )
=> ( D @ Z2 ) ) )
=> ( ( D @ X )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( D @ X2 )
=> ( ! [Z4: produc9003781588162623004on_val] :
( ( R3 @ Z4 @ X2 )
=> ( P6 @ Z4 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_395_accp__subset__induct,axiom,
! [D: ( produc1457211279475724562t_char > prat ) > $o,R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,P6: ( produc1457211279475724562t_char > prat ) > $o] :
( ( ord_le2830794348966088778prat_o @ D @ ( accp_P4765339447491148764r_prat @ R3 ) )
=> ( ! [X2: produc1457211279475724562t_char > prat,Z2: produc1457211279475724562t_char > prat] :
( ( D @ X2 )
=> ( ( R3 @ Z2 @ X2 )
=> ( D @ Z2 ) ) )
=> ( ( D @ X )
=> ( ! [X2: produc1457211279475724562t_char > prat] :
( ( D @ X2 )
=> ( ! [Z4: produc1457211279475724562t_char > prat] :
( ( R3 @ Z4 @ X2 )
=> ( P6 @ Z4 ) )
=> ( P6 @ X2 ) ) )
=> ( P6 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_396_ppos__add,axiom,
! [A4: prat,B2: prat] :
( ( ppos @ A4 )
=> ( ppos @ ( padd @ A4 @ B2 ) ) ) ).
% ppos_add
thf(fact_397_pmult__ppos,axiom,
! [A4: prat,B2: prat] :
( ( ppos @ A4 )
=> ( ( ppos @ B2 )
=> ( ppos @ ( pmult @ A4 @ B2 ) ) ) ) ).
% pmult_ppos
thf(fact_398_R__smaller,axiom,
! [W: state,A4: state] : ( greater @ W @ ( r @ A4 @ W ) ) ).
% R_smaller
thf(fact_399_R__compatible__same,axiom,
! [A4: state,W: state] :
( ( defined @ A4 @ W )
=> ( ( r @ A4 @ W )
= W ) ) ).
% R_compatible_same
thf(fact_400_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_state,Z3: set_state] : ( Y6 = Z3 ) )
= ( ^ [X3: set_state,Y4: set_state] :
( ( ord_le2494988322063910608_state @ X3 @ Y4 )
& ( ord_le2494988322063910608_state @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_401_ord__eq__le__trans,axiom,
! [A4: set_state,B2: set_state,C: set_state] :
( ( A4 = B2 )
=> ( ( ord_le2494988322063910608_state @ B2 @ C )
=> ( ord_le2494988322063910608_state @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_402_ord__le__eq__trans,axiom,
! [A4: set_state,B2: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_le2494988322063910608_state @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_403_order__antisym,axiom,
! [X: set_state,Y: set_state] :
( ( ord_le2494988322063910608_state @ X @ Y )
=> ( ( ord_le2494988322063910608_state @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_404_order_Otrans,axiom,
! [A4: set_state,B2: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B2 )
=> ( ( ord_le2494988322063910608_state @ B2 @ C )
=> ( ord_le2494988322063910608_state @ A4 @ C ) ) ) ).
% order.trans
thf(fact_405_order__trans,axiom,
! [X: set_state,Y: set_state,Z: set_state] :
( ( ord_le2494988322063910608_state @ X @ Y )
=> ( ( ord_le2494988322063910608_state @ Y @ Z )
=> ( ord_le2494988322063910608_state @ X @ Z ) ) ) ).
% order_trans
thf(fact_406_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_state,Z3: set_state] : ( Y6 = Z3 ) )
= ( ^ [A3: set_state,B: set_state] :
( ( ord_le2494988322063910608_state @ B @ A3 )
& ( ord_le2494988322063910608_state @ A3 @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_407_dual__order_Oantisym,axiom,
! [B2: set_state,A4: set_state] :
( ( ord_le2494988322063910608_state @ B2 @ A4 )
=> ( ( ord_le2494988322063910608_state @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_408_dual__order_Otrans,axiom,
! [B2: set_state,A4: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ B2 @ A4 )
=> ( ( ord_le2494988322063910608_state @ C @ B2 )
=> ( ord_le2494988322063910608_state @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_409_antisym,axiom,
! [A4: set_state,B2: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B2 )
=> ( ( ord_le2494988322063910608_state @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_410_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_state,Z3: set_state] : ( Y6 = Z3 ) )
= ( ^ [A3: set_state,B: set_state] :
( ( ord_le2494988322063910608_state @ A3 @ B )
& ( ord_le2494988322063910608_state @ B @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_411_order__subst1,axiom,
! [A4: set_state,F: set_state > set_state,B2: set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ ( F @ B2 ) )
=> ( ( ord_le2494988322063910608_state @ B2 @ C )
=> ( ! [X2: set_state,Y5: set_state] :
( ( ord_le2494988322063910608_state @ X2 @ Y5 )
=> ( ord_le2494988322063910608_state @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le2494988322063910608_state @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_412_order__subst2,axiom,
! [A4: set_state,B2: set_state,F: set_state > set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B2 )
=> ( ( ord_le2494988322063910608_state @ ( F @ B2 ) @ C )
=> ( ! [X2: set_state,Y5: set_state] :
( ( ord_le2494988322063910608_state @ X2 @ Y5 )
=> ( ord_le2494988322063910608_state @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le2494988322063910608_state @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_413_order__eq__refl,axiom,
! [X: set_state,Y: set_state] :
( ( X = Y )
=> ( ord_le2494988322063910608_state @ X @ Y ) ) ).
% order_eq_refl
thf(fact_414_ord__eq__le__subst,axiom,
! [A4: set_state,F: set_state > set_state,B2: set_state,C: set_state] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_le2494988322063910608_state @ B2 @ C )
=> ( ! [X2: set_state,Y5: set_state] :
( ( ord_le2494988322063910608_state @ X2 @ Y5 )
=> ( ord_le2494988322063910608_state @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le2494988322063910608_state @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_415_ord__le__eq__subst,axiom,
! [A4: set_state,B2: set_state,F: set_state > set_state,C: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_state,Y5: set_state] :
( ( ord_le2494988322063910608_state @ X2 @ Y5 )
=> ( ord_le2494988322063910608_state @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le2494988322063910608_state @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_416_order__antisym__conv,axiom,
! [Y: set_state,X: set_state] :
( ( ord_le2494988322063910608_state @ Y @ X )
=> ( ( ord_le2494988322063910608_state @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_417_non__scalable__R__charact,axiom,
! [W: state,A4: state] :
( ~ ( scalable @ W @ A4 )
=> ( ( ( get_m @ ( r @ A4 @ W ) )
= ( comp_min_mask @ ( get_m @ A4 ) @ ( get_m @ W ) ) )
& ( ( get_h @ ( r @ A4 @ W ) )
= ( get_h @ W ) ) ) ) ).
% non_scalable_R_charact
thf(fact_418_top__greatest,axiom,
! [A4: set_state] : ( ord_le2494988322063910608_state @ A4 @ top_top_set_state ) ).
% top_greatest
thf(fact_419_top_Oextremum__unique,axiom,
! [A4: set_state] :
( ( ord_le2494988322063910608_state @ top_top_set_state @ A4 )
= ( A4 = top_top_set_state ) ) ).
% top.extremum_unique
thf(fact_420_top_Oextremum__uniqueI,axiom,
! [A4: set_state] :
( ( ord_le2494988322063910608_state @ top_top_set_state @ A4 )
=> ( A4 = top_top_set_state ) ) ).
% top.extremum_uniqueI
thf(fact_421_pinv__pmult__ok,axiom,
! [P: prat] :
( ( ppos @ P )
=> ( ( pmult @ P @ ( pinv @ P ) )
= pwrite ) ) ).
% pinv_pmult_ok
thf(fact_422_subrelI,axiom,
! [R: set_Pr4231240917283394591_state,S: set_Pr4231240917283394591_state] :
( ! [X2: state,Y5: produc3882902684156697774_state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X2 @ Y5 ) @ R )
=> ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X2 @ Y5 ) @ S ) )
=> ( ord_le4032880777312351359_state @ R @ S ) ) ).
% subrelI
thf(fact_423_subrelI,axiom,
! [R: set_Pr2391701600042252046_state,S: set_Pr2391701600042252046_state] :
( ! [X2: state,Y5: state > state] :
( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X2 @ Y5 ) @ R )
=> ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X2 @ Y5 ) @ S ) )
=> ( ord_le6483787752022044334_state @ R @ S ) ) ).
% subrelI
thf(fact_424_subrelI,axiom,
! [R: set_Pr1688445902015331925_state,S: set_Pr1688445902015331925_state] :
( ! [X2: option_state,Y5: option_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X2 @ Y5 ) @ R )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X2 @ Y5 ) @ S ) )
=> ( ord_le6423325748750870005_state @ R @ S ) ) ).
% subrelI
thf(fact_425_subrelI,axiom,
! [R: set_Pr6039890582625207499_state,S: set_Pr6039890582625207499_state] :
( ! [X2: prat,Y5: state] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ R )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ S ) )
=> ( ord_le262197048489888875_state @ R @ S ) ) ).
% subrelI
thf(fact_426_subrelI,axiom,
! [R: set_Pr6847263425858009217_state,S: set_Pr6847263425858009217_state] :
( ! [X2: prat,Y5: set_state] :
( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X2 @ Y5 ) @ R )
=> ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X2 @ Y5 ) @ S ) )
=> ( ord_le5761494056910127137_state @ R @ S ) ) ).
% subrelI
thf(fact_427_subrelI,axiom,
! [R: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
( ! [X2: nat,Y5: list_char] :
( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ R )
=> ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ S ) )
=> ( ord_le6315511645215477266t_char @ R @ S ) ) ).
% subrelI
thf(fact_428_subrelI,axiom,
! [R: set_Pr7649655576376376530on_val,S: set_Pr7649655576376376530on_val] :
( ! [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] :
( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R )
=> ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ S ) )
=> ( ord_le6900622908513426738on_val @ R @ S ) ) ).
% subrelI
thf(fact_429_all__subset__image,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state,P6: set_Pr7649655576376376530on_val > $o] :
( ( ! [B6: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B6 @ ( image_4690842878458256517on_val @ F @ A ) )
=> ( P6 @ B6 ) ) )
= ( ! [B6: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ B6 @ A )
=> ( P6 @ ( image_4690842878458256517on_val @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_430_all__subset__image,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val,P6: set_Pr4231240917283394591_state > $o] :
( ( ! [B6: set_Pr4231240917283394591_state] :
( ( ord_le4032880777312351359_state @ B6 @ ( image_3204238854701872908_state @ F @ A ) )
=> ( P6 @ B6 ) ) )
= ( ! [B6: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B6 @ A )
=> ( P6 @ ( image_3204238854701872908_state @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_431_all__subset__image,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val,P6: set_Pr2391701600042252046_state > $o] :
( ( ! [B6: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ B6 @ ( image_4494208899119068747_state @ F @ A ) )
=> ( P6 @ B6 ) ) )
= ( ! [B6: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B6 @ A )
=> ( P6 @ ( image_4494208899119068747_state @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_432_all__subset__image,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,P6: set_Pr7649655576376376530on_val > $o] :
( ( ! [B6: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B6 @ ( image_4170889944715819327on_val @ F @ A ) )
=> ( P6 @ B6 ) ) )
= ( ! [B6: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B6 @ A )
=> ( P6 @ ( image_4170889944715819327on_val @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_433_all__subset__image,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,P6: set_state > $o] :
( ( ! [B6: set_state] :
( ( ord_le2494988322063910608_state @ B6 @ ( image_1205529093420515805_state @ F @ A ) )
=> ( P6 @ B6 ) ) )
= ( ! [B6: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ B6 @ A )
=> ( P6 @ ( image_1205529093420515805_state @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_434_all__subset__image,axiom,
! [F: state > state,A: set_state,P6: set_state > $o] :
( ( ! [B6: set_state] :
( ( ord_le2494988322063910608_state @ B6 @ ( image_state_state @ F @ A ) )
=> ( P6 @ B6 ) ) )
= ( ! [B6: set_state] :
( ( ord_le2494988322063910608_state @ B6 @ A )
=> ( P6 @ ( image_state_state @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_435_sum__coeff,axiom,
! [A4: prat,B2: prat] :
( ( ppos @ A4 )
=> ( ( ppos @ B2 )
=> ( ( padd @ ( pdiv @ A4 @ ( padd @ A4 @ B2 ) ) @ ( pdiv @ B2 @ ( padd @ A4 @ B2 ) ) )
= pwrite ) ) ) ).
% sum_coeff
thf(fact_436_surjD,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
( ( ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state )
= top_to1561438921946393474on_val )
=> ? [X2: produc3882902684156697774_state] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_437_surjD,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,Y: produc7272280185086996969_state] :
( ( ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val )
= top_to174962084833059535_state )
=> ? [X2: produc9003781588162623004on_val] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_438_surjD,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,Y: produc3882902684156697774_state] :
( ( ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val )
= top_to1744471583079920734_state )
=> ? [X2: produc9003781588162623004on_val] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_439_surjD,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
( ( ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val )
= top_to1561438921946393474on_val )
=> ? [X2: produc9003781588162623004on_val] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_440_surjD,axiom,
! [F: produc9003781588162623004on_val > state,Y: state] :
( ( ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val )
= top_top_set_state )
=> ? [X2: produc9003781588162623004on_val] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_441_surjD,axiom,
! [F: state > state,Y: state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ? [X2: state] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_442_surjE,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
( ( ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state )
= top_to1561438921946393474on_val )
=> ~ ! [X2: produc3882902684156697774_state] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_443_surjE,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,Y: produc7272280185086996969_state] :
( ( ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val )
= top_to174962084833059535_state )
=> ~ ! [X2: produc9003781588162623004on_val] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_444_surjE,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,Y: produc3882902684156697774_state] :
( ( ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val )
= top_to1744471583079920734_state )
=> ~ ! [X2: produc9003781588162623004on_val] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_445_surjE,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
( ( ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val )
= top_to1561438921946393474on_val )
=> ~ ! [X2: produc9003781588162623004on_val] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_446_surjE,axiom,
! [F: produc9003781588162623004on_val > state,Y: state] :
( ( ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val )
= top_top_set_state )
=> ~ ! [X2: produc9003781588162623004on_val] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_447_surjE,axiom,
! [F: state > state,Y: state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ~ ! [X2: state] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_448_surjI,axiom,
! [G: produc3882902684156697774_state > produc9003781588162623004on_val,F: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ! [X2: produc9003781588162623004on_val] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_4690842878458256517on_val @ G @ top_to1744471583079920734_state )
= top_to1561438921946393474on_val ) ) ).
% surjI
thf(fact_449_surjI,axiom,
! [G: produc9003781588162623004on_val > produc7272280185086996969_state,F: produc7272280185086996969_state > produc9003781588162623004on_val] :
( ! [X2: produc7272280185086996969_state] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_3204238854701872908_state @ G @ top_to1561438921946393474on_val )
= top_to174962084833059535_state ) ) ).
% surjI
thf(fact_450_surjI,axiom,
! [G: produc9003781588162623004on_val > produc3882902684156697774_state,F: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ! [X2: produc3882902684156697774_state] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_4494208899119068747_state @ G @ top_to1561438921946393474on_val )
= top_to1744471583079920734_state ) ) ).
% surjI
thf(fact_451_surjI,axiom,
! [G: produc9003781588162623004on_val > produc9003781588162623004on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
( ! [X2: produc9003781588162623004on_val] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_4170889944715819327on_val @ G @ top_to1561438921946393474on_val )
= top_to1561438921946393474on_val ) ) ).
% surjI
thf(fact_452_surjI,axiom,
! [G: produc9003781588162623004on_val > state,F: state > produc9003781588162623004on_val] :
( ! [X2: state] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_1205529093420515805_state @ G @ top_to1561438921946393474on_val )
= top_top_set_state ) ) ).
% surjI
thf(fact_453_surjI,axiom,
! [G: state > state,F: state > state] :
( ! [X2: state] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_state_state @ G @ top_top_set_state )
= top_top_set_state ) ) ).
% surjI
thf(fact_454_surj__def,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state )
= top_to1561438921946393474on_val )
= ( ! [Y4: produc9003781588162623004on_val] :
? [X3: produc3882902684156697774_state] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_455_surj__def,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state] :
( ( ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val )
= top_to174962084833059535_state )
= ( ! [Y4: produc7272280185086996969_state] :
? [X3: produc9003781588162623004on_val] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_456_surj__def,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val )
= top_to1744471583079920734_state )
= ( ! [Y4: produc3882902684156697774_state] :
? [X3: produc9003781588162623004on_val] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_457_surj__def,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
( ( ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val )
= top_to1561438921946393474on_val )
= ( ! [Y4: produc9003781588162623004on_val] :
? [X3: produc9003781588162623004on_val] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_458_surj__def,axiom,
! [F: produc9003781588162623004on_val > state] :
( ( ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val )
= top_top_set_state )
= ( ! [Y4: state] :
? [X3: produc9003781588162623004on_val] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_459_surj__def,axiom,
! [F: state > state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
= ( ! [Y4: state] :
? [X3: state] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_460_pinv__def,axiom,
( pinv
= ( pdiv @ pwrite ) ) ).
% pinv_def
thf(fact_461_pinv__pwrite,axiom,
( ( pinv @ pwrite )
= pwrite ) ).
% pinv_pwrite
thf(fact_462_ppos__inv,axiom,
! [P: prat] :
( ( ppos @ P )
=> ( ppos @ ( pinv @ P ) ) ) ).
% ppos_inv
thf(fact_463_top__empty__eq,axiom,
( top_to3660658859742706806tate_o
= ( ^ [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ top_to174962084833059535_state ) ) ) ).
% top_empty_eq
thf(fact_464_top__empty__eq,axiom,
( top_to9203468014822947715_val_o
= ( ^ [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ top_to1561438921946393474on_val ) ) ) ).
% top_empty_eq
thf(fact_465_top__empty__eq,axiom,
( top_to5055921138486834175tate_o
= ( ^ [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ top_to1744471583079920734_state ) ) ) ).
% top_empty_eq
thf(fact_466_top__empty__eq,axiom,
( top_top_state_o
= ( ^ [X3: state] : ( member_state @ X3 @ top_top_set_state ) ) ) ).
% top_empty_eq
thf(fact_467_pmult__pdiv__cancel,axiom,
! [A4: prat,X: prat] :
( ( ppos @ A4 )
=> ( ( pmult @ A4 @ ( pdiv @ X @ A4 ) )
= X ) ) ).
% pmult_pdiv_cancel
thf(fact_468_pinv__inverts,axiom,
! [A4: prat,B2: prat] :
( ( pgte @ A4 @ B2 )
=> ( ( ppos @ B2 )
=> ( pgte @ ( pinv @ B2 ) @ ( pinv @ A4 ) ) ) ) ).
% pinv_inverts
thf(fact_469_pdiv__smaller,axiom,
! [A4: prat,B2: prat] :
( ( pgte @ A4 @ B2 )
=> ( ( ppos @ A4 )
=> ( pgte @ pwrite @ ( pdiv @ B2 @ A4 ) ) ) ) ).
% pdiv_smaller
thf(fact_470_pinv__double__half,axiom,
! [P: prat] :
( ( ppos @ P )
=> ( ( pmult @ half @ ( pinv @ P ) )
= ( pinv @ ( padd @ P @ P ) ) ) ) ).
% pinv_double_half
thf(fact_471_comp__min__mask__def,axiom,
( comp_min_mask
= ( ^ [B: produc1457211279475724562t_char > prat,A3: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( pmin @ ( A3 @ Hl2 ) @ ( comp_one @ ( B @ Hl2 ) ) ) ) ) ).
% comp_min_mask_def
thf(fact_472_greater__mask__uni__equiv,axiom,
! [Pi4: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char,R: prat] :
( ( greate7680731506090375811t_char @ Pi4 @ ( uni_ma6578582744724537101t_char @ Hl @ R ) )
= ( pgte @ ( Pi4 @ Hl ) @ R ) ) ).
% greater_mask_uni_equiv
thf(fact_473_greater__mask__uniI,axiom,
! [Pi4: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char,R: prat] :
( ( pgte @ ( Pi4 @ Hl ) @ R )
=> ( greate7680731506090375811t_char @ Pi4 @ ( uni_ma6578582744724537101t_char @ Hl @ R ) ) ) ).
% greater_mask_uniI
thf(fact_474_half__plus__half,axiom,
( ( padd @ half @ half )
= pwrite ) ).
% half_plus_half
thf(fact_475_in__lex__prod,axiom,
! [A4: option_state,B2: option_state,A5: option_state,B3: option_state,R: set_Pr1688445902015331925_state,S: set_Pr1688445902015331925_state] :
( ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ A4 @ B2 ) @ ( produc9160152616361873709_state @ A5 @ B3 ) ) @ ( lex_pr8336311929016760645_state @ R @ S ) )
= ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A4 @ A5 ) @ R )
| ( ( A4 = A5 )
& ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_476_in__lex__prod,axiom,
! [A4: prat,B2: state,A5: prat,B3: state,R: set_Pr3298805672376727137t_prat,S: set_Pr1785066336555260981_state] :
( ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ A4 @ B2 ) @ ( produc6205719116230455971_state @ A5 @ B3 ) ) @ ( lex_prod_prat_state @ R @ S ) )
= ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A4 @ A5 ) @ R )
| ( ( A4 = A5 )
& ( member753036827967488894_state @ ( produc544653723139640077_state @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_477_in__lex__prod,axiom,
! [A4: prat,B2: set_state,A5: prat,B3: set_state,R: set_Pr3298805672376727137t_prat,S: set_Pr5618112136499017889_state] :
( ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ A4 @ B2 ) @ ( produc9080284787853255769_state @ A5 @ B3 ) ) @ ( lex_pr1787351029354341489_state @ R @ S ) )
= ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A4 @ A5 ) @ R )
| ( ( A4 = A5 )
& ( member1976142906092959210_state @ ( produc1834047914073684089_state @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_478_in__lex__prod,axiom,
! [A4: nat,B2: list_char,A5: nat,B3: list_char,R: set_Pr1261947904930325089at_nat,S: set_Pr4480099295215233283t_char] :
( ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ A4 @ B2 ) @ ( produc120197956887798346t_char @ A5 @ B3 ) ) @ ( lex_pr1681056561245511266t_char @ R @ S ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A5 ) @ R )
| ( ( A4 = A5 )
& ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_479_in__lex__prod,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val,A5: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr4698884584343881589on_val] :
( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A4 @ B2 ) @ ( produc74620467203698198on_val @ A5 @ B3 ) ) @ ( lex_pr8780192377291449342on_val @ R @ S ) )
= ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A4 @ A5 ) @ R )
| ( ( A4 = A5 )
& ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_480_pmin__comm,axiom,
( pmin
= ( ^ [A3: prat,B: prat] : ( pmin @ B @ A3 ) ) ) ).
% pmin_comm
thf(fact_481_pmin__is,axiom,
! [A4: prat,B2: prat] :
( ( pgte @ A4 @ B2 )
=> ( ( pmin @ A4 @ B2 )
= B2 ) ) ).
% pmin_is
thf(fact_482_pmin__greater,axiom,
! [A4: prat,B2: prat] : ( pgte @ A4 @ ( pmin @ A4 @ B2 ) ) ).
% pmin_greater
thf(fact_483_half__between__0__1,axiom,
( ( ppos @ half )
& ( pgt @ pwrite @ half ) ) ).
% half_between_0_1
thf(fact_484_pmin__pmax,axiom,
! [X: prat,A4: prat,B2: prat] :
( ( pgte @ X @ ( pmin @ A4 @ B2 ) )
=> ( X
= ( pmin @ ( pmax @ X @ A4 ) @ ( pmax @ X @ B2 ) ) ) ) ).
% pmin_pmax
thf(fact_485_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_486_same__fstI,axiom,
! [P6: prat > $o,X: prat,Y7: state,Y: state,R3: prat > set_Pr1785066336555260981_state] :
( ( P6 @ X )
=> ( ( member753036827967488894_state @ ( produc544653723139640077_state @ Y7 @ Y ) @ ( R3 @ X ) )
=> ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ X @ Y7 ) @ ( produc6205719116230455971_state @ X @ Y ) ) @ ( same_fst_prat_state @ P6 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_487_same__fstI,axiom,
! [P6: prat > $o,X: prat,Y7: set_state,Y: set_state,R3: prat > set_Pr5618112136499017889_state] :
( ( P6 @ X )
=> ( ( member1976142906092959210_state @ ( produc1834047914073684089_state @ Y7 @ Y ) @ ( R3 @ X ) )
=> ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ X @ Y7 ) @ ( produc9080284787853255769_state @ X @ Y ) ) @ ( same_f5333510722715586990_state @ P6 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_488_same__fstI,axiom,
! [P6: nat > $o,X: nat,Y7: list_char,Y: list_char,R3: nat > set_Pr4480099295215233283t_char] :
( ( P6 @ X )
=> ( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ Y7 @ Y ) @ ( R3 @ X ) )
=> ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ X @ Y7 ) @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( same_f652031872680891423t_char @ P6 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_489_same__fstI,axiom,
! [P6: ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val,R3: ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val] :
( ( P6 @ X )
=> ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y7 @ Y ) @ ( R3 @ X ) )
=> ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ X @ Y7 ) @ ( produc74620467203698198on_val @ X @ Y ) ) @ ( same_f8138205353876899841on_val @ P6 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_490_same__fstI,axiom,
! [P6: option_state > $o,X: option_state,Y7: option_state,Y: option_state,R3: option_state > set_Pr1688445902015331925_state] :
( ( P6 @ X )
=> ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Y7 @ Y ) @ ( R3 @ X ) )
=> ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ X @ Y7 ) @ ( produc9160152616361873709_state @ X @ Y ) ) @ ( same_f2704544105346032002_state @ P6 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_491_pgte__pgt,axiom,
! [A4: prat,B2: prat,C: prat,D2: prat] :
( ( pgt @ A4 @ B2 )
=> ( ( pgte @ C @ D2 )
=> ( pgt @ ( padd @ A4 @ C ) @ ( padd @ B2 @ D2 ) ) ) ) ).
% pgte_pgt
thf(fact_492_upper__bounded__smaller,axiom,
! [Pi4: produc1457211279475724562t_char > prat,A4: prat,P: prat] :
( ( upper_bounded @ Pi4 @ A4 )
=> ( upper_bounded @ ( multiply_mask @ P @ Pi4 ) @ ( pmult @ P @ A4 ) ) ) ).
% upper_bounded_smaller
thf(fact_493_pmax__comm,axiom,
( pmax
= ( ^ [A3: prat,B: prat] : ( pmax @ B @ A3 ) ) ) ).
% pmax_comm
thf(fact_494_padd__zero,axiom,
! [A4: prat,B2: prat] :
( ( pnone
= ( padd @ A4 @ B2 ) )
= ( ( A4 = pnone )
& ( B2 = pnone ) ) ) ).
% padd_zero
thf(fact_495_ppos__eq__pnone,axiom,
( ppos
= ( ^ [P2: prat] : ( P2 != pnone ) ) ) ).
% ppos_eq_pnone
thf(fact_496_pmult__special_I2_J,axiom,
! [X: prat] :
( ( pmult @ pnone @ X )
= pnone ) ).
% pmult_special(2)
thf(fact_497_upper__bounded__bigger,axiom,
! [Pi4: produc1457211279475724562t_char > prat,A4: prat,B2: prat] :
( ( upper_bounded @ Pi4 @ A4 )
=> ( ( pgte @ B2 @ A4 )
=> ( upper_bounded @ Pi4 @ B2 ) ) ) ).
% upper_bounded_bigger
thf(fact_498_upper__bounded__def,axiom,
( upper_bounded
= ( ^ [Pi3: produc1457211279475724562t_char > prat,P2: prat] :
! [Hl2: produc1457211279475724562t_char] : ( pgte @ P2 @ ( Pi3 @ Hl2 ) ) ) ) ).
% upper_bounded_def
thf(fact_499_upper__boundedI,axiom,
! [P: prat,Pi4: produc1457211279475724562t_char > prat] :
( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ P @ ( Pi4 @ Hl3 ) )
=> ( upper_bounded @ Pi4 @ P ) ) ).
% upper_boundedI
thf(fact_500_pmax__is__smaller,axiom,
! [X: prat,A4: prat,B2: prat] :
( ( pgte @ X @ A4 )
=> ( ( pgte @ X @ B2 )
=> ( pgte @ X @ ( pmax @ A4 @ B2 ) ) ) ) ).
% pmax_is_smaller
thf(fact_501_pmax__smaller,axiom,
! [A4: prat,B2: prat] : ( pgte @ ( pmax @ A4 @ B2 ) @ A4 ) ).
% pmax_smaller
thf(fact_502_pmax__is,axiom,
! [A4: prat,B2: prat] :
( ( pgte @ A4 @ B2 )
=> ( ( pmax @ A4 @ B2 )
= A4 ) ) ).
% pmax_is
thf(fact_503_pgt__implies__pgte,axiom,
! [A4: prat,B2: prat] :
( ( pgt @ A4 @ B2 )
=> ( pgte @ A4 @ B2 ) ) ).
% pgt_implies_pgte
thf(fact_504_not__pgte__charact,axiom,
! [A4: prat,B2: prat] :
( ( ~ ( pgte @ A4 @ B2 ) )
= ( pgt @ B2 @ A4 ) ) ).
% not_pgte_charact
thf(fact_505_valid__multiply,axiom,
! [A4: produc1457211279475724562t_char > prat,Ma: prat,P: prat] :
( ( valid_mask @ A4 )
=> ( ( upper_bounded @ A4 @ Ma )
=> ( ( pgte @ pwrite @ ( pmult @ Ma @ P ) )
=> ( valid_mask @ ( multiply_mask @ P @ A4 ) ) ) ) ) ).
% valid_multiply
thf(fact_506_valid__mult,axiom,
! [Pi4: produc1457211279475724562t_char > prat,P: prat] :
( ( valid_mask @ Pi4 )
=> ( ( pgte @ pwrite @ P )
=> ( valid_mask @ ( multiply_mask @ P @ Pi4 ) ) ) ) ).
% valid_mult
thf(fact_507_add__masks_Opelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) )
=> ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) ) ) ) ) ).
% add_masks.pelims
thf(fact_508_relChain__def,axiom,
( bNF_Ca2815325664443363616_state
= ( ^ [R2: set_Pr1688445902015331925_state,As: option_state > set_state] :
! [I2: option_state,J: option_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I2 @ J ) @ R2 )
=> ( ord_le2494988322063910608_state @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_509_image__Fpow__mono,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state,B5: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ ( image_4690842878458256517on_val @ F @ A ) @ B5 )
=> ( ord_le5660975535469346024on_val @ ( image_1939478397035811227on_val @ ( image_4690842878458256517on_val @ F ) @ ( finite6637889483708283156_state @ A ) ) @ ( finite7298901795774724150on_val @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_510_image__Fpow__mono,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val,B5: set_Pr4231240917283394591_state] :
( ( ord_le4032880777312351359_state @ ( image_3204238854701872908_state @ F @ A ) @ B5 )
=> ( ord_le8498320074568675253_state @ ( image_4960930050028392952_state @ ( image_3204238854701872908_state @ F ) @ ( finite7298901795774724150on_val @ A ) ) @ ( finite9006883799802114051_state @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_511_image__Fpow__mono,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val,B5: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ A ) @ B5 )
=> ( ord_le1960757061454972174_state @ ( image_2705476029607989621_state @ ( image_4494208899119068747_state @ F ) @ ( finite7298901795774724150on_val @ A ) ) @ ( finite6637889483708283156_state @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_512_image__Fpow__mono,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ ( image_4170889944715819327on_val @ F @ A ) @ B5 )
=> ( ord_le5660975535469346024on_val @ ( image_451893177406657707on_val @ ( image_4170889944715819327on_val @ F ) @ ( finite7298901795774724150on_val @ A ) ) @ ( finite7298901795774724150on_val @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_513_image__Fpow__mono,axiom,
! [F: state > state,A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A ) @ B5 )
=> ( ord_le5175021213330142598_state @ ( image_2476256681063834599_state @ ( image_state_state @ F ) @ ( finite_Fpow_state @ A ) ) @ ( finite_Fpow_state @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_514_image__Fpow__mono,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,B5: set_state] :
( ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ A ) @ B5 )
=> ( ord_le5175021213330142598_state @ ( image_8347066812701616201_state @ ( image_1205529093420515805_state @ F ) @ ( finite7298901795774724150on_val @ A ) ) @ ( finite_Fpow_state @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_515_compatible__same__mask,axiom,
! [A4: produc1457211279475724562t_char > prat,W: produc1457211279475724562t_char > prat] :
( ( valid_mask @ ( add_ma6586698345353345811t_char @ A4 @ W ) )
=> ( W
= ( comp_min_mask @ A4 @ W ) ) ) ).
% compatible_same_mask
thf(fact_516_mult__distr__masks,axiom,
! [A4: prat,B2: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( multiply_mask @ A4 @ ( add_ma6586698345353345811t_char @ B2 @ C ) )
= ( add_ma6586698345353345811t_char @ ( multiply_mask @ A4 @ B2 ) @ ( multiply_mask @ A4 @ C ) ) ) ).
% mult_distr_masks
thf(fact_517_add__masks_Oelims,axiom,
! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) ) ) ).
% add_masks.elims
thf(fact_518_add__masks_Osimps,axiom,
( add_ma6586698345353345811t_char
= ( ^ [Pi6: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( padd @ ( Pi6 @ Hl2 ) @ ( Pi3 @ Hl2 ) ) ) ) ).
% add_masks.simps
thf(fact_519_greater__mask__def,axiom,
( greate7680731506090375811t_char
= ( ^ [Pi6: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat] :
? [R2: produc1457211279475724562t_char > prat] :
( Pi6
= ( add_ma6586698345353345811t_char @ Pi3 @ R2 ) ) ) ) ).
% greater_mask_def
thf(fact_520_greater__mask__decomp,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
( ( greate7680731506090375811t_char @ A4 @ ( add_ma6586698345353345811t_char @ B2 @ C ) )
=> ? [A1: produc1457211279475724562t_char > prat,A22: produc1457211279475724562t_char > prat] :
( ( A4
= ( add_ma6586698345353345811t_char @ A1 @ A22 ) )
& ( greate7680731506090375811t_char @ A1 @ B2 )
& ( greate7680731506090375811t_char @ A22 @ C ) ) ) ).
% greater_mask_decomp
thf(fact_521_const__sum__mask__greater,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D2: produc1457211279475724562t_char > prat] :
( ( ( add_ma6586698345353345811t_char @ A4 @ B2 )
= ( add_ma6586698345353345811t_char @ C @ D2 ) )
=> ( ( greate7680731506090375811t_char @ A4 @ C )
=> ( greate7680731506090375811t_char @ D2 @ B2 ) ) ) ).
% const_sum_mask_greater
thf(fact_522_Fpow__mono,axiom,
! [A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ord_le5175021213330142598_state @ ( finite_Fpow_state @ A ) @ ( finite_Fpow_state @ B5 ) ) ) ).
% Fpow_mono
thf(fact_523_valid__sum_I1_J,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat,Ma: prat,Mb: prat] :
( ( valid_mask @ A4 )
=> ( ( valid_mask @ B2 )
=> ( ( upper_bounded @ A4 @ Ma )
=> ( ( upper_bounded @ B2 @ Mb )
=> ( ( pgte @ pwrite @ ( padd @ Ma @ Mb ) )
=> ( valid_mask @ ( add_ma6586698345353345811t_char @ A4 @ B2 ) ) ) ) ) ) ) ).
% valid_sum(1)
thf(fact_524_valid__sum_I2_J,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat,Ma: prat,Mb: prat] :
( ( valid_mask @ A4 )
=> ( ( valid_mask @ B2 )
=> ( ( upper_bounded @ A4 @ Ma )
=> ( ( upper_bounded @ B2 @ Mb )
=> ( ( pgte @ pwrite @ ( padd @ Ma @ Mb ) )
=> ( upper_bounded @ ( add_ma6586698345353345811t_char @ A4 @ B2 ) @ ( padd @ Ma @ Mb ) ) ) ) ) ) ) ).
% valid_sum(2)
thf(fact_525_valid__state__decompose,axiom,
! [A4: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
( ( valid_state @ ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ A4 @ B2 ) @ H ) )
=> ( valid_state @ ( produc74620467203698198on_val @ A4 @ H ) ) ) ).
% valid_state_decompose
thf(fact_526_mult__add__states,axiom,
! [A4: prat,B2: prat,Pi4: produc1457211279475724562t_char > prat] :
( ( multiply_mask @ ( padd @ A4 @ B2 ) @ Pi4 )
= ( add_ma6586698345353345811t_char @ ( multiply_mask @ A4 @ Pi4 ) @ ( multiply_mask @ B2 @ Pi4 ) ) ) ).
% mult_add_states
thf(fact_527_valid__larger__mask,axiom,
( valid_mask
= ( greate7680731506090375811t_char @ full_mask ) ) ).
% valid_larger_mask
thf(fact_528_valid__mask_Oelims_I3_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ~ ( valid_mask @ X )
=> ~ ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
& ! [F2: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
= pnone ) ) ) ).
% valid_mask.elims(3)
thf(fact_529_valid__mask_Oelims_I2_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ( valid_mask @ X )
=> ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl4 ) )
& ! [F3: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
= pnone ) ) ) ).
% valid_mask.elims(2)
thf(fact_530_valid__mask_Oelims_I1_J,axiom,
! [X: produc1457211279475724562t_char > prat,Y: $o] :
( ( ( valid_mask @ X )
= Y )
=> ( Y
= ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
& ! [F4: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
= pnone ) ) ) ) ).
% valid_mask.elims(1)
thf(fact_531_valid__mask_Osimps,axiom,
( valid_mask
= ( ^ [Pi3: produc1457211279475724562t_char > prat] :
( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi3 @ Hl2 ) )
& ! [F4: list_char] :
( ( Pi3 @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
= pnone ) ) ) ) ).
% valid_mask.simps
thf(fact_532_valid__maskI,axiom,
! [Pi4: produc1457211279475724562t_char > prat] :
( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi4 @ Hl3 ) )
=> ( ! [F2: list_char] :
( ( Pi4 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
= pnone )
=> ( valid_mask @ Pi4 ) ) ) ).
% valid_maskI
thf(fact_533_compatibleI,axiom,
! [A4: state,B2: state] :
( ( compatible_heaps @ ( get_h @ A4 ) @ ( get_h @ B2 ) )
=> ( ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ A4 ) @ ( get_m @ B2 ) ) )
=> ( defined @ A4 @ B2 ) ) ) ).
% compatibleI
thf(fact_534_larger__heap__comp,axiom,
! [A4: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A4 @ B2 )
=> ( ( compatible_heaps @ A4 @ C )
=> ( compatible_heaps @ B2 @ C ) ) ) ).
% larger_heap_comp
thf(fact_535_larger__compatible__sum__heaps,axiom,
! [A4: produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A4 @ X )
=> ( ( larger1532547405891354081ar_val @ B2 @ Y )
=> ( ( compatible_heaps @ A4 @ B2 )
=> ( compatible_heaps @ X @ Y ) ) ) ) ).
% larger_compatible_sum_heaps
thf(fact_536_valid__mask_Opelims_I1_J,axiom,
! [X: produc1457211279475724562t_char > prat,Y: $o] :
( ( ( valid_mask @ X )
= Y )
=> ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ( Y
= ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
& ! [F4: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
= pnone ) ) )
=> ~ ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X ) ) ) ) ).
% valid_mask.pelims(1)
thf(fact_537_valid__mask_Opelims_I2_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ( valid_mask @ X )
=> ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl4 ) )
& ! [F3: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
= pnone ) ) ) ) ) ).
% valid_mask.pelims(2)
thf(fact_538_valid__mask_Opelims_I3_J,axiom,
! [X: produc1457211279475724562t_char > prat] :
( ~ ( valid_mask @ X )
=> ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
=> ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
& ! [F2: list_char] :
( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
= pnone ) ) ) ) ) ).
% valid_mask.pelims(3)
thf(fact_539_valid__null__def,axiom,
( valid_null
= ( ^ [Pi3: produc1457211279475724562t_char > prat] :
! [F4: list_char] :
( ( Pi3 @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
= pnone ) ) ) ).
% valid_null_def
thf(fact_540_plus__ab__defined,axiom,
! [Phi5: state,Phi6: state] :
( ( ( plus @ Phi5 @ Phi6 )
!= none_state )
= ( ( compatible_heaps @ ( get_h @ Phi5 ) @ ( get_h @ Phi6 ) )
& ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ Phi5 ) @ ( get_m @ Phi6 ) ) ) ) ) ).
% plus_ab_defined
thf(fact_541_valid__state_Oelims_I3_J,axiom,
! [X: produc9003781588162623004on_val] :
( ~ ( valid_state @ X )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ( ( valid_mask @ Pi )
& ( valid_heap @ Pi @ H2 ) ) ) ) ).
% valid_state.elims(3)
thf(fact_542_PartialSA_Odefined__def,axiom,
( defined
= ( ^ [A3: state,B: state] :
( ( plus @ A3 @ B )
!= none_state ) ) ) ).
% PartialSA.defined_def
thf(fact_543_valid__state_Osimps,axiom,
! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
( ( valid_state @ ( produc74620467203698198on_val @ Pi4 @ H ) )
= ( ( valid_mask @ Pi4 )
& ( valid_heap @ Pi4 @ H ) ) ) ).
% valid_state.simps
thf(fact_544_valid__state_Oelims_I1_J,axiom,
! [X: produc9003781588162623004on_val,Y: $o] :
( ( ( valid_state @ X )
= Y )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ( Y
= ( ~ ( ( valid_mask @ Pi )
& ( valid_heap @ Pi @ H2 ) ) ) ) ) ) ).
% valid_state.elims(1)
thf(fact_545_valid__state_Oelims_I2_J,axiom,
! [X: produc9003781588162623004on_val] :
( ( valid_state @ X )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ~ ( ( valid_mask @ Pi )
& ( valid_heap @ Pi @ H2 ) ) ) ) ).
% valid_state.elims(2)
thf(fact_546_valid__state_Opelims_I1_J,axiom,
! [X: produc9003781588162623004on_val,Y: $o] :
( ( ( valid_state @ X )
= Y )
=> ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ( ( Y
= ( ( valid_mask @ Pi )
& ( valid_heap @ Pi @ H2 ) ) )
=> ~ ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H2 ) ) ) ) ) ) ).
% valid_state.pelims(1)
thf(fact_547_valid__state_Opelims_I2_J,axiom,
! [X: produc9003781588162623004on_val] :
( ( valid_state @ X )
=> ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ~ ( ( valid_mask @ Pi )
& ( valid_heap @ Pi @ H2 ) ) ) ) ) ) ).
% valid_state.pelims(2)
thf(fact_548_valid__state_Opelims_I3_J,axiom,
! [X: produc9003781588162623004on_val] :
( ~ ( valid_state @ X )
=> ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ( ( valid_mask @ Pi )
& ( valid_heap @ Pi @ H2 ) ) ) ) ) ) ).
% valid_state.pelims(3)
thf(fact_549_full__mask__def,axiom,
( full_mask
= ( ^ [Hl2: produc1457211279475724562t_char] :
( if_prat
@ ( ( produc6204653154488278070t_char @ Hl2 )
= null_nat )
@ pnone
@ pwrite ) ) ) ).
% full_mask_def
thf(fact_550_empty__heap__def,axiom,
( empty_7512026568161686585ar_val
= ( ^ [Hl2: produc1457211279475724562t_char] : none_val ) ) ).
% empty_heap_def
thf(fact_551_valid__stateI,axiom,
! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
( ( valid_mask @ Pi4 )
=> ( ! [Hl3: produc1457211279475724562t_char] :
( ( ppos @ ( Pi4 @ Hl3 ) )
=> ( ( H @ Hl3 )
!= none_val ) )
=> ( valid_state @ ( produc74620467203698198on_val @ Pi4 @ H ) ) ) ) ).
% valid_stateI
thf(fact_552_range__fst,axiom,
( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ top_to1250346684925019074t_char )
= top_top_set_nat ) ).
% range_fst
thf(fact_553_range__fst,axiom,
( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ top_to1561438921946393474on_val )
= top_to6849109582029272579r_prat ) ).
% range_fst
thf(fact_554_fst__conv,axiom,
! [X1: option_state,X22: option_state] :
( ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_555_fst__conv,axiom,
! [X1: prat,X22: state] :
( ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_556_fst__conv,axiom,
! [X1: prat,X22: set_state] :
( ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_557_fst__conv,axiom,
! [X1: nat,X22: list_char] :
( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_558_fst__conv,axiom,
! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] :
( ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_559_fst__eqD,axiom,
! [X: option_state,Y: option_state,A4: option_state] :
( ( ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_560_fst__eqD,axiom,
! [X: prat,Y: state,A4: prat] :
( ( ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_561_fst__eqD,axiom,
! [X: prat,Y: set_state,A4: prat] :
( ( ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_562_fst__eqD,axiom,
! [X: nat,Y: list_char,A4: nat] :
( ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_563_fst__eqD,axiom,
! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > prat] :
( ( ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X @ Y ) )
= A4 )
=> ( X = A4 ) ) ).
% fst_eqD
thf(fact_564_fstI,axiom,
! [X: produc3142500478612311029_state,Y: option_state,Z: option_state] :
( ( X
= ( produc9160152616361873709_state @ Y @ Z ) )
=> ( ( produc3900939990612242713_state @ X )
= Y ) ) ).
% fstI
thf(fact_565_fstI,axiom,
! [X: produc4844402304972151915_state,Y: prat,Z: state] :
( ( X
= ( produc6205719116230455971_state @ Y @ Z ) )
=> ( ( produc6030790797199838863_state @ X )
= Y ) ) ).
% fstI
thf(fact_566_fstI,axiom,
! [X: produc6251377953716854561_state,Y: prat,Z: set_state] :
( ( X
= ( produc9080284787853255769_state @ Y @ Z ) )
=> ( ( produc7279835715735703621_state @ X )
= Y ) ) ).
% fstI
thf(fact_567_fstI,axiom,
! [X: produc1457211279475724562t_char,Y: nat,Z: list_char] :
( ( X
= ( produc120197956887798346t_char @ Y @ Z ) )
=> ( ( produc6204653154488278070t_char @ X )
= Y ) ) ).
% fstI
thf(fact_568_fstI,axiom,
! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Y @ Z ) )
=> ( ( produc5239619271231976746on_val @ X )
= Y ) ) ).
% fstI
thf(fact_569_valid__heap__def,axiom,
( valid_heap
= ( ^ [Pi3: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
! [Hl2: produc1457211279475724562t_char] :
( ( ppos @ ( Pi3 @ Hl2 ) )
=> ( ( H4 @ Hl2 )
!= none_val ) ) ) ) ).
% valid_heap_def
thf(fact_570_valid__heapI,axiom,
! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
( ! [Hl3: produc1457211279475724562t_char] :
( ( ppos @ ( Pi4 @ Hl3 ) )
=> ( ( H @ Hl3 )
!= none_val ) )
=> ( valid_heap @ Pi4 @ H ) ) ).
% valid_heapI
thf(fact_571_eq__fst__iff,axiom,
! [A4: option_state,P: produc3142500478612311029_state] :
( ( A4
= ( produc3900939990612242713_state @ P ) )
= ( ? [B: option_state] :
( P
= ( produc9160152616361873709_state @ A4 @ B ) ) ) ) ).
% eq_fst_iff
thf(fact_572_eq__fst__iff,axiom,
! [A4: prat,P: produc4844402304972151915_state] :
( ( A4
= ( produc6030790797199838863_state @ P ) )
= ( ? [B: state] :
( P
= ( produc6205719116230455971_state @ A4 @ B ) ) ) ) ).
% eq_fst_iff
thf(fact_573_eq__fst__iff,axiom,
! [A4: prat,P: produc6251377953716854561_state] :
( ( A4
= ( produc7279835715735703621_state @ P ) )
= ( ? [B: set_state] :
( P
= ( produc9080284787853255769_state @ A4 @ B ) ) ) ) ).
% eq_fst_iff
thf(fact_574_eq__fst__iff,axiom,
! [A4: nat,P: produc1457211279475724562t_char] :
( ( A4
= ( produc6204653154488278070t_char @ P ) )
= ( ? [B: list_char] :
( P
= ( produc120197956887798346t_char @ A4 @ B ) ) ) ) ).
% eq_fst_iff
thf(fact_575_eq__fst__iff,axiom,
! [A4: produc1457211279475724562t_char > prat,P: produc9003781588162623004on_val] :
( ( A4
= ( produc5239619271231976746on_val @ P ) )
= ( ? [B: produc1457211279475724562t_char > option_val] :
( P
= ( produc74620467203698198on_val @ A4 @ B ) ) ) ) ).
% eq_fst_iff
thf(fact_576_PartialHeapSA_Ostable__def,axiom,
( stable
= ( ^ [Phi2: state] :
! [Hl2: produc1457211279475724562t_char] :
( ( ppos @ ( get_m @ Phi2 @ Hl2 ) )
= ( ( get_h @ Phi2 @ Hl2 )
!= none_val ) ) ) ) ).
% PartialHeapSA.stable_def
thf(fact_577_stableI,axiom,
! [Phi5: state] :
( ! [Hl3: produc1457211279475724562t_char] :
( ( ppos @ ( get_m @ Phi5 @ Hl3 ) )
= ( ( get_h @ Phi5 @ Hl3 )
!= none_val ) )
=> ( stable @ Phi5 ) ) ).
% stableI
thf(fact_578_image__Pow__mono,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state,B5: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ ( image_4690842878458256517on_val @ F @ A ) @ B5 )
=> ( ord_le5660975535469346024on_val @ ( image_1939478397035811227on_val @ ( image_4690842878458256517on_val @ F ) @ ( pow_Pr2430052815820169291_state @ A ) ) @ ( pow_Pr1687164190371123007on_val @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_579_image__Pow__mono,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val,B5: set_Pr4231240917283394591_state] :
( ( ord_le4032880777312351359_state @ ( image_3204238854701872908_state @ F @ A ) @ B5 )
=> ( ord_le8498320074568675253_state @ ( image_4960930050028392952_state @ ( image_3204238854701872908_state @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) ) @ ( pow_Pr3120271928052550924_state @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_580_image__Pow__mono,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val,B5: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ A ) @ B5 )
=> ( ord_le1960757061454972174_state @ ( image_2705476029607989621_state @ ( image_4494208899119068747_state @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) ) @ ( pow_Pr2430052815820169291_state @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_581_image__Pow__mono,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ ( image_4170889944715819327on_val @ F @ A ) @ B5 )
=> ( ord_le5660975535469346024on_val @ ( image_451893177406657707on_val @ ( image_4170889944715819327on_val @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) ) @ ( pow_Pr1687164190371123007on_val @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_582_image__Pow__mono,axiom,
! [F: state > state,A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A ) @ B5 )
=> ( ord_le5175021213330142598_state @ ( image_2476256681063834599_state @ ( image_state_state @ F ) @ ( pow_state @ A ) ) @ ( pow_state @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_583_image__Pow__mono,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,B5: set_state] :
( ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ A ) @ B5 )
=> ( ord_le5175021213330142598_state @ ( image_8347066812701616201_state @ ( image_1205529093420515805_state @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) ) @ ( pow_state @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_584_valid__pre__unit,axiom,
valid_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ).
% valid_pre_unit
thf(fact_585_pre__get__m_Opelims,axiom,
! [X: produc1317171644805857433on_val,Y: produc9003781588162623004on_val] :
( ( ( pre_ge6899797224320663452on_val @ X )
= Y )
=> ( ( accp_P6797916352740768290on_val @ pre_ge1694281482089598673on_val @ X )
=> ~ ( ( Y
= ( produc3129717363260770749on_val @ X ) )
=> ~ ( accp_P6797916352740768290on_val @ pre_ge1694281482089598673on_val @ X ) ) ) ) ).
% pre_get_m.pelims
thf(fact_586_pre__get__m_Opelims,axiom,
! [X: produc4844402304972151915_state,Y: prat] :
( ( ( pre_get_m_prat_state @ X )
= Y )
=> ( ( accp_P6233459008282737652_state @ pre_ge6835382860682234787_state @ X )
=> ~ ( ( Y
= ( produc6030790797199838863_state @ X ) )
=> ~ ( accp_P6233459008282737652_state @ pre_ge6835382860682234787_state @ X ) ) ) ) ).
% pre_get_m.pelims
thf(fact_587_pre__get__m_Opelims,axiom,
! [X: produc6251377953716854561_state,Y: prat] :
( ( ( pre_ge1435879308799382564_state @ X )
= Y )
=> ( ( accp_P6554740477754186410_state @ pre_ge4013444835548200281_state @ X )
=> ~ ( ( Y
= ( produc7279835715735703621_state @ X ) )
=> ~ ( accp_P6554740477754186410_state @ pre_ge4013444835548200281_state @ X ) ) ) ) ).
% pre_get_m.pelims
thf(fact_588_pre__get__m_Opelims,axiom,
! [X: produc1457211279475724562t_char,Y: nat] :
( ( ( pre_ge2740421702909910421t_char @ X )
= Y )
=> ( ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X )
=> ~ ( ( Y
= ( produc6204653154488278070t_char @ X ) )
=> ~ ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X ) ) ) ) ).
% pre_get_m.pelims
thf(fact_589_pre__get__m_Opelims,axiom,
! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat] :
( ( ( pre_ge8481574585653477131on_val @ X )
= Y )
=> ( ( accp_P4405556307444930259on_val @ pre_ge4462515927930180886on_val @ X )
=> ~ ( ( Y
= ( produc5239619271231976746on_val @ X ) )
=> ~ ( accp_P4405556307444930259on_val @ pre_ge4462515927930180886on_val @ X ) ) ) ) ).
% pre_get_m.pelims
thf(fact_590_Pow__UNIV,axiom,
( ( pow_state @ top_top_set_state )
= top_to5262587396890829782_state ) ).
% Pow_UNIV
thf(fact_591_Pow__iff,axiom,
! [A: set_state,B5: set_state] :
( ( member_set_state @ A @ ( pow_state @ B5 ) )
= ( ord_le2494988322063910608_state @ A @ B5 ) ) ).
% Pow_iff
thf(fact_592_PowI,axiom,
! [A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( member_set_state @ A @ ( pow_state @ B5 ) ) ) ).
% PowI
thf(fact_593_PowD,axiom,
! [A: set_state,B5: set_state] :
( ( member_set_state @ A @ ( pow_state @ B5 ) )
=> ( ord_le2494988322063910608_state @ A @ B5 ) ) ).
% PowD
thf(fact_594_get__m_Osimps,axiom,
( get_m
= ( ^ [A3: state] : ( produc5239619271231976746on_val @ ( rep_state @ A3 ) ) ) ) ).
% get_m.simps
thf(fact_595_get__m_Oelims,axiom,
! [X: state,Y: produc1457211279475724562t_char > prat] :
( ( ( get_m @ X )
= Y )
=> ( Y
= ( produc5239619271231976746on_val @ ( rep_state @ X ) ) ) ) ).
% get_m.elims
thf(fact_596_get__pre_I2_J,axiom,
( get_m
= ( ^ [X3: state] : ( produc5239619271231976746on_val @ ( rep_state @ X3 ) ) ) ) ).
% get_pre(2)
thf(fact_597_Pow__mono,axiom,
! [A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ord_le5175021213330142598_state @ ( pow_state @ A ) @ ( pow_state @ B5 ) ) ) ).
% Pow_mono
thf(fact_598_image__Pow__surj,axiom,
! [F: state > state,A: set_state,B5: set_state] :
( ( ( image_state_state @ F @ A )
= B5 )
=> ( ( image_2476256681063834599_state @ ( image_state_state @ F ) @ ( pow_state @ A ) )
= ( pow_state @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_599_image__Pow__surj,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state,B5: set_Pr7649655576376376530on_val] :
( ( ( image_4690842878458256517on_val @ F @ A )
= B5 )
=> ( ( image_1939478397035811227on_val @ ( image_4690842878458256517on_val @ F ) @ ( pow_Pr2430052815820169291_state @ A ) )
= ( pow_Pr1687164190371123007on_val @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_600_image__Pow__surj,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,B5: set_state] :
( ( ( image_1205529093420515805_state @ F @ A )
= B5 )
=> ( ( image_8347066812701616201_state @ ( image_1205529093420515805_state @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) )
= ( pow_state @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_601_image__Pow__surj,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val,B5: set_Pr4231240917283394591_state] :
( ( ( image_3204238854701872908_state @ F @ A )
= B5 )
=> ( ( image_4960930050028392952_state @ ( image_3204238854701872908_state @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) )
= ( pow_Pr3120271928052550924_state @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_602_image__Pow__surj,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val,B5: set_Pr2391701600042252046_state] :
( ( ( image_4494208899119068747_state @ F @ A )
= B5 )
=> ( ( image_2705476029607989621_state @ ( image_4494208899119068747_state @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) )
= ( pow_Pr2430052815820169291_state @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_603_image__Pow__surj,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ( ( image_4170889944715819327on_val @ F @ A )
= B5 )
=> ( ( image_451893177406657707on_val @ ( image_4170889944715819327on_val @ F ) @ ( pow_Pr1687164190371123007on_val @ A ) )
= ( pow_Pr1687164190371123007on_val @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_604_empty__mask_Opelims,axiom,
! [X: produc1457211279475724562t_char,Y: prat] :
( ( ( empty_3446695950879338768t_char @ X )
= Y )
=> ( ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_605_empty__mask_Opelims,axiom,
! [X: produc1317171644805857433on_val,Y: prat] :
( ( ( empty_4129289814623480215on_val @ X )
= Y )
=> ( ( accp_P6797916352740768290on_val @ empty_1698368654852286560on_val @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P6797916352740768290on_val @ empty_1698368654852286560on_val @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_606_empty__mask_Opelims,axiom,
! [X: produc4844402304972151915_state,Y: prat] :
( ( ( empty_3913122416104905065_state @ X )
= Y )
=> ( ( accp_P6233459008282737652_state @ empty_5283331674940636722_state @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P6233459008282737652_state @ empty_5283331674940636722_state @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_607_empty__mask_Opelims,axiom,
! [X: produc6251377953716854561_state,Y: prat] :
( ( ( empty_5541330147480151071_state @ X )
= Y )
=> ( ( accp_P6554740477754186410_state @ empty_7304051301540881640_state @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P6554740477754186410_state @ empty_7304051301540881640_state @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_608_empty__mask_Opelims,axiom,
! [X: produc9003781588162623004on_val,Y: prat] :
( ( ( empty_6102255041883331742on_val @ X )
= Y )
=> ( ( accp_P4405556307444930259on_val @ empty_5847456986941130773on_val @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P4405556307444930259on_val @ empty_5847456986941130773on_val @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_609_empty__mask_Opelims,axiom,
! [X: produc1457211279475724562t_char > prat,Y: prat] :
( ( ( empty_6616431780912394577r_prat @ X )
= Y )
=> ( ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X )
=> ~ ( ( Y = pnone )
=> ~ ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X ) ) ) ) ).
% empty_mask.pelims
thf(fact_610_get__m_Opelims,axiom,
! [X: state,Y: produc1457211279475724562t_char > prat] :
( ( ( get_m @ X )
= Y )
=> ( ( accp_state @ get_m_rel @ X )
=> ~ ( ( Y
= ( produc5239619271231976746on_val @ ( rep_state @ X ) ) )
=> ~ ( accp_state @ get_m_rel @ X ) ) ) ) ).
% get_m.pelims
thf(fact_611_unit__def,axiom,
( unit
= ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ) ) ).
% unit_def
thf(fact_612_core__def,axiom,
( core
= ( ^ [Phi2: state] : ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ ( get_h @ Phi2 ) ) ) ) ) ).
% core_def
thf(fact_613_unit__charact_I1_J,axiom,
( ( get_h @ unit )
= empty_7512026568161686585ar_val ) ).
% unit_charact(1)
thf(fact_614_range__snd,axiom,
( ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ top_to1561438921946393474on_val )
= top_to3326603054294128128on_val ) ).
% range_snd
thf(fact_615_prod_Ocollapse,axiom,
! [Prod: produc3142500478612311029_state] :
( ( produc9160152616361873709_state @ ( produc3900939990612242713_state @ Prod ) @ ( produc9117690093182453595_state @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_616_prod_Ocollapse,axiom,
! [Prod: produc4844402304972151915_state] :
( ( produc6205719116230455971_state @ ( produc6030790797199838863_state @ Prod ) @ ( produc8138835347214057169_state @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_617_prod_Ocollapse,axiom,
! [Prod: produc6251377953716854561_state] :
( ( produc9080284787853255769_state @ ( produc7279835715735703621_state @ Prod ) @ ( produc4944973370551335047_state @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_618_prod_Ocollapse,axiom,
! [Prod: produc1457211279475724562t_char] :
( ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_619_prod_Ocollapse,axiom,
! [Prod: produc9003781588162623004on_val] :
( ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_620_PartialSA_Ounit__core,axiom,
( ( core @ unit )
= unit ) ).
% PartialSA.unit_core
thf(fact_621_snd__conv,axiom,
! [X1: option_state,X22: option_state] :
( ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_622_snd__conv,axiom,
! [X1: prat,X22: state] :
( ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_623_snd__conv,axiom,
! [X1: prat,X22: set_state] :
( ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_624_snd__conv,axiom,
! [X1: nat,X22: list_char] :
( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_625_snd__conv,axiom,
! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] :
( ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_626_snd__eqD,axiom,
! [X: option_state,Y: option_state,A4: option_state] :
( ( ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ X @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_627_snd__eqD,axiom,
! [X: prat,Y: state,A4: state] :
( ( ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ X @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_628_snd__eqD,axiom,
! [X: prat,Y: set_state,A4: set_state] :
( ( ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ X @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_629_snd__eqD,axiom,
! [X: nat,Y: list_char,A4: list_char] :
( ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_630_snd__eqD,axiom,
! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > option_val] :
( ( ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_631_sndI,axiom,
! [X: produc3142500478612311029_state,Y: option_state,Z: option_state] :
( ( X
= ( produc9160152616361873709_state @ Y @ Z ) )
=> ( ( produc9117690093182453595_state @ X )
= Z ) ) ).
% sndI
thf(fact_632_sndI,axiom,
! [X: produc4844402304972151915_state,Y: prat,Z: state] :
( ( X
= ( produc6205719116230455971_state @ Y @ Z ) )
=> ( ( produc8138835347214057169_state @ X )
= Z ) ) ).
% sndI
thf(fact_633_sndI,axiom,
! [X: produc6251377953716854561_state,Y: prat,Z: set_state] :
( ( X
= ( produc9080284787853255769_state @ Y @ Z ) )
=> ( ( produc4944973370551335047_state @ X )
= Z ) ) ).
% sndI
thf(fact_634_sndI,axiom,
! [X: produc1457211279475724562t_char,Y: nat,Z: list_char] :
( ( X
= ( produc120197956887798346t_char @ Y @ Z ) )
=> ( ( produc1900778367047502200t_char @ X )
= Z ) ) ).
% sndI
thf(fact_635_sndI,axiom,
! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Y @ Z ) )
=> ( ( produc2319605628138516840on_val @ X )
= Z ) ) ).
% sndI
thf(fact_636_eq__snd__iff,axiom,
! [B2: option_state,P: produc3142500478612311029_state] :
( ( B2
= ( produc9117690093182453595_state @ P ) )
= ( ? [A3: option_state] :
( P
= ( produc9160152616361873709_state @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_637_eq__snd__iff,axiom,
! [B2: state,P: produc4844402304972151915_state] :
( ( B2
= ( produc8138835347214057169_state @ P ) )
= ( ? [A3: prat] :
( P
= ( produc6205719116230455971_state @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_638_eq__snd__iff,axiom,
! [B2: set_state,P: produc6251377953716854561_state] :
( ( B2
= ( produc4944973370551335047_state @ P ) )
= ( ? [A3: prat] :
( P
= ( produc9080284787853255769_state @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_639_eq__snd__iff,axiom,
! [B2: list_char,P: produc1457211279475724562t_char] :
( ( B2
= ( produc1900778367047502200t_char @ P ) )
= ( ? [A3: nat] :
( P
= ( produc120197956887798346t_char @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_640_eq__snd__iff,axiom,
! [B2: produc1457211279475724562t_char > option_val,P: produc9003781588162623004on_val] :
( ( B2
= ( produc2319605628138516840on_val @ P ) )
= ( ? [A3: produc1457211279475724562t_char > prat] :
( P
= ( produc74620467203698198on_val @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_641_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_642_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_643_prod__eqI,axiom,
! [P: produc1457211279475724562t_char,Q2: produc1457211279475724562t_char] :
( ( ( produc6204653154488278070t_char @ P )
= ( produc6204653154488278070t_char @ Q2 ) )
=> ( ( ( produc1900778367047502200t_char @ P )
= ( produc1900778367047502200t_char @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_644_prod__eqI,axiom,
! [P: produc9003781588162623004on_val,Q2: produc9003781588162623004on_val] :
( ( ( produc5239619271231976746on_val @ P )
= ( produc5239619271231976746on_val @ Q2 ) )
=> ( ( ( produc2319605628138516840on_val @ P )
= ( produc2319605628138516840on_val @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_645_prod__eq__iff,axiom,
( ( ^ [Y6: produc1457211279475724562t_char,Z3: produc1457211279475724562t_char] : ( Y6 = Z3 ) )
= ( ^ [S3: produc1457211279475724562t_char,T: produc1457211279475724562t_char] :
( ( ( produc6204653154488278070t_char @ S3 )
= ( produc6204653154488278070t_char @ T ) )
& ( ( produc1900778367047502200t_char @ S3 )
= ( produc1900778367047502200t_char @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_646_prod__eq__iff,axiom,
( ( ^ [Y6: produc9003781588162623004on_val,Z3: produc9003781588162623004on_val] : ( Y6 = Z3 ) )
= ( ^ [S3: produc9003781588162623004on_val,T: produc9003781588162623004on_val] :
( ( ( produc5239619271231976746on_val @ S3 )
= ( produc5239619271231976746on_val @ T ) )
& ( ( produc2319605628138516840on_val @ S3 )
= ( produc2319605628138516840on_val @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_647_PartialSA_Ocore__mono,axiom,
! [A4: state,B2: state] :
( ( greater @ A4 @ B2 )
=> ( greater @ ( core @ A4 ) @ ( core @ B2 ) ) ) ).
% PartialSA.core_mono
thf(fact_648_core__defined_I1_J,axiom,
! [Phi5: state] :
( ( get_h @ ( core @ Phi5 ) )
= ( get_h @ Phi5 ) ) ).
% core_defined(1)
thf(fact_649_PartialSA_Ounit__smaller,axiom,
! [Phi5: state] : ( greater @ Phi5 @ unit ) ).
% PartialSA.unit_smaller
thf(fact_650_stable__unit,axiom,
stable @ unit ).
% stable_unit
thf(fact_651_prod_Oexhaust__sel,axiom,
! [Prod: produc3142500478612311029_state] :
( Prod
= ( produc9160152616361873709_state @ ( produc3900939990612242713_state @ Prod ) @ ( produc9117690093182453595_state @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_652_prod_Oexhaust__sel,axiom,
! [Prod: produc4844402304972151915_state] :
( Prod
= ( produc6205719116230455971_state @ ( produc6030790797199838863_state @ Prod ) @ ( produc8138835347214057169_state @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_653_prod_Oexhaust__sel,axiom,
! [Prod: produc6251377953716854561_state] :
( Prod
= ( produc9080284787853255769_state @ ( produc7279835715735703621_state @ Prod ) @ ( produc4944973370551335047_state @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_654_prod_Oexhaust__sel,axiom,
! [Prod: produc1457211279475724562t_char] :
( Prod
= ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_655_prod_Oexhaust__sel,axiom,
! [Prod: produc9003781588162623004on_val] :
( Prod
= ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_656_surjective__pairing,axiom,
! [T2: produc3142500478612311029_state] :
( T2
= ( produc9160152616361873709_state @ ( produc3900939990612242713_state @ T2 ) @ ( produc9117690093182453595_state @ T2 ) ) ) ).
% surjective_pairing
thf(fact_657_surjective__pairing,axiom,
! [T2: produc4844402304972151915_state] :
( T2
= ( produc6205719116230455971_state @ ( produc6030790797199838863_state @ T2 ) @ ( produc8138835347214057169_state @ T2 ) ) ) ).
% surjective_pairing
thf(fact_658_surjective__pairing,axiom,
! [T2: produc6251377953716854561_state] :
( T2
= ( produc9080284787853255769_state @ ( produc7279835715735703621_state @ T2 ) @ ( produc4944973370551335047_state @ T2 ) ) ) ).
% surjective_pairing
thf(fact_659_surjective__pairing,axiom,
! [T2: produc1457211279475724562t_char] :
( T2
= ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ T2 ) @ ( produc1900778367047502200t_char @ T2 ) ) ) ).
% surjective_pairing
thf(fact_660_surjective__pairing,axiom,
! [T2: produc9003781588162623004on_val] :
( T2
= ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ T2 ) @ ( produc2319605628138516840on_val @ T2 ) ) ) ).
% surjective_pairing
thf(fact_661_get__pre_I1_J,axiom,
( get_h
= ( ^ [X3: state] : ( produc2319605628138516840on_val @ ( rep_state @ X3 ) ) ) ) ).
% get_pre(1)
thf(fact_662_get__h_Oelims,axiom,
! [X: state,Y: produc1457211279475724562t_char > option_val] :
( ( ( get_h @ X )
= Y )
=> ( Y
= ( produc2319605628138516840on_val @ ( rep_state @ X ) ) ) ) ).
% get_h.elims
thf(fact_663_get__h_Osimps,axiom,
( get_h
= ( ^ [A3: state] : ( produc2319605628138516840on_val @ ( rep_state @ A3 ) ) ) ) ).
% get_h.simps
thf(fact_664_PartialSA_Osmaller__compatible__core,axiom,
! [Y: state,X: state] :
( ( greater @ Y @ X )
=> ( defined @ X @ ( core @ Y ) ) ) ).
% PartialSA.smaller_compatible_core
thf(fact_665_core__defined_I2_J,axiom,
! [Phi5: state] :
( ( get_m @ ( core @ Phi5 ) )
= empty_3446695950879338768t_char ) ).
% core_defined(2)
thf(fact_666_unit__charact_I2_J,axiom,
( ( get_m @ unit )
= empty_3446695950879338768t_char ) ).
% unit_charact(2)
thf(fact_667_get__h_Opelims,axiom,
! [X: state,Y: produc1457211279475724562t_char > option_val] :
( ( ( get_h @ X )
= Y )
=> ( ( accp_state @ get_h_rel @ X )
=> ~ ( ( Y
= ( produc2319605628138516840on_val @ ( rep_state @ X ) ) )
=> ~ ( accp_state @ get_h_rel @ X ) ) ) ) ).
% get_h.pelims
thf(fact_668_compatible__def,axiom,
( compatible
= ( ^ [Phi2: produc9003781588162623004on_val,Phi: produc9003781588162623004on_val] :
( ( compatible_heaps @ ( produc2319605628138516840on_val @ Phi2 ) @ ( produc2319605628138516840on_val @ Phi ) )
& ( valid_mask @ ( add_ma6586698345353345811t_char @ ( produc5239619271231976746on_val @ Phi2 ) @ ( produc5239619271231976746on_val @ Phi ) ) ) ) ) ) ).
% compatible_def
thf(fact_669_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P6: option_state > option_state > $o,X: option_state,Y: option_state,A4: produc3142500478612311029_state] :
( ( P6 @ X @ Y )
=> ( ( A4
= ( produc9160152616361873709_state @ X @ Y ) )
=> ( P6 @ ( produc3900939990612242713_state @ A4 ) @ ( produc9117690093182453595_state @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_670_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P6: prat > state > $o,X: prat,Y: state,A4: produc4844402304972151915_state] :
( ( P6 @ X @ Y )
=> ( ( A4
= ( produc6205719116230455971_state @ X @ Y ) )
=> ( P6 @ ( produc6030790797199838863_state @ A4 ) @ ( produc8138835347214057169_state @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_671_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P6: prat > set_state > $o,X: prat,Y: set_state,A4: produc6251377953716854561_state] :
( ( P6 @ X @ Y )
=> ( ( A4
= ( produc9080284787853255769_state @ X @ Y ) )
=> ( P6 @ ( produc7279835715735703621_state @ A4 ) @ ( produc4944973370551335047_state @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_672_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P6: nat > list_char > $o,X: nat,Y: list_char,A4: produc1457211279475724562t_char] :
( ( P6 @ X @ Y )
=> ( ( A4
= ( produc120197956887798346t_char @ X @ Y ) )
=> ( P6 @ ( produc6204653154488278070t_char @ A4 ) @ ( produc1900778367047502200t_char @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_673_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P6: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A4: produc9003781588162623004on_val] :
( ( P6 @ X @ Y )
=> ( ( A4
= ( produc74620467203698198on_val @ X @ Y ) )
=> ( P6 @ ( produc5239619271231976746on_val @ A4 ) @ ( produc2319605628138516840on_val @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_674_conjI__realizer,axiom,
! [P6: option_state > $o,P: option_state,Q: option_state > $o,Q2: option_state] :
( ( P6 @ P )
=> ( ( Q @ Q2 )
=> ( ( P6 @ ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ P @ Q2 ) ) )
& ( Q @ ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_675_conjI__realizer,axiom,
! [P6: prat > $o,P: prat,Q: state > $o,Q2: state] :
( ( P6 @ P )
=> ( ( Q @ Q2 )
=> ( ( P6 @ ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ P @ Q2 ) ) )
& ( Q @ ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_676_conjI__realizer,axiom,
! [P6: prat > $o,P: prat,Q: set_state > $o,Q2: set_state] :
( ( P6 @ P )
=> ( ( Q @ Q2 )
=> ( ( P6 @ ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ P @ Q2 ) ) )
& ( Q @ ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_677_conjI__realizer,axiom,
! [P6: nat > $o,P: nat,Q: list_char > $o,Q2: list_char] :
( ( P6 @ P )
=> ( ( Q @ Q2 )
=> ( ( P6 @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ P @ Q2 ) ) )
& ( Q @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_678_conjI__realizer,axiom,
! [P6: ( produc1457211279475724562t_char > prat ) > $o,P: produc1457211279475724562t_char > prat,Q: ( produc1457211279475724562t_char > option_val ) > $o,Q2: produc1457211279475724562t_char > option_val] :
( ( P6 @ P )
=> ( ( Q @ Q2 )
=> ( ( P6 @ ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ P @ Q2 ) ) )
& ( Q @ ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_679_exI__realizer,axiom,
! [P6: option_state > option_state > $o,Y: option_state,X: option_state] :
( ( P6 @ Y @ X )
=> ( P6 @ ( produc9117690093182453595_state @ ( produc9160152616361873709_state @ X @ Y ) ) @ ( produc3900939990612242713_state @ ( produc9160152616361873709_state @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_680_exI__realizer,axiom,
! [P6: state > prat > $o,Y: state,X: prat] :
( ( P6 @ Y @ X )
=> ( P6 @ ( produc8138835347214057169_state @ ( produc6205719116230455971_state @ X @ Y ) ) @ ( produc6030790797199838863_state @ ( produc6205719116230455971_state @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_681_exI__realizer,axiom,
! [P6: set_state > prat > $o,Y: set_state,X: prat] :
( ( P6 @ Y @ X )
=> ( P6 @ ( produc4944973370551335047_state @ ( produc9080284787853255769_state @ X @ Y ) ) @ ( produc7279835715735703621_state @ ( produc9080284787853255769_state @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_682_exI__realizer,axiom,
! [P6: list_char > nat > $o,Y: list_char,X: nat] :
( ( P6 @ Y @ X )
=> ( P6 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_683_exI__realizer,axiom,
! [P6: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > $o,Y: produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char > prat] :
( ( P6 @ Y @ X )
=> ( P6 @ ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X @ Y ) ) @ ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_684_PartialSA_Omono__transformer__def,axiom,
! [T3: state > state] :
( ( packag4817583915586052229_state @ plus @ unit @ T3 )
= ( ! [Phi2: state,Phi: state] :
( ( greater @ Phi @ Phi2 )
=> ( greater @ ( T3 @ Phi ) @ ( T3 @ Phi2 ) ) )
& ( ( T3 @ unit )
= unit ) ) ) ).
% PartialSA.mono_transformer_def
thf(fact_685_package__logic_Omono__transformer_Ocong,axiom,
packag4817583915586052229_state = packag4817583915586052229_state ).
% package_logic.mono_transformer.cong
thf(fact_686_PartialSA_Ovalid__package__setI,axiom,
! [S4: set_Pr4231240917283394591_state,F: state] :
( ! [A6: state,U: state,T4: state > state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T4 ) ) @ S4 )
=> ( ( defined @ A6 @ U )
& ( greater @ ( core @ A6 ) @ ( core @ U ) )
& ( packag4817583915586052229_state @ plus @ unit @ T4 )
& ( greater @ A6 @ ( core @ ( T4 @ F ) ) ) ) )
=> ( packag3877571396724326311_state @ plus @ core @ unit @ S4 @ F ) ) ).
% PartialSA.valid_package_setI
thf(fact_687_compatibleI__old,axiom,
! [Phi5: produc9003781588162623004on_val,Phi6: produc9003781588162623004on_val] :
( ! [Hl3: produc1457211279475724562t_char,X2: val,Y5: val] :
( ( ( ( produc2319605628138516840on_val @ Phi5 @ Hl3 )
= ( some_val @ X2 ) )
& ( ( produc2319605628138516840on_val @ Phi6 @ Hl3 )
= ( some_val @ Y5 ) ) )
=> ( X2 = Y5 ) )
=> ( ( valid_mask @ ( add_ma6586698345353345811t_char @ ( produc5239619271231976746on_val @ Phi5 ) @ ( produc5239619271231976746on_val @ Phi6 ) ) )
=> ( compatible @ Phi5 @ Phi6 ) ) ) ).
% compatibleI_old
thf(fact_688_prod_Oswap__def,axiom,
( produc8425362190318354125_state
= ( ^ [P2: produc3142500478612311029_state] : ( produc9160152616361873709_state @ ( produc9117690093182453595_state @ P2 ) @ ( produc3900939990612242713_state @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_689_prod_Oswap__def,axiom,
( produc6343331096166649923e_prat
= ( ^ [P2: produc7334356684927367531e_prat] : ( produc6205719116230455971_state @ ( produc7795413381861543633e_prat @ P2 ) @ ( produc5687368831847325327e_prat @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_690_prod_Oswap__def,axiom,
( produc7016766326345971705e_prat
= ( ^ [P2: produc2654212508281724193e_prat] : ( produc9080284787853255769_state @ ( produc1175801883329052295e_prat @ P2 ) @ ( produc3510664228513420869e_prat @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_691_prod_Oswap__def,axiom,
( produc4410195085741832554ar_nat
= ( ^ [P2: produc7606469099237612818ar_nat] : ( produc120197956887798346t_char @ ( produc1259554233722553080ar_nat @ P2 ) @ ( produc5563429021163328950ar_nat @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_692_prod_Oswap__def,axiom,
( produc5819413633627552064r_prat
= ( ^ [P2: produc7776263825685906734r_prat] : ( produc74620467203698198on_val @ ( produc4142415063207305778r_prat @ P2 ) @ ( produc7062428706300765684r_prat @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_693_prod_Oswap__def,axiom,
( produc5051419219066781674t_char
= ( ^ [P2: produc1457211279475724562t_char] : ( produc8702345860417625034ar_nat @ ( produc1900778367047502200t_char @ P2 ) @ ( produc6204653154488278070t_char @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_694_prod_Oswap__def,axiom,
( produc3996604198558763126on_val
= ( ^ [P2: produc9003781588162623004on_val] : ( produc1897429902272487136r_prat @ ( produc2319605628138516840on_val @ P2 ) @ ( produc5239619271231976746on_val @ P2 ) ) ) ) ).
% prod.swap_def
thf(fact_695_plus__def,axiom,
( plus
= ( ^ [A3: state,B: state] : ( if_option_state @ ( compatible @ ( rep_state @ A3 ) @ ( rep_state @ B ) ) @ ( some_state @ ( abs_state @ ( add_states @ ( rep_state @ A3 ) @ ( rep_state @ B ) ) ) ) @ none_state ) ) ) ).
% plus_def
thf(fact_696_snd__swap,axiom,
! [X: produc1457211279475724562t_char] :
( ( produc1259554233722553080ar_nat @ ( produc5051419219066781674t_char @ X ) )
= ( produc6204653154488278070t_char @ X ) ) ).
% snd_swap
thf(fact_697_snd__swap,axiom,
! [X: produc9003781588162623004on_val] :
( ( produc4142415063207305778r_prat @ ( produc3996604198558763126on_val @ X ) )
= ( produc5239619271231976746on_val @ X ) ) ).
% snd_swap
thf(fact_698_snd__swap,axiom,
! [X: produc7776263825685906734r_prat] :
( ( produc2319605628138516840on_val @ ( produc5819413633627552064r_prat @ X ) )
= ( produc7062428706300765684r_prat @ X ) ) ).
% snd_swap
thf(fact_699_pair__in__swap__image,axiom,
! [Y: state,X: prat,A: set_Pr6039890582625207499_state] :
( ( member8716344949261458452e_prat @ ( produc5862297150877942435e_prat @ Y @ X ) @ ( image_2830361380300980677e_prat @ produc6686753061519163459_state @ A ) )
= ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_700_pair__in__swap__image,axiom,
! [Y: prat,X: state,A: set_Pr8996442262779224523e_prat] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Y @ X ) @ ( image_8107462086123161541_state @ produc6343331096166649923e_prat @ A ) )
= ( member8716344949261458452e_prat @ ( produc5862297150877942435e_prat @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_701_pair__in__swap__image,axiom,
! [Y: set_state,X: prat,A: set_Pr6847263425858009217_state] :
( ( member3111753566487124938e_prat @ ( produc5311113300630973017e_prat @ Y @ X ) @ ( image_8313154324233580357e_prat @ produc1562565776713478649_state @ A ) )
= ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_702_pair__in__swap__image,axiom,
! [Y: list_char,X: nat,A: set_Pr2507339680178222706t_char] :
( ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ Y @ X ) @ ( image_652198164289537669ar_nat @ produc5051419219066781674t_char @ A ) )
= ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_703_pair__in__swap__image,axiom,
! [Y: prat,X: set_state,A: set_Pr6566877984550221441e_prat] :
( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Y @ X ) @ ( image_8063195947988251461_state @ produc7016766326345971705e_prat @ A ) )
= ( member3111753566487124938e_prat @ ( produc5311113300630973017e_prat @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_704_pair__in__swap__image,axiom,
! [Y: nat,X: list_char,A: set_Pr8803067691650843250ar_nat] :
( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Y @ X ) @ ( image_688197443715608197t_char @ produc4410195085741832554ar_nat @ A ) )
= ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_705_pair__in__swap__image,axiom,
! [Y: state > state,X: state,A: set_Pr2391701600042252046_state] :
( ( member8708878012768793047_state @ ( produc4737362649269059174_state @ Y @ X ) @ ( image_2502765500263079301_state @ produc6438927260845192582_state @ A ) )
= ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_706_pair__in__swap__image,axiom,
! [Y: state,X: state > state,A: set_Pr2063381651806504974_state] :
( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Y @ X ) @ ( image_5540908381507763077_state @ produc2543546466205187078_state @ A ) )
= ( member8708878012768793047_state @ ( produc4737362649269059174_state @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_707_pair__in__swap__image,axiom,
! [Y: option_state,X: option_state,A: set_Pr1688445902015331925_state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Y @ X ) @ ( image_3543735668343923269_state @ produc8425362190318354125_state @ A ) )
= ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_708_pair__in__swap__image,axiom,
! [Y: produc3882902684156697774_state,X: state,A: set_Pr4231240917283394591_state] :
( ( member1546683425902591150_state @ ( produc7334084071316671761_state @ Y @ X ) @ ( image_6372718444620452615_state @ produc6564544480275723131_state @ A ) )
= ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_709_swap__simp,axiom,
! [X: state,Y: prat] :
( ( produc6343331096166649923e_prat @ ( produc5862297150877942435e_prat @ X @ Y ) )
= ( produc6205719116230455971_state @ Y @ X ) ) ).
% swap_simp
thf(fact_710_swap__simp,axiom,
! [X: set_state,Y: prat] :
( ( produc7016766326345971705e_prat @ ( produc5311113300630973017e_prat @ X @ Y ) )
= ( produc9080284787853255769_state @ Y @ X ) ) ).
% swap_simp
thf(fact_711_swap__simp,axiom,
! [X: list_char,Y: nat] :
( ( produc4410195085741832554ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y ) )
= ( produc120197956887798346t_char @ Y @ X ) ) ).
% swap_simp
thf(fact_712_swap__simp,axiom,
! [X: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > prat] :
( ( produc5819413633627552064r_prat @ ( produc1897429902272487136r_prat @ X @ Y ) )
= ( produc74620467203698198on_val @ Y @ X ) ) ).
% swap_simp
thf(fact_713_swap__simp,axiom,
! [X: option_state,Y: option_state] :
( ( produc8425362190318354125_state @ ( produc9160152616361873709_state @ X @ Y ) )
= ( produc9160152616361873709_state @ Y @ X ) ) ).
% swap_simp
thf(fact_714_swap__simp,axiom,
! [X: prat,Y: state] :
( ( produc6686753061519163459_state @ ( produc6205719116230455971_state @ X @ Y ) )
= ( produc5862297150877942435e_prat @ Y @ X ) ) ).
% swap_simp
thf(fact_715_swap__simp,axiom,
! [X: prat,Y: set_state] :
( ( produc1562565776713478649_state @ ( produc9080284787853255769_state @ X @ Y ) )
= ( produc5311113300630973017e_prat @ Y @ X ) ) ).
% swap_simp
thf(fact_716_swap__simp,axiom,
! [X: nat,Y: list_char] :
( ( produc5051419219066781674t_char @ ( produc120197956887798346t_char @ X @ Y ) )
= ( produc8702345860417625034ar_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_717_swap__simp,axiom,
! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
( ( produc3996604198558763126on_val @ ( produc74620467203698198on_val @ X @ Y ) )
= ( produc1897429902272487136r_prat @ Y @ X ) ) ).
% swap_simp
thf(fact_718_fst__swap,axiom,
! [X: produc7606469099237612818ar_nat] :
( ( produc6204653154488278070t_char @ ( produc4410195085741832554ar_nat @ X ) )
= ( produc1259554233722553080ar_nat @ X ) ) ).
% fst_swap
thf(fact_719_fst__swap,axiom,
! [X: produc7776263825685906734r_prat] :
( ( produc5239619271231976746on_val @ ( produc5819413633627552064r_prat @ X ) )
= ( produc4142415063207305778r_prat @ X ) ) ).
% fst_swap
thf(fact_720_fst__swap,axiom,
! [X: produc9003781588162623004on_val] :
( ( produc7062428706300765684r_prat @ ( produc3996604198558763126on_val @ X ) )
= ( produc2319605628138516840on_val @ X ) ) ).
% fst_swap
thf(fact_721_package__logic_Ovalid__package__set_Ocong,axiom,
packag3877571396724326311_state = packag3877571396724326311_state ).
% package_logic.valid_package_set.cong
thf(fact_722_compatible__options_Ocases,axiom,
! [X: produc5122021904355189735on_val] :
( ! [A6: val,B4: val] :
( X
!= ( produc904852209897547607on_val @ ( some_val @ A6 ) @ ( some_val @ B4 ) ) )
=> ( ! [Uv: option_val] :
( X
!= ( produc904852209897547607on_val @ none_val @ Uv ) )
=> ~ ! [Uu: option_val] :
( X
!= ( produc904852209897547607on_val @ Uu @ none_val ) ) ) ) ).
% compatible_options.cases
thf(fact_723_compatible__options_Ocases,axiom,
! [X: produc3142500478612311029_state] :
( ! [A6: state,B4: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B4 ) ) )
=> ( ! [Uv: option_state] :
( X
!= ( produc9160152616361873709_state @ none_state @ Uv ) )
=> ~ ! [Uu: option_state] :
( X
!= ( produc9160152616361873709_state @ Uu @ none_state ) ) ) ) ).
% compatible_options.cases
thf(fact_724_positivity,axiom,
! [A4: state,B2: state,C: state] :
( ( ( plus @ A4 @ B2 )
= ( some_state @ C ) )
=> ( ( ( some_state @ C )
= ( plus @ C @ C ) )
=> ( ( some_state @ A4 )
= ( plus @ A4 @ A4 ) ) ) ) ).
% positivity
thf(fact_725_asso1,axiom,
! [A4: state,B2: state,Ab: state,C: state,Bc: state] :
( ( ( ( plus @ A4 @ B2 )
= ( some_state @ Ab ) )
& ( ( plus @ B2 @ C )
= ( some_state @ Bc ) ) )
=> ( ( plus @ Ab @ C )
= ( plus @ A4 @ Bc ) ) ) ).
% asso1
thf(fact_726_PartialSA_Osplus_Ocases,axiom,
! [X: produc3142500478612311029_state] :
( ! [Uu: option_state] :
( X
!= ( produc9160152616361873709_state @ none_state @ Uu ) )
=> ( ! [V: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) )
=> ~ ! [A6: state,B4: state] :
( X
!= ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B4 ) ) ) ) ) ).
% PartialSA.splus.cases
thf(fact_727_larger__heapI,axiom,
! [H: produc1457211279475724562t_char > option_val,H5: produc1457211279475724562t_char > option_val] :
( ! [Hl3: produc1457211279475724562t_char,X2: val] :
( ( ( H @ Hl3 )
= ( some_val @ X2 ) )
=> ( ( H5 @ Hl3 )
= ( some_val @ X2 ) ) )
=> ( larger1532547405891354081ar_val @ H5 @ H ) ) ).
% larger_heapI
thf(fact_728_larger__heap__def,axiom,
( larger1532547405891354081ar_val
= ( ^ [H6: produc1457211279475724562t_char > option_val,H4: produc1457211279475724562t_char > option_val] :
! [Hl2: produc1457211279475724562t_char,X3: val] :
( ( ( H4 @ Hl2 )
= ( some_val @ X3 ) )
=> ( ( H6 @ Hl2 )
= ( some_val @ X3 ) ) ) ) ) ).
% larger_heap_def
thf(fact_729_compatible__heapsI,axiom,
! [H: produc1457211279475724562t_char > option_val,H5: produc1457211279475724562t_char > option_val] :
( ! [Hl3: produc1457211279475724562t_char,A6: val,B4: val] :
( ( ( H @ Hl3 )
= ( some_val @ A6 ) )
=> ( ( ( H5 @ Hl3 )
= ( some_val @ B4 ) )
=> ( A6 = B4 ) ) )
=> ( compatible_heaps @ H @ H5 ) ) ).
% compatible_heapsI
thf(fact_730_core__is__smaller,axiom,
( some_state
= ( ^ [X3: state] : ( plus @ X3 @ ( core @ X3 ) ) ) ) ).
% core_is_smaller
thf(fact_731_core__is__pure,axiom,
! [X: state] :
( ( some_state @ ( core @ X ) )
= ( plus @ ( core @ X ) @ ( core @ X ) ) ) ).
% core_is_pure
thf(fact_732_cancellative,axiom,
! [A4: state,B2: state,X: state,Y: state] :
( ( ( some_state @ A4 )
= ( plus @ B2 @ X ) )
=> ( ( ( some_state @ A4 )
= ( plus @ B2 @ Y ) )
=> ( ( ( core @ X )
= ( core @ Y ) )
=> ( X = Y ) ) ) ) ).
% cancellative
thf(fact_733_core__sum,axiom,
! [C: state,A4: state,B2: state] :
( ( ( some_state @ C )
= ( plus @ A4 @ B2 ) )
=> ( ( some_state @ ( core @ C ) )
= ( plus @ ( core @ A4 ) @ ( core @ B2 ) ) ) ) ).
% core_sum
thf(fact_734_core__max,axiom,
! [X: state,C: state] :
( ( ( some_state @ X )
= ( plus @ X @ C ) )
=> ? [R4: state] :
( ( some_state @ ( core @ X ) )
= ( plus @ C @ R4 ) ) ) ).
% core_max
thf(fact_735_PartialSA_Obigger__sum__smaller,axiom,
! [C: state,A4: state,B2: state,A5: state] :
( ( ( some_state @ C )
= ( plus @ A4 @ B2 ) )
=> ( ( greater @ A4 @ A5 )
=> ? [B7: state] :
( ( greater @ B7 @ B2 )
& ( ( some_state @ C )
= ( plus @ A5 @ B7 ) ) ) ) ) ).
% PartialSA.bigger_sum_smaller
thf(fact_736_PartialSA_Oaddition__bigger,axiom,
! [A5: state,A4: state,X5: state,B2: state,X: state] :
( ( greater @ A5 @ A4 )
=> ( ( ( some_state @ X5 )
= ( plus @ A5 @ B2 ) )
=> ( ( ( some_state @ X )
= ( plus @ A4 @ B2 ) )
=> ( greater @ X5 @ X ) ) ) ) ).
% PartialSA.addition_bigger
thf(fact_737_PartialSA_Ogreater__equiv,axiom,
( greater
= ( ^ [A3: state,B: state] :
? [C4: state] :
( ( some_state @ A3 )
= ( plus @ C4 @ B ) ) ) ) ).
% PartialSA.greater_equiv
thf(fact_738_PartialSA_Ogreater__def,axiom,
( greater
= ( ^ [A3: state,B: state] :
? [C4: state] :
( ( some_state @ A3 )
= ( plus @ B @ C4 ) ) ) ) ).
% PartialSA.greater_def
thf(fact_739_PartialSA_Obigger__sum,axiom,
! [Phi5: state,A4: state,B2: state,Phi6: state] :
( ( ( some_state @ Phi5 )
= ( plus @ A4 @ B2 ) )
=> ( ( greater @ Phi6 @ Phi5 )
=> ? [B7: state] :
( ( greater @ B7 @ B2 )
& ( ( some_state @ Phi6 )
= ( plus @ A4 @ B7 ) ) ) ) ) ).
% PartialSA.bigger_sum
thf(fact_740_stable__sum,axiom,
! [A4: state,B2: state,X: state] :
( ( stable @ A4 )
=> ( ( stable @ B2 )
=> ( ( ( some_state @ X )
= ( plus @ A4 @ B2 ) )
=> ( stable @ X ) ) ) ) ).
% stable_sum
thf(fact_741_unit__neutral,axiom,
( some_state
= ( ^ [A3: state] : ( plus @ A3 @ unit ) ) ) ).
% unit_neutral
thf(fact_742_PartialSA_Odefined__sum__move,axiom,
! [A4: state,B2: state,X: state,Y: state,A5: state] :
( ( defined @ A4 @ B2 )
=> ( ( ( some_state @ B2 )
= ( plus @ X @ Y ) )
=> ( ( ( some_state @ A5 )
= ( plus @ A4 @ X ) )
=> ( defined @ A5 @ Y ) ) ) ) ).
% PartialSA.defined_sum_move
thf(fact_743_PartialSA_Oadd__set__elem,axiom,
! [Phi5: state,A: set_state,B5: set_state] :
( ( member_state @ Phi5 @ ( add_set @ A @ B5 ) )
= ( ? [A3: state,B: state] :
( ( ( some_state @ Phi5 )
= ( plus @ A3 @ B ) )
& ( member_state @ A3 @ A )
& ( member_state @ B @ B5 ) ) ) ) ).
% PartialSA.add_set_elem
thf(fact_744_PartialSA_Ox__elem__set__product,axiom,
! [X: state,A: set_state,B5: set_state] :
( ( member_state @ X @ ( add_set @ A @ B5 ) )
= ( ? [A3: state,B: state] :
( ( member_state @ A3 @ A )
& ( member_state @ B @ B5 )
& ( ( some_state @ X )
= ( plus @ A3 @ B ) ) ) ) ) ).
% PartialSA.x_elem_set_product
thf(fact_745_asso2,axiom,
! [A4: state,B2: state,Ab: state,C: state] :
( ( ( ( plus @ A4 @ B2 )
= ( some_state @ Ab ) )
& ( ( plus @ B2 @ C )
= none_state ) )
=> ( ( plus @ Ab @ C )
= none_state ) ) ).
% asso2
thf(fact_746_PartialSA_Oasso3,axiom,
! [A4: state,B2: state,C: state,Bc: state] :
( ( ( plus @ A4 @ B2 )
= none_state )
=> ( ( ( plus @ B2 @ C )
= ( some_state @ Bc ) )
=> ( ( plus @ A4 @ Bc )
= none_state ) ) ) ).
% PartialSA.asso3
thf(fact_747_PartialSA_Ogreater__than__sum__exists,axiom,
! [A4: state,B2: state,B1: state,B22: state] :
( ( greater @ A4 @ B2 )
=> ( ( ( some_state @ B2 )
= ( plus @ B1 @ B22 ) )
=> ? [R4: state] :
( ( ( some_state @ A4 )
= ( plus @ R4 @ B22 ) )
& ( greater @ ( core @ R4 ) @ ( core @ A4 ) )
& ( greater @ R4 @ B1 ) ) ) ) ).
% PartialSA.greater_than_sum_exists
thf(fact_748_PartialSA_Obigger__core__sum__defined,axiom,
! [A4: state,B2: state] :
( ( greater @ ( core @ A4 ) @ B2 )
=> ( ( some_state @ A4 )
= ( plus @ A4 @ B2 ) ) ) ).
% PartialSA.bigger_core_sum_defined
thf(fact_749_PartialSA_Osmaller__than__core,axiom,
! [Y: state,X: state,Z: state] :
( ( greater @ Y @ X )
=> ( ( ( some_state @ Z )
= ( plus @ X @ ( core @ Y ) ) )
=> ( ( core @ Z )
= ( core @ Y ) ) ) ) ).
% PartialSA.smaller_than_core
thf(fact_750_PartialSA_Ominus__unique,axiom,
! [B2: state,A4: state,X: state,Y: state] :
( ( ( ( some_state @ B2 )
= ( plus @ A4 @ X ) )
& ( greater @ X @ ( core @ B2 ) ) )
=> ( ( ( ( some_state @ B2 )
= ( plus @ A4 @ Y ) )
& ( greater @ Y @ ( core @ B2 ) ) )
=> ( X = Y ) ) ) ).
% PartialSA.minus_unique
thf(fact_751_PartialSA_Ominus__exists,axiom,
! [B2: state,A4: state] :
( ( greater @ B2 @ A4 )
=> ? [X2: state] :
( ( ( some_state @ B2 )
= ( plus @ A4 @ X2 ) )
& ( greater @ X2 @ ( core @ B2 ) ) ) ) ).
% PartialSA.minus_exists
thf(fact_752_PartialSA_Oextract__core,axiom,
! [B2: state,A4: state,X: state] :
( ( ( ( some_state @ B2 )
= ( plus @ A4 @ X ) )
& ( greater @ X @ ( core @ B2 ) ) )
=> ( ( core @ X )
= ( core @ B2 ) ) ) ).
% PartialSA.extract_core
thf(fact_753_plus__charact_I1_J,axiom,
! [A4: state,B2: state,X: state] :
( ( ( plus @ A4 @ B2 )
= ( some_state @ X ) )
=> ( ( get_m @ X )
= ( add_ma6586698345353345811t_char @ ( get_m @ A4 ) @ ( get_m @ B2 ) ) ) ) ).
% plus_charact(1)
thf(fact_754_in__cwand,axiom,
! [A: set_state,W: state,B5: set_state] :
( ! [A6: state,X2: state] :
( ( ( member_state @ A6 @ A )
& ( ( some_state @ X2 )
= ( plus @ ( r @ A6 @ W ) @ A6 ) ) )
=> ( member_state @ X2 @ B5 ) )
=> ( member_state @ W @ ( cwand @ A @ B5 ) ) ) ).
% in_cwand
thf(fact_755_notin__range__Some,axiom,
! [X: option_val] :
( ( ~ ( member_option_val @ X @ ( image_val_option_val @ some_val @ top_top_set_val ) ) )
= ( X = none_val ) ) ).
% notin_range_Some
thf(fact_756_notin__range__Some,axiom,
! [X: option_state] :
( ( ~ ( member_option_state @ X @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) )
= ( X = none_state ) ) ).
% notin_range_Some
thf(fact_757_wandI,axiom,
! [A: set_state,W: state,B5: set_state] :
( ! [A6: state,X2: state] :
( ( ( member_state @ A6 @ A )
& ( ( some_state @ X2 )
= ( plus @ A6 @ W ) ) )
=> ( member_state @ X2 @ B5 ) )
=> ( member_state @ W @ ( wand @ A @ B5 ) ) ) ).
% wandI
thf(fact_758_PartialSA_OminusI,axiom,
! [B2: state,A4: state,X: state] :
( ( ( some_state @ B2 )
= ( plus @ A4 @ X ) )
=> ( ( greater @ X @ ( core @ B2 ) )
=> ( X
= ( minus @ B2 @ A4 ) ) ) ) ).
% PartialSA.minusI
thf(fact_759_PartialSA_Ominus__equiv__def,axiom,
! [B2: state,A4: state] :
( ( greater @ B2 @ A4 )
=> ( ( ( some_state @ B2 )
= ( plus @ A4 @ ( minus @ B2 @ A4 ) ) )
& ( greater @ ( minus @ B2 @ A4 ) @ ( core @ B2 ) ) ) ) ).
% PartialSA.minus_equiv_def
thf(fact_760_PartialSA_Ominus__core__weaker,axiom,
! [A4: state,B2: state] :
( ( core @ ( minus @ A4 @ B2 ) )
= ( minus @ ( core @ A4 ) @ ( core @ B2 ) ) ) ).
% PartialSA.minus_core_weaker
thf(fact_761_PartialSA_Ominus__core,axiom,
! [A4: state,B2: state] :
( ( core @ ( minus @ A4 @ B2 ) )
= ( core @ A4 ) ) ).
% PartialSA.minus_core
thf(fact_762_PartialSA_Ominus__default,axiom,
! [B2: state,A4: state] :
( ~ ( greater @ B2 @ A4 )
=> ( ( minus @ B2 @ A4 )
= B2 ) ) ).
% PartialSA.minus_default
thf(fact_763_PartialSA_Ominus__smaller,axiom,
! [X: state,A4: state] :
( ( greater @ X @ A4 )
=> ( greater @ X @ ( minus @ X @ A4 ) ) ) ).
% PartialSA.minus_smaller
thf(fact_764_PartialSA_Ogreater__minus__trans,axiom,
! [Y: state,X: state,A4: state] :
( ( greater @ Y @ X )
=> ( ( greater @ X @ A4 )
=> ( greater @ ( minus @ Y @ A4 ) @ ( minus @ X @ A4 ) ) ) ) ).
% PartialSA.greater_minus_trans
thf(fact_765_PartialSA_Ominus__equiv__def__any__elem,axiom,
! [X: state,A4: state,B2: state] :
( ( ( some_state @ X )
= ( plus @ A4 @ B2 ) )
=> ( ( some_state @ ( minus @ X @ A4 ) )
= ( plus @ B2 @ ( core @ X ) ) ) ) ).
% PartialSA.minus_equiv_def_any_elem
thf(fact_766_PartialSA_Ominus__sum,axiom,
! [A4: state,B2: state,C: state,X: state] :
( ( ( some_state @ A4 )
= ( plus @ B2 @ C ) )
=> ( ( greater @ X @ A4 )
=> ( ( minus @ X @ A4 )
= ( minus @ ( minus @ X @ B2 ) @ C ) ) ) ) ).
% PartialSA.minus_sum
thf(fact_767_PartialSA_Ominus__some,axiom,
! [A4: state,B2: state] :
( ( greater @ A4 @ B2 )
=> ( ( some_state @ A4 )
= ( plus @ B2 @ ( minus @ A4 @ B2 ) ) ) ) ).
% PartialSA.minus_some
thf(fact_768_PartialSA_Ominus__bigger,axiom,
! [X: state,A4: state,B2: state] :
( ( ( some_state @ X )
= ( plus @ A4 @ B2 ) )
=> ( greater @ ( minus @ X @ A4 ) @ B2 ) ) ).
% PartialSA.minus_bigger
thf(fact_769_PartialSA_Ominus__and__plus,axiom,
! [Omega: state,Omega2: state,R: state,A4: state] :
( ( ( some_state @ Omega )
= ( plus @ Omega2 @ R ) )
=> ( ( greater @ Omega2 @ A4 )
=> ( ( some_state @ ( minus @ Omega @ A4 ) )
= ( plus @ ( minus @ Omega2 @ A4 ) @ R ) ) ) ) ).
% PartialSA.minus_and_plus
thf(fact_770_PartialSA_Oprove__last__completeness,axiom,
! [A5: state,A4: state,Nf1: state,F22: state] :
( ( greater @ A5 @ A4 )
=> ( ( ( some_state @ A4 )
= ( plus @ Nf1 @ F22 ) )
=> ( greater @ ( minus @ A5 @ Nf1 ) @ F22 ) ) ) ).
% PartialSA.prove_last_completeness
thf(fact_771_PartialSA_Opackage__rhs__connection__instantiate,axiom,
! [Phi5: state,F: state,S4: set_Pr4231240917283394591_state,Pc: state > $o,A: aassertion_state,Phi6: state,F5: state,S5: set_Pr4231240917283394591_state,A4: state,U2: state,T3: state > state] :
( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi6 @ F5 @ S5 )
=> ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U2 @ T3 ) ) @ S4 )
=> ~ ! [Au: state] :
( ( ( some_state @ Au )
= ( plus @ A4 @ U2 ) )
=> ~ ( ( defined @ Au @ ( minus @ ( T3 @ F5 ) @ ( T3 @ F ) ) )
=> ? [A8: state,U3: state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A8 @ ( produc8632743443909064678_state @ U3 @ T3 ) ) @ S5 )
& ( greater @ ( core @ A8 ) @ ( core @ A4 ) )
& ( ( plus @ Au @ ( minus @ ( T3 @ F5 ) @ ( T3 @ F ) ) )
= ( plus @ A8 @ U3 ) )
& ( greater @ U3 @ U2 )
& ( packag8070727096299822599_state @ plus @ core @ Pc @ A @ A8 @ U3 @ U2 ) ) ) ) ) ) ).
% PartialSA.package_rhs_connection_instantiate
thf(fact_772_PartialSA_Opackage__rhs__connectionI,axiom,
! [Phi5: state,F: state,Phi6: state,F5: state,S4: set_Pr4231240917283394591_state,S5: set_Pr4231240917283394591_state,Pc: state > $o,A: aassertion_state] :
( ( ( plus @ Phi5 @ F )
= ( plus @ Phi6 @ F5 ) )
=> ( ( stable @ F5 )
=> ( ( defined @ Phi5 @ F )
=> ( ( greater @ F5 @ F )
=> ( ! [A6: state,U: state,T4: state > state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T4 ) ) @ S4 )
=> ? [Au2: state] :
( ( ( some_state @ Au2 )
= ( plus @ A6 @ U ) )
& ( ( defined @ Au2 @ ( minus @ ( T4 @ F5 ) @ ( T4 @ F ) ) )
=> ? [A9: state,U4: state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A9 @ ( produc8632743443909064678_state @ U4 @ T4 ) ) @ S5 )
& ( greater @ ( core @ A9 ) @ ( core @ A6 ) )
& ( ( plus @ Au2 @ ( minus @ ( T4 @ F5 ) @ ( T4 @ F ) ) )
= ( plus @ A9 @ U4 ) )
& ( greater @ U4 @ U )
& ( packag8070727096299822599_state @ plus @ core @ Pc @ A @ A9 @ U4 @ U ) ) ) ) )
=> ( packag2054414653763826108_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi6 @ F5 @ S5 ) ) ) ) ) ) ).
% PartialSA.package_rhs_connectionI
thf(fact_773_add__states_Opelims,axiom,
! [X: produc9003781588162623004on_val,Xa: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
( ( ( add_states @ X @ Xa )
= Y )
=> ( ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ X @ Xa ) )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ! [Pi2: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
( ( Xa
= ( produc74620467203698198on_val @ Pi2 @ H3 ) )
=> ( ( Y
= ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi2 ) @ ( map_ad2908576761495157028ar_val @ H2 @ H3 ) ) )
=> ~ ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H2 ) @ ( produc74620467203698198on_val @ Pi2 @ H3 ) ) ) ) ) ) ) ) ).
% add_states.pelims
thf(fact_774_PartialSA_Obigger__the,axiom,
! [A4: state,X5: state,Y: state,X: state] :
( ( ( some_state @ A4 )
= ( plus @ X5 @ Y ) )
=> ( ( greater @ X5 @ X )
=> ( greater @ ( the_state @ ( plus @ ( core @ A4 ) @ X5 ) ) @ ( the_state @ ( plus @ ( core @ A4 ) @ X ) ) ) ) ) ).
% PartialSA.bigger_the
thf(fact_775_map__prod__surj,axiom,
! [F: state > state,G: state > state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ( ( image_state_state @ G @ top_top_set_state )
= top_top_set_state )
=> ( ( image_1391693815708851269_state @ ( produc2703234997178046178_state @ F @ G ) @ top_to4183510833390158213_state )
= top_to4183510833390158213_state ) ) ) ).
% map_prod_surj
thf(fact_776_map__prod__surj,axiom,
! [F: ( produc1457211279475724562t_char > prat ) > state,G: ( produc1457211279475724562t_char > option_val ) > state > state] :
( ( ( image_8169909065618884296_state @ F @ top_to6849109582029272579r_prat )
= top_top_set_state )
=> ( ( ( image_5740063759318933428_state @ G @ top_to3326603054294128128on_val )
= top_to3504463554866254201_state )
=> ( ( image_4494208899119068747_state @ ( produc4331275009068178118_state @ F @ G ) @ top_to1561438921946393474on_val )
= top_to1744471583079920734_state ) ) ) ).
% map_prod_surj
thf(fact_777_map__prod__surj,axiom,
! [F: produc9003781588162623004on_val > state,G: state > state] :
( ( ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val )
= top_top_set_state )
=> ( ( ( image_state_state @ G @ top_top_set_state )
= top_top_set_state )
=> ( ( image_64025966464648291_state @ ( produc6573729364289813572_state @ F @ G ) @ top_to6920239243093810151_state )
= top_to4183510833390158213_state ) ) ) ).
% map_prod_surj
thf(fact_778_map__prod__surj,axiom,
! [F: state > produc1457211279475724562t_char > prat,G: ( state > state ) > produc1457211279475724562t_char > option_val] :
( ( ( image_3361215723131658834r_prat @ F @ top_top_set_state )
= top_to6849109582029272579r_prat )
=> ( ( ( image_3765377486457420852on_val @ G @ top_to3504463554866254201_state )
= top_to3326603054294128128on_val )
=> ( ( image_4690842878458256517on_val @ ( produc5229369175610244304on_val @ F @ G ) @ top_to1744471583079920734_state )
= top_to1561438921946393474on_val ) ) ) ).
% map_prod_surj
thf(fact_779_map__prod__surj,axiom,
! [F: state > state,G: produc9003781588162623004on_val > state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ( ( image_1205529093420515805_state @ G @ top_to1561438921946393474on_val )
= top_top_set_state )
=> ( ( image_1045922925412474211_state @ ( produc690352061171932228_state @ F @ G ) @ top_to8212523096849629927on_val )
= top_to4183510833390158213_state ) ) ) ).
% map_prod_surj
thf(fact_780_map__prod__surj,axiom,
! [F: ( produc1457211279475724562t_char > prat ) > state,G: ( produc1457211279475724562t_char > option_val ) > produc3882902684156697774_state] :
( ( ( image_8169909065618884296_state @ F @ top_to6849109582029272579r_prat )
= top_top_set_state )
=> ( ( ( image_9216765683938615757_state @ G @ top_to3326603054294128128on_val )
= top_to1744471583079920734_state )
=> ( ( image_3204238854701872908_state @ ( produc1473691456219299387_state @ F @ G ) @ top_to1561438921946393474on_val )
= top_to174962084833059535_state ) ) ) ).
% map_prod_surj
thf(fact_781_map__prod__surj,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,G: state > state] :
( ( ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state )
= top_to1561438921946393474on_val )
=> ( ( ( image_state_state @ G @ top_top_set_state )
= top_top_set_state )
=> ( ( image_7983644280233404825_state @ ( produc8455034571308914634_state @ F @ G ) @ top_to309837729776113405_state )
= top_to6920239243093810151_state ) ) ) ).
% map_prod_surj
thf(fact_782_map__prod__surj,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,G: state > state] :
( ( ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val )
= top_to1744471583079920734_state )
=> ( ( ( image_state_state @ G @ top_top_set_state )
= top_top_set_state )
=> ( ( image_10599206356837129_state @ ( produc220911464644599184_state @ F @ G ) @ top_to6920239243093810151_state )
= top_to309837729776113405_state ) ) ) ).
% map_prod_surj
thf(fact_783_map__prod__surj,axiom,
! [F: state > state,G: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ( ( image_4690842878458256517on_val @ G @ top_to1744471583079920734_state )
= top_to1561438921946393474on_val )
=> ( ( image_7904539918191850823on_val @ ( produc7625766271970964958on_val @ F @ G ) @ top_to174962084833059535_state )
= top_to8212523096849629927on_val ) ) ) ).
% map_prod_surj
thf(fact_784_map__prod__surj,axiom,
! [F: state > state,G: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( ( image_state_state @ F @ top_top_set_state )
= top_top_set_state )
=> ( ( ( image_4494208899119068747_state @ G @ top_to1561438921946393474on_val )
= top_to1744471583079920734_state )
=> ( ( image_7092850632495505115_state @ ( produc7429132292631777188_state @ F @ G ) @ top_to8212523096849629927on_val )
= top_to174962084833059535_state ) ) ) ).
% map_prod_surj
thf(fact_785_map__prod__simp,axiom,
! [F: prat > prat,G: state > state,A4: prat,B2: state] :
( ( produc3975281815818237454_state @ F @ G @ ( produc6205719116230455971_state @ A4 @ B2 ) )
= ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_786_map__prod__simp,axiom,
! [F: prat > prat,G: state > set_state,A4: prat,B2: state] :
( ( produc8819337198805138372_state @ F @ G @ ( produc6205719116230455971_state @ A4 @ B2 ) )
= ( produc9080284787853255769_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_787_map__prod__simp,axiom,
! [F: prat > nat,G: state > list_char,A4: prat,B2: state] :
( ( produc5873735731534387765t_char @ F @ G @ ( produc6205719116230455971_state @ A4 @ B2 ) )
= ( produc120197956887798346t_char @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_788_map__prod__simp,axiom,
! [F: prat > prat,G: set_state > state,A4: prat,B2: set_state] :
( ( produc8620925688948321732_state @ F @ G @ ( produc9080284787853255769_state @ A4 @ B2 ) )
= ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_789_map__prod__simp,axiom,
! [F: nat > prat,G: list_char > state,A4: nat,B2: list_char] :
( ( produc3247067958781840309_state @ F @ G @ ( produc120197956887798346t_char @ A4 @ B2 ) )
= ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_790_map__prod__simp,axiom,
! [F: option_state > prat,G: option_state > state,A4: option_state,B2: option_state] :
( ( produc4196963940547574808_state @ F @ G @ ( produc9160152616361873709_state @ A4 @ B2 ) )
= ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_791_map__prod__simp,axiom,
! [F: prat > option_state,G: state > option_state,A4: prat,B2: state] :
( ( produc2558317419863738904_state @ F @ G @ ( produc6205719116230455971_state @ A4 @ B2 ) )
= ( produc9160152616361873709_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_792_map__prod__simp,axiom,
! [F: prat > prat,G: set_state > set_state,A4: prat,B2: set_state] :
( ( produc7528155264795859834_state @ F @ G @ ( produc9080284787853255769_state @ A4 @ B2 ) )
= ( produc9080284787853255769_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_793_map__prod__simp,axiom,
! [F: prat > nat,G: set_state > list_char,A4: prat,B2: set_state] :
( ( produc427163445725915883t_char @ F @ G @ ( produc9080284787853255769_state @ A4 @ B2 ) )
= ( produc120197956887798346t_char @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_794_map__prod__simp,axiom,
! [F: nat > prat,G: list_char > set_state,A4: nat,B2: list_char] :
( ( produc5902993617565070699_state @ F @ G @ ( produc120197956887798346t_char @ A4 @ B2 ) )
= ( produc9080284787853255769_state @ ( F @ A4 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_795_fst__map__prod,axiom,
! [F: nat > nat,G: list_char > list_char,X: produc1457211279475724562t_char] :
( ( produc6204653154488278070t_char @ ( produc5890707981334396508t_char @ F @ G @ X ) )
= ( F @ ( produc6204653154488278070t_char @ X ) ) ) ).
% fst_map_prod
thf(fact_796_fst__map__prod,axiom,
! [F: ( produc1457211279475724562t_char > prat ) > nat,G: ( produc1457211279475724562t_char > option_val ) > list_char,X: produc9003781588162623004on_val] :
( ( produc6204653154488278070t_char @ ( produc4139938480765124394t_char @ F @ G @ X ) )
= ( F @ ( produc5239619271231976746on_val @ X ) ) ) ).
% fst_map_prod
thf(fact_797_fst__map__prod,axiom,
! [F: nat > produc1457211279475724562t_char > prat,G: list_char > produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char] :
( ( produc5239619271231976746on_val @ ( produc2605942250945026184on_val @ F @ G @ X ) )
= ( F @ ( produc6204653154488278070t_char @ X ) ) ) ).
% fst_map_prod
thf(fact_798_fst__map__prod,axiom,
! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc9003781588162623004on_val] :
( ( produc5239619271231976746on_val @ ( produc8229809728907007980on_val @ F @ G @ X ) )
= ( F @ ( produc5239619271231976746on_val @ X ) ) ) ).
% fst_map_prod
thf(fact_799_snd__map__prod,axiom,
! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc9003781588162623004on_val] :
( ( produc2319605628138516840on_val @ ( produc8229809728907007980on_val @ F @ G @ X ) )
= ( G @ ( produc2319605628138516840on_val @ X ) ) ) ).
% snd_map_prod
thf(fact_800_map__prod__imageI,axiom,
! [A4: prat,B2: state,R3: set_Pr6039890582625207499_state,F: prat > prat,G: state > state] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A4 @ B2 ) @ R3 )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_340407000345765061_state @ ( produc3975281815818237454_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_801_map__prod__imageI,axiom,
! [A4: prat,B2: state,R3: set_Pr6039890582625207499_state,F: prat > prat,G: state > set_state] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A4 @ B2 ) @ R3 )
=> ( member6708919011922255306_state @ ( produc9080284787853255769_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_1749495751380664187_state @ ( produc8819337198805138372_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_802_map__prod__imageI,axiom,
! [A4: prat,B2: state,R3: set_Pr6039890582625207499_state,F: prat > nat,G: state > list_char] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A4 @ B2 ) @ R3 )
=> ( member484235747325421115t_char @ ( produc120197956887798346t_char @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_7572051117385396844t_char @ ( produc5873735731534387765t_char @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_803_map__prod__imageI,axiom,
! [A4: prat,B2: set_state,R3: set_Pr6847263425858009217_state,F: prat > prat,G: set_state > state] :
( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A4 @ B2 ) @ R3 )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_1320018901640642191_state @ ( produc8620925688948321732_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_804_map__prod__imageI,axiom,
! [A4: nat,B2: list_char,R3: set_Pr2507339680178222706t_char,F: nat > prat,G: list_char > state] :
( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A4 @ B2 ) @ R3 )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_728606647374750302_state @ ( produc3247067958781840309_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_805_map__prod__imageI,axiom,
! [A4: state,B2: state > state,R3: set_Pr2391701600042252046_state,F: state > prat,G: ( state > state ) > state] :
( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A4 @ B2 ) @ R3 )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_5224139304855891394_state @ ( produc7645549320084024913_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_806_map__prod__imageI,axiom,
! [A4: option_state,B2: option_state,R3: set_Pr1688445902015331925_state,F: option_state > prat,G: option_state > state] :
( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A4 @ B2 ) @ R3 )
=> ( member6226390569306242836_state @ ( produc6205719116230455971_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_5268209060877288123_state @ ( produc4196963940547574808_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_807_map__prod__imageI,axiom,
! [A4: prat,B2: state,R3: set_Pr6039890582625207499_state,F: prat > state,G: state > state > state] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A4 @ B2 ) @ R3 )
=> ( member6715010751053357271_state @ ( produc8632743443909064678_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_1979208616201753608_state @ ( produc1236476099633551313_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_808_map__prod__imageI,axiom,
! [A4: prat,B2: state,R3: set_Pr6039890582625207499_state,F: prat > option_state,G: state > option_state] :
( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A4 @ B2 ) @ R3 )
=> ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_444406330706641487_state @ ( produc2558317419863738904_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_809_map__prod__imageI,axiom,
! [A4: prat,B2: set_state,R3: set_Pr6847263425858009217_state,F: prat > prat,G: set_state > set_state] :
( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A4 @ B2 ) @ R3 )
=> ( member6708919011922255306_state @ ( produc9080284787853255769_state @ ( F @ A4 ) @ ( G @ B2 ) ) @ ( image_2686947732813934917_state @ ( produc7528155264795859834_state @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_810_package__logic_Opackage__sat_Ocong,axiom,
packag8070727096299822599_state = packag8070727096299822599_state ).
% package_logic.package_sat.cong
thf(fact_811_package__logic_Opackage__rhs__connection_Ocong,axiom,
packag2054414653763826108_state = packag2054414653763826108_state ).
% package_logic.package_rhs_connection.cong
thf(fact_812_compatible__heaps__sum,axiom,
! [A4: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
( ( compatible_heaps @ A4 @ B2 )
=> ( ( compatible_heaps @ A4 @ C )
=> ( compatible_heaps @ A4 @ ( map_ad2908576761495157028ar_val @ B2 @ C ) ) ) ) ).
% compatible_heaps_sum
thf(fact_813_compatible__heaps__comm,axiom,
! [A4: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val] :
( ( compatible_heaps @ A4 @ B2 )
=> ( ( map_ad2908576761495157028ar_val @ A4 @ B2 )
= ( map_ad2908576761495157028ar_val @ B2 @ A4 ) ) ) ).
% compatible_heaps_comm
thf(fact_814_larger__heap__plus,axiom,
! [A4: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A4 @ B2 )
=> ( ( larger1532547405891354081ar_val @ A4 @ C )
=> ( larger1532547405891354081ar_val @ A4 @ ( map_ad2908576761495157028ar_val @ B2 @ C ) ) ) ) ).
% larger_heap_plus
thf(fact_815_empty__heap__neutral,axiom,
! [A4: produc1457211279475724562t_char > option_val] :
( ( map_ad2908576761495157028ar_val @ A4 @ empty_7512026568161686585ar_val )
= A4 ) ).
% empty_heap_neutral
thf(fact_816_prod__fun__imageE,axiom,
! [C: produc4844402304972151915_state,F: prat > prat,G: state > state,R3: set_Pr6039890582625207499_state] :
( ( member6226390569306242836_state @ C @ ( image_340407000345765061_state @ ( produc3975281815818237454_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: prat,Y5: state] :
( ( C
= ( produc6205719116230455971_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_817_prod__fun__imageE,axiom,
! [C: produc4844402304972151915_state,F: prat > prat,G: set_state > state,R3: set_Pr6847263425858009217_state] :
( ( member6226390569306242836_state @ C @ ( image_1320018901640642191_state @ ( produc8620925688948321732_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: prat,Y5: set_state] :
( ( C
= ( produc6205719116230455971_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_818_prod__fun__imageE,axiom,
! [C: produc4844402304972151915_state,F: nat > prat,G: list_char > state,R3: set_Pr2507339680178222706t_char] :
( ( member6226390569306242836_state @ C @ ( image_728606647374750302_state @ ( produc3247067958781840309_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: nat,Y5: list_char] :
( ( C
= ( produc6205719116230455971_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_819_prod__fun__imageE,axiom,
! [C: produc6251377953716854561_state,F: prat > prat,G: state > set_state,R3: set_Pr6039890582625207499_state] :
( ( member6708919011922255306_state @ C @ ( image_1749495751380664187_state @ ( produc8819337198805138372_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: prat,Y5: state] :
( ( C
= ( produc9080284787853255769_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_820_prod__fun__imageE,axiom,
! [C: produc1457211279475724562t_char,F: prat > nat,G: state > list_char,R3: set_Pr6039890582625207499_state] :
( ( member484235747325421115t_char @ C @ ( image_7572051117385396844t_char @ ( produc5873735731534387765t_char @ F @ G ) @ R3 ) )
=> ~ ! [X2: prat,Y5: state] :
( ( C
= ( produc120197956887798346t_char @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_821_prod__fun__imageE,axiom,
! [C: produc3882902684156697774_state,F: prat > state,G: state > state > state,R3: set_Pr6039890582625207499_state] :
( ( member6715010751053357271_state @ C @ ( image_1979208616201753608_state @ ( produc1236476099633551313_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: prat,Y5: state] :
( ( C
= ( produc8632743443909064678_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_822_prod__fun__imageE,axiom,
! [C: produc3142500478612311029_state,F: prat > option_state,G: state > option_state,R3: set_Pr6039890582625207499_state] :
( ( member3029510603097127326_state @ C @ ( image_444406330706641487_state @ ( produc2558317419863738904_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: prat,Y5: state] :
( ( C
= ( produc9160152616361873709_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_823_prod__fun__imageE,axiom,
! [C: produc4844402304972151915_state,F: state > prat,G: ( state > state ) > state,R3: set_Pr2391701600042252046_state] :
( ( member6226390569306242836_state @ C @ ( image_5224139304855891394_state @ ( produc7645549320084024913_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: state,Y5: state > state] :
( ( C
= ( produc6205719116230455971_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_824_prod__fun__imageE,axiom,
! [C: produc4844402304972151915_state,F: option_state > prat,G: option_state > state,R3: set_Pr1688445902015331925_state] :
( ( member6226390569306242836_state @ C @ ( image_5268209060877288123_state @ ( produc4196963940547574808_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: option_state,Y5: option_state] :
( ( C
= ( produc6205719116230455971_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_825_prod__fun__imageE,axiom,
! [C: produc6251377953716854561_state,F: prat > prat,G: set_state > set_state,R3: set_Pr6847263425858009217_state] :
( ( member6708919011922255306_state @ C @ ( image_2686947732813934917_state @ ( produc7528155264795859834_state @ F @ G ) @ R3 ) )
=> ~ ! [X2: prat,Y5: set_state] :
( ( C
= ( produc9080284787853255769_state @ ( F @ X2 ) @ ( G @ Y5 ) ) )
=> ~ ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X2 @ Y5 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_826_larger__heaps__sum__ineq,axiom,
! [A5: produc1457211279475724562t_char > option_val,A4: produc1457211279475724562t_char > option_val,B3: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val] :
( ( larger1532547405891354081ar_val @ A5 @ A4 )
=> ( ( larger1532547405891354081ar_val @ B3 @ B2 )
=> ( ( compatible_heaps @ A5 @ B3 )
=> ( larger1532547405891354081ar_val @ ( map_ad2908576761495157028ar_val @ A5 @ B3 ) @ ( map_ad2908576761495157028ar_val @ A4 @ B2 ) ) ) ) ) ).
% larger_heaps_sum_ineq
thf(fact_827_plus__charact_I2_J,axiom,
! [A4: state,B2: state,X: state] :
( ( ( plus @ A4 @ B2 )
= ( some_state @ X ) )
=> ( ( get_h @ X )
= ( map_ad2908576761495157028ar_val @ ( get_h @ A4 ) @ ( get_h @ B2 ) ) ) ) ).
% plus_charact(2)
thf(fact_828_add__states_Oelims,axiom,
! [X: produc9003781588162623004on_val,Xa: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
( ( ( add_states @ X @ Xa )
= Y )
=> ~ ! [Pi: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
( ( X
= ( produc74620467203698198on_val @ Pi @ H2 ) )
=> ! [Pi2: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
( ( Xa
= ( produc74620467203698198on_val @ Pi2 @ H3 ) )
=> ( Y
!= ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi2 ) @ ( map_ad2908576761495157028ar_val @ H2 @ H3 ) ) ) ) ) ) ).
% add_states.elims
thf(fact_829_add__states_Osimps,axiom,
! [Pi4: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val,Pi5: produc1457211279475724562t_char > prat,H5: produc1457211279475724562t_char > option_val] :
( ( add_states @ ( produc74620467203698198on_val @ Pi4 @ H ) @ ( produc74620467203698198on_val @ Pi5 @ H5 ) )
= ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi4 @ Pi5 ) @ ( map_ad2908576761495157028ar_val @ H @ H5 ) ) ) ).
% add_states.simps
thf(fact_830_PartialSA_Opackage__satI,axiom,
! [Pc: state > $o,A5: state,U5: state,U2: state,A: aassertion_state] :
( ( ( Pc @ ( core @ A5 ) )
=> ? [X4: state] :
( ( ( some_state @ X4 )
= ( plus @ ( core @ A5 ) @ ( minus @ U5 @ U2 ) ) )
& ( package_sat_state @ plus @ A @ X4 ) ) )
=> ( packag8070727096299822599_state @ plus @ core @ Pc @ A @ A5 @ U5 @ U2 ) ) ).
% PartialSA.package_satI
thf(fact_831_PartialSA_Opackage__sat__def,axiom,
! [Pc: state > $o,A: aassertion_state,A5: state,U5: state,U2: state] :
( ( packag8070727096299822599_state @ plus @ core @ Pc @ A @ A5 @ U5 @ U2 )
= ( ( Pc @ ( core @ A5 ) )
=> ? [X3: state] :
( ( ( some_state @ X3 )
= ( plus @ ( core @ A5 ) @ ( minus @ U5 @ U2 ) ) )
& ( package_sat_state @ plus @ A @ X3 ) ) ) ) ).
% PartialSA.package_sat_def
thf(fact_832_UNIV__option__conv,axiom,
( top_to853123400021307383on_val
= ( insert_option_val @ none_val @ ( image_val_option_val @ some_val @ top_top_set_val ) ) ) ).
% UNIV_option_conv
thf(fact_833_UNIV__option__conv,axiom,
( top_to7666338855062656496_state
= ( insert_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) ) ).
% UNIV_option_conv
thf(fact_834_PartialSA_Omax__projection__prop__pure__core,axiom,
sep_ma8214210560313151521_state @ plus @ ( sep_pure_state @ plus ) @ core ).
% PartialSA.max_projection_prop_pure_core
thf(fact_835_compatible__options_Oelims_I2_J,axiom,
! [X: option_state,Xa: option_state] :
( ( compat2278460363914054422_state @ X @ Xa )
=> ( ! [A6: state] :
( ( X
= ( some_state @ A6 ) )
=> ! [B4: state] :
( ( Xa
= ( some_state @ B4 ) )
=> ( A6 != B4 ) ) )
=> ( ( X != none_state )
=> ( Xa = none_state ) ) ) ) ).
% compatible_options.elims(2)
thf(fact_836_compatible__options_Oelims_I2_J,axiom,
! [X: option_val,Xa: option_val] :
( ( compat7209469420100717861ns_val @ X @ Xa )
=> ( ! [A6: val] :
( ( X
= ( some_val @ A6 ) )
=> ! [B4: val] :
( ( Xa
= ( some_val @ B4 ) )
=> ( A6 != B4 ) ) )
=> ( ( X != none_val )
=> ( Xa = none_val ) ) ) ) ).
% compatible_options.elims(2)
thf(fact_837_insert__absorb2,axiom,
! [X: state,A: set_state] :
( ( insert_state @ X @ ( insert_state @ X @ A ) )
= ( insert_state @ X @ A ) ) ).
% insert_absorb2
thf(fact_838_insert__iff,axiom,
! [A4: state,B2: state,A: set_state] :
( ( member_state @ A4 @ ( insert_state @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_state @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_839_insert__iff,axiom,
! [A4: produc7272280185086996969_state,B2: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( member7461423308196221312_state @ A4 @ ( insert175804457004744729_state @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member7461423308196221312_state @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_840_insert__iff,axiom,
! [A4: produc9003781588162623004on_val,B2: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( member2879030624002546611on_val @ A4 @ ( insert3030518898131756108on_val @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member2879030624002546611on_val @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_841_insert__iff,axiom,
! [A4: produc3882902684156697774_state,B2: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( member6715010751053357271_state @ A4 @ ( insert94804854012721790_state @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member6715010751053357271_state @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_842_insertCI,axiom,
! [A4: state,B5: set_state,B2: state] :
( ( ~ ( member_state @ A4 @ B5 )
=> ( A4 = B2 ) )
=> ( member_state @ A4 @ ( insert_state @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_843_insertCI,axiom,
! [A4: produc7272280185086996969_state,B5: set_Pr4231240917283394591_state,B2: produc7272280185086996969_state] :
( ( ~ ( member7461423308196221312_state @ A4 @ B5 )
=> ( A4 = B2 ) )
=> ( member7461423308196221312_state @ A4 @ ( insert175804457004744729_state @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_844_insertCI,axiom,
! [A4: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val,B2: produc9003781588162623004on_val] :
( ( ~ ( member2879030624002546611on_val @ A4 @ B5 )
=> ( A4 = B2 ) )
=> ( member2879030624002546611on_val @ A4 @ ( insert3030518898131756108on_val @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_845_insertCI,axiom,
! [A4: produc3882902684156697774_state,B5: set_Pr2391701600042252046_state,B2: produc3882902684156697774_state] :
( ( ~ ( member6715010751053357271_state @ A4 @ B5 )
=> ( A4 = B2 ) )
=> ( member6715010751053357271_state @ A4 @ ( insert94804854012721790_state @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_846_image__insert,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A4: produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ( image_4690842878458256517on_val @ F @ ( insert94804854012721790_state @ A4 @ B5 ) )
= ( insert3030518898131756108on_val @ ( F @ A4 ) @ ( image_4690842878458256517on_val @ F @ B5 ) ) ) ).
% image_insert
thf(fact_847_image__insert,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A4: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ( image_3204238854701872908_state @ F @ ( insert3030518898131756108on_val @ A4 @ B5 ) )
= ( insert175804457004744729_state @ ( F @ A4 ) @ ( image_3204238854701872908_state @ F @ B5 ) ) ) ).
% image_insert
thf(fact_848_image__insert,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A4: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ( image_4494208899119068747_state @ F @ ( insert3030518898131756108on_val @ A4 @ B5 ) )
= ( insert94804854012721790_state @ ( F @ A4 ) @ ( image_4494208899119068747_state @ F @ B5 ) ) ) ).
% image_insert
thf(fact_849_image__insert,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A4: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ( image_4170889944715819327on_val @ F @ ( insert3030518898131756108on_val @ A4 @ B5 ) )
= ( insert3030518898131756108on_val @ ( F @ A4 ) @ ( image_4170889944715819327on_val @ F @ B5 ) ) ) ).
% image_insert
thf(fact_850_image__insert,axiom,
! [F: produc9003781588162623004on_val > state,A4: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ( image_1205529093420515805_state @ F @ ( insert3030518898131756108on_val @ A4 @ B5 ) )
= ( insert_state @ ( F @ A4 ) @ ( image_1205529093420515805_state @ F @ B5 ) ) ) ).
% image_insert
thf(fact_851_image__insert,axiom,
! [F: state > state,A4: state,B5: set_state] :
( ( image_state_state @ F @ ( insert_state @ A4 @ B5 ) )
= ( insert_state @ ( F @ A4 ) @ ( image_state_state @ F @ B5 ) ) ) ).
% image_insert
thf(fact_852_insert__image,axiom,
! [X: state,A: set_state,F: state > state] :
( ( member_state @ X @ A )
=> ( ( insert_state @ ( F @ X ) @ ( image_state_state @ F @ A ) )
= ( image_state_state @ F @ A ) ) ) ).
% insert_image
thf(fact_853_insert__image,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state] :
( ( member7461423308196221312_state @ X @ A )
=> ( ( insert_state @ ( F @ X ) @ ( image_2710133271811360426_state @ F @ A ) )
= ( image_2710133271811360426_state @ F @ A ) ) ) ).
% insert_image
thf(fact_854_insert__image,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc7272280185086996969_state] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( insert175804457004744729_state @ ( F @ X ) @ ( image_3204238854701872908_state @ F @ A ) )
= ( image_3204238854701872908_state @ F @ A ) ) ) ).
% insert_image
thf(fact_855_insert__image,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( insert94804854012721790_state @ ( F @ X ) @ ( image_4494208899119068747_state @ F @ A ) )
= ( image_4494208899119068747_state @ F @ A ) ) ) ).
% insert_image
thf(fact_856_insert__image,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( insert3030518898131756108on_val @ ( F @ X ) @ ( image_4170889944715819327on_val @ F @ A ) )
= ( image_4170889944715819327on_val @ F @ A ) ) ) ).
% insert_image
thf(fact_857_insert__image,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > state] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( insert_state @ ( F @ X ) @ ( image_1205529093420515805_state @ F @ A ) )
= ( image_1205529093420515805_state @ F @ A ) ) ) ).
% insert_image
thf(fact_858_insert__image,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( insert3030518898131756108on_val @ ( F @ X ) @ ( image_4690842878458256517on_val @ F @ A ) )
= ( image_4690842878458256517on_val @ F @ A ) ) ) ).
% insert_image
thf(fact_859_insert__image,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( insert_state @ ( F @ X ) @ ( image_6759111139246981411_state @ F @ A ) )
= ( image_6759111139246981411_state @ F @ A ) ) ) ).
% insert_image
thf(fact_860_insert__subset,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
( ( ord_le4032880777312351359_state @ ( insert175804457004744729_state @ X @ A ) @ B5 )
= ( ( member7461423308196221312_state @ X @ B5 )
& ( ord_le4032880777312351359_state @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_861_insert__subset,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X @ A ) @ B5 )
= ( ( member2879030624002546611on_val @ X @ B5 )
& ( ord_le6900622908513426738on_val @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_862_insert__subset,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
( ( ord_le6483787752022044334_state @ ( insert94804854012721790_state @ X @ A ) @ B5 )
= ( ( member6715010751053357271_state @ X @ B5 )
& ( ord_le6483787752022044334_state @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_863_insert__subset,axiom,
! [X: state,A: set_state,B5: set_state] :
( ( ord_le2494988322063910608_state @ ( insert_state @ X @ A ) @ B5 )
= ( ( member_state @ X @ B5 )
& ( ord_le2494988322063910608_state @ A @ B5 ) ) ) ).
% insert_subset
thf(fact_864_package__logic_Osat_Ocong,axiom,
package_sat_state = package_sat_state ).
% package_logic.sat.cong
thf(fact_865_mk__disjoint__insert,axiom,
! [A4: state,A: set_state] :
( ( member_state @ A4 @ A )
=> ? [B8: set_state] :
( ( A
= ( insert_state @ A4 @ B8 ) )
& ~ ( member_state @ A4 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_866_mk__disjoint__insert,axiom,
! [A4: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( member7461423308196221312_state @ A4 @ A )
=> ? [B8: set_Pr4231240917283394591_state] :
( ( A
= ( insert175804457004744729_state @ A4 @ B8 ) )
& ~ ( member7461423308196221312_state @ A4 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_867_mk__disjoint__insert,axiom,
! [A4: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( member2879030624002546611on_val @ A4 @ A )
=> ? [B8: set_Pr7649655576376376530on_val] :
( ( A
= ( insert3030518898131756108on_val @ A4 @ B8 ) )
& ~ ( member2879030624002546611on_val @ A4 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_868_mk__disjoint__insert,axiom,
! [A4: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( member6715010751053357271_state @ A4 @ A )
=> ? [B8: set_Pr2391701600042252046_state] :
( ( A
= ( insert94804854012721790_state @ A4 @ B8 ) )
& ~ ( member6715010751053357271_state @ A4 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_869_insert__commute,axiom,
! [X: state,Y: state,A: set_state] :
( ( insert_state @ X @ ( insert_state @ Y @ A ) )
= ( insert_state @ Y @ ( insert_state @ X @ A ) ) ) ).
% insert_commute
thf(fact_870_insert__eq__iff,axiom,
! [A4: state,A: set_state,B2: state,B5: set_state] :
( ~ ( member_state @ A4 @ A )
=> ( ~ ( member_state @ B2 @ B5 )
=> ( ( ( insert_state @ A4 @ A )
= ( insert_state @ B2 @ B5 ) )
= ( ( ( A4 = B2 )
=> ( A = B5 ) )
& ( ( A4 != B2 )
=> ? [C5: set_state] :
( ( A
= ( insert_state @ B2 @ C5 ) )
& ~ ( member_state @ B2 @ C5 )
& ( B5
= ( insert_state @ A4 @ C5 ) )
& ~ ( member_state @ A4 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_871_insert__eq__iff,axiom,
! [A4: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,B2: produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] :
( ~ ( member7461423308196221312_state @ A4 @ A )
=> ( ~ ( member7461423308196221312_state @ B2 @ B5 )
=> ( ( ( insert175804457004744729_state @ A4 @ A )
= ( insert175804457004744729_state @ B2 @ B5 ) )
= ( ( ( A4 = B2 )
=> ( A = B5 ) )
& ( ( A4 != B2 )
=> ? [C5: set_Pr4231240917283394591_state] :
( ( A
= ( insert175804457004744729_state @ B2 @ C5 ) )
& ~ ( member7461423308196221312_state @ B2 @ C5 )
& ( B5
= ( insert175804457004744729_state @ A4 @ C5 ) )
& ~ ( member7461423308196221312_state @ A4 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_872_insert__eq__iff,axiom,
! [A4: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B2: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ~ ( member2879030624002546611on_val @ A4 @ A )
=> ( ~ ( member2879030624002546611on_val @ B2 @ B5 )
=> ( ( ( insert3030518898131756108on_val @ A4 @ A )
= ( insert3030518898131756108on_val @ B2 @ B5 ) )
= ( ( ( A4 = B2 )
=> ( A = B5 ) )
& ( ( A4 != B2 )
=> ? [C5: set_Pr7649655576376376530on_val] :
( ( A
= ( insert3030518898131756108on_val @ B2 @ C5 ) )
& ~ ( member2879030624002546611on_val @ B2 @ C5 )
& ( B5
= ( insert3030518898131756108on_val @ A4 @ C5 ) )
& ~ ( member2879030624002546611on_val @ A4 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_873_insert__eq__iff,axiom,
! [A4: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,B2: produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ~ ( member6715010751053357271_state @ A4 @ A )
=> ( ~ ( member6715010751053357271_state @ B2 @ B5 )
=> ( ( ( insert94804854012721790_state @ A4 @ A )
= ( insert94804854012721790_state @ B2 @ B5 ) )
= ( ( ( A4 = B2 )
=> ( A = B5 ) )
& ( ( A4 != B2 )
=> ? [C5: set_Pr2391701600042252046_state] :
( ( A
= ( insert94804854012721790_state @ B2 @ C5 ) )
& ~ ( member6715010751053357271_state @ B2 @ C5 )
& ( B5
= ( insert94804854012721790_state @ A4 @ C5 ) )
& ~ ( member6715010751053357271_state @ A4 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_874_insert__absorb,axiom,
! [A4: state,A: set_state] :
( ( member_state @ A4 @ A )
=> ( ( insert_state @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_875_insert__absorb,axiom,
! [A4: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( member7461423308196221312_state @ A4 @ A )
=> ( ( insert175804457004744729_state @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_876_insert__absorb,axiom,
! [A4: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( member2879030624002546611on_val @ A4 @ A )
=> ( ( insert3030518898131756108on_val @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_877_insert__absorb,axiom,
! [A4: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( member6715010751053357271_state @ A4 @ A )
=> ( ( insert94804854012721790_state @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_878_insert__ident,axiom,
! [X: state,A: set_state,B5: set_state] :
( ~ ( member_state @ X @ A )
=> ( ~ ( member_state @ X @ B5 )
=> ( ( ( insert_state @ X @ A )
= ( insert_state @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_879_insert__ident,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
( ~ ( member7461423308196221312_state @ X @ A )
=> ( ~ ( member7461423308196221312_state @ X @ B5 )
=> ( ( ( insert175804457004744729_state @ X @ A )
= ( insert175804457004744729_state @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_880_insert__ident,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ~ ( member2879030624002546611on_val @ X @ A )
=> ( ~ ( member2879030624002546611on_val @ X @ B5 )
=> ( ( ( insert3030518898131756108on_val @ X @ A )
= ( insert3030518898131756108on_val @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_881_insert__ident,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
( ~ ( member6715010751053357271_state @ X @ A )
=> ( ~ ( member6715010751053357271_state @ X @ B5 )
=> ( ( ( insert94804854012721790_state @ X @ A )
= ( insert94804854012721790_state @ X @ B5 ) )
= ( A = B5 ) ) ) ) ).
% insert_ident
thf(fact_882_Set_Oset__insert,axiom,
! [X: state,A: set_state] :
( ( member_state @ X @ A )
=> ~ ! [B8: set_state] :
( ( A
= ( insert_state @ X @ B8 ) )
=> ( member_state @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_883_Set_Oset__insert,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( member7461423308196221312_state @ X @ A )
=> ~ ! [B8: set_Pr4231240917283394591_state] :
( ( A
= ( insert175804457004744729_state @ X @ B8 ) )
=> ( member7461423308196221312_state @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_884_Set_Oset__insert,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( member2879030624002546611on_val @ X @ A )
=> ~ ! [B8: set_Pr7649655576376376530on_val] :
( ( A
= ( insert3030518898131756108on_val @ X @ B8 ) )
=> ( member2879030624002546611on_val @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_885_Set_Oset__insert,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( member6715010751053357271_state @ X @ A )
=> ~ ! [B8: set_Pr2391701600042252046_state] :
( ( A
= ( insert94804854012721790_state @ X @ B8 ) )
=> ( member6715010751053357271_state @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_886_insertI2,axiom,
! [A4: state,B5: set_state,B2: state] :
( ( member_state @ A4 @ B5 )
=> ( member_state @ A4 @ ( insert_state @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_887_insertI2,axiom,
! [A4: produc7272280185086996969_state,B5: set_Pr4231240917283394591_state,B2: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ A4 @ B5 )
=> ( member7461423308196221312_state @ A4 @ ( insert175804457004744729_state @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_888_insertI2,axiom,
! [A4: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val,B2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ A4 @ B5 )
=> ( member2879030624002546611on_val @ A4 @ ( insert3030518898131756108on_val @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_889_insertI2,axiom,
! [A4: produc3882902684156697774_state,B5: set_Pr2391701600042252046_state,B2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ A4 @ B5 )
=> ( member6715010751053357271_state @ A4 @ ( insert94804854012721790_state @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_890_insertI1,axiom,
! [A4: state,B5: set_state] : ( member_state @ A4 @ ( insert_state @ A4 @ B5 ) ) ).
% insertI1
thf(fact_891_insertI1,axiom,
! [A4: produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] : ( member7461423308196221312_state @ A4 @ ( insert175804457004744729_state @ A4 @ B5 ) ) ).
% insertI1
thf(fact_892_insertI1,axiom,
! [A4: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] : ( member2879030624002546611on_val @ A4 @ ( insert3030518898131756108on_val @ A4 @ B5 ) ) ).
% insertI1
thf(fact_893_insertI1,axiom,
! [A4: produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] : ( member6715010751053357271_state @ A4 @ ( insert94804854012721790_state @ A4 @ B5 ) ) ).
% insertI1
thf(fact_894_insertE,axiom,
! [A4: state,B2: state,A: set_state] :
( ( member_state @ A4 @ ( insert_state @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_state @ A4 @ A ) ) ) ).
% insertE
thf(fact_895_insertE,axiom,
! [A4: produc7272280185086996969_state,B2: produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( member7461423308196221312_state @ A4 @ ( insert175804457004744729_state @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member7461423308196221312_state @ A4 @ A ) ) ) ).
% insertE
thf(fact_896_insertE,axiom,
! [A4: produc9003781588162623004on_val,B2: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( member2879030624002546611on_val @ A4 @ ( insert3030518898131756108on_val @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member2879030624002546611on_val @ A4 @ A ) ) ) ).
% insertE
thf(fact_897_insertE,axiom,
! [A4: produc3882902684156697774_state,B2: produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( member6715010751053357271_state @ A4 @ ( insert94804854012721790_state @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member6715010751053357271_state @ A4 @ A ) ) ) ).
% insertE
thf(fact_898_sep__algebra_Opure_Ocong,axiom,
sep_pure_state = sep_pure_state ).
% sep_algebra.pure.cong
thf(fact_899_insert__subsetI,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,X6: set_Pr4231240917283394591_state] :
( ( member7461423308196221312_state @ X @ A )
=> ( ( ord_le4032880777312351359_state @ X6 @ A )
=> ( ord_le4032880777312351359_state @ ( insert175804457004744729_state @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_900_insert__subsetI,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,X6: set_Pr7649655576376376530on_val] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( ord_le6900622908513426738on_val @ X6 @ A )
=> ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_901_insert__subsetI,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,X6: set_Pr2391701600042252046_state] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( ord_le6483787752022044334_state @ X6 @ A )
=> ( ord_le6483787752022044334_state @ ( insert94804854012721790_state @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_902_insert__subsetI,axiom,
! [X: state,A: set_state,X6: set_state] :
( ( member_state @ X @ A )
=> ( ( ord_le2494988322063910608_state @ X6 @ A )
=> ( ord_le2494988322063910608_state @ ( insert_state @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_903_insert__mono,axiom,
! [C2: set_state,D: set_state,A4: state] :
( ( ord_le2494988322063910608_state @ C2 @ D )
=> ( ord_le2494988322063910608_state @ ( insert_state @ A4 @ C2 ) @ ( insert_state @ A4 @ D ) ) ) ).
% insert_mono
thf(fact_904_subset__insert,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
( ~ ( member7461423308196221312_state @ X @ A )
=> ( ( ord_le4032880777312351359_state @ A @ ( insert175804457004744729_state @ X @ B5 ) )
= ( ord_le4032880777312351359_state @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_905_subset__insert,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
( ~ ( member2879030624002546611on_val @ X @ A )
=> ( ( ord_le6900622908513426738on_val @ A @ ( insert3030518898131756108on_val @ X @ B5 ) )
= ( ord_le6900622908513426738on_val @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_906_subset__insert,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
( ~ ( member6715010751053357271_state @ X @ A )
=> ( ( ord_le6483787752022044334_state @ A @ ( insert94804854012721790_state @ X @ B5 ) )
= ( ord_le6483787752022044334_state @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_907_subset__insert,axiom,
! [X: state,A: set_state,B5: set_state] :
( ~ ( member_state @ X @ A )
=> ( ( ord_le2494988322063910608_state @ A @ ( insert_state @ X @ B5 ) )
= ( ord_le2494988322063910608_state @ A @ B5 ) ) ) ).
% subset_insert
thf(fact_908_subset__insertI,axiom,
! [B5: set_state,A4: state] : ( ord_le2494988322063910608_state @ B5 @ ( insert_state @ A4 @ B5 ) ) ).
% subset_insertI
thf(fact_909_subset__insertI2,axiom,
! [A: set_state,B5: set_state,B2: state] :
( ( ord_le2494988322063910608_state @ A @ B5 )
=> ( ord_le2494988322063910608_state @ A @ ( insert_state @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_910_insert__UNIV,axiom,
! [X: state] :
( ( insert_state @ X @ top_top_set_state )
= top_top_set_state ) ).
% insert_UNIV
thf(fact_911_compatible__options_Osimps_I1_J,axiom,
! [A4: state,B2: state] :
( ( compat2278460363914054422_state @ ( some_state @ A4 ) @ ( some_state @ B2 ) )
= ( A4 = B2 ) ) ).
% compatible_options.simps(1)
thf(fact_912_compatible__options_Osimps_I1_J,axiom,
! [A4: val,B2: val] :
( ( compat7209469420100717861ns_val @ ( some_val @ A4 ) @ ( some_val @ B2 ) )
= ( A4 = B2 ) ) ).
% compatible_options.simps(1)
thf(fact_913_compatible__options_Oelims_I3_J,axiom,
! [X: option_state,Xa: option_state] :
( ~ ( compat2278460363914054422_state @ X @ Xa )
=> ~ ! [A6: state] :
( ( X
= ( some_state @ A6 ) )
=> ! [B4: state] :
( ( Xa
= ( some_state @ B4 ) )
=> ( A6 = B4 ) ) ) ) ).
% compatible_options.elims(3)
thf(fact_914_compatible__options_Oelims_I3_J,axiom,
! [X: option_val,Xa: option_val] :
( ~ ( compat7209469420100717861ns_val @ X @ Xa )
=> ~ ! [A6: val] :
( ( X
= ( some_val @ A6 ) )
=> ! [B4: val] :
( ( Xa
= ( some_val @ B4 ) )
=> ( A6 = B4 ) ) ) ) ).
% compatible_options.elims(3)
thf(fact_915_compatible__options_Osimps_I2_J,axiom,
! [Uv2: option_state] : ( compat2278460363914054422_state @ none_state @ Uv2 ) ).
% compatible_options.simps(2)
thf(fact_916_compatible__options_Osimps_I2_J,axiom,
! [Uv2: option_val] : ( compat7209469420100717861ns_val @ none_val @ Uv2 ) ).
% compatible_options.simps(2)
thf(fact_917_compatible__options_Osimps_I3_J,axiom,
! [Uu2: option_state] : ( compat2278460363914054422_state @ Uu2 @ none_state ) ).
% compatible_options.simps(3)
thf(fact_918_compatible__options_Osimps_I3_J,axiom,
! [Uu2: option_val] : ( compat7209469420100717861ns_val @ Uu2 @ none_val ) ).
% compatible_options.simps(3)
thf(fact_919_PartialSA_Opure__def,axiom,
! [A4: state] :
( ( sep_pure_state @ plus @ A4 )
= ( ( some_state @ A4 )
= ( plus @ A4 @ A4 ) ) ) ).
% PartialSA.pure_def
thf(fact_920_PartialSA_Opure__stable,axiom,
! [A4: state,B2: state,C: state] :
( ( sep_pure_state @ plus @ A4 )
=> ( ( sep_pure_state @ plus @ B2 )
=> ( ( ( some_state @ C )
= ( plus @ A4 @ B2 ) )
=> ( sep_pure_state @ plus @ C ) ) ) ) ).
% PartialSA.pure_stable
thf(fact_921_PartialSA_Opure__smaller,axiom,
! [A4: state,B2: state] :
( ( sep_pure_state @ plus @ A4 )
=> ( ( greater @ A4 @ B2 )
=> ( sep_pure_state @ plus @ B2 ) ) ) ).
% PartialSA.pure_smaller
thf(fact_922_compatible__heaps__def,axiom,
( compatible_heaps
= ( ^ [H4: produc1457211279475724562t_char > option_val,H6: produc1457211279475724562t_char > option_val] :
! [Hl2: produc1457211279475724562t_char] : ( compat7209469420100717861ns_val @ ( H4 @ Hl2 ) @ ( H6 @ Hl2 ) ) ) ) ).
% compatible_heaps_def
thf(fact_923_PartialSA_Osmaller__pure__sum__smaller,axiom,
! [Y: state,A4: state,B2: state,X: state] :
( ( greater @ Y @ A4 )
=> ( ( greater @ Y @ B2 )
=> ( ( ( some_state @ X )
= ( plus @ A4 @ B2 ) )
=> ( ( sep_pure_state @ plus @ B2 )
=> ( greater @ Y @ X ) ) ) ) ) ).
% PartialSA.smaller_pure_sum_smaller
thf(fact_924_compatible__options_Oelims_I1_J,axiom,
! [X: option_state,Xa: option_state,Y: $o] :
( ( ( compat2278460363914054422_state @ X @ Xa )
= Y )
=> ( ! [A6: state] :
( ( X
= ( some_state @ A6 ) )
=> ! [B4: state] :
( ( Xa
= ( some_state @ B4 ) )
=> ( Y
= ( A6 != B4 ) ) ) )
=> ( ( ( X = none_state )
=> ~ Y )
=> ~ ( ( Xa = none_state )
=> ~ Y ) ) ) ) ).
% compatible_options.elims(1)
thf(fact_925_compatible__options_Oelims_I1_J,axiom,
! [X: option_val,Xa: option_val,Y: $o] :
( ( ( compat7209469420100717861ns_val @ X @ Xa )
= Y )
=> ( ! [A6: val] :
( ( X
= ( some_val @ A6 ) )
=> ! [B4: val] :
( ( Xa
= ( some_val @ B4 ) )
=> ( Y
= ( A6 != B4 ) ) ) )
=> ( ( ( X = none_val )
=> ~ Y )
=> ~ ( ( Xa = none_val )
=> ~ Y ) ) ) ) ).
% compatible_options.elims(1)
thf(fact_926_compatible__options_Opelims_I1_J,axiom,
! [X: option_state,Xa: option_state,Y: $o] :
( ( ( compat2278460363914054422_state @ X @ Xa )
= Y )
=> ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ Xa ) )
=> ( ! [A6: state] :
( ( X
= ( some_state @ A6 ) )
=> ! [B4: state] :
( ( Xa
= ( some_state @ B4 ) )
=> ( ( Y
= ( A6 = B4 ) )
=> ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B4 ) ) ) ) ) )
=> ( ( ( X = none_state )
=> ( Y
=> ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) ) )
=> ~ ( ( Xa = none_state )
=> ( Y
=> ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ none_state ) ) ) ) ) ) ) ) ).
% compatible_options.pelims(1)
thf(fact_927_compatible__options_Opelims_I1_J,axiom,
! [X: option_val,Xa: option_val,Y: $o] :
( ( ( compat7209469420100717861ns_val @ X @ Xa )
= Y )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ Xa ) )
=> ( ! [A6: val] :
( ( X
= ( some_val @ A6 ) )
=> ! [B4: val] :
( ( Xa
= ( some_val @ B4 ) )
=> ( ( Y
= ( A6 = B4 ) )
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A6 ) @ ( some_val @ B4 ) ) ) ) ) )
=> ( ( ( X = none_val )
=> ( Y
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) ) )
=> ~ ( ( Xa = none_val )
=> ( Y
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ none_val ) ) ) ) ) ) ) ) ).
% compatible_options.pelims(1)
thf(fact_928_compatible__options_Opelims_I2_J,axiom,
! [X: option_state,Xa: option_state] :
( ( compat2278460363914054422_state @ X @ Xa )
=> ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ Xa ) )
=> ( ! [A6: state] :
( ( X
= ( some_state @ A6 ) )
=> ! [B4: state] :
( ( Xa
= ( some_state @ B4 ) )
=> ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B4 ) ) )
=> ( A6 != B4 ) ) ) )
=> ( ( ( X = none_state )
=> ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa ) ) )
=> ~ ( ( Xa = none_state )
=> ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ none_state ) ) ) ) ) ) ) ).
% compatible_options.pelims(2)
thf(fact_929_compatible__options_Opelims_I2_J,axiom,
! [X: option_val,Xa: option_val] :
( ( compat7209469420100717861ns_val @ X @ Xa )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ Xa ) )
=> ( ! [A6: val] :
( ( X
= ( some_val @ A6 ) )
=> ! [B4: val] :
( ( Xa
= ( some_val @ B4 ) )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A6 ) @ ( some_val @ B4 ) ) )
=> ( A6 != B4 ) ) ) )
=> ( ( ( X = none_val )
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa ) ) )
=> ~ ( ( Xa = none_val )
=> ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ none_val ) ) ) ) ) ) ) ).
% compatible_options.pelims(2)
thf(fact_930_compatible__options_Opelims_I3_J,axiom,
! [X: option_state,Xa: option_state] :
( ~ ( compat2278460363914054422_state @ X @ Xa )
=> ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ Xa ) )
=> ~ ! [A6: state] :
( ( X
= ( some_state @ A6 ) )
=> ! [B4: state] :
( ( Xa
= ( some_state @ B4 ) )
=> ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B4 ) ) )
=> ( A6 = B4 ) ) ) ) ) ) ).
% compatible_options.pelims(3)
thf(fact_931_compatible__options_Opelims_I3_J,axiom,
! [X: option_val,Xa: option_val] :
( ~ ( compat7209469420100717861ns_val @ X @ Xa )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ Xa ) )
=> ~ ! [A6: val] :
( ( X
= ( some_val @ A6 ) )
=> ! [B4: val] :
( ( Xa
= ( some_val @ B4 ) )
=> ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A6 ) @ ( some_val @ B4 ) ) )
=> ( A6 = B4 ) ) ) ) ) ) ).
% compatible_options.pelims(3)
thf(fact_932_PartialSA_Ois__footprint__standard__def,axiom,
! [W: state,A: aassertion_state,B5: aassertion_state] :
( ( packag7096211993527611489_state @ plus @ W @ A @ B5 )
= ( ! [A3: state,B: state] :
( ( ( package_sat_state @ plus @ A @ A3 )
& ( ( some_state @ B )
= ( plus @ A3 @ W ) ) )
=> ( package_sat_state @ plus @ B5 @ B ) ) ) ) ).
% PartialSA.is_footprint_standard_def
thf(fact_933_PartialSA_Ois__footprint__general__def,axiom,
! [W: state,R3: state > state > state,A: aassertion_state,B5: aassertion_state] :
( ( packag6192661653625646554_state @ plus @ W @ R3 @ A @ B5 )
= ( ! [A3: state,B: state] :
( ( ( package_sat_state @ plus @ A @ A3 )
& ( ( some_state @ B )
= ( plus @ A3 @ ( R3 @ A3 @ W ) ) ) )
=> ( package_sat_state @ plus @ B5 @ B ) ) ) ) ).
% PartialSA.is_footprint_general_def
thf(fact_934_package__logic_Ois__footprint__general_Ocong,axiom,
packag6192661653625646554_state = packag6192661653625646554_state ).
% package_logic.is_footprint_general.cong
thf(fact_935_package__logic_Ois__footprint__standard_Ocong,axiom,
packag7096211993527611489_state = packag7096211993527611489_state ).
% package_logic.is_footprint_standard.cong
thf(fact_936_PartialSA_Ois__footprint__generalI,axiom,
! [A: aassertion_state,R3: state > state > state,W: state,B5: aassertion_state] :
( ! [A6: state,B4: state] :
( ( ( package_sat_state @ plus @ A @ A6 )
& ( ( some_state @ B4 )
= ( plus @ A6 @ ( R3 @ A6 @ W ) ) ) )
=> ( package_sat_state @ plus @ B5 @ B4 ) )
=> ( packag6192661653625646554_state @ plus @ W @ R3 @ A @ B5 ) ) ).
% PartialSA.is_footprint_generalI
thf(fact_937_PartialSA_Ois__footprint__standardI,axiom,
! [A: aassertion_state,W: state,B5: aassertion_state] :
( ! [A6: state,B4: state] :
( ( ( package_sat_state @ plus @ A @ A6 )
& ( ( some_state @ B4 )
= ( plus @ A6 @ W ) ) )
=> ( package_sat_state @ plus @ B5 @ B4 ) )
=> ( packag7096211993527611489_state @ plus @ W @ A @ B5 ) ) ).
% PartialSA.is_footprint_standardI
thf(fact_938_PartialSA_Omono__pure__condI,axiom,
! [B2: state > $o] :
( ! [Phi4: state] :
( ( B2 @ Phi4 )
= ( B2 @ ( core @ Phi4 ) ) )
=> ( ! [Phi4: state,Phi3: state,R4: state] :
( ( ( sep_pure_state @ plus @ R4 )
& ( ( some_state @ Phi3 )
= ( plus @ Phi4 @ R4 ) )
& ~ ( B2 @ Phi4 ) )
=> ~ ( B2 @ Phi3 ) )
=> ( packag6595153354952283300_state @ plus @ core @ B2 ) ) ) ).
% PartialSA.mono_pure_condI
thf(fact_939_PartialSA_Omono__pure__cond__def,axiom,
! [B2: state > $o] :
( ( packag6595153354952283300_state @ plus @ core @ B2 )
= ( ! [Phi2: state] :
( ( B2 @ Phi2 )
= ( B2 @ ( core @ Phi2 ) ) )
& ! [Phi: state,Phi2: state,R2: state] :
( ( ( sep_pure_state @ plus @ R2 )
& ( ( some_state @ Phi )
= ( plus @ Phi2 @ R2 ) )
& ~ ( B2 @ Phi2 ) )
=> ~ ( B2 @ Phi ) ) ) ) ).
% PartialSA.mono_pure_cond_def
thf(fact_940_PartialSA_Owf__assertion__and__the,axiom,
! [A4: state,B2: state,A: aassertion_state] :
( ( defined @ ( core @ A4 ) @ B2 )
=> ( ( package_sat_state @ plus @ A @ B2 )
=> ( ( packag7298478212905792770_state @ plus @ core @ A )
=> ( package_sat_state @ plus @ A @ ( the_state @ ( plus @ ( core @ A4 ) @ B2 ) ) ) ) ) ) ).
% PartialSA.wf_assertion_and_the
thf(fact_941_PartialSA_Owf__assertion__sat__larger__pure,axiom,
! [A: aassertion_state,Phi5: state,Phi6: state,R: state] :
( ( packag7298478212905792770_state @ plus @ core @ A )
=> ( ( package_sat_state @ plus @ A @ Phi5 )
=> ( ( ( some_state @ Phi6 )
= ( plus @ Phi5 @ R ) )
=> ( ( sep_pure_state @ plus @ R )
=> ( package_sat_state @ plus @ A @ Phi6 ) ) ) ) ) ).
% PartialSA.wf_assertion_sat_larger_pure
thf(fact_942_package__logic_Omono__pure__cond_Ocong,axiom,
packag6595153354952283300_state = packag6595153354952283300_state ).
% package_logic.mono_pure_cond.cong
thf(fact_943_package__logic_Owf__assertion_Ocong,axiom,
packag7298478212905792770_state = packag7298478212905792770_state ).
% package_logic.wf_assertion.cong
thf(fact_944_PartialSA_Opackage__rhs__proof,axiom,
! [Phi5: state,F: state,S4: set_Pr4231240917283394591_state,Pc: state > $o,A: aassertion_state,Phi6: state,F5: state,S5: set_Pr4231240917283394591_state] :
( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi6 @ F5 @ S5 )
=> ( ( packag3877571396724326311_state @ plus @ core @ unit @ S4 @ F )
=> ( ( packag7298478212905792770_state @ plus @ core @ A )
=> ( ( packag6595153354952283300_state @ plus @ core @ Pc )
=> ( ( stable @ F )
=> ( ( defined @ Phi5 @ F )
=> ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi6 @ F5 @ S5 )
& ( packag3877571396724326311_state @ plus @ core @ unit @ S5 @ F5 ) ) ) ) ) ) ) ) ).
% PartialSA.package_rhs_proof
thf(fact_945_PartialSA_Osat__star__exists__bigger,axiom,
! [A: aassertion_state,B5: aassertion_state,Phi5: state] :
( ( package_sat_state @ plus @ ( aStar_state @ A @ B5 ) @ Phi5 )
=> ( ( packag7298478212905792770_state @ plus @ core @ A )
=> ( ( packag7298478212905792770_state @ plus @ core @ B5 )
=> ? [A6: state,B4: state] :
( ( greater @ ( core @ A6 ) @ ( core @ Phi5 ) )
& ( greater @ ( core @ B4 ) @ ( core @ Phi5 ) )
& ( ( some_state @ Phi5 )
= ( plus @ A6 @ B4 ) )
& ( package_sat_state @ plus @ A @ A6 )
& ( package_sat_state @ plus @ B5 @ B4 ) ) ) ) ) ).
% PartialSA.sat_star_exists_bigger
thf(fact_946_PartialSA_Omono__pure__cond__conj,axiom,
! [Pc: state > $o,B2: state > $o] :
( ( packag6595153354952283300_state @ plus @ core @ Pc )
=> ( ( packag6595153354952283300_state @ plus @ core @ B2 )
=> ( packag6595153354952283300_state @ plus @ core @ ( packag1330488657391168505_state @ Pc @ B2 ) ) ) ) ).
% PartialSA.mono_pure_cond_conj
thf(fact_947_PartialSA_Omono__pruner__def,axiom,
! [P: state > $o] :
( ( packag2456304381420842418_state @ plus @ P )
= ( ! [Phi: state,Phi2: state,R2: state] :
( ( ( sep_pure_state @ plus @ R2 )
& ( P @ Phi2 )
& ( ( some_state @ Phi )
= ( plus @ Phi2 @ R2 ) ) )
=> ( P @ Phi ) ) ) ) ).
% PartialSA.mono_pruner_def
thf(fact_948_aassertion_Oinject_I1_J,axiom,
! [X11: aassertion_state,X12: aassertion_state,Y11: aassertion_state,Y12: aassertion_state] :
( ( ( aStar_state @ X11 @ X12 )
= ( aStar_state @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% aassertion.inject(1)
thf(fact_949_PartialSA_Obool__conj__def,axiom,
( packag1330488657391168505_state
= ( ^ [A3: state > $o,B: state > $o,X3: state] :
( ( A3 @ X3 )
& ( B @ X3 ) ) ) ) ).
% PartialSA.bool_conj_def
thf(fact_950_package__logic_Omono__pruner_Ocong,axiom,
packag2456304381420842418_state = packag2456304381420842418_state ).
% package_logic.mono_pruner.cong
thf(fact_951_package__logic_Opackage__rhs_Ocong,axiom,
packag3171892606229212990_state = packag3171892606229212990_state ).
% package_logic.package_rhs.cong
thf(fact_952_PartialSA_Opackage__rhs_Ointros_I1_J,axiom,
! [Phi5: state,F: state,S4: set_Pr4231240917283394591_state,Pc: state > $o,A: aassertion_state,Phi6: state,F5: state,S5: set_Pr4231240917283394591_state,Ba: aassertion_state,Phi7: state,F6: state,S6: set_Pr4231240917283394591_state] :
( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi6 @ F5 @ S5 )
=> ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F5 @ S5 @ Pc @ Ba @ Phi7 @ F6 @ S6 )
=> ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ ( aStar_state @ A @ Ba ) @ Phi7 @ F6 @ S6 ) ) ) ).
% PartialSA.package_rhs.intros(1)
thf(fact_953_PartialSA_OAStar,axiom,
! [Phi5: state,F: state,S4: set_Pr4231240917283394591_state,Pc: state > $o,A: aassertion_state,Phi6: state,F5: state,S5: set_Pr4231240917283394591_state,B5: aassertion_state,Phi7: state,F6: state,S6: set_Pr4231240917283394591_state] :
( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi6 @ F5 @ S5 )
=> ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F5 @ S5 @ Pc @ B5 @ Phi7 @ F6 @ S6 )
=> ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ ( aStar_state @ A @ B5 ) @ Phi7 @ F6 @ S6 ) ) ) ).
% PartialSA.AStar
thf(fact_954_PartialSA_Osat_Osimps_I1_J,axiom,
! [Aa2: aassertion_state,B5: aassertion_state,Phi5: state] :
( ( package_sat_state @ plus @ ( aStar_state @ Aa2 @ B5 ) @ Phi5 )
= ( ? [A3: state,B: state] :
( ( ( some_state @ Phi5 )
= ( plus @ A3 @ B ) )
& ( package_sat_state @ plus @ Aa2 @ A3 )
& ( package_sat_state @ plus @ B5 @ B ) ) ) ) ).
% PartialSA.sat.simps(1)
thf(fact_955_PartialSA_Owf__assertion_Osimps_I1_J,axiom,
! [Aa2: aassertion_state,B5: aassertion_state] :
( ( packag7298478212905792770_state @ plus @ core @ ( aStar_state @ Aa2 @ B5 ) )
= ( ( packag7298478212905792770_state @ plus @ core @ Aa2 )
& ( packag7298478212905792770_state @ plus @ core @ B5 ) ) ) ).
% PartialSA.wf_assertion.simps(1)
thf(fact_956_PartialSA_OAImp,axiom,
! [Phi5: state,F: state,S4: set_Pr4231240917283394591_state,Pc: state > $o,B2: state > $o,A: aassertion_state,Phi6: state,F5: state,S5: set_Pr4231240917283394591_state] :
( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ ( packag1330488657391168505_state @ Pc @ B2 ) @ A @ Phi6 @ F5 @ S5 )
=> ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ ( aImp_state @ B2 @ A ) @ Phi6 @ F5 @ S5 ) ) ).
% PartialSA.AImp
thf(fact_957_PartialSA_Ogeneral__completeness,axiom,
! [S4: set_Pr4231240917283394591_state,F: state,A: aassertion_state,Phi5: state,Phi6: state] :
( ! [A6: state,U: state,T4: state > state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T4 ) ) @ S4 )
=> ! [X2: state] :
( ( ( some_state @ X2 )
= ( plus @ A6 @ ( T4 @ F ) ) )
=> ( package_sat_state @ plus @ A @ X2 ) ) )
=> ( ( ( some_state @ Phi5 )
= ( plus @ Phi6 @ F ) )
=> ( ( stable @ F )
=> ( ( packag3877571396724326311_state @ plus @ core @ unit @ S4 @ unit )
=> ( ( packag7298478212905792770_state @ plus @ core @ A )
=> ? [X_1: set_Pr4231240917283394591_state] :
( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ unit @ S4
@ ^ [Uu3: state] : $true
@ A
@ Phi6
@ F
@ X_1 ) ) ) ) ) ) ).
% PartialSA.general_completeness
thf(fact_958_PartialSA_Osetify__sum__image,axiom,
! [P6: state > $o,F: state > state,A: set_state,B5: set_state] :
( ( sep_setify_state @ P6 @ ( add_set @ ( image_state_state @ F @ A ) @ B5 ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ A )
=> ( sep_setify_state @ P6 @ ( add_set @ ( insert_state @ ( F @ X3 ) @ bot_bot_set_state ) @ B5 ) ) ) ) ) ).
% PartialSA.setify_sum_image
thf(fact_959_PartialSA_Osetify__sum__image,axiom,
! [P6: state > $o,F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,B5: set_state] :
( ( sep_setify_state @ P6 @ ( add_set @ ( image_1205529093420515805_state @ F @ A ) @ B5 ) )
= ( ! [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
=> ( sep_setify_state @ P6 @ ( add_set @ ( insert_state @ ( F @ X3 ) @ bot_bot_set_state ) @ B5 ) ) ) ) ) ).
% PartialSA.setify_sum_image
thf(fact_960_PartialSA_Owf__assertion_Osimps_I3_J,axiom,
! [A: state > $o] :
( ( packag7298478212905792770_state @ plus @ core @ ( aSem_state @ A ) )
= ( packag2456304381420842418_state @ plus @ A ) ) ).
% PartialSA.wf_assertion.simps(3)
thf(fact_961_empty__Collect__eq,axiom,
! [P6: produc7272280185086996969_state > $o] :
( ( bot_bo6430881413863817907_state
= ( collec906845016635587262_state @ P6 ) )
= ( ! [X3: produc7272280185086996969_state] :
~ ( P6 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_962_empty__Collect__eq,axiom,
! [P6: produc9003781588162623004on_val > $o] :
( ( bot_bo6722054719108133734on_val
= ( collec7475294502966882545on_val @ P6 ) )
= ( ! [X3: produc9003781588162623004on_val] :
~ ( P6 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_963_empty__Collect__eq,axiom,
! [P6: state > $o] :
( ( bot_bot_set_state
= ( collect_state @ P6 ) )
= ( ! [X3: state] :
~ ( P6 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_964_Collect__empty__eq,axiom,
! [P6: produc7272280185086996969_state > $o] :
( ( ( collec906845016635587262_state @ P6 )
= bot_bo6430881413863817907_state )
= ( ! [X3: produc7272280185086996969_state] :
~ ( P6 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_965_Collect__empty__eq,axiom,
! [P6: produc9003781588162623004on_val > $o] :
( ( ( collec7475294502966882545on_val @ P6 )
= bot_bo6722054719108133734on_val )
= ( ! [X3: produc9003781588162623004on_val] :
~ ( P6 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_966_Collect__empty__eq,axiom,
! [P6: state > $o] :
( ( ( collect_state @ P6 )
= bot_bot_set_state )
= ( ! [X3: state] :
~ ( P6 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_967_all__not__in__conv,axiom,
! [A: set_Pr4231240917283394591_state] :
( ( ! [X3: produc7272280185086996969_state] :
~ ( member7461423308196221312_state @ X3 @ A ) )
= ( A = bot_bo6430881413863817907_state ) ) ).
% all_not_in_conv
thf(fact_968_all__not__in__conv,axiom,
! [A: set_Pr7649655576376376530on_val] :
( ( ! [X3: produc9003781588162623004on_val] :
~ ( member2879030624002546611on_val @ X3 @ A ) )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% all_not_in_conv
thf(fact_969_all__not__in__conv,axiom,
! [A: set_Pr2391701600042252046_state] :
( ( ! [X3: produc3882902684156697774_state] :
~ ( member6715010751053357271_state @ X3 @ A ) )
= ( A = bot_bo3524941057552913786_state ) ) ).
% all_not_in_conv
thf(fact_970_all__not__in__conv,axiom,
! [A: set_state] :
( ( ! [X3: state] :
~ ( member_state @ X3 @ A ) )
= ( A = bot_bot_set_state ) ) ).
% all_not_in_conv
thf(fact_971_empty__iff,axiom,
! [C: produc7272280185086996969_state] :
~ ( member7461423308196221312_state @ C @ bot_bo6430881413863817907_state ) ).
% empty_iff
thf(fact_972_empty__iff,axiom,
! [C: produc9003781588162623004on_val] :
~ ( member2879030624002546611on_val @ C @ bot_bo6722054719108133734on_val ) ).
% empty_iff
thf(fact_973_empty__iff,axiom,
! [C: produc3882902684156697774_state] :
~ ( member6715010751053357271_state @ C @ bot_bo3524941057552913786_state ) ).
% empty_iff
thf(fact_974_empty__iff,axiom,
! [C: state] :
~ ( member_state @ C @ bot_bot_set_state ) ).
% empty_iff
thf(fact_975_aassertion_Oinject_I2_J,axiom,
! [X21: state > $o,X222: aassertion_state,Y21: state > $o,Y22: aassertion_state] :
( ( ( aImp_state @ X21 @ X222 )
= ( aImp_state @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% aassertion.inject(2)
thf(fact_976_aassertion_Oinject_I3_J,axiom,
! [X32: state > $o,Y32: state > $o] :
( ( ( aSem_state @ X32 )
= ( aSem_state @ Y32 ) )
= ( X32 = Y32 ) ) ).
% aassertion.inject(3)
thf(fact_977_image__ident,axiom,
! [Y8: set_state] :
( ( image_state_state
@ ^ [X3: state] : X3
@ Y8 )
= Y8 ) ).
% image_ident
thf(fact_978_image__ident,axiom,
! [Y8: set_Pr7649655576376376530on_val] :
( ( image_4170889944715819327on_val
@ ^ [X3: produc9003781588162623004on_val] : X3
@ Y8 )
= Y8 ) ).
% image_ident
thf(fact_979_image__empty,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( image_4690842878458256517on_val @ F @ bot_bo3524941057552913786_state )
= bot_bo6722054719108133734on_val ) ).
% image_empty
thf(fact_980_image__empty,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state] :
( ( image_3204238854701872908_state @ F @ bot_bo6722054719108133734on_val )
= bot_bo6430881413863817907_state ) ).
% image_empty
thf(fact_981_image__empty,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( image_4494208899119068747_state @ F @ bot_bo6722054719108133734on_val )
= bot_bo3524941057552913786_state ) ).
% image_empty
thf(fact_982_image__empty,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
( ( image_4170889944715819327on_val @ F @ bot_bo6722054719108133734on_val )
= bot_bo6722054719108133734on_val ) ).
% image_empty
thf(fact_983_image__empty,axiom,
! [F: produc9003781588162623004on_val > state] :
( ( image_1205529093420515805_state @ F @ bot_bo6722054719108133734on_val )
= bot_bot_set_state ) ).
% image_empty
thf(fact_984_image__empty,axiom,
! [F: state > state] :
( ( image_state_state @ F @ bot_bot_set_state )
= bot_bot_set_state ) ).
% image_empty
thf(fact_985_empty__is__image,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state] :
( ( bot_bo6722054719108133734on_val
= ( image_4690842878458256517on_val @ F @ A ) )
= ( A = bot_bo3524941057552913786_state ) ) ).
% empty_is_image
thf(fact_986_empty__is__image,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val] :
( ( bot_bo6430881413863817907_state
= ( image_3204238854701872908_state @ F @ A ) )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% empty_is_image
thf(fact_987_empty__is__image,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val] :
( ( bot_bo3524941057552913786_state
= ( image_4494208899119068747_state @ F @ A ) )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% empty_is_image
thf(fact_988_empty__is__image,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( bot_bo6722054719108133734on_val
= ( image_4170889944715819327on_val @ F @ A ) )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% empty_is_image
thf(fact_989_empty__is__image,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( bot_bot_set_state
= ( image_1205529093420515805_state @ F @ A ) )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% empty_is_image
thf(fact_990_empty__is__image,axiom,
! [F: state > state,A: set_state] :
( ( bot_bot_set_state
= ( image_state_state @ F @ A ) )
= ( A = bot_bot_set_state ) ) ).
% empty_is_image
thf(fact_991_image__is__empty,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state] :
( ( ( image_4690842878458256517on_val @ F @ A )
= bot_bo6722054719108133734on_val )
= ( A = bot_bo3524941057552913786_state ) ) ).
% image_is_empty
thf(fact_992_image__is__empty,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val] :
( ( ( image_3204238854701872908_state @ F @ A )
= bot_bo6430881413863817907_state )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% image_is_empty
thf(fact_993_image__is__empty,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val] :
( ( ( image_4494208899119068747_state @ F @ A )
= bot_bo3524941057552913786_state )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% image_is_empty
thf(fact_994_image__is__empty,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( ( image_4170889944715819327on_val @ F @ A )
= bot_bo6722054719108133734on_val )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% image_is_empty
thf(fact_995_image__is__empty,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( ( image_1205529093420515805_state @ F @ A )
= bot_bot_set_state )
= ( A = bot_bo6722054719108133734on_val ) ) ).
% image_is_empty
thf(fact_996_image__is__empty,axiom,
! [F: state > state,A: set_state] :
( ( ( image_state_state @ F @ A )
= bot_bot_set_state )
= ( A = bot_bot_set_state ) ) ).
% image_is_empty
thf(fact_997_empty__subsetI,axiom,
! [A: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A ) ).
% empty_subsetI
thf(fact_998_subset__empty,axiom,
! [A: set_state] :
( ( ord_le2494988322063910608_state @ A @ bot_bot_set_state )
= ( A = bot_bot_set_state ) ) ).
% subset_empty
thf(fact_999_singletonI,axiom,
! [A4: produc7272280185086996969_state] : ( member7461423308196221312_state @ A4 @ ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) ) ).
% singletonI
thf(fact_1000_singletonI,axiom,
! [A4: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ A4 @ ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) ) ).
% singletonI
thf(fact_1001_singletonI,axiom,
! [A4: produc3882902684156697774_state] : ( member6715010751053357271_state @ A4 @ ( insert94804854012721790_state @ A4 @ bot_bo3524941057552913786_state ) ) ).
% singletonI
thf(fact_1002_singletonI,axiom,
! [A4: state] : ( member_state @ A4 @ ( insert_state @ A4 @ bot_bot_set_state ) ) ).
% singletonI
thf(fact_1003_Pow__empty,axiom,
( ( pow_state @ bot_bot_set_state )
= ( insert_set_state @ bot_bot_set_state @ bot_bo2271482359692755898_state ) ) ).
% Pow_empty
thf(fact_1004_Pow__singleton__iff,axiom,
! [X6: set_state,Y8: set_state] :
( ( ( pow_state @ X6 )
= ( insert_set_state @ Y8 @ bot_bo2271482359692755898_state ) )
= ( ( X6 = bot_bot_set_state )
& ( Y8 = bot_bot_set_state ) ) ) ).
% Pow_singleton_iff
thf(fact_1005_Collect__const,axiom,
! [P6: $o] :
( ( P6
=> ( ( collec906845016635587262_state
@ ^ [S3: produc7272280185086996969_state] : P6 )
= top_to174962084833059535_state ) )
& ( ~ P6
=> ( ( collec906845016635587262_state
@ ^ [S3: produc7272280185086996969_state] : P6 )
= bot_bo6430881413863817907_state ) ) ) ).
% Collect_const
thf(fact_1006_Collect__const,axiom,
! [P6: $o] :
( ( P6
=> ( ( collec7475294502966882545on_val
@ ^ [S3: produc9003781588162623004on_val] : P6 )
= top_to1561438921946393474on_val ) )
& ( ~ P6
=> ( ( collec7475294502966882545on_val
@ ^ [S3: produc9003781588162623004on_val] : P6 )
= bot_bo6722054719108133734on_val ) ) ) ).
% Collect_const
thf(fact_1007_Collect__const,axiom,
! [P6: $o] :
( ( P6
=> ( ( collect_state
@ ^ [S3: state] : P6 )
= top_top_set_state ) )
& ( ~ P6
=> ( ( collect_state
@ ^ [S3: state] : P6 )
= bot_bot_set_state ) ) ) ).
% Collect_const
thf(fact_1008_singleton__conv2,axiom,
! [A4: produc7272280185086996969_state] :
( ( collec906845016635587262_state
@ ( ^ [Y6: produc7272280185086996969_state,Z3: produc7272280185086996969_state] : ( Y6 = Z3 )
@ A4 ) )
= ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) ) ).
% singleton_conv2
thf(fact_1009_singleton__conv2,axiom,
! [A4: produc9003781588162623004on_val] :
( ( collec7475294502966882545on_val
@ ( ^ [Y6: produc9003781588162623004on_val,Z3: produc9003781588162623004on_val] : ( Y6 = Z3 )
@ A4 ) )
= ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) ) ).
% singleton_conv2
thf(fact_1010_singleton__conv2,axiom,
! [A4: state] :
( ( collect_state
@ ( ^ [Y6: state,Z3: state] : ( Y6 = Z3 )
@ A4 ) )
= ( insert_state @ A4 @ bot_bot_set_state ) ) ).
% singleton_conv2
thf(fact_1011_singleton__conv,axiom,
! [A4: produc7272280185086996969_state] :
( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] : ( X3 = A4 ) )
= ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) ) ).
% singleton_conv
thf(fact_1012_singleton__conv,axiom,
! [A4: produc9003781588162623004on_val] :
( ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] : ( X3 = A4 ) )
= ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) ) ).
% singleton_conv
thf(fact_1013_singleton__conv,axiom,
! [A4: state] :
( ( collect_state
@ ^ [X3: state] : ( X3 = A4 ) )
= ( insert_state @ A4 @ bot_bot_set_state ) ) ).
% singleton_conv
thf(fact_1014_singleton__insert__inj__eq_H,axiom,
! [A4: state,A: set_state,B2: state] :
( ( ( insert_state @ A4 @ A )
= ( insert_state @ B2 @ bot_bot_set_state ) )
= ( ( A4 = B2 )
& ( ord_le2494988322063910608_state @ A @ ( insert_state @ B2 @ bot_bot_set_state ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1015_singleton__insert__inj__eq,axiom,
! [B2: state,A4: state,A: set_state] :
( ( ( insert_state @ B2 @ bot_bot_set_state )
= ( insert_state @ A4 @ A ) )
= ( ( A4 = B2 )
& ( ord_le2494988322063910608_state @ A @ ( insert_state @ B2 @ bot_bot_set_state ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1016_range__constant,axiom,
! [X: produc9003781588162623004on_val] :
( ( image_4690842878458256517on_val
@ ^ [Uu3: produc3882902684156697774_state] : X
@ top_to1744471583079920734_state )
= ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) ) ).
% range_constant
thf(fact_1017_range__constant,axiom,
! [X: produc7272280185086996969_state] :
( ( image_3204238854701872908_state
@ ^ [Uu3: produc9003781588162623004on_val] : X
@ top_to1561438921946393474on_val )
= ( insert175804457004744729_state @ X @ bot_bo6430881413863817907_state ) ) ).
% range_constant
thf(fact_1018_range__constant,axiom,
! [X: produc3882902684156697774_state] :
( ( image_4494208899119068747_state
@ ^ [Uu3: produc9003781588162623004on_val] : X
@ top_to1561438921946393474on_val )
= ( insert94804854012721790_state @ X @ bot_bo3524941057552913786_state ) ) ).
% range_constant
thf(fact_1019_range__constant,axiom,
! [X: produc9003781588162623004on_val] :
( ( image_4170889944715819327on_val
@ ^ [Uu3: produc9003781588162623004on_val] : X
@ top_to1561438921946393474on_val )
= ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) ) ).
% range_constant
thf(fact_1020_range__constant,axiom,
! [X: state] :
( ( image_1205529093420515805_state
@ ^ [Uu3: produc9003781588162623004on_val] : X
@ top_to1561438921946393474on_val )
= ( insert_state @ X @ bot_bot_set_state ) ) ).
% range_constant
thf(fact_1021_range__constant,axiom,
! [X: state] :
( ( image_state_state
@ ^ [Uu3: state] : X
@ top_top_set_state )
= ( insert_state @ X @ bot_bot_set_state ) ) ).
% range_constant
thf(fact_1022_PartialSA_Osat_Ocases,axiom,
! [X: produc2756261550280135502_state] :
( ! [A10: aassertion_state,B8: aassertion_state,Phi4: state] :
( X
!= ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Phi4 ) )
=> ( ! [B4: state > $o,A10: aassertion_state,Phi4: state] :
( X
!= ( produc6654088501289988998_state @ ( aImp_state @ B4 @ A10 ) @ Phi4 ) )
=> ~ ! [A10: state > $o,Phi4: state] :
( X
!= ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Phi4 ) ) ) ) ).
% PartialSA.sat.cases
thf(fact_1023_PartialSA_Owf__assertion_Ocases,axiom,
! [X: aassertion_state] :
( ! [A10: aassertion_state,B8: aassertion_state] :
( X
!= ( aStar_state @ A10 @ B8 ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( X
!= ( aImp_state @ B4 @ A10 ) )
=> ~ ! [A10: state > $o] :
( X
!= ( aSem_state @ A10 ) ) ) ) ).
% PartialSA.wf_assertion.cases
thf(fact_1024_aassertion_Odistinct_I3_J,axiom,
! [X11: aassertion_state,X12: aassertion_state,X32: state > $o] :
( ( aStar_state @ X11 @ X12 )
!= ( aSem_state @ X32 ) ) ).
% aassertion.distinct(3)
thf(fact_1025_aassertion_Odistinct_I1_J,axiom,
! [X11: aassertion_state,X12: aassertion_state,X21: state > $o,X222: aassertion_state] :
( ( aStar_state @ X11 @ X12 )
!= ( aImp_state @ X21 @ X222 ) ) ).
% aassertion.distinct(1)
thf(fact_1026_aassertion_Oexhaust,axiom,
! [Y: aassertion_state] :
( ! [X112: aassertion_state,X122: aassertion_state] :
( Y
!= ( aStar_state @ X112 @ X122 ) )
=> ( ! [X212: state > $o,X223: aassertion_state] :
( Y
!= ( aImp_state @ X212 @ X223 ) )
=> ~ ! [X33: state > $o] :
( Y
!= ( aSem_state @ X33 ) ) ) ) ).
% aassertion.exhaust
thf(fact_1027_singleton__inject,axiom,
! [A4: state,B2: state] :
( ( ( insert_state @ A4 @ bot_bot_set_state )
= ( insert_state @ B2 @ bot_bot_set_state ) )
=> ( A4 = B2 ) ) ).
% singleton_inject
thf(fact_1028_insert__not__empty,axiom,
! [A4: state,A: set_state] :
( ( insert_state @ A4 @ A )
!= bot_bot_set_state ) ).
% insert_not_empty
thf(fact_1029_doubleton__eq__iff,axiom,
! [A4: state,B2: state,C: state,D2: state] :
( ( ( insert_state @ A4 @ ( insert_state @ B2 @ bot_bot_set_state ) )
= ( insert_state @ C @ ( insert_state @ D2 @ bot_bot_set_state ) ) )
= ( ( ( A4 = C )
& ( B2 = D2 ) )
| ( ( A4 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1030_Collect__conv__if2,axiom,
! [P6: produc7272280185086996969_state > $o,A4: produc7272280185086996969_state] :
( ( ( P6 @ A4 )
=> ( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( A4 = X3 )
& ( P6 @ X3 ) ) )
= ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) ) )
& ( ~ ( P6 @ A4 )
=> ( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( A4 = X3 )
& ( P6 @ X3 ) ) )
= bot_bo6430881413863817907_state ) ) ) ).
% Collect_conv_if2
thf(fact_1031_Collect__conv__if2,axiom,
! [P6: produc9003781588162623004on_val > $o,A4: produc9003781588162623004on_val] :
( ( ( P6 @ A4 )
=> ( ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( A4 = X3 )
& ( P6 @ X3 ) ) )
= ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) ) )
& ( ~ ( P6 @ A4 )
=> ( ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( A4 = X3 )
& ( P6 @ X3 ) ) )
= bot_bo6722054719108133734on_val ) ) ) ).
% Collect_conv_if2
thf(fact_1032_Collect__conv__if2,axiom,
! [P6: state > $o,A4: state] :
( ( ( P6 @ A4 )
=> ( ( collect_state
@ ^ [X3: state] :
( ( A4 = X3 )
& ( P6 @ X3 ) ) )
= ( insert_state @ A4 @ bot_bot_set_state ) ) )
& ( ~ ( P6 @ A4 )
=> ( ( collect_state
@ ^ [X3: state] :
( ( A4 = X3 )
& ( P6 @ X3 ) ) )
= bot_bot_set_state ) ) ) ).
% Collect_conv_if2
thf(fact_1033_Collect__conv__if,axiom,
! [P6: produc7272280185086996969_state > $o,A4: produc7272280185086996969_state] :
( ( ( P6 @ A4 )
=> ( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( X3 = A4 )
& ( P6 @ X3 ) ) )
= ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) ) )
& ( ~ ( P6 @ A4 )
=> ( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( X3 = A4 )
& ( P6 @ X3 ) ) )
= bot_bo6430881413863817907_state ) ) ) ).
% Collect_conv_if
thf(fact_1034_Collect__conv__if,axiom,
! [P6: produc9003781588162623004on_val > $o,A4: produc9003781588162623004on_val] :
( ( ( P6 @ A4 )
=> ( ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( X3 = A4 )
& ( P6 @ X3 ) ) )
= ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) ) )
& ( ~ ( P6 @ A4 )
=> ( ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( X3 = A4 )
& ( P6 @ X3 ) ) )
= bot_bo6722054719108133734on_val ) ) ) ).
% Collect_conv_if
thf(fact_1035_Collect__conv__if,axiom,
! [P6: state > $o,A4: state] :
( ( ( P6 @ A4 )
=> ( ( collect_state
@ ^ [X3: state] :
( ( X3 = A4 )
& ( P6 @ X3 ) ) )
= ( insert_state @ A4 @ bot_bot_set_state ) ) )
& ( ~ ( P6 @ A4 )
=> ( ( collect_state
@ ^ [X3: state] :
( ( X3 = A4 )
& ( P6 @ X3 ) ) )
= bot_bot_set_state ) ) ) ).
% Collect_conv_if
thf(fact_1036_insert__Collect,axiom,
! [A4: produc7272280185086996969_state,P6: produc7272280185086996969_state > $o] :
( ( insert175804457004744729_state @ A4 @ ( collec906845016635587262_state @ P6 ) )
= ( collec906845016635587262_state
@ ^ [U6: produc7272280185086996969_state] :
( ( U6 != A4 )
=> ( P6 @ U6 ) ) ) ) ).
% insert_Collect
thf(fact_1037_insert__Collect,axiom,
! [A4: state,P6: state > $o] :
( ( insert_state @ A4 @ ( collect_state @ P6 ) )
= ( collect_state
@ ^ [U6: state] :
( ( U6 != A4 )
=> ( P6 @ U6 ) ) ) ) ).
% insert_Collect
thf(fact_1038_insert__Collect,axiom,
! [A4: produc9003781588162623004on_val,P6: produc9003781588162623004on_val > $o] :
( ( insert3030518898131756108on_val @ A4 @ ( collec7475294502966882545on_val @ P6 ) )
= ( collec7475294502966882545on_val
@ ^ [U6: produc9003781588162623004on_val] :
( ( U6 != A4 )
=> ( P6 @ U6 ) ) ) ) ).
% insert_Collect
thf(fact_1039_singleton__iff,axiom,
! [B2: produc7272280185086996969_state,A4: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ B2 @ ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_1040_singleton__iff,axiom,
! [B2: produc9003781588162623004on_val,A4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ B2 @ ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_1041_singleton__iff,axiom,
! [B2: produc3882902684156697774_state,A4: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ B2 @ ( insert94804854012721790_state @ A4 @ bot_bo3524941057552913786_state ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_1042_singleton__iff,axiom,
! [B2: state,A4: state] :
( ( member_state @ B2 @ ( insert_state @ A4 @ bot_bot_set_state ) )
= ( B2 = A4 ) ) ).
% singleton_iff
thf(fact_1043_insert__compr,axiom,
( insert94804854012721790_state
= ( ^ [A3: produc3882902684156697774_state,B6: set_Pr2391701600042252046_state] :
( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( X3 = A3 )
| ( member6715010751053357271_state @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_1044_insert__compr,axiom,
( insert175804457004744729_state
= ( ^ [A3: produc7272280185086996969_state,B6: set_Pr4231240917283394591_state] :
( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( X3 = A3 )
| ( member7461423308196221312_state @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_1045_insert__compr,axiom,
( insert_state
= ( ^ [A3: state,B6: set_state] :
( collect_state
@ ^ [X3: state] :
( ( X3 = A3 )
| ( member_state @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_1046_insert__compr,axiom,
( insert3030518898131756108on_val
= ( ^ [A3: produc9003781588162623004on_val,B6: set_Pr7649655576376376530on_val] :
( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( X3 = A3 )
| ( member2879030624002546611on_val @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_1047_singletonD,axiom,
! [B2: produc7272280185086996969_state,A4: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ B2 @ ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_1048_singletonD,axiom,
! [B2: produc9003781588162623004on_val,A4: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ B2 @ ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_1049_singletonD,axiom,
! [B2: produc3882902684156697774_state,A4: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ B2 @ ( insert94804854012721790_state @ A4 @ bot_bo3524941057552913786_state ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_1050_singletonD,axiom,
! [B2: state,A4: state] :
( ( member_state @ B2 @ ( insert_state @ A4 @ bot_bot_set_state ) )
=> ( B2 = A4 ) ) ).
% singletonD
thf(fact_1051_image__constant,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,C: produc7272280185086996969_state] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( image_3204238854701872908_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert175804457004744729_state @ C @ bot_bo6430881413863817907_state ) ) ) ).
% image_constant
thf(fact_1052_image__constant,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,C: produc3882902684156697774_state] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( image_4494208899119068747_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert94804854012721790_state @ C @ bot_bo3524941057552913786_state ) ) ) ).
% image_constant
thf(fact_1053_image__constant,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,C: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( image_4170889944715819327on_val
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert3030518898131756108on_val @ C @ bot_bo6722054719108133734on_val ) ) ) ).
% image_constant
thf(fact_1054_image__constant,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,C: produc9003781588162623004on_val] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( image_4690842878458256517on_val
@ ^ [X3: produc3882902684156697774_state] : C
@ A )
= ( insert3030518898131756108on_val @ C @ bot_bo6722054719108133734on_val ) ) ) ).
% image_constant
thf(fact_1055_image__constant,axiom,
! [X: state,A: set_state,C: state] :
( ( member_state @ X @ A )
=> ( ( image_state_state
@ ^ [X3: state] : C
@ A )
= ( insert_state @ C @ bot_bot_set_state ) ) ) ).
% image_constant
thf(fact_1056_image__constant,axiom,
! [X: produc7272280185086996969_state,A: set_Pr4231240917283394591_state,C: state] :
( ( member7461423308196221312_state @ X @ A )
=> ( ( image_2710133271811360426_state
@ ^ [X3: produc7272280185086996969_state] : C
@ A )
= ( insert_state @ C @ bot_bot_set_state ) ) ) ).
% image_constant
thf(fact_1057_image__constant,axiom,
! [X: produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val,C: state] :
( ( member2879030624002546611on_val @ X @ A )
=> ( ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert_state @ C @ bot_bot_set_state ) ) ) ).
% image_constant
thf(fact_1058_image__constant,axiom,
! [X: produc3882902684156697774_state,A: set_Pr2391701600042252046_state,C: state] :
( ( member6715010751053357271_state @ X @ A )
=> ( ( image_6759111139246981411_state
@ ^ [X3: produc3882902684156697774_state] : C
@ A )
= ( insert_state @ C @ bot_bot_set_state ) ) ) ).
% image_constant
thf(fact_1059_image__constant__conv,axiom,
! [A: set_Pr2391701600042252046_state,C: produc9003781588162623004on_val] :
( ( ( A = bot_bo3524941057552913786_state )
=> ( ( image_4690842878458256517on_val
@ ^ [X3: produc3882902684156697774_state] : C
@ A )
= bot_bo6722054719108133734on_val ) )
& ( ( A != bot_bo3524941057552913786_state )
=> ( ( image_4690842878458256517on_val
@ ^ [X3: produc3882902684156697774_state] : C
@ A )
= ( insert3030518898131756108on_val @ C @ bot_bo6722054719108133734on_val ) ) ) ) ).
% image_constant_conv
thf(fact_1060_image__constant__conv,axiom,
! [A: set_Pr7649655576376376530on_val,C: produc7272280185086996969_state] :
( ( ( A = bot_bo6722054719108133734on_val )
=> ( ( image_3204238854701872908_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= bot_bo6430881413863817907_state ) )
& ( ( A != bot_bo6722054719108133734on_val )
=> ( ( image_3204238854701872908_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert175804457004744729_state @ C @ bot_bo6430881413863817907_state ) ) ) ) ).
% image_constant_conv
thf(fact_1061_image__constant__conv,axiom,
! [A: set_Pr7649655576376376530on_val,C: produc3882902684156697774_state] :
( ( ( A = bot_bo6722054719108133734on_val )
=> ( ( image_4494208899119068747_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= bot_bo3524941057552913786_state ) )
& ( ( A != bot_bo6722054719108133734on_val )
=> ( ( image_4494208899119068747_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert94804854012721790_state @ C @ bot_bo3524941057552913786_state ) ) ) ) ).
% image_constant_conv
thf(fact_1062_image__constant__conv,axiom,
! [A: set_Pr7649655576376376530on_val,C: produc9003781588162623004on_val] :
( ( ( A = bot_bo6722054719108133734on_val )
=> ( ( image_4170889944715819327on_val
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= bot_bo6722054719108133734on_val ) )
& ( ( A != bot_bo6722054719108133734on_val )
=> ( ( image_4170889944715819327on_val
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert3030518898131756108on_val @ C @ bot_bo6722054719108133734on_val ) ) ) ) ).
% image_constant_conv
thf(fact_1063_image__constant__conv,axiom,
! [A: set_Pr7649655576376376530on_val,C: state] :
( ( ( A = bot_bo6722054719108133734on_val )
=> ( ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= bot_bot_set_state ) )
& ( ( A != bot_bo6722054719108133734on_val )
=> ( ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : C
@ A )
= ( insert_state @ C @ bot_bot_set_state ) ) ) ) ).
% image_constant_conv
thf(fact_1064_image__constant__conv,axiom,
! [A: set_state,C: state] :
( ( ( A = bot_bot_set_state )
=> ( ( image_state_state
@ ^ [X3: state] : C
@ A )
= bot_bot_set_state ) )
& ( ( A != bot_bot_set_state )
=> ( ( image_state_state
@ ^ [X3: state] : C
@ A )
= ( insert_state @ C @ bot_bot_set_state ) ) ) ) ).
% image_constant_conv
thf(fact_1065_pred__subset__eq2,axiom,
! [R3: set_Pr4231240917283394591_state,S4: set_Pr4231240917283394591_state] :
( ( ord_le6704609105599929480tate_o
@ ^ [X3: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ Y4 ) @ R3 )
@ ^ [X3: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ Y4 ) @ S4 ) )
= ( ord_le4032880777312351359_state @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_1066_pred__subset__eq2,axiom,
! [R3: set_Pr2391701600042252046_state,S4: set_Pr2391701600042252046_state] :
( ( ord_le1106678366802930969tate_o
@ ^ [X3: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X3 @ Y4 ) @ R3 )
@ ^ [X3: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X3 @ Y4 ) @ S4 ) )
= ( ord_le6483787752022044334_state @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_1067_pred__subset__eq2,axiom,
! [R3: set_Pr1688445902015331925_state,S4: set_Pr1688445902015331925_state] :
( ( ord_le2052508915978315794tate_o
@ ^ [X3: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ R3 )
@ ^ [X3: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ S4 ) )
= ( ord_le6423325748750870005_state @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_1068_pred__subset__eq2,axiom,
! [R3: set_Pr6039890582625207499_state,S4: set_Pr6039890582625207499_state] :
( ( ord_le9040380983013460636tate_o
@ ^ [X3: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X3 @ Y4 ) @ R3 )
@ ^ [X3: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X3 @ Y4 ) @ S4 ) )
= ( ord_le262197048489888875_state @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_1069_pred__subset__eq2,axiom,
! [R3: set_Pr6847263425858009217_state,S4: set_Pr6847263425858009217_state] :
( ( ord_le8830457724327557734tate_o
@ ^ [X3: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X3 @ Y4 ) @ R3 )
@ ^ [X3: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X3 @ Y4 ) @ S4 ) )
= ( ord_le5761494056910127137_state @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_1070_pred__subset__eq2,axiom,
! [R3: set_Pr2507339680178222706t_char,S4: set_Pr2507339680178222706t_char] :
( ( ord_le2712619360209801397char_o
@ ^ [X3: nat,Y4: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y4 ) @ R3 )
@ ^ [X3: nat,Y4: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y4 ) @ S4 ) )
= ( ord_le6315511645215477266t_char @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_1071_pred__subset__eq2,axiom,
! [R3: set_Pr7649655576376376530on_val,S4: set_Pr7649655576376376530on_val] :
( ( ord_le3852884270828607079_val_o
@ ^ [X3: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y4 ) @ R3 )
@ ^ [X3: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y4 ) @ S4 ) )
= ( ord_le6900622908513426738on_val @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_1072_Pow__bottom,axiom,
! [B5: set_state] : ( member_set_state @ bot_bot_set_state @ ( pow_state @ B5 ) ) ).
% Pow_bottom
thf(fact_1073_subset__emptyI,axiom,
! [A: set_Pr4231240917283394591_state] :
( ! [X2: produc7272280185086996969_state] :
~ ( member7461423308196221312_state @ X2 @ A )
=> ( ord_le4032880777312351359_state @ A @ bot_bo6430881413863817907_state ) ) ).
% subset_emptyI
thf(fact_1074_subset__emptyI,axiom,
! [A: set_Pr7649655576376376530on_val] :
( ! [X2: produc9003781588162623004on_val] :
~ ( member2879030624002546611on_val @ X2 @ A )
=> ( ord_le6900622908513426738on_val @ A @ bot_bo6722054719108133734on_val ) ) ).
% subset_emptyI
thf(fact_1075_subset__emptyI,axiom,
! [A: set_Pr2391701600042252046_state] :
( ! [X2: produc3882902684156697774_state] :
~ ( member6715010751053357271_state @ X2 @ A )
=> ( ord_le6483787752022044334_state @ A @ bot_bo3524941057552913786_state ) ) ).
% subset_emptyI
thf(fact_1076_subset__emptyI,axiom,
! [A: set_state] :
( ! [X2: state] :
~ ( member_state @ X2 @ A )
=> ( ord_le2494988322063910608_state @ A @ bot_bot_set_state ) ) ).
% subset_emptyI
thf(fact_1077_aassertion_Odistinct_I5_J,axiom,
! [X21: state > $o,X222: aassertion_state,X32: state > $o] :
( ( aImp_state @ X21 @ X222 )
!= ( aSem_state @ X32 ) ) ).
% aassertion.distinct(5)
thf(fact_1078_ex__in__conv,axiom,
! [A: set_Pr4231240917283394591_state] :
( ( ? [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ A ) )
= ( A != bot_bo6430881413863817907_state ) ) ).
% ex_in_conv
thf(fact_1079_ex__in__conv,axiom,
! [A: set_Pr7649655576376376530on_val] :
( ( ? [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ A ) )
= ( A != bot_bo6722054719108133734on_val ) ) ).
% ex_in_conv
thf(fact_1080_ex__in__conv,axiom,
! [A: set_Pr2391701600042252046_state] :
( ( ? [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ A ) )
= ( A != bot_bo3524941057552913786_state ) ) ).
% ex_in_conv
thf(fact_1081_ex__in__conv,axiom,
! [A: set_state] :
( ( ? [X3: state] : ( member_state @ X3 @ A ) )
= ( A != bot_bot_set_state ) ) ).
% ex_in_conv
thf(fact_1082_empty__def,axiom,
( bot_bo6430881413863817907_state
= ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] : $false ) ) ).
% empty_def
thf(fact_1083_empty__def,axiom,
( bot_bo6722054719108133734on_val
= ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] : $false ) ) ).
% empty_def
thf(fact_1084_empty__def,axiom,
( bot_bot_set_state
= ( collect_state
@ ^ [X3: state] : $false ) ) ).
% empty_def
thf(fact_1085_equals0I,axiom,
! [A: set_Pr4231240917283394591_state] :
( ! [Y5: produc7272280185086996969_state] :
~ ( member7461423308196221312_state @ Y5 @ A )
=> ( A = bot_bo6430881413863817907_state ) ) ).
% equals0I
thf(fact_1086_equals0I,axiom,
! [A: set_Pr7649655576376376530on_val] :
( ! [Y5: produc9003781588162623004on_val] :
~ ( member2879030624002546611on_val @ Y5 @ A )
=> ( A = bot_bo6722054719108133734on_val ) ) ).
% equals0I
thf(fact_1087_equals0I,axiom,
! [A: set_Pr2391701600042252046_state] :
( ! [Y5: produc3882902684156697774_state] :
~ ( member6715010751053357271_state @ Y5 @ A )
=> ( A = bot_bo3524941057552913786_state ) ) ).
% equals0I
thf(fact_1088_equals0I,axiom,
! [A: set_state] :
( ! [Y5: state] :
~ ( member_state @ Y5 @ A )
=> ( A = bot_bot_set_state ) ) ).
% equals0I
thf(fact_1089_equals0D,axiom,
! [A: set_Pr4231240917283394591_state,A4: produc7272280185086996969_state] :
( ( A = bot_bo6430881413863817907_state )
=> ~ ( member7461423308196221312_state @ A4 @ A ) ) ).
% equals0D
thf(fact_1090_equals0D,axiom,
! [A: set_Pr7649655576376376530on_val,A4: produc9003781588162623004on_val] :
( ( A = bot_bo6722054719108133734on_val )
=> ~ ( member2879030624002546611on_val @ A4 @ A ) ) ).
% equals0D
thf(fact_1091_equals0D,axiom,
! [A: set_Pr2391701600042252046_state,A4: produc3882902684156697774_state] :
( ( A = bot_bo3524941057552913786_state )
=> ~ ( member6715010751053357271_state @ A4 @ A ) ) ).
% equals0D
thf(fact_1092_equals0D,axiom,
! [A: set_state,A4: state] :
( ( A = bot_bot_set_state )
=> ~ ( member_state @ A4 @ A ) ) ).
% equals0D
thf(fact_1093_emptyE,axiom,
! [A4: produc7272280185086996969_state] :
~ ( member7461423308196221312_state @ A4 @ bot_bo6430881413863817907_state ) ).
% emptyE
thf(fact_1094_emptyE,axiom,
! [A4: produc9003781588162623004on_val] :
~ ( member2879030624002546611on_val @ A4 @ bot_bo6722054719108133734on_val ) ).
% emptyE
thf(fact_1095_emptyE,axiom,
! [A4: produc3882902684156697774_state] :
~ ( member6715010751053357271_state @ A4 @ bot_bo3524941057552913786_state ) ).
% emptyE
thf(fact_1096_emptyE,axiom,
! [A4: state] :
~ ( member_state @ A4 @ bot_bot_set_state ) ).
% emptyE
thf(fact_1097_prop__restrict,axiom,
! [X: produc3882902684156697774_state,Z5: set_Pr2391701600042252046_state,X6: set_Pr2391701600042252046_state,P6: produc3882902684156697774_state > $o] :
( ( member6715010751053357271_state @ X @ Z5 )
=> ( ( ord_le6483787752022044334_state @ Z5
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ X6 )
& ( P6 @ X3 ) ) ) )
=> ( P6 @ X ) ) ) ).
% prop_restrict
thf(fact_1098_prop__restrict,axiom,
! [X: produc7272280185086996969_state,Z5: set_Pr4231240917283394591_state,X6: set_Pr4231240917283394591_state,P6: produc7272280185086996969_state > $o] :
( ( member7461423308196221312_state @ X @ Z5 )
=> ( ( ord_le4032880777312351359_state @ Z5
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ X6 )
& ( P6 @ X3 ) ) ) )
=> ( P6 @ X ) ) ) ).
% prop_restrict
thf(fact_1099_prop__restrict,axiom,
! [X: produc9003781588162623004on_val,Z5: set_Pr7649655576376376530on_val,X6: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] :
( ( member2879030624002546611on_val @ X @ Z5 )
=> ( ( ord_le6900622908513426738on_val @ Z5
@ ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ X6 )
& ( P6 @ X3 ) ) ) )
=> ( P6 @ X ) ) ) ).
% prop_restrict
thf(fact_1100_prop__restrict,axiom,
! [X: state,Z5: set_state,X6: set_state,P6: state > $o] :
( ( member_state @ X @ Z5 )
=> ( ( ord_le2494988322063910608_state @ Z5
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ X6 )
& ( P6 @ X3 ) ) ) )
=> ( P6 @ X ) ) ) ).
% prop_restrict
thf(fact_1101_Collect__restrict,axiom,
! [X6: set_Pr2391701600042252046_state,P6: produc3882902684156697774_state > $o] :
( ord_le6483787752022044334_state
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ X6 )
& ( P6 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_1102_Collect__restrict,axiom,
! [X6: set_Pr4231240917283394591_state,P6: produc7272280185086996969_state > $o] :
( ord_le4032880777312351359_state
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ X6 )
& ( P6 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_1103_Collect__restrict,axiom,
! [X6: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] :
( ord_le6900622908513426738on_val
@ ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ X6 )
& ( P6 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_1104_Collect__restrict,axiom,
! [X6: set_state,P6: state > $o] :
( ord_le2494988322063910608_state
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ X6 )
& ( P6 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_1105_Collect__subset,axiom,
! [A: set_Pr2391701600042252046_state,P6: produc3882902684156697774_state > $o] :
( ord_le6483787752022044334_state
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
& ( P6 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1106_Collect__subset,axiom,
! [A: set_Pr4231240917283394591_state,P6: produc7272280185086996969_state > $o] :
( ord_le4032880777312351359_state
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ A )
& ( P6 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1107_Collect__subset,axiom,
! [A: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] :
( ord_le6900622908513426738on_val
@ ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( P6 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1108_Collect__subset,axiom,
! [A: set_state,P6: state > $o] :
( ord_le2494988322063910608_state
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ A )
& ( P6 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1109_less__eq__set__def,axiom,
( ord_le4032880777312351359_state
= ( ^ [A2: set_Pr4231240917283394591_state,B6: set_Pr4231240917283394591_state] :
( ord_le7281427340513060550tate_o
@ ^ [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ A2 )
@ ^ [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1110_less__eq__set__def,axiom,
( ord_le6900622908513426738on_val
= ( ^ [A2: set_Pr7649655576376376530on_val,B6: set_Pr7649655576376376530on_val] :
( ord_le6140313152057869779_val_o
@ ^ [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ A2 )
@ ^ [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1111_less__eq__set__def,axiom,
( ord_le6483787752022044334_state
= ( ^ [A2: set_Pr2391701600042252046_state,B6: set_Pr2391701600042252046_state] :
( ord_le8077393285996101039tate_o
@ ^ [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ A2 )
@ ^ [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1112_less__eq__set__def,axiom,
( ord_le2494988322063910608_state
= ( ^ [A2: set_state,B6: set_state] :
( ord_less_eq_state_o
@ ^ [X3: state] : ( member_state @ X3 @ A2 )
@ ^ [X3: state] : ( member_state @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1113_pred__subset__eq,axiom,
! [R3: set_Pr4231240917283394591_state,S4: set_Pr4231240917283394591_state] :
( ( ord_le7281427340513060550tate_o
@ ^ [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ R3 )
@ ^ [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ S4 ) )
= ( ord_le4032880777312351359_state @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_1114_pred__subset__eq,axiom,
! [R3: set_Pr7649655576376376530on_val,S4: set_Pr7649655576376376530on_val] :
( ( ord_le6140313152057869779_val_o
@ ^ [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ R3 )
@ ^ [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ S4 ) )
= ( ord_le6900622908513426738on_val @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_1115_pred__subset__eq,axiom,
! [R3: set_Pr2391701600042252046_state,S4: set_Pr2391701600042252046_state] :
( ( ord_le8077393285996101039tate_o
@ ^ [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ R3 )
@ ^ [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ S4 ) )
= ( ord_le6483787752022044334_state @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_1116_pred__subset__eq,axiom,
! [R3: set_state,S4: set_state] :
( ( ord_less_eq_state_o
@ ^ [X3: state] : ( member_state @ X3 @ R3 )
@ ^ [X3: state] : ( member_state @ X3 @ S4 ) )
= ( ord_le2494988322063910608_state @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_1117_Pow__def,axiom,
( pow_state
= ( ^ [A2: set_state] :
( collect_set_state
@ ^ [B6: set_state] : ( ord_le2494988322063910608_state @ B6 @ A2 ) ) ) ) ).
% Pow_def
thf(fact_1118_bot_Oextremum__uniqueI,axiom,
! [A4: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ bot_bot_set_state )
=> ( A4 = bot_bot_set_state ) ) ).
% bot.extremum_uniqueI
thf(fact_1119_bot_Oextremum__unique,axiom,
! [A4: set_state] :
( ( ord_le2494988322063910608_state @ A4 @ bot_bot_set_state )
= ( A4 = bot_bot_set_state ) ) ).
% bot.extremum_unique
thf(fact_1120_bot_Oextremum,axiom,
! [A4: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A4 ) ).
% bot.extremum
thf(fact_1121_subset__CollectI,axiom,
! [B5: set_Pr2391701600042252046_state,A: set_Pr2391701600042252046_state,Q: produc3882902684156697774_state > $o,P6: produc3882902684156697774_state > $o] :
( ( ord_le6483787752022044334_state @ B5 @ A )
=> ( ! [X2: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X2 @ B5 )
=> ( ( Q @ X2 )
=> ( P6 @ X2 ) ) )
=> ( ord_le6483787752022044334_state
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
& ( P6 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1122_subset__CollectI,axiom,
! [B5: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,Q: produc7272280185086996969_state > $o,P6: produc7272280185086996969_state > $o] :
( ( ord_le4032880777312351359_state @ B5 @ A )
=> ( ! [X2: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X2 @ B5 )
=> ( ( Q @ X2 )
=> ( P6 @ X2 ) ) )
=> ( ord_le4032880777312351359_state
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ A )
& ( P6 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1123_subset__CollectI,axiom,
! [B5: set_Pr7649655576376376530on_val,A: set_Pr7649655576376376530on_val,Q: produc9003781588162623004on_val > $o,P6: produc9003781588162623004on_val > $o] :
( ( ord_le6900622908513426738on_val @ B5 @ A )
=> ( ! [X2: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X2 @ B5 )
=> ( ( Q @ X2 )
=> ( P6 @ X2 ) ) )
=> ( ord_le6900622908513426738on_val
@ ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( P6 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1124_subset__CollectI,axiom,
! [B5: set_state,A: set_state,Q: state > $o,P6: state > $o] :
( ( ord_le2494988322063910608_state @ B5 @ A )
=> ( ! [X2: state] :
( ( member_state @ X2 @ B5 )
=> ( ( Q @ X2 )
=> ( P6 @ X2 ) ) )
=> ( ord_le2494988322063910608_state
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ A )
& ( P6 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1125_subset__Collect__iff,axiom,
! [B5: set_Pr2391701600042252046_state,A: set_Pr2391701600042252046_state,P6: produc3882902684156697774_state > $o] :
( ( ord_le6483787752022044334_state @ B5 @ A )
=> ( ( ord_le6483787752022044334_state @ B5
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
& ( P6 @ X3 ) ) ) )
= ( ! [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ B5 )
=> ( P6 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1126_subset__Collect__iff,axiom,
! [B5: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,P6: produc7272280185086996969_state > $o] :
( ( ord_le4032880777312351359_state @ B5 @ A )
=> ( ( ord_le4032880777312351359_state @ B5
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ A )
& ( P6 @ X3 ) ) ) )
= ( ! [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ B5 )
=> ( P6 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1127_subset__Collect__iff,axiom,
! [B5: set_Pr7649655576376376530on_val,A: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] :
( ( ord_le6900622908513426738on_val @ B5 @ A )
=> ( ( ord_le6900622908513426738on_val @ B5
@ ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( P6 @ X3 ) ) ) )
= ( ! [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ B5 )
=> ( P6 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1128_subset__Collect__iff,axiom,
! [B5: set_state,A: set_state,P6: state > $o] :
( ( ord_le2494988322063910608_state @ B5 @ A )
=> ( ( ord_le2494988322063910608_state @ B5
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ A )
& ( P6 @ X3 ) ) ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ B5 )
=> ( P6 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1129_imageE,axiom,
! [B2: state,F: state > state,A: set_state] :
( ( member_state @ B2 @ ( image_state_state @ F @ A ) )
=> ~ ! [X2: state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1130_imageE,axiom,
! [B2: state,F: produc3882902684156697774_state > state,A: set_Pr2391701600042252046_state] :
( ( member_state @ B2 @ ( image_6759111139246981411_state @ F @ A ) )
=> ~ ! [X2: produc3882902684156697774_state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member6715010751053357271_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1131_imageE,axiom,
! [B2: produc3882902684156697774_state,F: state > produc3882902684156697774_state,A: set_state] :
( ( member6715010751053357271_state @ B2 @ ( image_5311902671526399277_state @ F @ A ) )
=> ~ ! [X2: state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1132_imageE,axiom,
! [B2: state,F: produc7272280185086996969_state > state,A: set_Pr4231240917283394591_state] :
( ( member_state @ B2 @ ( image_2710133271811360426_state @ F @ A ) )
=> ~ ! [X2: produc7272280185086996969_state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member7461423308196221312_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1133_imageE,axiom,
! [B2: produc7272280185086996969_state,F: state > produc7272280185086996969_state,A: set_state] :
( ( member7461423308196221312_state @ B2 @ ( image_632992669292831658_state @ F @ A ) )
=> ~ ! [X2: state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1134_imageE,axiom,
! [B2: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( member6715010751053357271_state @ B2 @ ( image_508898238547643525_state @ F @ A ) )
=> ~ ! [X2: produc3882902684156697774_state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member6715010751053357271_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1135_imageE,axiom,
! [B2: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state,A: set_Pr2391701600042252046_state] :
( ( member7461423308196221312_state @ B2 @ ( image_4364510454419742290_state @ F @ A ) )
=> ~ ! [X2: produc3882902684156697774_state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member6715010751053357271_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1136_imageE,axiom,
! [B2: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state,A: set_Pr4231240917283394591_state] :
( ( member6715010751053357271_state @ B2 @ ( image_7466395622266389182_state @ F @ A ) )
=> ~ ! [X2: produc7272280185086996969_state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member7461423308196221312_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1137_imageE,axiom,
! [B2: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state,A: set_Pr4231240917283394591_state] :
( ( member7461423308196221312_state @ B2 @ ( image_3064086290059306969_state @ F @ A ) )
=> ~ ! [X2: produc7272280185086996969_state] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member7461423308196221312_state @ X2 @ A ) ) ) ).
% imageE
thf(fact_1138_imageE,axiom,
! [B2: state,F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( member_state @ B2 @ ( image_1205529093420515805_state @ F @ A ) )
=> ~ ! [X2: produc9003781588162623004on_val] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member2879030624002546611on_val @ X2 @ A ) ) ) ).
% imageE
thf(fact_1139_image__image,axiom,
! [F: state > state,G: state > state,A: set_state] :
( ( image_state_state @ F @ ( image_state_state @ G @ A ) )
= ( image_state_state
@ ^ [X3: state] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1140_image__image,axiom,
! [F: state > state,G: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( image_state_state @ F @ ( image_1205529093420515805_state @ G @ A ) )
= ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1141_image__image,axiom,
! [F: produc9003781588162623004on_val > state,G: state > produc9003781588162623004on_val,A: set_state] :
( ( image_1205529093420515805_state @ F @ ( image_4772210114854293981on_val @ G @ A ) )
= ( image_state_state
@ ^ [X3: state] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1142_image__image,axiom,
! [F: state > produc3882902684156697774_state,G: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( image_5311902671526399277_state @ F @ ( image_1205529093420515805_state @ G @ A ) )
= ( image_4494208899119068747_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1143_image__image,axiom,
! [F: produc3882902684156697774_state > state,G: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val] :
( ( image_6759111139246981411_state @ F @ ( image_4494208899119068747_state @ G @ A ) )
= ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1144_image__image,axiom,
! [F: produc9003781588162623004on_val > state,G: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state] :
( ( image_1205529093420515805_state @ F @ ( image_4690842878458256517on_val @ G @ A ) )
= ( image_6759111139246981411_state
@ ^ [X3: produc3882902684156697774_state] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1145_image__image,axiom,
! [F: state > produc7272280185086996969_state,G: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val] :
( ( image_632992669292831658_state @ F @ ( image_1205529093420515805_state @ G @ A ) )
= ( image_3204238854701872908_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1146_image__image,axiom,
! [F: produc7272280185086996969_state > state,G: produc9003781588162623004on_val > produc7272280185086996969_state,A: set_Pr7649655576376376530on_val] :
( ( image_2710133271811360426_state @ F @ ( image_3204238854701872908_state @ G @ A ) )
= ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1147_image__image,axiom,
! [F: produc3882902684156697774_state > produc3882902684156697774_state,G: produc9003781588162623004on_val > produc3882902684156697774_state,A: set_Pr7649655576376376530on_val] :
( ( image_508898238547643525_state @ F @ ( image_4494208899119068747_state @ G @ A ) )
= ( image_4494208899119068747_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1148_image__image,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,G: produc3882902684156697774_state > produc3882902684156697774_state,A: set_Pr2391701600042252046_state] :
( ( image_4690842878458256517on_val @ F @ ( image_508898238547643525_state @ G @ A ) )
= ( image_4690842878458256517on_val
@ ^ [X3: produc3882902684156697774_state] : ( F @ ( G @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1149_Compr__image__eq,axiom,
! [F: state > state,A: set_state,P6: state > $o] :
( ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ ( image_state_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_state_state @ F
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1150_Compr__image__eq,axiom,
! [F: state > produc3882902684156697774_state,A: set_state,P6: produc3882902684156697774_state > $o] :
( ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ ( image_5311902671526399277_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_5311902671526399277_state @ F
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1151_Compr__image__eq,axiom,
! [F: produc3882902684156697774_state > state,A: set_Pr2391701600042252046_state,P6: state > $o] :
( ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ ( image_6759111139246981411_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_6759111139246981411_state @ F
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1152_Compr__image__eq,axiom,
! [F: state > produc7272280185086996969_state,A: set_state,P6: produc7272280185086996969_state > $o] :
( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ ( image_632992669292831658_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_632992669292831658_state @ F
@ ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1153_Compr__image__eq,axiom,
! [F: produc7272280185086996969_state > state,A: set_Pr4231240917283394591_state,P6: state > $o] :
( ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ ( image_2710133271811360426_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_2710133271811360426_state @ F
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1154_Compr__image__eq,axiom,
! [F: produc3882902684156697774_state > produc3882902684156697774_state,A: set_Pr2391701600042252046_state,P6: produc3882902684156697774_state > $o] :
( ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ ( image_508898238547643525_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_508898238547643525_state @ F
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1155_Compr__image__eq,axiom,
! [F: produc7272280185086996969_state > produc3882902684156697774_state,A: set_Pr4231240917283394591_state,P6: produc3882902684156697774_state > $o] :
( ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ ( image_7466395622266389182_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_7466395622266389182_state @ F
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1156_Compr__image__eq,axiom,
! [F: produc3882902684156697774_state > produc7272280185086996969_state,A: set_Pr2391701600042252046_state,P6: produc7272280185086996969_state > $o] :
( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ ( image_4364510454419742290_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_4364510454419742290_state @ F
@ ( collec394671971739345689_state
@ ^ [X3: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1157_Compr__image__eq,axiom,
! [F: produc7272280185086996969_state > produc7272280185086996969_state,A: set_Pr4231240917283394591_state,P6: produc7272280185086996969_state > $o] :
( ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ ( image_3064086290059306969_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_3064086290059306969_state @ F
@ ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1158_Compr__image__eq,axiom,
! [F: produc9003781588162623004on_val > state,A: set_Pr7649655576376376530on_val,P6: state > $o] :
( ( collect_state
@ ^ [X3: state] :
( ( member_state @ X3 @ ( image_1205529093420515805_state @ F @ A ) )
& ( P6 @ X3 ) ) )
= ( image_1205529093420515805_state @ F
@ ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X3 @ A )
& ( P6 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1159_Sup_OSUP__identity__eq,axiom,
! [Sup: set_state > state,A: set_state] :
( ( Sup
@ ( image_state_state
@ ^ [X3: state] : X3
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_1160_Sup_OSUP__identity__eq,axiom,
! [Sup: set_Pr7649655576376376530on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( Sup
@ ( image_4170889944715819327on_val
@ ^ [X3: produc9003781588162623004on_val] : X3
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_1161_Inf_OINF__identity__eq,axiom,
! [Inf: set_state > state,A: set_state] :
( ( Inf
@ ( image_state_state
@ ^ [X3: state] : X3
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_1162_Inf_OINF__identity__eq,axiom,
! [Inf: set_Pr7649655576376376530on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( Inf
@ ( image_4170889944715819327on_val
@ ^ [X3: produc9003781588162623004on_val] : X3
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_1163_image__Collect__subsetI,axiom,
! [P6: state > $o,F: state > state,B5: set_state] :
( ! [X2: state] :
( ( P6 @ X2 )
=> ( member_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le2494988322063910608_state @ ( image_state_state @ F @ ( collect_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1164_image__Collect__subsetI,axiom,
! [P6: state > $o,F: state > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ! [X2: state] :
( ( P6 @ X2 )
=> ( member6715010751053357271_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ ( collect_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1165_image__Collect__subsetI,axiom,
! [P6: state > $o,F: state > produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] :
( ! [X2: state] :
( ( P6 @ X2 )
=> ( member7461423308196221312_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ ( collect_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1166_image__Collect__subsetI,axiom,
! [P6: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > state,B5: set_state] :
( ! [X2: produc7272280185086996969_state] :
( ( P6 @ X2 )
=> ( member_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le2494988322063910608_state @ ( image_2710133271811360426_state @ F @ ( collec906845016635587262_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1167_image__Collect__subsetI,axiom,
! [P6: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ! [X2: produc7272280185086996969_state] :
( ( P6 @ X2 )
=> ( member6715010751053357271_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6483787752022044334_state @ ( image_7466395622266389182_state @ F @ ( collec906845016635587262_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1168_image__Collect__subsetI,axiom,
! [P6: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] :
( ! [X2: produc7272280185086996969_state] :
( ( P6 @ X2 )
=> ( member7461423308196221312_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le4032880777312351359_state @ ( image_3064086290059306969_state @ F @ ( collec906845016635587262_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1169_image__Collect__subsetI,axiom,
! [P6: state > $o,F: state > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ! [X2: state] :
( ( P6 @ X2 )
=> ( member2879030624002546611on_val @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ ( collect_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1170_image__Collect__subsetI,axiom,
! [P6: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > state,B5: set_state] :
( ! [X2: produc9003781588162623004on_val] :
( ( P6 @ X2 )
=> ( member_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ ( collec7475294502966882545on_val @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1171_image__Collect__subsetI,axiom,
! [P6: produc3882902684156697774_state > $o,F: produc3882902684156697774_state > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
( ! [X2: produc3882902684156697774_state] :
( ( P6 @ X2 )
=> ( member2879030624002546611on_val @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6900622908513426738on_val @ ( image_4690842878458256517on_val @ F @ ( collec394671971739345689_state @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1172_image__Collect__subsetI,axiom,
! [P6: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
( ! [X2: produc9003781588162623004on_val] :
( ( P6 @ X2 )
=> ( member6715010751053357271_state @ ( F @ X2 ) @ B5 ) )
=> ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ ( collec7475294502966882545on_val @ P6 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1173_pred__equals__eq2,axiom,
! [R3: set_Pr4231240917283394591_state,S4: set_Pr4231240917283394591_state] :
( ( ( ^ [X3: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ Y4 ) @ R3 ) )
= ( ^ [X3: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ Y4 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_1174_pred__equals__eq2,axiom,
! [R3: set_Pr2391701600042252046_state,S4: set_Pr2391701600042252046_state] :
( ( ( ^ [X3: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X3 @ Y4 ) @ R3 ) )
= ( ^ [X3: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X3 @ Y4 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_1175_pred__equals__eq2,axiom,
! [R3: set_Pr1688445902015331925_state,S4: set_Pr1688445902015331925_state] :
( ( ( ^ [X3: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ R3 ) )
= ( ^ [X3: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_1176_pred__equals__eq2,axiom,
! [R3: set_Pr6039890582625207499_state,S4: set_Pr6039890582625207499_state] :
( ( ( ^ [X3: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X3 @ Y4 ) @ R3 ) )
= ( ^ [X3: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X3 @ Y4 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_1177_pred__equals__eq2,axiom,
! [R3: set_Pr6847263425858009217_state,S4: set_Pr6847263425858009217_state] :
( ( ( ^ [X3: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X3 @ Y4 ) @ R3 ) )
= ( ^ [X3: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X3 @ Y4 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_1178_pred__equals__eq2,axiom,
! [R3: set_Pr2507339680178222706t_char,S4: set_Pr2507339680178222706t_char] :
( ( ( ^ [X3: nat,Y4: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y4 ) @ R3 ) )
= ( ^ [X3: nat,Y4: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y4 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_1179_pred__equals__eq2,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 ) )
= ( ^ [X3: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y4 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_1180_top__empty__eq2,axiom,
( top_to8386588197433551576tate_o
= ( ^ [X3: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ Y4 ) @ top_to174962084833059535_state ) ) ) ).
% top_empty_eq2
thf(fact_1181_top__empty__eq2,axiom,
( top_to53189150443868873tate_o
= ( ^ [X3: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X3 @ Y4 ) @ top_to1744471583079920734_state ) ) ) ).
% top_empty_eq2
thf(fact_1182_top__empty__eq2,axiom,
( top_to2009358410527348162tate_o
= ( ^ [X3: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X3 @ Y4 ) @ top_to5248747511432852389_state ) ) ) ).
% top_empty_eq2
thf(fact_1183_top__empty__eq2,axiom,
( top_top_prat_state_o
= ( ^ [X3: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X3 @ Y4 ) @ top_to5813313822981523995_state ) ) ) ).
% top_empty_eq2
thf(fact_1184_top__empty__eq2,axiom,
( top_to8783976926004287510tate_o
= ( ^ [X3: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X3 @ Y4 ) @ top_to4399397253935103441_state ) ) ) ).
% top_empty_eq2
thf(fact_1185_top__empty__eq2,axiom,
( top_to1537031869075220069char_o
= ( ^ [X3: nat,Y4: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y4 ) @ top_to1250346684925019074t_char ) ) ) ).
% top_empty_eq2
thf(fact_1186_top__empty__eq2,axiom,
( top_to1211379171580150967_val_o
= ( ^ [X3: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y4 ) @ top_to1561438921946393474on_val ) ) ) ).
% top_empty_eq2
thf(fact_1187_range__composition,axiom,
! [F: state > state,G: state > state] :
( ( image_state_state
@ ^ [X3: state] : ( F @ ( G @ X3 ) )
@ top_top_set_state )
= ( image_state_state @ F @ ( image_state_state @ G @ top_top_set_state ) ) ) ).
% range_composition
thf(fact_1188_range__composition,axiom,
! [F: state > state,G: produc9003781588162623004on_val > state] :
( ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ top_to1561438921946393474on_val )
= ( image_state_state @ F @ ( image_1205529093420515805_state @ G @ top_to1561438921946393474on_val ) ) ) ).
% range_composition
thf(fact_1189_range__composition,axiom,
! [F: produc9003781588162623004on_val > state,G: state > produc9003781588162623004on_val] :
( ( image_state_state
@ ^ [X3: state] : ( F @ ( G @ X3 ) )
@ top_top_set_state )
= ( image_1205529093420515805_state @ F @ ( image_4772210114854293981on_val @ G @ top_top_set_state ) ) ) ).
% range_composition
thf(fact_1190_range__composition,axiom,
! [F: produc9003781588162623004on_val > state,G: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( image_6759111139246981411_state
@ ^ [X3: produc3882902684156697774_state] : ( F @ ( G @ X3 ) )
@ top_to1744471583079920734_state )
= ( image_1205529093420515805_state @ F @ ( image_4690842878458256517on_val @ G @ top_to1744471583079920734_state ) ) ) ).
% range_composition
thf(fact_1191_range__composition,axiom,
! [F: produc3882902684156697774_state > state,G: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ top_to1561438921946393474on_val )
= ( image_6759111139246981411_state @ F @ ( image_4494208899119068747_state @ G @ top_to1561438921946393474on_val ) ) ) ).
% range_composition
thf(fact_1192_range__composition,axiom,
! [F: state > produc3882902684156697774_state,G: produc9003781588162623004on_val > state] :
( ( image_4494208899119068747_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ top_to1561438921946393474on_val )
= ( image_5311902671526399277_state @ F @ ( image_1205529093420515805_state @ G @ top_to1561438921946393474on_val ) ) ) ).
% range_composition
thf(fact_1193_range__composition,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,G: state > produc3882902684156697774_state] :
( ( image_4772210114854293981on_val
@ ^ [X3: state] : ( F @ ( G @ X3 ) )
@ top_top_set_state )
= ( image_4690842878458256517on_val @ F @ ( image_5311902671526399277_state @ G @ top_top_set_state ) ) ) ).
% range_composition
thf(fact_1194_range__composition,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,G: state > produc9003781588162623004on_val] :
( ( image_5311902671526399277_state
@ ^ [X3: state] : ( F @ ( G @ X3 ) )
@ top_top_set_state )
= ( image_4494208899119068747_state @ F @ ( image_4772210114854293981on_val @ G @ top_top_set_state ) ) ) ).
% range_composition
thf(fact_1195_range__composition,axiom,
! [F: produc7272280185086996969_state > state,G: produc9003781588162623004on_val > produc7272280185086996969_state] :
( ( image_1205529093420515805_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ top_to1561438921946393474on_val )
= ( image_2710133271811360426_state @ F @ ( image_3204238854701872908_state @ G @ top_to1561438921946393474on_val ) ) ) ).
% range_composition
thf(fact_1196_range__composition,axiom,
! [F: state > produc7272280185086996969_state,G: produc9003781588162623004on_val > state] :
( ( image_3204238854701872908_state
@ ^ [X3: produc9003781588162623004on_val] : ( F @ ( G @ X3 ) )
@ top_to1561438921946393474on_val )
= ( image_632992669292831658_state @ F @ ( image_1205529093420515805_state @ G @ top_to1561438921946393474on_val ) ) ) ).
% range_composition
thf(fact_1197_rangeE,axiom,
! [B2: state,F: produc9003781588162623004on_val > state] :
( ( member_state @ B2 @ ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val ) )
=> ~ ! [X2: produc9003781588162623004on_val] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1198_rangeE,axiom,
! [B2: produc7272280185086996969_state,F: produc9003781588162623004on_val > produc7272280185086996969_state] :
( ( member7461423308196221312_state @ B2 @ ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val ) )
=> ~ ! [X2: produc9003781588162623004on_val] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1199_rangeE,axiom,
! [B2: produc9003781588162623004on_val,F: produc3882902684156697774_state > produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ B2 @ ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state ) )
=> ~ ! [X2: produc3882902684156697774_state] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1200_rangeE,axiom,
! [B2: produc9003781588162623004on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ B2 @ ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val ) )
=> ~ ! [X2: produc9003781588162623004on_val] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1201_rangeE,axiom,
! [B2: produc3882902684156697774_state,F: produc9003781588162623004on_val > produc3882902684156697774_state] :
( ( member6715010751053357271_state @ B2 @ ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val ) )
=> ~ ! [X2: produc9003781588162623004on_val] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1202_rangeE,axiom,
! [B2: state,F: state > state] :
( ( member_state @ B2 @ ( image_state_state @ F @ top_top_set_state ) )
=> ~ ! [X2: state] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1203_rangeE,axiom,
! [B2: produc7272280185086996969_state,F: state > produc7272280185086996969_state] :
( ( member7461423308196221312_state @ B2 @ ( image_632992669292831658_state @ F @ top_top_set_state ) )
=> ~ ! [X2: state] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1204_rangeE,axiom,
! [B2: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ B2 @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) )
=> ~ ! [X2: state] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1205_rangeE,axiom,
! [B2: produc3882902684156697774_state,F: state > produc3882902684156697774_state] :
( ( member6715010751053357271_state @ B2 @ ( image_5311902671526399277_state @ F @ top_top_set_state ) )
=> ~ ! [X2: state] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1206_UNIV__def,axiom,
( top_to174962084833059535_state
= ( collec906845016635587262_state
@ ^ [X3: produc7272280185086996969_state] : $true ) ) ).
% UNIV_def
thf(fact_1207_UNIV__def,axiom,
( top_to1561438921946393474on_val
= ( collec7475294502966882545on_val
@ ^ [X3: produc9003781588162623004on_val] : $true ) ) ).
% UNIV_def
thf(fact_1208_UNIV__def,axiom,
( top_top_set_state
= ( collect_state
@ ^ [X3: state] : $true ) ) ).
% UNIV_def
thf(fact_1209_empty__not__UNIV,axiom,
bot_bot_set_state != top_top_set_state ).
% empty_not_UNIV
thf(fact_1210_PartialSA_Oempty__set__sum,axiom,
! [A: set_state] :
( ( add_set @ bot_bot_set_state @ A )
= bot_bot_set_state ) ).
% PartialSA.empty_set_sum
thf(fact_1211_PartialSA_Osat_Oelims_I3_J,axiom,
! [X: aassertion_state,Xa: state] :
( ~ ( package_sat_state @ plus @ X @ Xa )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( 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] :
( ( X
= ( aImp_state @ B4 @ A10 ) )
=> ( ( B4 @ Xa )
=> ( package_sat_state @ plus @ A10 @ Xa ) ) )
=> ~ ! [A10: state > $o] :
( ( X
= ( aSem_state @ A10 ) )
=> ( A10 @ Xa ) ) ) ) ) ).
% PartialSA.sat.elims(3)
thf(fact_1212_PartialSA_Osat_Oelims_I2_J,axiom,
! [X: aassertion_state,Xa: state] :
( ( package_sat_state @ plus @ X @ Xa )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( aStar_state @ A10 @ B8 ) )
=> ~ ? [A6: state,B4: state] :
( ( ( some_state @ Xa )
= ( plus @ A6 @ B4 ) )
& ( package_sat_state @ plus @ A10 @ A6 )
& ( package_sat_state @ plus @ B8 @ B4 ) ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( ( X
= ( aImp_state @ B4 @ A10 ) )
=> ~ ( ( B4 @ Xa )
=> ( package_sat_state @ plus @ A10 @ Xa ) ) )
=> ~ ! [A10: state > $o] :
( ( X
= ( aSem_state @ A10 ) )
=> ~ ( A10 @ Xa ) ) ) ) ) ).
% PartialSA.sat.elims(2)
thf(fact_1213_PartialSA_Osat_Oelims_I1_J,axiom,
! [X: aassertion_state,Xa: state,Y: $o] :
( ( ( package_sat_state @ plus @ X @ Xa )
= Y )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( aStar_state @ A10 @ B8 ) )
=> ( Y
= ( ~ ? [A3: state,B: state] :
( ( ( some_state @ Xa )
= ( plus @ A3 @ B ) )
& ( package_sat_state @ plus @ A10 @ A3 )
& ( package_sat_state @ plus @ B8 @ B ) ) ) ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( ( X
= ( aImp_state @ B4 @ A10 ) )
=> ( Y
= ( ~ ( ( B4 @ Xa )
=> ( package_sat_state @ plus @ A10 @ Xa ) ) ) ) )
=> ~ ! [A10: state > $o] :
( ( X
= ( aSem_state @ A10 ) )
=> ( Y
= ( ~ ( A10 @ Xa ) ) ) ) ) ) ) ).
% PartialSA.sat.elims(1)
thf(fact_1214_PartialSA_Osat_Osimps_I3_J,axiom,
! [A: state > $o,Phi5: state] :
( ( package_sat_state @ plus @ ( aSem_state @ A ) @ Phi5 )
= ( A @ Phi5 ) ) ).
% PartialSA.sat.simps(3)
thf(fact_1215_PartialSA_Osat_Osimps_I2_J,axiom,
! [B2: state > $o,Aa2: aassertion_state,Phi5: state] :
( ( package_sat_state @ plus @ ( aImp_state @ B2 @ Aa2 ) @ Phi5 )
= ( ( B2 @ Phi5 )
=> ( package_sat_state @ plus @ Aa2 @ Phi5 ) ) ) ).
% PartialSA.sat.simps(2)
thf(fact_1216_subset__singletonD,axiom,
! [A: set_state,X: state] :
( ( ord_le2494988322063910608_state @ A @ ( insert_state @ X @ bot_bot_set_state ) )
=> ( ( A = bot_bot_set_state )
| ( A
= ( insert_state @ X @ bot_bot_set_state ) ) ) ) ).
% subset_singletonD
thf(fact_1217_subset__singleton__iff,axiom,
! [X6: set_state,A4: state] :
( ( ord_le2494988322063910608_state @ X6 @ ( insert_state @ A4 @ bot_bot_set_state ) )
= ( ( X6 = bot_bot_set_state )
| ( X6
= ( insert_state @ A4 @ bot_bot_set_state ) ) ) ) ).
% subset_singleton_iff
thf(fact_1218_PartialSA_Obigger__singleton,axiom,
! [Phi6: state,Phi5: state] :
( ( greater @ Phi6 @ Phi5 )
=> ( greater_set @ ( insert_state @ Phi6 @ bot_bot_set_state ) @ ( insert_state @ Phi5 @ bot_bot_set_state ) ) ) ).
% PartialSA.bigger_singleton
thf(fact_1219_PartialSA_Osetify__sum,axiom,
! [P6: state > $o,A: set_state,B5: set_state] :
( ( sep_setify_state @ P6 @ ( add_set @ A @ B5 ) )
= ( ! [X3: state] :
( ( member_state @ X3 @ A )
=> ( sep_setify_state @ P6 @ ( add_set @ ( insert_state @ X3 @ bot_bot_set_state ) @ B5 ) ) ) ) ) ).
% PartialSA.setify_sum
thf(fact_1220_range__eq__singletonD,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A4: produc9003781588162623004on_val,X: produc3882902684156697774_state] :
( ( ( image_4690842878458256517on_val @ F @ top_to1744471583079920734_state )
= ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1221_range__eq__singletonD,axiom,
! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A4: produc7272280185086996969_state,X: produc9003781588162623004on_val] :
( ( ( image_3204238854701872908_state @ F @ top_to1561438921946393474on_val )
= ( insert175804457004744729_state @ A4 @ bot_bo6430881413863817907_state ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1222_range__eq__singletonD,axiom,
! [F: produc9003781588162623004on_val > produc3882902684156697774_state,A4: produc3882902684156697774_state,X: produc9003781588162623004on_val] :
( ( ( image_4494208899119068747_state @ F @ top_to1561438921946393474on_val )
= ( insert94804854012721790_state @ A4 @ bot_bo3524941057552913786_state ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1223_range__eq__singletonD,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A4: produc9003781588162623004on_val,X: produc9003781588162623004on_val] :
( ( ( image_4170889944715819327on_val @ F @ top_to1561438921946393474on_val )
= ( insert3030518898131756108on_val @ A4 @ bot_bo6722054719108133734on_val ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1224_range__eq__singletonD,axiom,
! [F: produc9003781588162623004on_val > state,A4: state,X: produc9003781588162623004on_val] :
( ( ( image_1205529093420515805_state @ F @ top_to1561438921946393474on_val )
= ( insert_state @ A4 @ bot_bot_set_state ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1225_range__eq__singletonD,axiom,
! [F: state > state,A4: state,X: state] :
( ( ( image_state_state @ F @ top_top_set_state )
= ( insert_state @ A4 @ bot_bot_set_state ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1226_PartialSA_Owf__assertion_Oelims_I3_J,axiom,
! [X: aassertion_state] :
( ~ ( packag7298478212905792770_state @ plus @ core @ X )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( aStar_state @ A10 @ B8 ) )
=> ( ( packag7298478212905792770_state @ plus @ core @ A10 )
& ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( ( X
= ( aImp_state @ B4 @ A10 ) )
=> ( ( packag6595153354952283300_state @ plus @ core @ B4 )
& ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
=> ~ ! [A10: state > $o] :
( ( X
= ( aSem_state @ A10 ) )
=> ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ).
% PartialSA.wf_assertion.elims(3)
thf(fact_1227_PartialSA_Owf__assertion_Oelims_I2_J,axiom,
! [X: aassertion_state] :
( ( packag7298478212905792770_state @ plus @ core @ X )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( aStar_state @ A10 @ B8 ) )
=> ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
& ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( ( X
= ( aImp_state @ B4 @ A10 ) )
=> ~ ( ( packag6595153354952283300_state @ plus @ core @ B4 )
& ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
=> ~ ! [A10: state > $o] :
( ( X
= ( aSem_state @ A10 ) )
=> ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ).
% PartialSA.wf_assertion.elims(2)
thf(fact_1228_PartialSA_Owf__assertion_Oelims_I1_J,axiom,
! [X: aassertion_state,Y: $o] :
( ( ( packag7298478212905792770_state @ plus @ core @ X )
= Y )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( aStar_state @ A10 @ B8 ) )
=> ( Y
= ( ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
& ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( ( X
= ( aImp_state @ B4 @ A10 ) )
=> ( Y
= ( ~ ( ( packag6595153354952283300_state @ plus @ core @ B4 )
& ( packag7298478212905792770_state @ plus @ core @ A10 ) ) ) ) )
=> ~ ! [A10: state > $o] :
( ( X
= ( aSem_state @ A10 ) )
=> ( Y
= ( ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).
% PartialSA.wf_assertion.elims(1)
thf(fact_1229_PartialSA_Osum__then__singleton,axiom,
! [A4: state,B2: state,C: state] :
( ( ( some_state @ A4 )
= ( plus @ B2 @ C ) )
= ( ( insert_state @ A4 @ bot_bot_set_state )
= ( add_set @ ( insert_state @ B2 @ bot_bot_set_state ) @ ( insert_state @ C @ bot_bot_set_state ) ) ) ) ).
% PartialSA.sum_then_singleton
thf(fact_1230_PartialSA_Owf__assertion_Osimps_I2_J,axiom,
! [B2: state > $o,Aa2: aassertion_state] :
( ( packag7298478212905792770_state @ plus @ core @ ( aImp_state @ B2 @ Aa2 ) )
= ( ( packag6595153354952283300_state @ plus @ core @ B2 )
& ( packag7298478212905792770_state @ plus @ core @ Aa2 ) ) ) ).
% PartialSA.wf_assertion.simps(2)
thf(fact_1231_PartialSA_Ocompleteness,axiom,
! [B5: aassertion_state,W: state,R3: state > state > state,A: aassertion_state,Sigma: state,Sigma2: state] :
( ( packag7298478212905792770_state @ plus @ core @ B5 )
=> ( ( ( stable @ W )
& ( packag6192661653625646554_state @ plus @ W @ R3 @ A @ B5 ) )
=> ( ( ( some_state @ Sigma )
= ( plus @ Sigma2 @ W ) )
=> ( ! [A6: state] :
( ( package_sat_state @ plus @ A @ A6 )
=> ( packag4817583915586052229_state @ plus @ unit @ ( R3 @ A6 ) ) )
=> ? [X_1: set_Pr4231240917283394591_state] :
( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ unit @ ( R3 @ A3 ) ) ) )
& ( package_sat_state @ plus @ A @ A3 ) ) )
@ ^ [Uu3: state] : $true
@ B5
@ Sigma2
@ W
@ X_1 ) ) ) ) ) ).
% PartialSA.completeness
thf(fact_1232_PartialSA_Owf__assertion_Opelims_I1_J,axiom,
! [X: aassertion_state,Y: $o] :
( ( ( packag7298478212905792770_state @ plus @ core @ X )
= Y )
=> ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( 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] :
( ( X
= ( 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] :
( ( X
= ( aSem_state @ A10 ) )
=> ( ( Y
= ( packag2456304381420842418_state @ plus @ A10 ) )
=> ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) ) ) ) ) ) ) ) ).
% PartialSA.wf_assertion.pelims(1)
thf(fact_1233_PartialSA_Owf__assertion_Opelims_I2_J,axiom,
! [X: aassertion_state] :
( ( packag7298478212905792770_state @ plus @ core @ X )
=> ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( 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] :
( ( X
= ( 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] :
( ( X
= ( aSem_state @ A10 ) )
=> ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) )
=> ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).
% PartialSA.wf_assertion.pelims(2)
thf(fact_1234_PartialSA_Owf__assertion_Opelims_I3_J,axiom,
! [X: aassertion_state] :
( ~ ( packag7298478212905792770_state @ plus @ core @ X )
=> ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( 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] :
( ( X
= ( 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] :
( ( X
= ( aSem_state @ A10 ) )
=> ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) )
=> ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).
% PartialSA.wf_assertion.pelims(3)
thf(fact_1235_bot__set__def,axiom,
( bot_bo6430881413863817907_state
= ( collec906845016635587262_state @ bot_bo4916705512949510546tate_o ) ) ).
% bot_set_def
thf(fact_1236_bot__set__def,axiom,
( bot_bo6722054719108133734on_val
= ( collec7475294502966882545on_val @ bot_bo6370234372831922335_val_o ) ) ).
% bot_set_def
thf(fact_1237_bot__set__def,axiom,
( bot_bot_set_state
= ( collect_state @ bot_bot_state_o ) ) ).
% bot_set_def
thf(fact_1238_Setcompr__eq__image,axiom,
! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,A: set_Pr7649655576376376530on_val] :
( ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [X3: produc9003781588162623004on_val] :
( ( Uu3
= ( F @ X3 ) )
& ( member2879030624002546611on_val @ X3 @ A ) ) )
= ( image_4170889944715819327on_val @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1239_Setcompr__eq__image,axiom,
! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A: set_Pr2391701600042252046_state] :
( ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [X3: produc3882902684156697774_state] :
( ( Uu3
= ( F @ X3 ) )
& ( member6715010751053357271_state @ X3 @ A ) ) )
= ( image_4690842878458256517on_val @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1240_PartialSA_Oadd__set__def,axiom,
( add_set
= ( ^ [A2: set_state,B6: set_state] :
( collect_state
@ ^ [Uu3: state] :
? [Phi2: state,A3: state,B: state] :
( ( Uu3 = Phi2 )
& ( member_state @ A3 @ A2 )
& ( member_state @ B @ B6 )
& ( ( some_state @ Phi2 )
= ( plus @ A3 @ B ) ) ) ) ) ) ).
% PartialSA.add_set_def
thf(fact_1241_PartialSA_Oupper__closure__def,axiom,
! [A: set_state] :
( ( sep_up1246176804924251236_state @ plus @ A )
= ( collect_state
@ ^ [Uu3: state] :
? [Phi: state,Phi2: state] :
( ( Uu3 = Phi )
& ( greater @ Phi @ Phi2 )
& ( member_state @ Phi2 @ A ) ) ) ) ).
% PartialSA.upper_closure_def
thf(fact_1242_wand__equiv__def,axiom,
( wand
= ( ^ [A2: set_state,B6: set_state] :
( collect_state
@ ^ [Uu3: state] :
? [Phi2: state] :
( ( Uu3 = Phi2 )
& ( ord_le2494988322063910608_state @ ( add_set @ A2 @ ( insert_state @ Phi2 @ bot_bot_set_state ) ) @ B6 ) ) ) ) ) ).
% wand_equiv_def
thf(fact_1243_PartialSA_Osoundness,axiom,
! [B5: aassertion_state,A: aassertion_state,S4: set_state,R3: state > state > state,Sigma: state,Sigma2: state,W: state,S5: set_Pr4231240917283394591_state] :
( ( packag7298478212905792770_state @ plus @ core @ B5 )
=> ( ! [A6: state] :
( ( package_sat_state @ plus @ A @ A6 )
=> ( member_state @ A6 @ S4 ) )
=> ( ! [A6: state] :
( ( member_state @ A6 @ S4 )
=> ( packag4817583915586052229_state @ plus @ unit @ ( R3 @ A6 ) ) )
=> ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ unit @ ( R3 @ A3 ) ) ) )
& ( member_state @ A3 @ S4 ) ) )
@ ^ [Uu3: state] : $true
@ B5
@ Sigma2
@ W
@ S5 )
=> ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ B5 ) )
| ( packag4152378406607043437_state @ plus @ S5 ) )
=> ( ( stable @ W )
& ( ( some_state @ Sigma )
= ( plus @ Sigma2 @ W ) )
& ( packag6192661653625646554_state @ plus @ W @ R3 @ A @ B5 ) ) ) ) ) ) ) ).
% PartialSA.soundness
thf(fact_1244_PartialSA_Ocompleteness__paper,axiom,
! [B5: aassertion_state,W: state,A: aassertion_state,Sigma: state,Sigma2: state] :
( ( packag7298478212905792770_state @ plus @ core @ B5 )
=> ( ( ( stable @ W )
& ( packag7096211993527611489_state @ plus @ W @ A @ B5 ) )
=> ( ( ( some_state @ Sigma )
= ( plus @ Sigma2 @ W ) )
=> ? [X_1: set_Pr4231240917283394591_state] :
( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
& ( package_sat_state @ plus @ A @ A3 ) ) )
@ ^ [Uu3: state] : $true
@ B5
@ Sigma2
@ W
@ X_1 ) ) ) ) ).
% PartialSA.completeness_paper
thf(fact_1245_PartialSA_OAddFromOutside,axiom,
! [Phi5: state,Phi6: state,M2: state,F5: state,S5: set_Pr4231240917283394591_state,Pc: state > $o,A: aassertion_state,Phi7: state,F6: state,S6: set_Pr4231240917283394591_state,F: state,S4: set_Pr4231240917283394591_state] :
( ( ( some_state @ Phi5 )
= ( plus @ Phi6 @ M2 ) )
=> ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F5 @ S5 @ Pc @ A @ Phi7 @ F6 @ S6 )
=> ( ( stable @ M2 )
=> ( ( ( some_state @ F5 )
= ( plus @ F @ M2 ) )
=> ( ( S5
= ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state,R2: state] :
( ( Uu3
= ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S4 )
& ( ( some_state @ R2 )
= ( plus @ A3 @ ( minus @ ( T5 @ F5 ) @ ( T5 @ F ) ) ) )
& ( defined @ R2 @ U6 ) ) ) )
=> ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi7 @ F6 @ S6 ) ) ) ) ) ) ).
% PartialSA.AddFromOutside
thf(fact_1246_Rep__state,axiom,
! [X: state] :
( member2879030624002546611on_val @ ( rep_state @ X )
@ ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [Phi2: produc9003781588162623004on_val] :
( ( Uu3 = Phi2 )
& ( valid_state @ Phi2 ) ) ) ) ).
% Rep_state
thf(fact_1247_Rep__state__cases,axiom,
! [Y: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ Y @ ( collec7475294502966882545on_val @ valid_state ) )
=> ~ ! [X2: state] :
( Y
!= ( rep_state @ X2 ) ) ) ).
% Rep_state_cases
thf(fact_1248_Rep__state__induct,axiom,
! [Y: produc9003781588162623004on_val,P6: produc9003781588162623004on_val > $o] :
( ( member2879030624002546611on_val @ Y
@ ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [Phi2: produc9003781588162623004on_val] :
( ( Uu3 = Phi2 )
& ( valid_state @ Phi2 ) ) ) )
=> ( ! [X2: state] : ( P6 @ ( rep_state @ X2 ) )
=> ( P6 @ Y ) ) ) ).
% Rep_state_induct
thf(fact_1249_Abs__state__inject,axiom,
! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ X
@ ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [Phi2: produc9003781588162623004on_val] :
( ( Uu3 = Phi2 )
& ( valid_state @ Phi2 ) ) ) )
=> ( ( member2879030624002546611on_val @ Y
@ ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [Phi2: produc9003781588162623004on_val] :
( ( Uu3 = Phi2 )
& ( valid_state @ Phi2 ) ) ) )
=> ( ( ( abs_state @ X )
= ( abs_state @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_state_inject
thf(fact_1250_Abs__state__induct,axiom,
! [P6: state > $o,X: state] :
( ! [Y5: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ Y5
@ ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [Phi2: produc9003781588162623004on_val] :
( ( Uu3 = Phi2 )
& ( valid_state @ Phi2 ) ) ) )
=> ( P6 @ ( abs_state @ Y5 ) ) )
=> ( P6 @ X ) ) ).
% Abs_state_induct
thf(fact_1251_Abs__state__cases,axiom,
! [X: state] :
~ ! [Y5: produc9003781588162623004on_val] :
( ( X
= ( abs_state @ Y5 ) )
=> ~ ( member2879030624002546611on_val @ Y5 @ ( collec7475294502966882545on_val @ valid_state ) ) ) ).
% Abs_state_cases
thf(fact_1252_Abs__state__inverse,axiom,
! [Y: produc9003781588162623004on_val] :
( ( member2879030624002546611on_val @ Y
@ ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [Phi2: produc9003781588162623004on_val] :
( ( Uu3 = Phi2 )
& ( valid_state @ Phi2 ) ) ) )
=> ( ( rep_state @ ( abs_state @ Y ) )
= Y ) ) ).
% Abs_state_inverse
thf(fact_1253_scaled__def,axiom,
( scaled
= ( ^ [Phi2: state] :
( collect_state
@ ^ [Uu3: state] :
? [P2: prat] :
( ( Uu3
= ( multiply @ P2 @ Phi2 ) )
& ( ppos @ P2 )
& ( pgte @ pwrite @ P2 ) ) ) ) ) ).
% scaled_def
thf(fact_1254_PartialSA_Osoundness__paper,axiom,
! [B5: aassertion_state,A: aassertion_state,S4: set_state,Sigma: state,Sigma2: state,W: state,S5: set_Pr4231240917283394591_state] :
( ( packag7298478212905792770_state @ plus @ core @ B5 )
=> ( ! [A6: state] :
( ( package_sat_state @ plus @ A @ A6 )
=> ( member_state @ A6 @ S4 ) )
=> ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
& ( member_state @ A3 @ S4 ) ) )
@ ^ [Uu3: state] : $true
@ B5
@ Sigma2
@ W
@ S5 )
=> ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ B5 ) )
| ( packag4152378406607043437_state @ plus @ S5 ) )
=> ( ( stable @ W )
& ( ( some_state @ Sigma )
= ( plus @ Sigma2 @ W ) )
& ( packag7096211993527611489_state @ plus @ W @ A @ B5 ) ) ) ) ) ) ).
% PartialSA.soundness_paper
thf(fact_1255_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,Phi3: state,F7: state,S7: set_Pr4231240917283394591_state,B8: aassertion_state] :
( ( A52
= ( aStar_state @ A10 @ B8 ) )
=> ( ( packag3171892606229212990_state @ plus @ core @ stable @ A12 @ A23 @ A32 @ A42 @ A10 @ Phi3 @ F7 @ S7 )
=> ~ ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F7 @ S7 @ 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] :
? [A3: state,U6: state,T5: state > state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
& ~ ( A42 @ A3 ) ) )
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state] :
( ( Uu3
= ( produc5886875603596089627_state @ ( minus @ A3 @ ( Witness @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ ( Witness @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
& ( A42 @ A3 ) ) ) ) ) ) ) ) )
=> ~ ! [Phi3: state,M3: state] :
( ( ( some_state @ A12 )
= ( plus @ Phi3 @ M3 ) )
=> ! [F7: state,S7: set_Pr4231240917283394591_state] :
( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F7 @ S7 @ A42 @ A52 @ A62 @ A72 @ A82 )
=> ( ( stable @ M3 )
=> ( ( ( some_state @ F7 )
= ( plus @ A23 @ M3 ) )
=> ( S7
!= ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state,R2: state] :
( ( Uu3
= ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ A32 )
& ( ( some_state @ R2 )
= ( plus @ A3 @ ( minus @ ( T5 @ F7 ) @ ( T5 @ A23 ) ) ) )
& ( defined @ R2 @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% PartialSA.package_rhs.cases
thf(fact_1256_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 )
= ( ? [Phi2: state,F4: state,S8: set_Pr4231240917283394591_state,Pc2: state > $o,A2: aassertion_state,Phi: state,F8: state,S9: set_Pr4231240917283394591_state,B6: aassertion_state,Phi8: state,F9: state,S10: set_Pr4231240917283394591_state] :
( ( A12 = Phi2 )
& ( A23 = F4 )
& ( A32 = S8 )
& ( A42 = Pc2 )
& ( A52
= ( aStar_state @ A2 @ B6 ) )
& ( A62 = Phi8 )
& ( A72 = F9 )
& ( A82 = S10 )
& ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S8 @ Pc2 @ A2 @ Phi @ F8 @ S9 )
& ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F8 @ S9 @ Pc2 @ B6 @ Phi8 @ F9 @ S10 ) )
| ? [Phi2: state,F4: state,S8: set_Pr4231240917283394591_state,Pc2: state > $o,B: state > $o,A2: aassertion_state,Phi: state,F8: state,S9: set_Pr4231240917283394591_state] :
( ( A12 = Phi2 )
& ( A23 = F4 )
& ( A32 = S8 )
& ( A42 = Pc2 )
& ( A52
= ( aImp_state @ B @ A2 ) )
& ( A62 = Phi )
& ( A72 = F8 )
& ( A82 = S9 )
& ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F4 @ S8 @ ( packag1330488657391168505_state @ Pc2 @ B ) @ A2 @ Phi @ F8 @ S9 ) )
| ? [S8: set_Pr4231240917283394591_state,Pc2: state > $o,Witness2: produc7272280185086996969_state > state,B6: state > $o,S9: set_Pr4231240917283394591_state,Phi2: state,F4: state] :
( ( A12 = Phi2 )
& ( A23 = F4 )
& ( A32 = S8 )
& ( A42 = Pc2 )
& ( A52
= ( aSem_state @ B6 ) )
& ( A62 = Phi2 )
& ( A72 = F4 )
& ( A82 = S9 )
& ! [X3: state,Y4: state,Z6: state > state,Aa3: state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X3 @ ( produc8632743443909064678_state @ Y4 @ Z6 ) ) @ S8 )
=> ( ( Pc2 @ X3 )
=> ( ( Aa3
= ( Witness2 @ ( produc5886875603596089627_state @ X3 @ ( produc8632743443909064678_state @ Y4 @ Z6 ) ) ) )
=> ( ( greater @ X3 @ Aa3 )
& ( B6 @ Aa3 ) ) ) ) )
& ( S9
= ( sup_su3837536599209510219_state
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S8 )
& ~ ( Pc2 @ A3 ) ) )
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state,B: state] :
( ( Uu3
= ( produc5886875603596089627_state @ ( minus @ A3 @ B ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B ) ) @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S8 )
& ( Pc2 @ A3 )
& ( B
= ( Witness2 @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) ) ) ) )
| ? [Phi2: state,Phi: state,M4: state,F8: state,S9: set_Pr4231240917283394591_state,Pc2: state > $o,A2: aassertion_state,Phi8: state,F9: state,S10: set_Pr4231240917283394591_state,F4: state,S8: set_Pr4231240917283394591_state] :
( ( A12 = Phi2 )
& ( A23 = F4 )
& ( A32 = S8 )
& ( A42 = Pc2 )
& ( A52 = A2 )
& ( A62 = Phi8 )
& ( A72 = F9 )
& ( A82 = S10 )
& ( ( some_state @ Phi2 )
= ( plus @ Phi @ M4 ) )
& ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F8 @ S9 @ Pc2 @ A2 @ Phi8 @ F9 @ S10 )
& ( stable @ M4 )
& ( ( some_state @ F8 )
= ( plus @ F4 @ M4 ) )
& ( S9
= ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state,R2: state] :
( ( Uu3
= ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S8 )
& ( ( some_state @ R2 )
= ( plus @ A3 @ ( minus @ ( T5 @ F8 ) @ ( T5 @ F4 ) ) ) )
& ( defined @ R2 @ U6 ) ) ) ) ) ) ) ).
% PartialSA.package_rhs.simps
thf(fact_1257_PartialSA_OASem,axiom,
! [S4: set_Pr4231240917283394591_state,Pc: state > $o,Witness3: produc7272280185086996969_state > state,B5: state > $o,S5: set_Pr4231240917283394591_state,Phi5: state,F: state] :
( ! [A6: state,U: state,T4: state > state,B4: state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T4 ) ) @ S4 )
=> ( ( Pc @ A6 )
=> ( ( B4
= ( Witness3 @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T4 ) ) ) )
=> ( ( greater @ A6 @ B4 )
& ( B5 @ B4 ) ) ) ) )
=> ( ( S5
= ( sup_su3837536599209510219_state
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S4 )
& ~ ( Pc @ A3 ) ) )
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,U6: state,T5: state > state,B: state] :
( ( Uu3
= ( produc5886875603596089627_state @ ( minus @ A3 @ B ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B ) ) @ T5 ) ) )
& ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S4 )
& ( Pc @ A3 )
& ( B
= ( Witness3 @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) ) ) ) ) ) )
=> ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ ( aSem_state @ B5 ) @ Phi5 @ F @ S5 ) ) ) ).
% PartialSA.ASem
thf(fact_1258_PartialSA_Osat_Opelims_I3_J,axiom,
! [X: aassertion_state,Xa: state] :
( ~ ( package_sat_state @ plus @ X @ Xa )
=> ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X @ Xa ) )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( 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] :
( ( X
= ( 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] :
( ( X
= ( aSem_state @ A10 ) )
=> ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) )
=> ( A10 @ Xa ) ) ) ) ) ) ) ).
% PartialSA.sat.pelims(3)
thf(fact_1259_PartialSA_Opackage__logic__axioms,axiom,
package_logic_state @ plus @ core @ unit @ stable ).
% PartialSA.package_logic_axioms
thf(fact_1260_PartialSA_Osat_Opelims_I1_J,axiom,
! [X: aassertion_state,Xa: state,Y: $o] :
( ( ( package_sat_state @ plus @ X @ Xa )
= Y )
=> ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X @ Xa ) )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( aStar_state @ A10 @ B8 ) )
=> ( ( Y
= ( ? [A3: state,B: state] :
( ( ( some_state @ Xa )
= ( plus @ A3 @ B ) )
& ( package_sat_state @ plus @ A10 @ A3 )
& ( package_sat_state @ plus @ B8 @ B ) ) ) )
=> ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) ) ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( ( X
= ( 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] :
( ( X
= ( aSem_state @ A10 ) )
=> ( ( Y
= ( A10 @ Xa ) )
=> ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) ) ) ) ) ) ) ) ).
% PartialSA.sat.pelims(1)
thf(fact_1261_PartialSA_Osat_Opelims_I2_J,axiom,
! [X: aassertion_state,Xa: state] :
( ( package_sat_state @ plus @ X @ Xa )
=> ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X @ Xa ) )
=> ( ! [A10: aassertion_state,B8: aassertion_state] :
( ( X
= ( aStar_state @ A10 @ B8 ) )
=> ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa ) )
=> ~ ? [A6: state,B4: state] :
( ( ( some_state @ Xa )
= ( plus @ A6 @ B4 ) )
& ( package_sat_state @ plus @ A10 @ A6 )
& ( package_sat_state @ plus @ B8 @ B4 ) ) ) )
=> ( ! [B4: state > $o,A10: aassertion_state] :
( ( X
= ( 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] :
( ( X
= ( aSem_state @ A10 ) )
=> ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa ) )
=> ~ ( A10 @ Xa ) ) ) ) ) ) ) ).
% PartialSA.sat.pelims(2)
thf(fact_1262_PartialSA_Ominus__def,axiom,
( minus
= ( ^ [B: state,A3: state] :
( fun_TH8221542053827138557_state @ B
@ ^ [X3: state] :
( ( ( some_state @ B )
= ( plus @ A3 @ X3 ) )
& ( greater @ X3 @ ( core @ B ) ) ) ) ) ) ).
% PartialSA.minus_def
thf(fact_1263_PartialSA_Ogeneral__soundness,axiom,
! [Phi5: state,S4: set_Pr2391701600042252046_state,A: aassertion_state,Phi6: state,F: state,S5: set_Pr4231240917283394591_state] :
( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ unit
@ ( collec906845016635587262_state
@ ^ [Uu3: produc7272280185086996969_state] :
? [A3: state,T5: state > state] :
( ( Uu3
= ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ unit @ T5 ) ) )
& ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A3 @ T5 ) @ S4 ) ) )
@ ^ [Uu3: state] : $true
@ A
@ Phi6
@ F
@ S5 )
=> ( ! [A6: state,T4: state > state] :
( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A6 @ T4 ) @ S4 )
=> ( packag4817583915586052229_state @ plus @ unit @ T4 ) )
=> ( ( packag7298478212905792770_state @ plus @ core @ A )
=> ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ A ) )
| ( packag4152378406607043437_state @ plus @ S5 ) )
=> ( ( ( some_state @ Phi5 )
= ( plus @ Phi6 @ F ) )
& ( stable @ F )
& ! [X4: produc3882902684156697774_state] :
( ( member6715010751053357271_state @ X4 @ S4 )
=> ( produc3645877548210629341tate_o
@ ^ [A3: state,T5: state > state] :
( ( defined @ A3 @ ( T5 @ F ) )
=> ( package_sat_state @ plus @ A @ ( the_state @ ( plus @ A3 @ ( T5 @ F ) ) ) ) )
@ X4 ) ) ) ) ) ) ) ).
% PartialSA.general_soundness
thf(fact_1264_PartialSA_Ounder__def,axiom,
! [A: set_state,Omega2: state] :
( ( sep_under_state @ plus @ A @ Omega2 )
= ( collect_state
@ ^ [Uu3: state] :
? [Omega3: state] :
( ( Uu3 = Omega3 )
& ( member_state @ Omega3 @ A )
& ( greater @ Omega2 @ Omega3 ) ) ) ) ).
% PartialSA.under_def
thf(fact_1265_PartialSA_Ocompleteness__aux,axiom,
! [S4: set_Pr4231240917283394591_state,F1: state > state > ( state > state ) > state,F22: state > state > ( state > state ) > state,Pc: state > $o,A: aassertion_state,F: state,Phi5: state] :
( ! [A6: state,U: state,T4: state > state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T4 ) ) @ S4 )
=> ( ( greater @ ( core @ ( F1 @ A6 @ U @ T4 ) ) @ ( core @ A6 ) )
& ( ( some_state @ A6 )
= ( plus @ ( F1 @ A6 @ U @ T4 ) @ ( F22 @ A6 @ U @ T4 ) ) )
& ( ( Pc @ ( core @ A6 ) )
=> ( package_sat_state @ plus @ A @ ( the_state @ ( plus @ ( core @ A6 ) @ ( F1 @ A6 @ U @ T4 ) ) ) ) ) ) )
=> ( ( packag3877571396724326311_state @ plus @ core @ unit @ S4 @ F )
=> ( ( packag7298478212905792770_state @ plus @ core @ A )
=> ( ( packag6595153354952283300_state @ plus @ core @ Pc )
=> ( ( ( stable @ F )
& ( defined @ Phi5 @ F ) )
=> ? [S7: set_Pr4231240917283394591_state] :
( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi5 @ F @ S7 )
& ! [X4: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X4 @ S7 )
=> ( produc1745419677418529872tate_o
@ ^ [A13: state] :
( produc3645877548210629341tate_o
@ ^ [U8: state,T5: state > state] :
? [A3: state,U6: state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U6 @ T5 ) ) @ S4 )
& ( greater @ A13 @ ( F22 @ A3 @ U6 @ T5 ) )
& ( ( core @ A13 )
= ( core @ A3 ) ) ) )
@ X4 ) ) ) ) ) ) ) ) ).
% PartialSA.completeness_aux
thf(fact_1266_defined__def,axiom,
( defined
= ( sep_defined_state @ plus ) ) ).
% defined_def
thf(fact_1267_PartialSA_Ovalid__package__set__def,axiom,
! [S4: set_Pr4231240917283394591_state,F: state] :
( ( packag3877571396724326311_state @ plus @ core @ unit @ S4 @ F )
= ( ! [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ S4 )
=> ( produc1745419677418529872tate_o
@ ^ [A3: state] :
( produc3645877548210629341tate_o
@ ^ [U6: state,T5: state > state] :
( ( defined @ A3 @ U6 )
& ( greater @ ( core @ A3 ) @ ( core @ U6 ) )
& ( packag4817583915586052229_state @ plus @ unit @ T5 )
& ( greater @ A3 @ ( core @ ( T5 @ F ) ) ) ) )
@ X3 ) ) ) ) ).
% PartialSA.valid_package_set_def
thf(fact_1268_PartialSA_Opackage__rhs__connection__def,axiom,
! [Phi5: state,F: state,S4: set_Pr4231240917283394591_state,Pc: state > $o,A: aassertion_state,Phi6: state,F5: state,S5: set_Pr4231240917283394591_state] :
( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi5 @ F @ S4 @ Pc @ A @ Phi6 @ F5 @ S5 )
= ( ( greater @ F5 @ F )
& ( defined @ Phi5 @ F )
& ( ( plus @ Phi5 @ F )
= ( plus @ Phi6 @ F5 ) )
& ( stable @ F5 )
& ! [X3: produc7272280185086996969_state] :
( ( member7461423308196221312_state @ X3 @ S4 )
=> ( produc1745419677418529872tate_o
@ ^ [A3: state] :
( produc3645877548210629341tate_o
@ ^ [U6: state,T5: state > state] :
? [Au3: state] :
( ( ( some_state @ Au3 )
= ( plus @ A3 @ U6 ) )
& ( ( defined @ Au3 @ ( minus @ ( T5 @ F5 ) @ ( T5 @ F ) ) )
=> ? [A13: state,U8: state] :
( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A13 @ ( produc8632743443909064678_state @ U8 @ T5 ) ) @ S5 )
& ( greater @ ( core @ A13 ) @ ( core @ A3 ) )
& ( ( plus @ Au3 @ ( minus @ ( T5 @ F5 ) @ ( T5 @ F ) ) )
= ( plus @ A13 @ U8 ) )
& ( greater @ U8 @ U6 )
& ( packag8070727096299822599_state @ plus @ core @ Pc @ A @ A13 @ U8 @ U6 ) ) ) ) )
@ X3 ) ) ) ) ).
% PartialSA.package_rhs_connection_def
thf(fact_1269_greater__def,axiom,
( greater
= ( sep_greater_state @ plus ) ) ).
% greater_def
thf(fact_1270_minus__def,axiom,
( minus
= ( sep_minus_state @ plus @ core ) ) ).
% minus_def
thf(fact_1271_cwand__def,axiom,
( cwand
= ( ^ [A2: set_state,B6: set_state] :
( collect_state
@ ^ [Uu3: state] :
? [W2: state] :
( ( Uu3 = W2 )
& ! [A3: state,X3: state] :
( ( ( member_state @ A3 @ A2 )
& ( ( some_state @ X3 )
= ( plus @ ( r @ A3 @ W2 ) @ A3 ) ) )
=> ( member_state @ X3 @ B6 ) ) ) ) ) ) ).
% cwand_def
thf(fact_1272_wand__def,axiom,
( wand
= ( ^ [A2: set_state,B6: set_state] :
( collect_state
@ ^ [Uu3: state] :
? [W2: state] :
( ( Uu3 = W2 )
& ! [A3: state,X3: state] :
( ( ( member_state @ A3 @ A2 )
& ( ( some_state @ X3 )
= ( plus @ W2 @ A3 ) ) )
=> ( member_state @ X3 @ B6 ) ) ) ) ) ) ).
% wand_def
thf(fact_1273_type__definition__state,axiom,
( type_d7385716227342696708on_val @ rep_state @ abs_state
@ ( collec7475294502966882545on_val
@ ^ [Uu3: produc9003781588162623004on_val] :
? [Phi2: produc9003781588162623004on_val] :
( ( Uu3 = Phi2 )
& ( valid_state @ Phi2 ) ) ) ) ).
% type_definition_state
thf(fact_1274_comp__one__def,axiom,
( comp_one
= ( ^ [P2: prat] :
( fChoice_prat
@ ^ [R2: prat] :
( ( padd @ P2 @ R2 )
= pwrite ) ) ) ) ).
% comp_one_def
% Helper facts (8)
thf(help_If_2_1_If_001t__PosRat__Oprat_T,axiom,
! [X: prat,Y: prat] :
( ( if_prat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__PosRat__Oprat_T,axiom,
! [X: prat,Y: prat] :
( ( if_prat @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001t__PosRat__Oprat_T,axiom,
! [P6: prat > $o] :
( ( P6 @ ( fChoice_prat @ P6 ) )
= ( ? [X7: prat] : ( P6 @ X7 ) ) ) ).
thf(help_If_2_1_If_001t__PartialHeapSA__Ostate_T,axiom,
! [X: state,Y: state] :
( ( if_state @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__PartialHeapSA__Ostate_T,axiom,
! [X: state,Y: state] :
( ( if_state @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
! [P6: $o] :
( ( P6 = $true )
| ( P6 = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
! [X: option_state,Y: option_state] :
( ( if_option_state @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
! [X: option_state,Y: option_state] :
( ( if_option_state @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_state @ ( multiply @ p @ x ) @ ( multip8064567061438756306ertion @ p @ a ) ).
%------------------------------------------------------------------------------