TPTP Problem File: SWW471^2.p
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%------------------------------------------------------------------------------
% File : SWW471^2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 269, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_500_thf_l269 [Bla11]
% Status : ContradictoryAxioms
% Rating : 1.00 v7.2.0, 0.88 v7.1.0, 1.00 v5.3.0
% Syntax : Number of formulae : 849 ( 298 unt; 135 typ; 0 def)
% Number of atoms : 3194 ( 697 equ; 48 cnn)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 6398 ( 258 ~; 46 |; 101 &;5209 @)
% ( 150 <=>; 634 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 1823 (1823 >; 0 *; 0 +; 0 <<)
% Number of symbols : 134 ( 127 usr; 12 con; 0-5 aty)
% Number of variables : 2050 ( 106 ^;1909 !; 35 ?;2050 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:25:05
%------------------------------------------------------------------------------
%----Should-be-implicit typings (8)
thf(ty_ty_t__a,type,
x_a: $tType ).
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Opname,type,
pname: $tType ).
thf(ty_ty_tc__Com__Ostate,type,
state: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
hoare_1775062406iple_a: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
hoare_1167836817_state: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
option_com: $tType ).
%----Explicit typings (127)
thf(sy_c_Big__Operators_Olattice__class_OInf__fin_000_062_Itc__Com__Opname_M_Eo_J,type,
big_la1126801287name_o: ( ( pname > $o ) > $o ) > pname > $o ).
thf(sy_c_Big__Operators_Olattice__class_OInf__fin_000_062_Itc__Hoare____Mirabelle___,type,
big_la447547205le_a_o: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Big__Operators_Olattice__class_OInf__fin_000_062_Itc__Hoare____Mirabelle____001,type,
big_la831793456tate_o: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Big__Operators_Olattice__class_OInf__fin_000_Eo,type,
big_la1690136417_fin_o: ( $o > $o ) > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_Itc__Com__Opname_M_Eo_J,type,
big_la1286884090name_o: ( ( pname > $o ) > $o ) > pname > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_Itc__Hoare____Mirabelle___,type,
big_la1843772984le_a_o: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_Itc__Hoare____Mirabelle____002,type,
big_la1138507389tate_o: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_Eo,type,
big_la727467310_fin_o: ( $o > $o ) > $o ).
thf(sy_c_Com_Obody,type,
body_1: pname > option_com ).
thf(sy_c_Com_Ocom_OBODY,type,
body: pname > com ).
thf(sy_c_Com_Ocom_OCond,type,
cond: ( state > $o ) > com > com > com ).
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_OSemi,type,
semi: com > com > com ).
thf(sy_c_Com_Ocom_OWhile,type,
while: ( state > $o ) > com > com ).
thf(sy_c_Com_Ocom_Ocom__size,type,
com_size: com > nat ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Com__Opname_M_Eo_J,type,
finite297249702name_o: ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_,type,
finite789576932le_a_o: ( ( hoare_1775062406iple_a > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple__003,type,
finite1380128977tate_o: ( ( hoare_1167836817_state > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_Eo,type,
finite_finite_o: ( $o > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
finite_finite_pname: ( pname > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_,type,
finite2063573081iple_a: ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__C,type,
finite1084549118_state: ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otr,type,
finite1805141964_pname: ( ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > ( pname > hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > ( pname > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otr_004,type,
finite1068437657_pname: ( ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( pname > hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > ( pname > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,
finite1282449217_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple_,type,
finite2078349315iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple__005,type,
finite1074406356_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,
finite89670078_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
finite1358382848iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot_006,type,
finite806517911_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Com__Opname_M_Eo_J,type,
minus_minus_pname_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx__,type,
minus_1944206118le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx___007,type,
minus_2107060239tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_OThe_000tc__Com__Opname,type,
the_pname: ( pname > $o ) > pname ).
thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
the_Ho1155011127iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_,type,
the_Ho310147232_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_OMGT,type,
hoare_Mirabelle_MGT: com > hoare_1167836817_state ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000t__a,type,
hoare_1508237396rivs_a: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000tc__Com__Ostate,type,
hoare_123228589_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__valids_000t__a,type,
hoare_1846070742lids_a: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__valids_000tc__Com__Ostate,type,
hoare_529639851_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000t__a,type,
hoare_1766022166iple_a: ( x_a > state > $o ) > com > ( x_a > state > $o ) > hoare_1775062406iple_a ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000tc__Com__Ostate,type,
hoare_908217195_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1167836817_state ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple__size_000t__a,type,
hoare_1118907895size_a: ( x_a > nat ) > hoare_1775062406iple_a > nat ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple__size_000tc__Com__Ostate,type,
hoare_545207370_state: ( state > nat ) > hoare_1167836817_state > nat ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple__valid_000t__a,type,
hoare_1462269968alid_a: nat > hoare_1775062406iple_a > $o ).
thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple__valid_000tc__Com__Ostate,type,
hoare_56934129_state: nat > hoare_1167836817_state > $o ).
thf(sy_c_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
if_Hoa1047340790iple_a: $o > hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ).
thf(sy_c_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
if_Hoa833675553_state: $o > hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_I_062_Itc__Com__Opname_M_Eo_,type,
semila2013987940me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_I_062_Itc__Hoare____Mirabell,type,
semila1691990438_a_o_o: ( ( hoare_1775062406iple_a > $o ) > $o ) > ( ( hoare_1775062406iple_a > $o ) > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_I_062_Itc__Hoare____Mirabell_008,type,
semila1758709489te_o_o: ( ( hoare_1167836817_state > $o ) > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_I_Eo_M_Eo_J,type,
semila232696320nf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Com__Opname_M_Eo_J,type,
semila1673364395name_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____s,type,
semila966743401le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____s_009,type,
semila179895820tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_Eo,type,
semila854092349_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Com__Opname_M_Eo_,type,
semila181081674me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Hoare____Mirabell,type,
semila2069193356_a_o_o: ( ( hoare_1775062406iple_a > $o ) > $o ) > ( ( hoare_1775062406iple_a > $o ) > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Hoare____Mirabell_010,type,
semila866907787te_o_o: ( ( hoare_1167836817_state > $o ) > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_Eo_M_Eo_J,type,
semila2062604954up_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Com__Opname_M_Eo_J,type,
semila1780557381name_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____s,type,
semila13410563le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____s_011,type,
semila1172322802tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_Eo,type,
semila10642723_sup_o: $o > $o > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
nat_case_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Com__Ocom,type,
size_size_com: com > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It_,type,
size_s724313756iple_a: hoare_1775062406iple_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc,type,
size_s645941755_state: hoare_1167836817_state > nat ).
thf(sy_c_Natural_Oevalc,type,
evalc: com > state > state > $o ).
thf(sy_c_Natural_Oevaln,type,
evaln: com > state > nat > state > $o ).
thf(sy_c_Option_Othe_000tc__Com__Ocom,type,
the_com: option_com > com ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
bot_bot_pname_o_o: ( pname > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Hoare____Mirabelle____srushsu,type,
bot_bo1976773294_a_o_o: ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Hoare____Mirabelle____srushsu_012,type,
bot_bo691907561te_o_o: ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J,type,
bot_bot_pname_o: pname > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O,type,
bot_bo751897185le_a_o: hoare_1775062406iple_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O_013,type,
bot_bo70021908tate_o: hoare_1167836817_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Set_OCollect_000tc__Com__Opname,type,
collect_pname: ( pname > $o ) > pname > $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
collec676402587iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,
collec1027672124_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000_062_Itc__Com__Opname_M_Eo_J,type,
image_1085733413name_o: ( ( pname > $o ) > pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_M,type,
image_2014247585le_a_o: ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > ( ( hoare_1775062406iple_a > $o ) > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com_,type,
image_1488525317tate_o: ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Set_Oimage_000_Eo_000_Eo,type,
image_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
image_pname_pname: ( pname > pname ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____srushsumbx__Otri,type,
image_2063119815iple_a: ( pname > hoare_1775062406iple_a ) > ( pname > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____srushsumbx__Otri_014,type,
image_575578384_state: ( pname > hoare_1167836817_state ) > ( pname > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc__,type,
image_51246659_pname: ( hoare_1775062406iple_a > pname ) > ( hoare_1775062406iple_a > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc___015,type,
image_1170193413iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc___016,type,
image_1021683026_state: ( hoare_1775062406iple_a > hoare_1167836817_state ) > ( hoare_1775062406iple_a > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat,type,
image_1802845250iple_a: ( hoare_1167836817_state > hoare_1775062406iple_a ) > ( hoare_1167836817_state > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_017,type,
image_31595733_state: ( hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Com__Opname_M_Eo_J,type,
insert_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,type,
insert1210049533le_a_o: ( hoare_1775062406iple_a > $o ) > ( ( hoare_1775062406iple_a > $o ) > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,type,
insert999278200tate_o: ( hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_Eo,type,
insert_o: $o > ( $o > $o ) > $o > $o ).
thf(sy_c_Set_Oinsert_000tc__Com__Opname,type,
insert_pname: pname > ( pname > $o ) > pname > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
insert1281456128iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Osta,type,
insert2134838167_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Othe__elem_000tc__Com__Opname,type,
the_elem_pname: ( pname > $o ) > pname ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
the_el1844711461iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__O,type,
the_el323660082_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_fequal_000tc__Com__Opname,type,
fequal_pname: pname > pname > $o ).
thf(sy_c_fequal_000tc__Com__Ostate,type,
fequal_state: state > state > $o ).
thf(sy_c_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
fequal1288209029iple_a: hoare_1775062406iple_a > hoare_1775062406iple_a > $o ).
thf(sy_c_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
fequal1831255762_state: hoare_1167836817_state > hoare_1167836817_state > $o ).
thf(sy_c_member_000_062_Itc__Com__Opname_M_Eo_J,type,
member_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_M_Eo_,type,
member1207314404le_a_o: ( hoare_1775062406iple_a > $o ) > ( ( hoare_1775062406iple_a > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,
member864234961tate_o: ( hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > $o ).
thf(sy_c_member_000_Eo,type,
member_o: $o > ( $o > $o ) > $o ).
thf(sy_c_member_000tc__Com__Opname,type,
member_pname: pname > ( pname > $o ) > $o ).
thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
member2122167641iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
member2058392318_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_v_G,type,
g: hoare_1775062406iple_a > $o ).
thf(sy_v_P,type,
p: pname > x_a > state > $o ).
thf(sy_v_Procs,type,
procs: pname > $o ).
thf(sy_v_Q,type,
q: pname > x_a > state > $o ).
thf(sy_v_n,type,
n: nat ).
%----Relevant facts (698)
thf(fact_0_triple_Oinject,axiom,
! [Fun1_4: x_a > state > $o,Com: com,Fun2_4: x_a > state > $o,Fun1_3: x_a > state > $o,Com_1: com,Fun2_3: x_a > state > $o] :
( ( ( hoare_1766022166iple_a @ Fun1_4 @ Com @ Fun2_4 )
= ( hoare_1766022166iple_a @ Fun1_3 @ Com_1 @ Fun2_3 ) )
<=> ( ( Fun1_4 = Fun1_3 )
& ( Com = Com_1 )
& ( Fun2_4 = Fun2_3 ) ) ) ).
thf(fact_1_triple_Oinject,axiom,
! [Fun1_4: state > state > $o,Com: com,Fun2_4: state > state > $o,Fun1_3: state > state > $o,Com_1: com,Fun2_3: state > state > $o] :
( ( ( hoare_908217195_state @ Fun1_4 @ Com @ Fun2_4 )
= ( hoare_908217195_state @ Fun1_3 @ Com_1 @ Fun2_3 ) )
<=> ( ( Fun1_4 = Fun1_3 )
& ( Com = Com_1 )
& ( Fun2_4 = Fun2_3 ) ) ) ).
thf(fact_2_hoare__valids__def,axiom,
! [G_25: hoare_1167836817_state > $o,Ts_4: hoare_1167836817_state > $o] :
( ( hoare_529639851_state @ G_25 @ Ts_4 )
<=> ! [N: nat] :
( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ G_25 )
=> ( hoare_56934129_state @ N @ X ) )
=> ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ Ts_4 )
=> ( hoare_56934129_state @ N @ X ) ) ) ) ).
thf(fact_3_hoare__valids__def,axiom,
! [G_25: hoare_1775062406iple_a > $o,Ts_4: hoare_1775062406iple_a > $o] :
( ( hoare_1846070742lids_a @ G_25 @ Ts_4 )
<=> ! [N: nat] :
( ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ G_25 )
=> ( hoare_1462269968alid_a @ N @ X ) )
=> ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ Ts_4 )
=> ( hoare_1462269968alid_a @ N @ X ) ) ) ) ).
thf(fact_4_hoare__derivs_OBody,axiom,
! [G_24: hoare_1167836817_state > $o,P_37: pname > state > state > $o,Q_20: pname > state > state > $o,Procs_1: pname > $o] :
( ( hoare_123228589_state
@ ( semila1172322802tate_o @ G_24
@ ( image_575578384_state
@ ^ [P_10: pname] : ( hoare_908217195_state @ ( P_37 @ P_10 ) @ ( body @ P_10 ) @ ( Q_20 @ P_10 ) )
@ Procs_1 ) )
@ ( image_575578384_state
@ ^ [P_10: pname] : ( hoare_908217195_state @ ( P_37 @ P_10 ) @ ( the_com @ ( body_1 @ P_10 ) ) @ ( Q_20 @ P_10 ) )
@ Procs_1 ) )
=> ( hoare_123228589_state @ G_24
@ ( image_575578384_state
@ ^ [P_10: pname] : ( hoare_908217195_state @ ( P_37 @ P_10 ) @ ( body @ P_10 ) @ ( Q_20 @ P_10 ) )
@ Procs_1 ) ) ) ).
thf(fact_5_hoare__derivs_OBody,axiom,
! [G_24: hoare_1775062406iple_a > $o,P_37: pname > x_a > state > $o,Q_20: pname > x_a > state > $o,Procs_1: pname > $o] :
( ( hoare_1508237396rivs_a
@ ( semila13410563le_a_o @ G_24
@ ( image_2063119815iple_a
@ ^ [P_10: pname] : ( hoare_1766022166iple_a @ ( P_37 @ P_10 ) @ ( body @ P_10 ) @ ( Q_20 @ P_10 ) )
@ Procs_1 ) )
@ ( image_2063119815iple_a
@ ^ [P_10: pname] : ( hoare_1766022166iple_a @ ( P_37 @ P_10 ) @ ( the_com @ ( body_1 @ P_10 ) ) @ ( Q_20 @ P_10 ) )
@ Procs_1 ) )
=> ( hoare_1508237396rivs_a @ G_24
@ ( image_2063119815iple_a
@ ^ [P_10: pname] : ( hoare_1766022166iple_a @ ( P_37 @ P_10 ) @ ( body @ P_10 ) @ ( Q_20 @ P_10 ) )
@ Procs_1 ) ) ) ).
thf(fact_6_UnE,axiom,
! [C_42: hoare_1167836817_state,A_138: hoare_1167836817_state > $o,B_79: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_42 @ ( semila1172322802tate_o @ A_138 @ B_79 ) )
=> ( ~ ( member2058392318_state @ C_42 @ A_138 )
=> ( member2058392318_state @ C_42 @ B_79 ) ) ) ).
thf(fact_7_UnE,axiom,
! [C_42: hoare_1775062406iple_a,A_138: hoare_1775062406iple_a > $o,B_79: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_42 @ ( semila13410563le_a_o @ A_138 @ B_79 ) )
=> ( ~ ( member2122167641iple_a @ C_42 @ A_138 )
=> ( member2122167641iple_a @ C_42 @ B_79 ) ) ) ).
thf(fact_8_UnE,axiom,
! [C_42: pname,A_138: pname > $o,B_79: pname > $o] :
( ( member_pname @ C_42 @ ( semila1780557381name_o @ A_138 @ B_79 ) )
=> ( ~ ( member_pname @ C_42 @ A_138 )
=> ( member_pname @ C_42 @ B_79 ) ) ) ).
thf(fact_9_sup1E,axiom,
! [A_137: hoare_1167836817_state > $o,B_78: hoare_1167836817_state > $o,X_51: hoare_1167836817_state] :
( ( semila1172322802tate_o @ A_137 @ B_78 @ X_51 )
=> ( ~ ( A_137 @ X_51 )
=> ( B_78 @ X_51 ) ) ) ).
thf(fact_10_sup1E,axiom,
! [A_137: pname > $o,B_78: pname > $o,X_51: pname] :
( ( semila1780557381name_o @ A_137 @ B_78 @ X_51 )
=> ( ~ ( A_137 @ X_51 )
=> ( B_78 @ X_51 ) ) ) ).
thf(fact_11_sup1E,axiom,
! [A_137: hoare_1775062406iple_a > $o,B_78: hoare_1775062406iple_a > $o,X_51: hoare_1775062406iple_a] :
( ( semila13410563le_a_o @ A_137 @ B_78 @ X_51 )
=> ( ~ ( A_137 @ X_51 )
=> ( B_78 @ X_51 ) ) ) ).
thf(fact_12_sup1CI,axiom,
! [A_136: hoare_1167836817_state > $o,B_77: hoare_1167836817_state > $o,X_50: hoare_1167836817_state] :
( ( ~ ( B_77 @ X_50 )
=> ( A_136 @ X_50 ) )
=> ( semila1172322802tate_o @ A_136 @ B_77 @ X_50 ) ) ).
thf(fact_13_sup1CI,axiom,
! [A_136: pname > $o,B_77: pname > $o,X_50: pname] :
( ( ~ ( B_77 @ X_50 )
=> ( A_136 @ X_50 ) )
=> ( semila1780557381name_o @ A_136 @ B_77 @ X_50 ) ) ).
thf(fact_14_sup1CI,axiom,
! [A_136: hoare_1775062406iple_a > $o,B_77: hoare_1775062406iple_a > $o,X_50: hoare_1775062406iple_a] :
( ( ~ ( B_77 @ X_50 )
=> ( A_136 @ X_50 ) )
=> ( semila13410563le_a_o @ A_136 @ B_77 @ X_50 ) ) ).
thf(fact_15_UnCI,axiom,
! [A_135: hoare_1167836817_state > $o,C_41: hoare_1167836817_state,B_76: hoare_1167836817_state > $o] :
( ( ~ ( member2058392318_state @ C_41 @ B_76 )
=> ( member2058392318_state @ C_41 @ A_135 ) )
=> ( member2058392318_state @ C_41 @ ( semila1172322802tate_o @ A_135 @ B_76 ) ) ) ).
thf(fact_16_UnCI,axiom,
! [A_135: hoare_1775062406iple_a > $o,C_41: hoare_1775062406iple_a,B_76: hoare_1775062406iple_a > $o] :
( ( ~ ( member2122167641iple_a @ C_41 @ B_76 )
=> ( member2122167641iple_a @ C_41 @ A_135 ) )
=> ( member2122167641iple_a @ C_41 @ ( semila13410563le_a_o @ A_135 @ B_76 ) ) ) ).
thf(fact_17_UnCI,axiom,
! [A_135: pname > $o,C_41: pname,B_76: pname > $o] :
( ( ~ ( member_pname @ C_41 @ B_76 )
=> ( member_pname @ C_41 @ A_135 ) )
=> ( member_pname @ C_41 @ ( semila1780557381name_o @ A_135 @ B_76 ) ) ) ).
thf(fact_18_image__eqI,axiom,
! [A_134: pname > $o,B_75: hoare_1167836817_state,F_41: pname > hoare_1167836817_state,X_49: pname] :
( ( B_75
= ( F_41 @ X_49 ) )
=> ( ( member_pname @ X_49 @ A_134 )
=> ( member2058392318_state @ B_75 @ ( image_575578384_state @ F_41 @ A_134 ) ) ) ) ).
thf(fact_19_image__eqI,axiom,
! [A_134: hoare_1775062406iple_a > $o,B_75: pname,F_41: hoare_1775062406iple_a > pname,X_49: hoare_1775062406iple_a] :
( ( B_75
= ( F_41 @ X_49 ) )
=> ( ( member2122167641iple_a @ X_49 @ A_134 )
=> ( member_pname @ B_75 @ ( image_51246659_pname @ F_41 @ A_134 ) ) ) ) ).
thf(fact_20_image__eqI,axiom,
! [A_134: pname > $o,B_75: hoare_1775062406iple_a,F_41: pname > hoare_1775062406iple_a,X_49: pname] :
( ( B_75
= ( F_41 @ X_49 ) )
=> ( ( member_pname @ X_49 @ A_134 )
=> ( member2122167641iple_a @ B_75 @ ( image_2063119815iple_a @ F_41 @ A_134 ) ) ) ) ).
thf(fact_21_image__Un,axiom,
! [F_40: pname > hoare_1167836817_state,A_133: pname > $o,B_74: pname > $o] :
( ( image_575578384_state @ F_40 @ ( semila1780557381name_o @ A_133 @ B_74 ) )
= ( semila1172322802tate_o @ ( image_575578384_state @ F_40 @ A_133 ) @ ( image_575578384_state @ F_40 @ B_74 ) ) ) ).
thf(fact_22_image__Un,axiom,
! [F_40: hoare_1775062406iple_a > hoare_1167836817_state,A_133: hoare_1775062406iple_a > $o,B_74: hoare_1775062406iple_a > $o] :
( ( image_1021683026_state @ F_40 @ ( semila13410563le_a_o @ A_133 @ B_74 ) )
= ( semila1172322802tate_o @ ( image_1021683026_state @ F_40 @ A_133 ) @ ( image_1021683026_state @ F_40 @ B_74 ) ) ) ).
thf(fact_23_image__Un,axiom,
! [F_40: hoare_1775062406iple_a > pname,A_133: hoare_1775062406iple_a > $o,B_74: hoare_1775062406iple_a > $o] :
( ( image_51246659_pname @ F_40 @ ( semila13410563le_a_o @ A_133 @ B_74 ) )
= ( semila1780557381name_o @ ( image_51246659_pname @ F_40 @ A_133 ) @ ( image_51246659_pname @ F_40 @ B_74 ) ) ) ).
thf(fact_24_image__Un,axiom,
! [F_40: hoare_1167836817_state > hoare_1775062406iple_a,A_133: hoare_1167836817_state > $o,B_74: hoare_1167836817_state > $o] :
( ( image_1802845250iple_a @ F_40 @ ( semila1172322802tate_o @ A_133 @ B_74 ) )
= ( semila13410563le_a_o @ ( image_1802845250iple_a @ F_40 @ A_133 ) @ ( image_1802845250iple_a @ F_40 @ B_74 ) ) ) ).
thf(fact_25_image__Un,axiom,
! [F_40: pname > hoare_1775062406iple_a,A_133: pname > $o,B_74: pname > $o] :
( ( image_2063119815iple_a @ F_40 @ ( semila1780557381name_o @ A_133 @ B_74 ) )
= ( semila13410563le_a_o @ ( image_2063119815iple_a @ F_40 @ A_133 ) @ ( image_2063119815iple_a @ F_40 @ B_74 ) ) ) ).
thf(fact_26_sup__fun__def,axiom,
! [F_39: hoare_1167836817_state > $o,G_23: hoare_1167836817_state > $o,X: hoare_1167836817_state] :
( ( semila1172322802tate_o @ F_39 @ G_23 @ X )
<=> ( semila10642723_sup_o @ ( F_39 @ X ) @ ( G_23 @ X ) ) ) ).
thf(fact_27_sup__fun__def,axiom,
! [F_39: pname > $o,G_23: pname > $o,X: pname] :
( ( semila1780557381name_o @ F_39 @ G_23 @ X )
<=> ( semila10642723_sup_o @ ( F_39 @ X ) @ ( G_23 @ X ) ) ) ).
thf(fact_28_sup__fun__def,axiom,
! [F_39: hoare_1775062406iple_a > $o,G_23: hoare_1775062406iple_a > $o,X: hoare_1775062406iple_a] :
( ( semila13410563le_a_o @ F_39 @ G_23 @ X )
<=> ( semila10642723_sup_o @ ( F_39 @ X ) @ ( G_23 @ X ) ) ) ).
thf(fact_29_sup__apply,axiom,
! [F_38: hoare_1167836817_state > $o,G_22: hoare_1167836817_state > $o,X_48: hoare_1167836817_state] :
( ( semila1172322802tate_o @ F_38 @ G_22 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_38 @ X_48 ) @ ( G_22 @ X_48 ) ) ) ).
thf(fact_30_sup__apply,axiom,
! [F_38: pname > $o,G_22: pname > $o,X_48: pname] :
( ( semila1780557381name_o @ F_38 @ G_22 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_38 @ X_48 ) @ ( G_22 @ X_48 ) ) ) ).
thf(fact_31_sup__apply,axiom,
! [F_38: hoare_1775062406iple_a > $o,G_22: hoare_1775062406iple_a > $o,X_48: hoare_1775062406iple_a] :
( ( semila13410563le_a_o @ F_38 @ G_22 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_38 @ X_48 ) @ ( G_22 @ X_48 ) ) ) ).
thf(fact_32_cut,axiom,
! [G_21: hoare_1167836817_state > $o,G_20: hoare_1167836817_state > $o,Ts_3: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_20 @ Ts_3 )
=> ( ( hoare_123228589_state @ G_21 @ G_20 )
=> ( hoare_123228589_state @ G_21 @ Ts_3 ) ) ) ).
thf(fact_33_cut,axiom,
! [G_21: hoare_1775062406iple_a > $o,G_20: hoare_1775062406iple_a > $o,Ts_3: hoare_1775062406iple_a > $o] :
( ( hoare_1508237396rivs_a @ G_20 @ Ts_3 )
=> ( ( hoare_1508237396rivs_a @ G_21 @ G_20 )
=> ( hoare_1508237396rivs_a @ G_21 @ Ts_3 ) ) ) ).
thf(fact_34_sup__assoc,axiom,
! [X_47: hoare_1167836817_state > $o,Y_21: hoare_1167836817_state > $o,Z_14: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( semila1172322802tate_o @ X_47 @ Y_21 ) @ Z_14 )
= ( semila1172322802tate_o @ X_47 @ ( semila1172322802tate_o @ Y_21 @ Z_14 ) ) ) ).
thf(fact_35_sup__assoc,axiom,
! [X_47: pname > $o,Y_21: pname > $o,Z_14: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ X_47 @ Y_21 ) @ Z_14 )
= ( semila1780557381name_o @ X_47 @ ( semila1780557381name_o @ Y_21 @ Z_14 ) ) ) ).
thf(fact_36_sup__assoc,axiom,
! [X_47: $o,Y_21: $o,Z_14: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ X_47 @ Y_21 ) @ Z_14 )
<=> ( semila10642723_sup_o @ X_47 @ ( semila10642723_sup_o @ Y_21 @ Z_14 ) ) ) ).
thf(fact_37_sup__assoc,axiom,
! [X_47: hoare_1775062406iple_a > $o,Y_21: hoare_1775062406iple_a > $o,Z_14: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( semila13410563le_a_o @ X_47 @ Y_21 ) @ Z_14 )
= ( semila13410563le_a_o @ X_47 @ ( semila13410563le_a_o @ Y_21 @ Z_14 ) ) ) ).
thf(fact_38_inf__sup__aci_I6_J,axiom,
! [X_46: hoare_1167836817_state > $o,Y_20: hoare_1167836817_state > $o,Z_13: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( semila1172322802tate_o @ X_46 @ Y_20 ) @ Z_13 )
= ( semila1172322802tate_o @ X_46 @ ( semila1172322802tate_o @ Y_20 @ Z_13 ) ) ) ).
thf(fact_39_inf__sup__aci_I6_J,axiom,
! [X_46: pname > $o,Y_20: pname > $o,Z_13: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ X_46 @ Y_20 ) @ Z_13 )
= ( semila1780557381name_o @ X_46 @ ( semila1780557381name_o @ Y_20 @ Z_13 ) ) ) ).
thf(fact_40_inf__sup__aci_I6_J,axiom,
! [X_46: $o,Y_20: $o,Z_13: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ X_46 @ Y_20 ) @ Z_13 )
<=> ( semila10642723_sup_o @ X_46 @ ( semila10642723_sup_o @ Y_20 @ Z_13 ) ) ) ).
thf(fact_41_inf__sup__aci_I6_J,axiom,
! [X_46: hoare_1775062406iple_a > $o,Y_20: hoare_1775062406iple_a > $o,Z_13: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( semila13410563le_a_o @ X_46 @ Y_20 ) @ Z_13 )
= ( semila13410563le_a_o @ X_46 @ ( semila13410563le_a_o @ Y_20 @ Z_13 ) ) ) ).
thf(fact_42_sup_Oassoc,axiom,
! [A_132: hoare_1167836817_state > $o,B_73: hoare_1167836817_state > $o,C_40: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( semila1172322802tate_o @ A_132 @ B_73 ) @ C_40 )
= ( semila1172322802tate_o @ A_132 @ ( semila1172322802tate_o @ B_73 @ C_40 ) ) ) ).
thf(fact_43_sup_Oassoc,axiom,
! [A_132: pname > $o,B_73: pname > $o,C_40: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ A_132 @ B_73 ) @ C_40 )
= ( semila1780557381name_o @ A_132 @ ( semila1780557381name_o @ B_73 @ C_40 ) ) ) ).
thf(fact_44_sup_Oassoc,axiom,
! [A_132: $o,B_73: $o,C_40: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ A_132 @ B_73 ) @ C_40 )
<=> ( semila10642723_sup_o @ A_132 @ ( semila10642723_sup_o @ B_73 @ C_40 ) ) ) ).
thf(fact_45_sup_Oassoc,axiom,
! [A_132: hoare_1775062406iple_a > $o,B_73: hoare_1775062406iple_a > $o,C_40: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( semila13410563le_a_o @ A_132 @ B_73 ) @ C_40 )
= ( semila13410563le_a_o @ A_132 @ ( semila13410563le_a_o @ B_73 @ C_40 ) ) ) ).
thf(fact_46_sup__left__commute,axiom,
! [X_45: hoare_1167836817_state > $o,Y_19: hoare_1167836817_state > $o,Z_12: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_45 @ ( semila1172322802tate_o @ Y_19 @ Z_12 ) )
= ( semila1172322802tate_o @ Y_19 @ ( semila1172322802tate_o @ X_45 @ Z_12 ) ) ) ).
thf(fact_47_sup__left__commute,axiom,
! [X_45: pname > $o,Y_19: pname > $o,Z_12: pname > $o] :
( ( semila1780557381name_o @ X_45 @ ( semila1780557381name_o @ Y_19 @ Z_12 ) )
= ( semila1780557381name_o @ Y_19 @ ( semila1780557381name_o @ X_45 @ Z_12 ) ) ) ).
thf(fact_48_sup__left__commute,axiom,
! [X_45: $o,Y_19: $o,Z_12: $o] :
( ( semila10642723_sup_o @ X_45 @ ( semila10642723_sup_o @ Y_19 @ Z_12 ) )
<=> ( semila10642723_sup_o @ Y_19 @ ( semila10642723_sup_o @ X_45 @ Z_12 ) ) ) ).
thf(fact_49_sup__left__commute,axiom,
! [X_45: hoare_1775062406iple_a > $o,Y_19: hoare_1775062406iple_a > $o,Z_12: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_45 @ ( semila13410563le_a_o @ Y_19 @ Z_12 ) )
= ( semila13410563le_a_o @ Y_19 @ ( semila13410563le_a_o @ X_45 @ Z_12 ) ) ) ).
thf(fact_50_inf__sup__aci_I7_J,axiom,
! [X_44: hoare_1167836817_state > $o,Y_18: hoare_1167836817_state > $o,Z_11: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_44 @ ( semila1172322802tate_o @ Y_18 @ Z_11 ) )
= ( semila1172322802tate_o @ Y_18 @ ( semila1172322802tate_o @ X_44 @ Z_11 ) ) ) ).
thf(fact_51_inf__sup__aci_I7_J,axiom,
! [X_44: pname > $o,Y_18: pname > $o,Z_11: pname > $o] :
( ( semila1780557381name_o @ X_44 @ ( semila1780557381name_o @ Y_18 @ Z_11 ) )
= ( semila1780557381name_o @ Y_18 @ ( semila1780557381name_o @ X_44 @ Z_11 ) ) ) ).
thf(fact_52_inf__sup__aci_I7_J,axiom,
! [X_44: $o,Y_18: $o,Z_11: $o] :
( ( semila10642723_sup_o @ X_44 @ ( semila10642723_sup_o @ Y_18 @ Z_11 ) )
<=> ( semila10642723_sup_o @ Y_18 @ ( semila10642723_sup_o @ X_44 @ Z_11 ) ) ) ).
thf(fact_53_inf__sup__aci_I7_J,axiom,
! [X_44: hoare_1775062406iple_a > $o,Y_18: hoare_1775062406iple_a > $o,Z_11: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_44 @ ( semila13410563le_a_o @ Y_18 @ Z_11 ) )
= ( semila13410563le_a_o @ Y_18 @ ( semila13410563le_a_o @ X_44 @ Z_11 ) ) ) ).
thf(fact_54_sup_Oleft__commute,axiom,
! [B_72: hoare_1167836817_state > $o,A_131: hoare_1167836817_state > $o,C_39: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ B_72 @ ( semila1172322802tate_o @ A_131 @ C_39 ) )
= ( semila1172322802tate_o @ A_131 @ ( semila1172322802tate_o @ B_72 @ C_39 ) ) ) ).
thf(fact_55_sup_Oleft__commute,axiom,
! [B_72: pname > $o,A_131: pname > $o,C_39: pname > $o] :
( ( semila1780557381name_o @ B_72 @ ( semila1780557381name_o @ A_131 @ C_39 ) )
= ( semila1780557381name_o @ A_131 @ ( semila1780557381name_o @ B_72 @ C_39 ) ) ) ).
thf(fact_56_sup_Oleft__commute,axiom,
! [B_72: $o,A_131: $o,C_39: $o] :
( ( semila10642723_sup_o @ B_72 @ ( semila10642723_sup_o @ A_131 @ C_39 ) )
<=> ( semila10642723_sup_o @ A_131 @ ( semila10642723_sup_o @ B_72 @ C_39 ) ) ) ).
thf(fact_57_sup_Oleft__commute,axiom,
! [B_72: hoare_1775062406iple_a > $o,A_131: hoare_1775062406iple_a > $o,C_39: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ B_72 @ ( semila13410563le_a_o @ A_131 @ C_39 ) )
= ( semila13410563le_a_o @ A_131 @ ( semila13410563le_a_o @ B_72 @ C_39 ) ) ) ).
thf(fact_58_sup__left__idem,axiom,
! [X_43: hoare_1167836817_state > $o,Y_17: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_43 @ ( semila1172322802tate_o @ X_43 @ Y_17 ) )
= ( semila1172322802tate_o @ X_43 @ Y_17 ) ) ).
thf(fact_59_sup__left__idem,axiom,
! [X_43: pname > $o,Y_17: pname > $o] :
( ( semila1780557381name_o @ X_43 @ ( semila1780557381name_o @ X_43 @ Y_17 ) )
= ( semila1780557381name_o @ X_43 @ Y_17 ) ) ).
thf(fact_60_sup__left__idem,axiom,
! [X_43: $o,Y_17: $o] :
( ( semila10642723_sup_o @ X_43 @ ( semila10642723_sup_o @ X_43 @ Y_17 ) )
<=> ( semila10642723_sup_o @ X_43 @ Y_17 ) ) ).
thf(fact_61_sup__left__idem,axiom,
! [X_43: hoare_1775062406iple_a > $o,Y_17: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_43 @ ( semila13410563le_a_o @ X_43 @ Y_17 ) )
= ( semila13410563le_a_o @ X_43 @ Y_17 ) ) ).
thf(fact_62_inf__sup__aci_I8_J,axiom,
! [X_42: hoare_1167836817_state > $o,Y_16: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_42 @ ( semila1172322802tate_o @ X_42 @ Y_16 ) )
= ( semila1172322802tate_o @ X_42 @ Y_16 ) ) ).
thf(fact_63_inf__sup__aci_I8_J,axiom,
! [X_42: pname > $o,Y_16: pname > $o] :
( ( semila1780557381name_o @ X_42 @ ( semila1780557381name_o @ X_42 @ Y_16 ) )
= ( semila1780557381name_o @ X_42 @ Y_16 ) ) ).
thf(fact_64_inf__sup__aci_I8_J,axiom,
! [X_42: $o,Y_16: $o] :
( ( semila10642723_sup_o @ X_42 @ ( semila10642723_sup_o @ X_42 @ Y_16 ) )
<=> ( semila10642723_sup_o @ X_42 @ Y_16 ) ) ).
thf(fact_65_inf__sup__aci_I8_J,axiom,
! [X_42: hoare_1775062406iple_a > $o,Y_16: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_42 @ ( semila13410563le_a_o @ X_42 @ Y_16 ) )
= ( semila13410563le_a_o @ X_42 @ Y_16 ) ) ).
thf(fact_66_sup_Oleft__idem,axiom,
! [A_130: hoare_1167836817_state > $o,B_71: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_130 @ ( semila1172322802tate_o @ A_130 @ B_71 ) )
= ( semila1172322802tate_o @ A_130 @ B_71 ) ) ).
thf(fact_67_sup_Oleft__idem,axiom,
! [A_130: pname > $o,B_71: pname > $o] :
( ( semila1780557381name_o @ A_130 @ ( semila1780557381name_o @ A_130 @ B_71 ) )
= ( semila1780557381name_o @ A_130 @ B_71 ) ) ).
thf(fact_68_sup_Oleft__idem,axiom,
! [A_130: $o,B_71: $o] :
( ( semila10642723_sup_o @ A_130 @ ( semila10642723_sup_o @ A_130 @ B_71 ) )
<=> ( semila10642723_sup_o @ A_130 @ B_71 ) ) ).
thf(fact_69_sup_Oleft__idem,axiom,
! [A_130: hoare_1775062406iple_a > $o,B_71: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_130 @ ( semila13410563le_a_o @ A_130 @ B_71 ) )
= ( semila13410563le_a_o @ A_130 @ B_71 ) ) ).
thf(fact_70_sup__commute,axiom,
! [X_41: hoare_1167836817_state > $o,Y_15: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_41 @ Y_15 )
= ( semila1172322802tate_o @ Y_15 @ X_41 ) ) ).
thf(fact_71_sup__commute,axiom,
! [X_41: pname > $o,Y_15: pname > $o] :
( ( semila1780557381name_o @ X_41 @ Y_15 )
= ( semila1780557381name_o @ Y_15 @ X_41 ) ) ).
thf(fact_72_sup__commute,axiom,
! [X_41: $o,Y_15: $o] :
( ( semila10642723_sup_o @ X_41 @ Y_15 )
<=> ( semila10642723_sup_o @ Y_15 @ X_41 ) ) ).
thf(fact_73_sup__commute,axiom,
! [X_41: hoare_1775062406iple_a > $o,Y_15: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_41 @ Y_15 )
= ( semila13410563le_a_o @ Y_15 @ X_41 ) ) ).
thf(fact_74_inf__sup__aci_I5_J,axiom,
! [X_40: hoare_1167836817_state > $o,Y_14: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_40 @ Y_14 )
= ( semila1172322802tate_o @ Y_14 @ X_40 ) ) ).
thf(fact_75_inf__sup__aci_I5_J,axiom,
! [X_40: pname > $o,Y_14: pname > $o] :
( ( semila1780557381name_o @ X_40 @ Y_14 )
= ( semila1780557381name_o @ Y_14 @ X_40 ) ) ).
thf(fact_76_inf__sup__aci_I5_J,axiom,
! [X_40: $o,Y_14: $o] :
( ( semila10642723_sup_o @ X_40 @ Y_14 )
<=> ( semila10642723_sup_o @ Y_14 @ X_40 ) ) ).
thf(fact_77_inf__sup__aci_I5_J,axiom,
! [X_40: hoare_1775062406iple_a > $o,Y_14: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_40 @ Y_14 )
= ( semila13410563le_a_o @ Y_14 @ X_40 ) ) ).
thf(fact_78_sup_Ocommute,axiom,
! [A_129: hoare_1167836817_state > $o,B_70: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_129 @ B_70 )
= ( semila1172322802tate_o @ B_70 @ A_129 ) ) ).
thf(fact_79_sup_Ocommute,axiom,
! [A_129: pname > $o,B_70: pname > $o] :
( ( semila1780557381name_o @ A_129 @ B_70 )
= ( semila1780557381name_o @ B_70 @ A_129 ) ) ).
thf(fact_80_sup_Ocommute,axiom,
! [A_129: $o,B_70: $o] :
( ( semila10642723_sup_o @ A_129 @ B_70 )
<=> ( semila10642723_sup_o @ B_70 @ A_129 ) ) ).
thf(fact_81_sup_Ocommute,axiom,
! [A_129: hoare_1775062406iple_a > $o,B_70: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_129 @ B_70 )
= ( semila13410563le_a_o @ B_70 @ A_129 ) ) ).
thf(fact_82_sup__idem,axiom,
! [X_39: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_83_sup__idem,axiom,
! [X_39: pname > $o] :
( ( semila1780557381name_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_84_sup__idem,axiom,
! [X_39: $o] :
( ( semila10642723_sup_o @ X_39 @ X_39 )
<=> X_39 ) ).
thf(fact_85_sup__idem,axiom,
! [X_39: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_86_sup_Oidem,axiom,
! [A_128: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_128 @ A_128 )
= A_128 ) ).
thf(fact_87_sup_Oidem,axiom,
! [A_128: pname > $o] :
( ( semila1780557381name_o @ A_128 @ A_128 )
= A_128 ) ).
thf(fact_88_sup_Oidem,axiom,
! [A_128: $o] :
( ( semila10642723_sup_o @ A_128 @ A_128 )
<=> A_128 ) ).
thf(fact_89_sup_Oidem,axiom,
! [A_128: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_128 @ A_128 )
= A_128 ) ).
thf(fact_90_rev__image__eqI,axiom,
! [B_69: hoare_1167836817_state,F_37: pname > hoare_1167836817_state,X_38: pname,A_127: pname > $o] :
( ( member_pname @ X_38 @ A_127 )
=> ( ( B_69
= ( F_37 @ X_38 ) )
=> ( member2058392318_state @ B_69 @ ( image_575578384_state @ F_37 @ A_127 ) ) ) ) ).
thf(fact_91_rev__image__eqI,axiom,
! [B_69: pname,F_37: hoare_1775062406iple_a > pname,X_38: hoare_1775062406iple_a,A_127: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_38 @ A_127 )
=> ( ( B_69
= ( F_37 @ X_38 ) )
=> ( member_pname @ B_69 @ ( image_51246659_pname @ F_37 @ A_127 ) ) ) ) ).
thf(fact_92_rev__image__eqI,axiom,
! [B_69: hoare_1775062406iple_a,F_37: pname > hoare_1775062406iple_a,X_38: pname,A_127: pname > $o] :
( ( member_pname @ X_38 @ A_127 )
=> ( ( B_69
= ( F_37 @ X_38 ) )
=> ( member2122167641iple_a @ B_69 @ ( image_2063119815iple_a @ F_37 @ A_127 ) ) ) ) ).
thf(fact_93_imageI,axiom,
! [F_36: pname > hoare_1167836817_state,X_37: pname,A_126: pname > $o] :
( ( member_pname @ X_37 @ A_126 )
=> ( member2058392318_state @ ( F_36 @ X_37 ) @ ( image_575578384_state @ F_36 @ A_126 ) ) ) ).
thf(fact_94_imageI,axiom,
! [F_36: hoare_1775062406iple_a > pname,X_37: hoare_1775062406iple_a,A_126: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_37 @ A_126 )
=> ( member_pname @ ( F_36 @ X_37 ) @ ( image_51246659_pname @ F_36 @ A_126 ) ) ) ).
thf(fact_95_imageI,axiom,
! [F_36: pname > hoare_1775062406iple_a,X_37: pname,A_126: pname > $o] :
( ( member_pname @ X_37 @ A_126 )
=> ( member2122167641iple_a @ ( F_36 @ X_37 ) @ ( image_2063119815iple_a @ F_36 @ A_126 ) ) ) ).
thf(fact_96_image__iff,axiom,
! [Z_10: hoare_1167836817_state,F_35: pname > hoare_1167836817_state,A_125: pname > $o] :
( ( member2058392318_state @ Z_10 @ ( image_575578384_state @ F_35 @ A_125 ) )
<=> ? [X: pname] :
( ( member_pname @ X @ A_125 )
& ( Z_10
= ( F_35 @ X ) ) ) ) ).
thf(fact_97_image__iff,axiom,
! [Z_10: hoare_1775062406iple_a,F_35: pname > hoare_1775062406iple_a,A_125: pname > $o] :
( ( member2122167641iple_a @ Z_10 @ ( image_2063119815iple_a @ F_35 @ A_125 ) )
<=> ? [X: pname] :
( ( member_pname @ X @ A_125 )
& ( Z_10
= ( F_35 @ X ) ) ) ) ).
thf(fact_98_UnI2,axiom,
! [A_124: hoare_1167836817_state > $o,C_38: hoare_1167836817_state,B_68: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_38 @ B_68 )
=> ( member2058392318_state @ C_38 @ ( semila1172322802tate_o @ A_124 @ B_68 ) ) ) ).
thf(fact_99_UnI2,axiom,
! [A_124: hoare_1775062406iple_a > $o,C_38: hoare_1775062406iple_a,B_68: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_38 @ B_68 )
=> ( member2122167641iple_a @ C_38 @ ( semila13410563le_a_o @ A_124 @ B_68 ) ) ) ).
thf(fact_100_UnI2,axiom,
! [A_124: pname > $o,C_38: pname,B_68: pname > $o] :
( ( member_pname @ C_38 @ B_68 )
=> ( member_pname @ C_38 @ ( semila1780557381name_o @ A_124 @ B_68 ) ) ) ).
thf(fact_101_UnI1,axiom,
! [B_67: hoare_1167836817_state > $o,C_37: hoare_1167836817_state,A_123: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_37 @ A_123 )
=> ( member2058392318_state @ C_37 @ ( semila1172322802tate_o @ A_123 @ B_67 ) ) ) ).
thf(fact_102_UnI1,axiom,
! [B_67: hoare_1775062406iple_a > $o,C_37: hoare_1775062406iple_a,A_123: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_37 @ A_123 )
=> ( member2122167641iple_a @ C_37 @ ( semila13410563le_a_o @ A_123 @ B_67 ) ) ) ).
thf(fact_103_UnI1,axiom,
! [B_67: pname > $o,C_37: pname,A_123: pname > $o] :
( ( member_pname @ C_37 @ A_123 )
=> ( member_pname @ C_37 @ ( semila1780557381name_o @ A_123 @ B_67 ) ) ) ).
thf(fact_104_sup1I2,axiom,
! [A_122: hoare_1167836817_state > $o,B_66: hoare_1167836817_state > $o,X_36: hoare_1167836817_state] :
( ( B_66 @ X_36 )
=> ( semila1172322802tate_o @ A_122 @ B_66 @ X_36 ) ) ).
thf(fact_105_sup1I2,axiom,
! [A_122: pname > $o,B_66: pname > $o,X_36: pname] :
( ( B_66 @ X_36 )
=> ( semila1780557381name_o @ A_122 @ B_66 @ X_36 ) ) ).
thf(fact_106_sup1I2,axiom,
! [A_122: hoare_1775062406iple_a > $o,B_66: hoare_1775062406iple_a > $o,X_36: hoare_1775062406iple_a] :
( ( B_66 @ X_36 )
=> ( semila13410563le_a_o @ A_122 @ B_66 @ X_36 ) ) ).
thf(fact_107_sup1I1,axiom,
! [B_65: hoare_1167836817_state > $o,A_121: hoare_1167836817_state > $o,X_35: hoare_1167836817_state] :
( ( A_121 @ X_35 )
=> ( semila1172322802tate_o @ A_121 @ B_65 @ X_35 ) ) ).
thf(fact_108_sup1I1,axiom,
! [B_65: pname > $o,A_121: pname > $o,X_35: pname] :
( ( A_121 @ X_35 )
=> ( semila1780557381name_o @ A_121 @ B_65 @ X_35 ) ) ).
thf(fact_109_sup1I1,axiom,
! [B_65: hoare_1775062406iple_a > $o,A_121: hoare_1775062406iple_a > $o,X_35: hoare_1775062406iple_a] :
( ( A_121 @ X_35 )
=> ( semila13410563le_a_o @ A_121 @ B_65 @ X_35 ) ) ).
thf(fact_110_ball__Un,axiom,
! [P_36: hoare_1167836817_state > $o,A_120: hoare_1167836817_state > $o,B_64: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ ( semila1172322802tate_o @ A_120 @ B_64 ) )
=> ( P_36 @ X ) )
<=> ( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ A_120 )
=> ( P_36 @ X ) )
& ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ B_64 )
=> ( P_36 @ X ) ) ) ) ).
thf(fact_111_ball__Un,axiom,
! [P_36: pname > $o,A_120: pname > $o,B_64: pname > $o] :
( ! [X: pname] :
( ( member_pname @ X @ ( semila1780557381name_o @ A_120 @ B_64 ) )
=> ( P_36 @ X ) )
<=> ( ! [X: pname] :
( ( member_pname @ X @ A_120 )
=> ( P_36 @ X ) )
& ! [X: pname] :
( ( member_pname @ X @ B_64 )
=> ( P_36 @ X ) ) ) ) ).
thf(fact_112_ball__Un,axiom,
! [P_36: hoare_1775062406iple_a > $o,A_120: hoare_1775062406iple_a > $o,B_64: hoare_1775062406iple_a > $o] :
( ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ ( semila13410563le_a_o @ A_120 @ B_64 ) )
=> ( P_36 @ X ) )
<=> ( ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ A_120 )
=> ( P_36 @ X ) )
& ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ B_64 )
=> ( P_36 @ X ) ) ) ) ).
thf(fact_113_bex__Un,axiom,
! [P_35: hoare_1167836817_state > $o,A_119: hoare_1167836817_state > $o,B_63: hoare_1167836817_state > $o] :
( ? [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ ( semila1172322802tate_o @ A_119 @ B_63 ) )
& ( P_35 @ X ) )
<=> ( ? [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ A_119 )
& ( P_35 @ X ) )
| ? [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ B_63 )
& ( P_35 @ X ) ) ) ) ).
thf(fact_114_bex__Un,axiom,
! [P_35: pname > $o,A_119: pname > $o,B_63: pname > $o] :
( ? [X: pname] :
( ( member_pname @ X @ ( semila1780557381name_o @ A_119 @ B_63 ) )
& ( P_35 @ X ) )
<=> ( ? [X: pname] :
( ( member_pname @ X @ A_119 )
& ( P_35 @ X ) )
| ? [X: pname] :
( ( member_pname @ X @ B_63 )
& ( P_35 @ X ) ) ) ) ).
thf(fact_115_bex__Un,axiom,
! [P_35: hoare_1775062406iple_a > $o,A_119: hoare_1775062406iple_a > $o,B_63: hoare_1775062406iple_a > $o] :
( ? [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ ( semila13410563le_a_o @ A_119 @ B_63 ) )
& ( P_35 @ X ) )
<=> ( ? [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ A_119 )
& ( P_35 @ X ) )
| ? [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ B_63 )
& ( P_35 @ X ) ) ) ) ).
thf(fact_116_Un__assoc,axiom,
! [A_118: hoare_1167836817_state > $o,B_62: hoare_1167836817_state > $o,C_36: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( semila1172322802tate_o @ A_118 @ B_62 ) @ C_36 )
= ( semila1172322802tate_o @ A_118 @ ( semila1172322802tate_o @ B_62 @ C_36 ) ) ) ).
thf(fact_117_Un__assoc,axiom,
! [A_118: pname > $o,B_62: pname > $o,C_36: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ A_118 @ B_62 ) @ C_36 )
= ( semila1780557381name_o @ A_118 @ ( semila1780557381name_o @ B_62 @ C_36 ) ) ) ).
thf(fact_118_Un__assoc,axiom,
! [A_118: hoare_1775062406iple_a > $o,B_62: hoare_1775062406iple_a > $o,C_36: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( semila13410563le_a_o @ A_118 @ B_62 ) @ C_36 )
= ( semila13410563le_a_o @ A_118 @ ( semila13410563le_a_o @ B_62 @ C_36 ) ) ) ).
thf(fact_119_Un__iff,axiom,
! [C_35: hoare_1167836817_state,A_117: hoare_1167836817_state > $o,B_61: hoare_1167836817_state > $o] :
( ( member2058392318_state @ C_35 @ ( semila1172322802tate_o @ A_117 @ B_61 ) )
<=> ( ( member2058392318_state @ C_35 @ A_117 )
| ( member2058392318_state @ C_35 @ B_61 ) ) ) ).
thf(fact_120_Un__iff,axiom,
! [C_35: hoare_1775062406iple_a,A_117: hoare_1775062406iple_a > $o,B_61: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_35 @ ( semila13410563le_a_o @ A_117 @ B_61 ) )
<=> ( ( member2122167641iple_a @ C_35 @ A_117 )
| ( member2122167641iple_a @ C_35 @ B_61 ) ) ) ).
thf(fact_121_Un__iff,axiom,
! [C_35: pname,A_117: pname > $o,B_61: pname > $o] :
( ( member_pname @ C_35 @ ( semila1780557381name_o @ A_117 @ B_61 ) )
<=> ( ( member_pname @ C_35 @ A_117 )
| ( member_pname @ C_35 @ B_61 ) ) ) ).
thf(fact_122_Un__left__commute,axiom,
! [A_116: hoare_1167836817_state > $o,B_60: hoare_1167836817_state > $o,C_34: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_116 @ ( semila1172322802tate_o @ B_60 @ C_34 ) )
= ( semila1172322802tate_o @ B_60 @ ( semila1172322802tate_o @ A_116 @ C_34 ) ) ) ).
thf(fact_123_Un__left__commute,axiom,
! [A_116: pname > $o,B_60: pname > $o,C_34: pname > $o] :
( ( semila1780557381name_o @ A_116 @ ( semila1780557381name_o @ B_60 @ C_34 ) )
= ( semila1780557381name_o @ B_60 @ ( semila1780557381name_o @ A_116 @ C_34 ) ) ) ).
thf(fact_124_Un__left__commute,axiom,
! [A_116: hoare_1775062406iple_a > $o,B_60: hoare_1775062406iple_a > $o,C_34: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_116 @ ( semila13410563le_a_o @ B_60 @ C_34 ) )
= ( semila13410563le_a_o @ B_60 @ ( semila13410563le_a_o @ A_116 @ C_34 ) ) ) ).
thf(fact_125_Un__left__absorb,axiom,
! [A_115: hoare_1167836817_state > $o,B_59: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_115 @ ( semila1172322802tate_o @ A_115 @ B_59 ) )
= ( semila1172322802tate_o @ A_115 @ B_59 ) ) ).
thf(fact_126_Un__left__absorb,axiom,
! [A_115: pname > $o,B_59: pname > $o] :
( ( semila1780557381name_o @ A_115 @ ( semila1780557381name_o @ A_115 @ B_59 ) )
= ( semila1780557381name_o @ A_115 @ B_59 ) ) ).
thf(fact_127_Un__left__absorb,axiom,
! [A_115: hoare_1775062406iple_a > $o,B_59: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_115 @ ( semila13410563le_a_o @ A_115 @ B_59 ) )
= ( semila13410563le_a_o @ A_115 @ B_59 ) ) ).
thf(fact_128_Un__commute,axiom,
! [A_114: hoare_1167836817_state > $o,B_58: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_114 @ B_58 )
= ( semila1172322802tate_o @ B_58 @ A_114 ) ) ).
thf(fact_129_Un__commute,axiom,
! [A_114: pname > $o,B_58: pname > $o] :
( ( semila1780557381name_o @ A_114 @ B_58 )
= ( semila1780557381name_o @ B_58 @ A_114 ) ) ).
thf(fact_130_Un__commute,axiom,
! [A_114: hoare_1775062406iple_a > $o,B_58: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_114 @ B_58 )
= ( semila13410563le_a_o @ B_58 @ A_114 ) ) ).
thf(fact_131_Un__def,axiom,
! [A_113: hoare_1167836817_state > $o,B_57: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_113 @ B_57 )
= ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (|) @ ( member2058392318_state @ X @ A_113 ) @ ( member2058392318_state @ X @ B_57 ) ) ) ) ).
thf(fact_132_Un__def,axiom,
! [A_113: hoare_1775062406iple_a > $o,B_57: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_113 @ B_57 )
= ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (|) @ ( member2122167641iple_a @ X @ A_113 ) @ ( member2122167641iple_a @ X @ B_57 ) ) ) ) ).
thf(fact_133_Un__def,axiom,
! [A_113: pname > $o,B_57: pname > $o] :
( ( semila1780557381name_o @ A_113 @ B_57 )
= ( collect_pname
@ ^ [X: pname] : ( (|) @ ( member_pname @ X @ A_113 ) @ ( member_pname @ X @ B_57 ) ) ) ) ).
thf(fact_134_Un__absorb,axiom,
! [A_112: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_112 @ A_112 )
= A_112 ) ).
thf(fact_135_Un__absorb,axiom,
! [A_112: pname > $o] :
( ( semila1780557381name_o @ A_112 @ A_112 )
= A_112 ) ).
thf(fact_136_Un__absorb,axiom,
! [A_112: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_112 @ A_112 )
= A_112 ) ).
thf(fact_137_image__image,axiom,
! [F_34: hoare_1775062406iple_a > hoare_1167836817_state,G_19: pname > hoare_1775062406iple_a,A_111: pname > $o] :
( ( image_1021683026_state @ F_34 @ ( image_2063119815iple_a @ G_19 @ A_111 ) )
= ( image_575578384_state
@ ^ [X: pname] : ( F_34 @ ( G_19 @ X ) )
@ A_111 ) ) ).
thf(fact_138_image__image,axiom,
! [F_34: hoare_1167836817_state > hoare_1775062406iple_a,G_19: pname > hoare_1167836817_state,A_111: pname > $o] :
( ( image_1802845250iple_a @ F_34 @ ( image_575578384_state @ G_19 @ A_111 ) )
= ( image_2063119815iple_a
@ ^ [X: pname] : ( F_34 @ ( G_19 @ X ) )
@ A_111 ) ) ).
thf(fact_139_sup__Un__eq,axiom,
! [R_2: hoare_1167836817_state > $o,S_6: hoare_1167836817_state > $o,X: hoare_1167836817_state] :
( ( semila1172322802tate_o
@ ^ [Y_2: hoare_1167836817_state] : ( member2058392318_state @ Y_2 @ R_2 )
@ ^ [Y_2: hoare_1167836817_state] : ( member2058392318_state @ Y_2 @ S_6 )
@ X )
<=> ( member2058392318_state @ X @ ( semila1172322802tate_o @ R_2 @ S_6 ) ) ) ).
thf(fact_140_sup__Un__eq,axiom,
! [R_2: hoare_1775062406iple_a > $o,S_6: hoare_1775062406iple_a > $o,X: hoare_1775062406iple_a] :
( ( semila13410563le_a_o
@ ^ [Y_2: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_2 @ R_2 )
@ ^ [Y_2: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_2 @ S_6 )
@ X )
<=> ( member2122167641iple_a @ X @ ( semila13410563le_a_o @ R_2 @ S_6 ) ) ) ).
thf(fact_141_sup__Un__eq,axiom,
! [R_2: pname > $o,S_6: pname > $o,X: pname] :
( ( semila1780557381name_o
@ ^ [Y_2: pname] : ( member_pname @ Y_2 @ R_2 )
@ ^ [Y_2: pname] : ( member_pname @ Y_2 @ S_6 )
@ X )
<=> ( member_pname @ X @ ( semila1780557381name_o @ R_2 @ S_6 ) ) ) ).
thf(fact_142_Collect__disj__eq,axiom,
! [P_34: pname > $o,Q_19: pname > $o] :
( ( collect_pname
@ ^ [X: pname] : ( (|) @ ( P_34 @ X ) @ ( Q_19 @ X ) ) )
= ( semila1780557381name_o @ ( collect_pname @ P_34 ) @ ( collect_pname @ Q_19 ) ) ) ).
thf(fact_143_Collect__disj__eq,axiom,
! [P_34: hoare_1167836817_state > $o,Q_19: hoare_1167836817_state > $o] :
( ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (|) @ ( P_34 @ X ) @ ( Q_19 @ X ) ) )
= ( semila1172322802tate_o @ ( collec1027672124_state @ P_34 ) @ ( collec1027672124_state @ Q_19 ) ) ) ).
thf(fact_144_Collect__disj__eq,axiom,
! [P_34: hoare_1775062406iple_a > $o,Q_19: hoare_1775062406iple_a > $o] :
( ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (|) @ ( P_34 @ X ) @ ( Q_19 @ X ) ) )
= ( semila13410563le_a_o @ ( collec676402587iple_a @ P_34 ) @ ( collec676402587iple_a @ Q_19 ) ) ) ).
thf(fact_145_imageE,axiom,
! [B_56: pname,F_33: hoare_1775062406iple_a > pname,A_110: hoare_1775062406iple_a > $o] :
( ( member_pname @ B_56 @ ( image_51246659_pname @ F_33 @ A_110 ) )
=> ~ ! [X: hoare_1775062406iple_a] :
( ( B_56
= ( F_33 @ X ) )
=> ~ ( member2122167641iple_a @ X @ A_110 ) ) ) ).
thf(fact_146_imageE,axiom,
! [B_56: hoare_1167836817_state,F_33: pname > hoare_1167836817_state,A_110: pname > $o] :
( ( member2058392318_state @ B_56 @ ( image_575578384_state @ F_33 @ A_110 ) )
=> ~ ! [X: pname] :
( ( B_56
= ( F_33 @ X ) )
=> ~ ( member_pname @ X @ A_110 ) ) ) ).
thf(fact_147_imageE,axiom,
! [B_56: hoare_1775062406iple_a,F_33: pname > hoare_1775062406iple_a,A_110: pname > $o] :
( ( member2122167641iple_a @ B_56 @ ( image_2063119815iple_a @ F_33 @ A_110 ) )
=> ~ ! [X: pname] :
( ( B_56
= ( F_33 @ X ) )
=> ~ ( member_pname @ X @ A_110 ) ) ) ).
thf(fact_148_Body__triple__valid__Suc,axiom,
! [N_8: nat,P_33: state > state > $o,Pn_6: pname,Q_18: state > state > $o] :
( ( hoare_56934129_state @ N_8 @ ( hoare_908217195_state @ P_33 @ ( the_com @ ( body_1 @ Pn_6 ) ) @ Q_18 ) )
<=> ( hoare_56934129_state @ ( suc @ N_8 ) @ ( hoare_908217195_state @ P_33 @ ( body @ Pn_6 ) @ Q_18 ) ) ) ).
thf(fact_149_Body__triple__valid__Suc,axiom,
! [N_8: nat,P_33: x_a > state > $o,Pn_6: pname,Q_18: x_a > state > $o] :
( ( hoare_1462269968alid_a @ N_8 @ ( hoare_1766022166iple_a @ P_33 @ ( the_com @ ( body_1 @ Pn_6 ) ) @ Q_18 ) )
<=> ( hoare_1462269968alid_a @ ( suc @ N_8 ) @ ( hoare_1766022166iple_a @ P_33 @ ( body @ Pn_6 ) @ Q_18 ) ) ) ).
thf(fact_150_triple_Oexhaust,axiom,
! [Y_13: hoare_1775062406iple_a] :
~ ! [Fun1_2: x_a > state > $o,Com_4: com,Fun2_2: x_a > state > $o] :
( Y_13
!= ( hoare_1766022166iple_a @ Fun1_2 @ Com_4 @ Fun2_2 ) ) ).
thf(fact_151_triple_Oexhaust,axiom,
! [Y_13: hoare_1167836817_state] :
~ ! [Fun1_2: state > state > $o,Com_4: com,Fun2_2: state > state > $o] :
( Y_13
!= ( hoare_908217195_state @ Fun1_2 @ Com_4 @ Fun2_2 ) ) ).
thf(fact_152_Body1,axiom,
! [Pn_5: pname,G_18: hoare_1167836817_state > $o,P_32: pname > state > state > $o,Q_17: pname > state > state > $o,Procs: pname > $o] :
( ( hoare_123228589_state
@ ( semila1172322802tate_o @ G_18
@ ( image_575578384_state
@ ^ [P_10: pname] : ( hoare_908217195_state @ ( P_32 @ P_10 ) @ ( body @ P_10 ) @ ( Q_17 @ P_10 ) )
@ Procs ) )
@ ( image_575578384_state
@ ^ [P_10: pname] : ( hoare_908217195_state @ ( P_32 @ P_10 ) @ ( the_com @ ( body_1 @ P_10 ) ) @ ( Q_17 @ P_10 ) )
@ Procs ) )
=> ( ( member_pname @ Pn_5 @ Procs )
=> ( hoare_123228589_state @ G_18 @ ( insert2134838167_state @ ( hoare_908217195_state @ ( P_32 @ Pn_5 ) @ ( body @ Pn_5 ) @ ( Q_17 @ Pn_5 ) ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_153_Body1,axiom,
! [Pn_5: pname,G_18: hoare_1775062406iple_a > $o,P_32: pname > x_a > state > $o,Q_17: pname > x_a > state > $o,Procs: pname > $o] :
( ( hoare_1508237396rivs_a
@ ( semila13410563le_a_o @ G_18
@ ( image_2063119815iple_a
@ ^ [P_10: pname] : ( hoare_1766022166iple_a @ ( P_32 @ P_10 ) @ ( body @ P_10 ) @ ( Q_17 @ P_10 ) )
@ Procs ) )
@ ( image_2063119815iple_a
@ ^ [P_10: pname] : ( hoare_1766022166iple_a @ ( P_32 @ P_10 ) @ ( the_com @ ( body_1 @ P_10 ) ) @ ( Q_17 @ P_10 ) )
@ Procs ) )
=> ( ( member_pname @ Pn_5 @ Procs )
=> ( hoare_1508237396rivs_a @ G_18 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ ( P_32 @ Pn_5 ) @ ( body @ Pn_5 ) @ ( Q_17 @ Pn_5 ) ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_154_image__cong,axiom,
! [F_32: pname > hoare_1167836817_state,G_17: pname > hoare_1167836817_state,M_2: pname > $o,N_7: pname > $o] :
( ( M_2 = N_7 )
=> ( ! [X: pname] :
( ( member_pname @ X @ N_7 )
=> ( ( F_32 @ X )
= ( G_17 @ X ) ) )
=> ( ( image_575578384_state @ F_32 @ M_2 )
= ( image_575578384_state @ G_17 @ N_7 ) ) ) ) ).
thf(fact_155_image__cong,axiom,
! [F_32: pname > hoare_1775062406iple_a,G_17: pname > hoare_1775062406iple_a,M_2: pname > $o,N_7: pname > $o] :
( ( M_2 = N_7 )
=> ( ! [X: pname] :
( ( member_pname @ X @ N_7 )
=> ( ( F_32 @ X )
= ( G_17 @ X ) ) )
=> ( ( image_2063119815iple_a @ F_32 @ M_2 )
= ( image_2063119815iple_a @ G_17 @ N_7 ) ) ) ) ).
thf(fact_156_Body__triple__valid__0,axiom,
! [P_31: state > state > $o,Pn_4: pname,Q_16: state > state > $o] : ( hoare_56934129_state @ zero_zero_nat @ ( hoare_908217195_state @ P_31 @ ( body @ Pn_4 ) @ Q_16 ) ) ).
thf(fact_157_Body__triple__valid__0,axiom,
! [P_31: x_a > state > $o,Pn_4: pname,Q_16: x_a > state > $o] : ( hoare_1462269968alid_a @ zero_zero_nat @ ( hoare_1766022166iple_a @ P_31 @ ( body @ Pn_4 ) @ Q_16 ) ) ).
thf(fact_158_com_Osimps_I6_J,axiom,
! [Pname: pname,Pname_1: pname] :
( ( ( body @ Pname )
= ( body @ Pname_1 ) )
<=> ( Pname = Pname_1 ) ) ).
thf(fact_159_evalc_OBody,axiom,
! [Pn_1: pname,S0: state,S1: state] :
( ( evalc @ ( the_com @ ( body_1 @ Pn_1 ) ) @ S0 @ S1 )
=> ( evalc @ ( body @ Pn_1 ) @ S0 @ S1 ) ) ).
thf(fact_160_evalc__elim__cases_I6_J,axiom,
! [P: pname,S: state,S1: state] :
( ( evalc @ ( body @ P ) @ S @ S1 )
=> ( evalc @ ( the_com @ ( body_1 @ P ) ) @ S @ S1 ) ) ).
thf(fact_161_Sup__fin_Oidem,axiom,
! [X_34: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_162_Sup__fin_Oidem,axiom,
! [X_34: pname > $o] :
( ( semila1780557381name_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_163_Sup__fin_Oidem,axiom,
! [X_34: $o] :
( ( semila10642723_sup_o @ X_34 @ X_34 )
<=> X_34 ) ).
thf(fact_164_Sup__fin_Oidem,axiom,
! [X_34: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_165_emptyE,axiom,
! [A_109: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ A_109 @ bot_bo751897185le_a_o ) ).
thf(fact_166_emptyE,axiom,
! [A_109: hoare_1167836817_state] :
~ ( member2058392318_state @ A_109 @ bot_bo70021908tate_o ) ).
thf(fact_167_emptyE,axiom,
! [A_109: pname] :
~ ( member_pname @ A_109 @ bot_bot_pname_o ) ).
thf(fact_168_insertE,axiom,
! [A_108: hoare_1167836817_state,B_55: hoare_1167836817_state,A_107: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_108 @ ( insert2134838167_state @ B_55 @ A_107 ) )
=> ( ( A_108 != B_55 )
=> ( member2058392318_state @ A_108 @ A_107 ) ) ) ).
thf(fact_169_insertE,axiom,
! [A_108: hoare_1775062406iple_a,B_55: hoare_1775062406iple_a,A_107: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_108 @ ( insert1281456128iple_a @ B_55 @ A_107 ) )
=> ( ( A_108 != B_55 )
=> ( member2122167641iple_a @ A_108 @ A_107 ) ) ) ).
thf(fact_170_insertE,axiom,
! [A_108: pname,B_55: pname,A_107: pname > $o] :
( ( member_pname @ A_108 @ ( insert_pname @ B_55 @ A_107 ) )
=> ( ( A_108 != B_55 )
=> ( member_pname @ A_108 @ A_107 ) ) ) ).
thf(fact_171_insertCI,axiom,
! [B_54: hoare_1167836817_state,A_106: hoare_1167836817_state,B_53: hoare_1167836817_state > $o] :
( ( ~ ( member2058392318_state @ A_106 @ B_53 )
=> ( A_106 = B_54 ) )
=> ( member2058392318_state @ A_106 @ ( insert2134838167_state @ B_54 @ B_53 ) ) ) ).
thf(fact_172_insertCI,axiom,
! [B_54: hoare_1775062406iple_a,A_106: hoare_1775062406iple_a,B_53: hoare_1775062406iple_a > $o] :
( ( ~ ( member2122167641iple_a @ A_106 @ B_53 )
=> ( A_106 = B_54 ) )
=> ( member2122167641iple_a @ A_106 @ ( insert1281456128iple_a @ B_54 @ B_53 ) ) ) ).
thf(fact_173_insertCI,axiom,
! [B_54: pname,A_106: pname,B_53: pname > $o] :
( ( ~ ( member_pname @ A_106 @ B_53 )
=> ( A_106 = B_54 ) )
=> ( member_pname @ A_106 @ ( insert_pname @ B_54 @ B_53 ) ) ) ).
thf(fact_174_empty__not__insert,axiom,
! [A_105: hoare_1167836817_state,A_104: hoare_1167836817_state > $o] :
( bot_bo70021908tate_o
!= ( insert2134838167_state @ A_105 @ A_104 ) ) ).
thf(fact_175_empty__not__insert,axiom,
! [A_105: hoare_1775062406iple_a,A_104: hoare_1775062406iple_a > $o] :
( bot_bo751897185le_a_o
!= ( insert1281456128iple_a @ A_105 @ A_104 ) ) ).
thf(fact_176_empty__not__insert,axiom,
! [A_105: pname,A_104: pname > $o] :
( bot_bot_pname_o
!= ( insert_pname @ A_105 @ A_104 ) ) ).
thf(fact_177_insert__not__empty,axiom,
! [A_103: hoare_1167836817_state,A_102: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_103 @ A_102 )
!= bot_bo70021908tate_o ) ).
thf(fact_178_insert__not__empty,axiom,
! [A_103: hoare_1775062406iple_a,A_102: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_103 @ A_102 )
!= bot_bo751897185le_a_o ) ).
thf(fact_179_insert__not__empty,axiom,
! [A_103: pname,A_102: pname > $o] :
( ( insert_pname @ A_103 @ A_102 )
!= bot_bot_pname_o ) ).
thf(fact_180_bot__empty__eq,axiom,
! [X: hoare_1775062406iple_a] :
( ( bot_bo751897185le_a_o @ X )
<=> ( member2122167641iple_a @ X @ bot_bo751897185le_a_o ) ) ).
thf(fact_181_bot__empty__eq,axiom,
! [X: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X )
<=> ( member2058392318_state @ X @ bot_bo70021908tate_o ) ) ).
thf(fact_182_bot__empty__eq,axiom,
! [X: pname] :
( ( bot_bot_pname_o @ X )
<=> ( member_pname @ X @ bot_bot_pname_o ) ) ).
thf(fact_183_empty__def,axiom,
( bot_bo751897185le_a_o
= ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : $false ) ) ).
thf(fact_184_empty__def,axiom,
( bot_bot_pname_o
= ( collect_pname
@ ^ [X: pname] : $false ) ) ).
thf(fact_185_empty__def,axiom,
( bot_bo70021908tate_o
= ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : $false ) ) ).
thf(fact_186_insertI1,axiom,
! [A_101: hoare_1167836817_state,B_52: hoare_1167836817_state > $o] : ( member2058392318_state @ A_101 @ ( insert2134838167_state @ A_101 @ B_52 ) ) ).
thf(fact_187_insertI1,axiom,
! [A_101: hoare_1775062406iple_a,B_52: hoare_1775062406iple_a > $o] : ( member2122167641iple_a @ A_101 @ ( insert1281456128iple_a @ A_101 @ B_52 ) ) ).
thf(fact_188_insertI1,axiom,
! [A_101: pname,B_52: pname > $o] : ( member_pname @ A_101 @ ( insert_pname @ A_101 @ B_52 ) ) ).
thf(fact_189_all__not__in__conv,axiom,
! [A_100: hoare_1775062406iple_a > $o] :
( ! [X: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ X @ A_100 )
<=> ( A_100 = bot_bo751897185le_a_o ) ) ).
thf(fact_190_all__not__in__conv,axiom,
! [A_100: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
~ ( member2058392318_state @ X @ A_100 )
<=> ( A_100 = bot_bo70021908tate_o ) ) ).
thf(fact_191_all__not__in__conv,axiom,
! [A_100: pname > $o] :
( ! [X: pname] :
~ ( member_pname @ X @ A_100 )
<=> ( A_100 = bot_bot_pname_o ) ) ).
thf(fact_192_singleton__conv2,axiom,
! [A_99: hoare_1167836817_state] :
( ( collec1027672124_state @ ( fequal1831255762_state @ A_99 ) )
= ( insert2134838167_state @ A_99 @ bot_bo70021908tate_o ) ) ).
thf(fact_193_singleton__conv2,axiom,
! [A_99: hoare_1775062406iple_a] :
( ( collec676402587iple_a @ ( fequal1288209029iple_a @ A_99 ) )
= ( insert1281456128iple_a @ A_99 @ bot_bo751897185le_a_o ) ) ).
thf(fact_194_singleton__conv2,axiom,
! [A_99: pname] :
( ( collect_pname @ ( fequal_pname @ A_99 ) )
= ( insert_pname @ A_99 @ bot_bot_pname_o ) ) ).
thf(fact_195_ex__in__conv,axiom,
! [A_98: hoare_1775062406iple_a > $o] :
( ? [X: hoare_1775062406iple_a] : ( member2122167641iple_a @ X @ A_98 )
<=> ( A_98 != bot_bo751897185le_a_o ) ) ).
thf(fact_196_ex__in__conv,axiom,
! [A_98: hoare_1167836817_state > $o] :
( ? [X: hoare_1167836817_state] : ( member2058392318_state @ X @ A_98 )
<=> ( A_98 != bot_bo70021908tate_o ) ) ).
thf(fact_197_ex__in__conv,axiom,
! [A_98: pname > $o] :
( ? [X: pname] : ( member_pname @ X @ A_98 )
<=> ( A_98 != bot_bot_pname_o ) ) ).
thf(fact_198_singleton__conv,axiom,
! [A_97: hoare_1167836817_state] :
( ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : X = A_97 )
= ( insert2134838167_state @ A_97 @ bot_bo70021908tate_o ) ) ).
thf(fact_199_singleton__conv,axiom,
! [A_97: hoare_1775062406iple_a] :
( ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : X = A_97 )
= ( insert1281456128iple_a @ A_97 @ bot_bo751897185le_a_o ) ) ).
thf(fact_200_singleton__conv,axiom,
! [A_97: pname] :
( ( collect_pname
@ ^ [X: pname] : X = A_97 )
= ( insert_pname @ A_97 @ bot_bot_pname_o ) ) ).
thf(fact_201_Collect__conv__if2,axiom,
! [P_30: hoare_1167836817_state > $o,A_96: hoare_1167836817_state] :
( ( ( P_30 @ A_96 )
=> ( ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (&) @ ( A_96 = X ) @ ( P_30 @ X ) ) )
= ( insert2134838167_state @ A_96 @ bot_bo70021908tate_o ) ) )
& ( ~ ( P_30 @ A_96 )
=> ( ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (&) @ ( A_96 = X ) @ ( P_30 @ X ) ) )
= bot_bo70021908tate_o ) ) ) ).
thf(fact_202_Collect__conv__if2,axiom,
! [P_30: hoare_1775062406iple_a > $o,A_96: hoare_1775062406iple_a] :
( ( ( P_30 @ A_96 )
=> ( ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (&) @ ( A_96 = X ) @ ( P_30 @ X ) ) )
= ( insert1281456128iple_a @ A_96 @ bot_bo751897185le_a_o ) ) )
& ( ~ ( P_30 @ A_96 )
=> ( ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (&) @ ( A_96 = X ) @ ( P_30 @ X ) ) )
= bot_bo751897185le_a_o ) ) ) ).
thf(fact_203_Collect__conv__if2,axiom,
! [P_30: pname > $o,A_96: pname] :
( ( ( P_30 @ A_96 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( A_96 = X ) @ ( P_30 @ X ) ) )
= ( insert_pname @ A_96 @ bot_bot_pname_o ) ) )
& ( ~ ( P_30 @ A_96 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( A_96 = X ) @ ( P_30 @ X ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_204_Collect__conv__if,axiom,
! [P_29: hoare_1167836817_state > $o,A_95: hoare_1167836817_state] :
( ( ( P_29 @ A_95 )
=> ( ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (&) @ ( X = A_95 ) @ ( P_29 @ X ) ) )
= ( insert2134838167_state @ A_95 @ bot_bo70021908tate_o ) ) )
& ( ~ ( P_29 @ A_95 )
=> ( ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (&) @ ( X = A_95 ) @ ( P_29 @ X ) ) )
= bot_bo70021908tate_o ) ) ) ).
thf(fact_205_Collect__conv__if,axiom,
! [P_29: hoare_1775062406iple_a > $o,A_95: hoare_1775062406iple_a] :
( ( ( P_29 @ A_95 )
=> ( ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (&) @ ( X = A_95 ) @ ( P_29 @ X ) ) )
= ( insert1281456128iple_a @ A_95 @ bot_bo751897185le_a_o ) ) )
& ( ~ ( P_29 @ A_95 )
=> ( ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (&) @ ( X = A_95 ) @ ( P_29 @ X ) ) )
= bot_bo751897185le_a_o ) ) ) ).
thf(fact_206_Collect__conv__if,axiom,
! [P_29: pname > $o,A_95: pname] :
( ( ( P_29 @ A_95 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( X = A_95 ) @ ( P_29 @ X ) ) )
= ( insert_pname @ A_95 @ bot_bot_pname_o ) ) )
& ( ~ ( P_29 @ A_95 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( X = A_95 ) @ ( P_29 @ X ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_207_empty__Collect__eq,axiom,
! [P_28: hoare_1775062406iple_a > $o] :
( ( bot_bo751897185le_a_o
= ( collec676402587iple_a @ P_28 ) )
<=> ! [X: hoare_1775062406iple_a] :
~ ( P_28 @ X ) ) ).
thf(fact_208_empty__Collect__eq,axiom,
! [P_28: pname > $o] :
( ( bot_bot_pname_o
= ( collect_pname @ P_28 ) )
<=> ! [X: pname] :
~ ( P_28 @ X ) ) ).
thf(fact_209_empty__Collect__eq,axiom,
! [P_28: hoare_1167836817_state > $o] :
( ( bot_bo70021908tate_o
= ( collec1027672124_state @ P_28 ) )
<=> ! [X: hoare_1167836817_state] :
~ ( P_28 @ X ) ) ).
thf(fact_210_mem__def,axiom,
! [X_33: hoare_1775062406iple_a,A_94: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_33 @ A_94 )
<=> ( A_94 @ X_33 ) ) ).
thf(fact_211_mem__def,axiom,
! [X_33: pname,A_94: pname > $o] :
( ( member_pname @ X_33 @ A_94 )
<=> ( A_94 @ X_33 ) ) ).
thf(fact_212_Collect__def,axiom,
! [P_27: hoare_1775062406iple_a > $o] :
( ( collec676402587iple_a @ P_27 )
= P_27 ) ).
thf(fact_213_Collect__def,axiom,
! [P_27: pname > $o] :
( ( collect_pname @ P_27 )
= P_27 ) ).
thf(fact_214_empty__iff,axiom,
! [C_33: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ C_33 @ bot_bo751897185le_a_o ) ).
thf(fact_215_empty__iff,axiom,
! [C_33: hoare_1167836817_state] :
~ ( member2058392318_state @ C_33 @ bot_bo70021908tate_o ) ).
thf(fact_216_empty__iff,axiom,
! [C_33: pname] :
~ ( member_pname @ C_33 @ bot_bot_pname_o ) ).
thf(fact_217_insert__compr,axiom,
! [A_93: hoare_1167836817_state,B_51: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_93 @ B_51 )
= ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : ( (|) @ ( X = A_93 ) @ ( member2058392318_state @ X @ B_51 ) ) ) ) ).
thf(fact_218_insert__compr,axiom,
! [A_93: hoare_1775062406iple_a,B_51: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_93 @ B_51 )
= ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (|) @ ( X = A_93 ) @ ( member2122167641iple_a @ X @ B_51 ) ) ) ) ).
thf(fact_219_insert__compr,axiom,
! [A_93: pname,B_51: pname > $o] :
( ( insert_pname @ A_93 @ B_51 )
= ( collect_pname
@ ^ [X: pname] : ( (|) @ ( X = A_93 ) @ ( member_pname @ X @ B_51 ) ) ) ) ).
thf(fact_220_insert__Collect,axiom,
! [A_92: hoare_1167836817_state,P_26: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_92 @ ( collec1027672124_state @ P_26 ) )
= ( collec1027672124_state
@ ^ [U_2: hoare_1167836817_state] : ( (=>) @ ( (~) @ ( U_2 = A_92 ) ) @ ( P_26 @ U_2 ) ) ) ) ).
thf(fact_221_insert__Collect,axiom,
! [A_92: hoare_1775062406iple_a,P_26: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_92 @ ( collec676402587iple_a @ P_26 ) )
= ( collec676402587iple_a
@ ^ [U_2: hoare_1775062406iple_a] : ( (=>) @ ( (~) @ ( U_2 = A_92 ) ) @ ( P_26 @ U_2 ) ) ) ) ).
thf(fact_222_insert__Collect,axiom,
! [A_92: pname,P_26: pname > $o] :
( ( insert_pname @ A_92 @ ( collect_pname @ P_26 ) )
= ( collect_pname
@ ^ [U_2: pname] : ( (=>) @ ( (~) @ ( U_2 = A_92 ) ) @ ( P_26 @ U_2 ) ) ) ) ).
thf(fact_223_singleton__iff,axiom,
! [B_50: hoare_1167836817_state,A_91: hoare_1167836817_state] :
( ( member2058392318_state @ B_50 @ ( insert2134838167_state @ A_91 @ bot_bo70021908tate_o ) )
<=> ( B_50 = A_91 ) ) ).
thf(fact_224_singleton__iff,axiom,
! [B_50: hoare_1775062406iple_a,A_91: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ B_50 @ ( insert1281456128iple_a @ A_91 @ bot_bo751897185le_a_o ) )
<=> ( B_50 = A_91 ) ) ).
thf(fact_225_singleton__iff,axiom,
! [B_50: pname,A_91: pname] :
( ( member_pname @ B_50 @ ( insert_pname @ A_91 @ bot_bot_pname_o ) )
<=> ( B_50 = A_91 ) ) ).
thf(fact_226_insert__absorb2,axiom,
! [X_32: hoare_1167836817_state,A_90: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X_32 @ ( insert2134838167_state @ X_32 @ A_90 ) )
= ( insert2134838167_state @ X_32 @ A_90 ) ) ).
thf(fact_227_insert__absorb2,axiom,
! [X_32: hoare_1775062406iple_a,A_90: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ X_32 @ ( insert1281456128iple_a @ X_32 @ A_90 ) )
= ( insert1281456128iple_a @ X_32 @ A_90 ) ) ).
thf(fact_228_insert__absorb2,axiom,
! [X_32: pname,A_90: pname > $o] :
( ( insert_pname @ X_32 @ ( insert_pname @ X_32 @ A_90 ) )
= ( insert_pname @ X_32 @ A_90 ) ) ).
thf(fact_229_insert__commute,axiom,
! [X_31: hoare_1167836817_state,Y_12: hoare_1167836817_state,A_89: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X_31 @ ( insert2134838167_state @ Y_12 @ A_89 ) )
= ( insert2134838167_state @ Y_12 @ ( insert2134838167_state @ X_31 @ A_89 ) ) ) ).
thf(fact_230_insert__commute,axiom,
! [X_31: hoare_1775062406iple_a,Y_12: hoare_1775062406iple_a,A_89: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ X_31 @ ( insert1281456128iple_a @ Y_12 @ A_89 ) )
= ( insert1281456128iple_a @ Y_12 @ ( insert1281456128iple_a @ X_31 @ A_89 ) ) ) ).
thf(fact_231_insert__commute,axiom,
! [X_31: pname,Y_12: pname,A_89: pname > $o] :
( ( insert_pname @ X_31 @ ( insert_pname @ Y_12 @ A_89 ) )
= ( insert_pname @ Y_12 @ ( insert_pname @ X_31 @ A_89 ) ) ) ).
thf(fact_232_insert__iff,axiom,
! [A_88: hoare_1167836817_state,B_49: hoare_1167836817_state,A_87: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_88 @ ( insert2134838167_state @ B_49 @ A_87 ) )
<=> ( ( A_88 = B_49 )
| ( member2058392318_state @ A_88 @ A_87 ) ) ) ).
thf(fact_233_insert__iff,axiom,
! [A_88: hoare_1775062406iple_a,B_49: hoare_1775062406iple_a,A_87: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_88 @ ( insert1281456128iple_a @ B_49 @ A_87 ) )
<=> ( ( A_88 = B_49 )
| ( member2122167641iple_a @ A_88 @ A_87 ) ) ) ).
thf(fact_234_insert__iff,axiom,
! [A_88: pname,B_49: pname,A_87: pname > $o] :
( ( member_pname @ A_88 @ ( insert_pname @ B_49 @ A_87 ) )
<=> ( ( A_88 = B_49 )
| ( member_pname @ A_88 @ A_87 ) ) ) ).
thf(fact_235_Collect__empty__eq,axiom,
! [P_25: hoare_1775062406iple_a > $o] :
( ( ( collec676402587iple_a @ P_25 )
= bot_bo751897185le_a_o )
<=> ! [X: hoare_1775062406iple_a] :
~ ( P_25 @ X ) ) ).
thf(fact_236_Collect__empty__eq,axiom,
! [P_25: pname > $o] :
( ( ( collect_pname @ P_25 )
= bot_bot_pname_o )
<=> ! [X: pname] :
~ ( P_25 @ X ) ) ).
thf(fact_237_Collect__empty__eq,axiom,
! [P_25: hoare_1167836817_state > $o] :
( ( ( collec1027672124_state @ P_25 )
= bot_bo70021908tate_o )
<=> ! [X: hoare_1167836817_state] :
~ ( P_25 @ X ) ) ).
thf(fact_238_doubleton__eq__iff,axiom,
! [A_86: hoare_1167836817_state,B_48: hoare_1167836817_state,C_32: hoare_1167836817_state,D_1: hoare_1167836817_state] :
( ( ( insert2134838167_state @ A_86 @ ( insert2134838167_state @ B_48 @ bot_bo70021908tate_o ) )
= ( insert2134838167_state @ C_32 @ ( insert2134838167_state @ D_1 @ bot_bo70021908tate_o ) ) )
<=> ( ( ( A_86 = C_32 )
& ( B_48 = D_1 ) )
| ( ( A_86 = D_1 )
& ( B_48 = C_32 ) ) ) ) ).
thf(fact_239_doubleton__eq__iff,axiom,
! [A_86: hoare_1775062406iple_a,B_48: hoare_1775062406iple_a,C_32: hoare_1775062406iple_a,D_1: hoare_1775062406iple_a] :
( ( ( insert1281456128iple_a @ A_86 @ ( insert1281456128iple_a @ B_48 @ bot_bo751897185le_a_o ) )
= ( insert1281456128iple_a @ C_32 @ ( insert1281456128iple_a @ D_1 @ bot_bo751897185le_a_o ) ) )
<=> ( ( ( A_86 = C_32 )
& ( B_48 = D_1 ) )
| ( ( A_86 = D_1 )
& ( B_48 = C_32 ) ) ) ) ).
thf(fact_240_doubleton__eq__iff,axiom,
! [A_86: pname,B_48: pname,C_32: pname,D_1: pname] :
( ( ( insert_pname @ A_86 @ ( insert_pname @ B_48 @ bot_bot_pname_o ) )
= ( insert_pname @ C_32 @ ( insert_pname @ D_1 @ bot_bot_pname_o ) ) )
<=> ( ( ( A_86 = C_32 )
& ( B_48 = D_1 ) )
| ( ( A_86 = D_1 )
& ( B_48 = C_32 ) ) ) ) ).
thf(fact_241_insert__code,axiom,
! [Y_11: hoare_1167836817_state,A_85: hoare_1167836817_state > $o,X_30: hoare_1167836817_state] :
( ( insert2134838167_state @ Y_11 @ A_85 @ X_30 )
<=> ( ( Y_11 = X_30 )
| ( A_85 @ X_30 ) ) ) ).
thf(fact_242_insert__code,axiom,
! [Y_11: hoare_1775062406iple_a,A_85: hoare_1775062406iple_a > $o,X_30: hoare_1775062406iple_a] :
( ( insert1281456128iple_a @ Y_11 @ A_85 @ X_30 )
<=> ( ( Y_11 = X_30 )
| ( A_85 @ X_30 ) ) ) ).
thf(fact_243_insert__code,axiom,
! [Y_11: pname,A_85: pname > $o,X_30: pname] :
( ( insert_pname @ Y_11 @ A_85 @ X_30 )
<=> ( ( Y_11 = X_30 )
| ( A_85 @ X_30 ) ) ) ).
thf(fact_244_insert__ident,axiom,
! [B_47: hoare_1167836817_state > $o,X_29: hoare_1167836817_state,A_84: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_29 @ A_84 )
=> ( ~ ( member2058392318_state @ X_29 @ B_47 )
=> ( ( ( insert2134838167_state @ X_29 @ A_84 )
= ( insert2134838167_state @ X_29 @ B_47 ) )
<=> ( A_84 = B_47 ) ) ) ) ).
thf(fact_245_insert__ident,axiom,
! [B_47: hoare_1775062406iple_a > $o,X_29: hoare_1775062406iple_a,A_84: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ X_29 @ A_84 )
=> ( ~ ( member2122167641iple_a @ X_29 @ B_47 )
=> ( ( ( insert1281456128iple_a @ X_29 @ A_84 )
= ( insert1281456128iple_a @ X_29 @ B_47 ) )
<=> ( A_84 = B_47 ) ) ) ) ).
thf(fact_246_insert__ident,axiom,
! [B_47: pname > $o,X_29: pname,A_84: pname > $o] :
( ~ ( member_pname @ X_29 @ A_84 )
=> ( ~ ( member_pname @ X_29 @ B_47 )
=> ( ( ( insert_pname @ X_29 @ A_84 )
= ( insert_pname @ X_29 @ B_47 ) )
<=> ( A_84 = B_47 ) ) ) ) ).
thf(fact_247_equals0D,axiom,
! [A_83: hoare_1775062406iple_a,A_82: hoare_1775062406iple_a > $o] :
( ( A_82 = bot_bo751897185le_a_o )
=> ~ ( member2122167641iple_a @ A_83 @ A_82 ) ) ).
thf(fact_248_equals0D,axiom,
! [A_83: hoare_1167836817_state,A_82: hoare_1167836817_state > $o] :
( ( A_82 = bot_bo70021908tate_o )
=> ~ ( member2058392318_state @ A_83 @ A_82 ) ) ).
thf(fact_249_equals0D,axiom,
! [A_83: pname,A_82: pname > $o] :
( ( A_82 = bot_bot_pname_o )
=> ~ ( member_pname @ A_83 @ A_82 ) ) ).
thf(fact_250_insertI2,axiom,
! [B_46: hoare_1167836817_state,A_81: hoare_1167836817_state,B_45: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_81 @ B_45 )
=> ( member2058392318_state @ A_81 @ ( insert2134838167_state @ B_46 @ B_45 ) ) ) ).
thf(fact_251_insertI2,axiom,
! [B_46: hoare_1775062406iple_a,A_81: hoare_1775062406iple_a,B_45: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_81 @ B_45 )
=> ( member2122167641iple_a @ A_81 @ ( insert1281456128iple_a @ B_46 @ B_45 ) ) ) ).
thf(fact_252_insertI2,axiom,
! [B_46: pname,A_81: pname,B_45: pname > $o] :
( ( member_pname @ A_81 @ B_45 )
=> ( member_pname @ A_81 @ ( insert_pname @ B_46 @ B_45 ) ) ) ).
thf(fact_253_insert__absorb,axiom,
! [A_80: hoare_1167836817_state,A_79: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_80 @ A_79 )
=> ( ( insert2134838167_state @ A_80 @ A_79 )
= A_79 ) ) ).
thf(fact_254_insert__absorb,axiom,
! [A_80: hoare_1775062406iple_a,A_79: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_80 @ A_79 )
=> ( ( insert1281456128iple_a @ A_80 @ A_79 )
= A_79 ) ) ).
thf(fact_255_insert__absorb,axiom,
! [A_80: pname,A_79: pname > $o] :
( ( member_pname @ A_80 @ A_79 )
=> ( ( insert_pname @ A_80 @ A_79 )
= A_79 ) ) ).
thf(fact_256_singletonE,axiom,
! [B_44: hoare_1167836817_state,A_78: hoare_1167836817_state] :
( ( member2058392318_state @ B_44 @ ( insert2134838167_state @ A_78 @ bot_bo70021908tate_o ) )
=> ( B_44 = A_78 ) ) ).
thf(fact_257_singletonE,axiom,
! [B_44: hoare_1775062406iple_a,A_78: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ B_44 @ ( insert1281456128iple_a @ A_78 @ bot_bo751897185le_a_o ) )
=> ( B_44 = A_78 ) ) ).
thf(fact_258_singletonE,axiom,
! [B_44: pname,A_78: pname] :
( ( member_pname @ B_44 @ ( insert_pname @ A_78 @ bot_bot_pname_o ) )
=> ( B_44 = A_78 ) ) ).
thf(fact_259_singleton__inject,axiom,
! [A_77: hoare_1167836817_state,B_43: hoare_1167836817_state] :
( ( ( insert2134838167_state @ A_77 @ bot_bo70021908tate_o )
= ( insert2134838167_state @ B_43 @ bot_bo70021908tate_o ) )
=> ( A_77 = B_43 ) ) ).
thf(fact_260_singleton__inject,axiom,
! [A_77: hoare_1775062406iple_a,B_43: hoare_1775062406iple_a] :
( ( ( insert1281456128iple_a @ A_77 @ bot_bo751897185le_a_o )
= ( insert1281456128iple_a @ B_43 @ bot_bo751897185le_a_o ) )
=> ( A_77 = B_43 ) ) ).
thf(fact_261_singleton__inject,axiom,
! [A_77: pname,B_43: pname] :
( ( ( insert_pname @ A_77 @ bot_bot_pname_o )
= ( insert_pname @ B_43 @ bot_bot_pname_o ) )
=> ( A_77 = B_43 ) ) ).
thf(fact_262_com__det,axiom,
! [U_1: state,C_19: com,S: state,T: state] :
( ( evalc @ C_19 @ S @ T )
=> ( ( evalc @ C_19 @ S @ U_1 )
=> ( U_1 = T ) ) ) ).
thf(fact_263_insert__is__Un,axiom,
! [A_76: hoare_1167836817_state,A_75: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_76 @ A_75 )
= ( semila1172322802tate_o @ ( insert2134838167_state @ A_76 @ bot_bo70021908tate_o ) @ A_75 ) ) ).
thf(fact_264_insert__is__Un,axiom,
! [A_76: pname,A_75: pname > $o] :
( ( insert_pname @ A_76 @ A_75 )
= ( semila1780557381name_o @ ( insert_pname @ A_76 @ bot_bot_pname_o ) @ A_75 ) ) ).
thf(fact_265_insert__is__Un,axiom,
! [A_76: hoare_1775062406iple_a,A_75: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_76 @ A_75 )
= ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_76 @ bot_bo751897185le_a_o ) @ A_75 ) ) ).
thf(fact_266_insert__compr__raw,axiom,
! [X: hoare_1167836817_state,Xa: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X @ Xa )
= ( collec1027672124_state
@ ^ [Y_2: hoare_1167836817_state] : ( (|) @ ( Y_2 = X ) @ ( member2058392318_state @ Y_2 @ Xa ) ) ) ) ).
thf(fact_267_insert__compr__raw,axiom,
! [X: hoare_1775062406iple_a,Xa: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ X @ Xa )
= ( collec676402587iple_a
@ ^ [Y_2: hoare_1775062406iple_a] : ( (|) @ ( Y_2 = X ) @ ( member2122167641iple_a @ Y_2 @ Xa ) ) ) ) ).
thf(fact_268_insert__compr__raw,axiom,
! [X: pname,Xa: pname > $o] :
( ( insert_pname @ X @ Xa )
= ( collect_pname
@ ^ [Y_2: pname] : ( (|) @ ( Y_2 = X ) @ ( member_pname @ Y_2 @ Xa ) ) ) ) ).
thf(fact_269_derivs__insertD,axiom,
! [G_16: hoare_1167836817_state > $o,T_3: hoare_1167836817_state,Ts_2: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_3 @ Ts_2 ) )
=> ( ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_3 @ bot_bo70021908tate_o ) )
& ( hoare_123228589_state @ G_16 @ Ts_2 ) ) ) ).
thf(fact_270_derivs__insertD,axiom,
! [G_16: hoare_1775062406iple_a > $o,T_3: hoare_1775062406iple_a,Ts_2: hoare_1775062406iple_a > $o] :
( ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_3 @ Ts_2 ) )
=> ( ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_3 @ bot_bo751897185le_a_o ) )
& ( hoare_1508237396rivs_a @ G_16 @ Ts_2 ) ) ) ).
thf(fact_271_hoare__derivs_Oinsert,axiom,
! [Ts_1: hoare_1167836817_state > $o,G_15: hoare_1167836817_state > $o,T_2: hoare_1167836817_state] :
( ( hoare_123228589_state @ G_15 @ ( insert2134838167_state @ T_2 @ bot_bo70021908tate_o ) )
=> ( ( hoare_123228589_state @ G_15 @ Ts_1 )
=> ( hoare_123228589_state @ G_15 @ ( insert2134838167_state @ T_2 @ Ts_1 ) ) ) ) ).
thf(fact_272_hoare__derivs_Oinsert,axiom,
! [Ts_1: hoare_1775062406iple_a > $o,G_15: hoare_1775062406iple_a > $o,T_2: hoare_1775062406iple_a] :
( ( hoare_1508237396rivs_a @ G_15 @ ( insert1281456128iple_a @ T_2 @ bot_bo751897185le_a_o ) )
=> ( ( hoare_1508237396rivs_a @ G_15 @ Ts_1 )
=> ( hoare_1508237396rivs_a @ G_15 @ ( insert1281456128iple_a @ T_2 @ Ts_1 ) ) ) ) ).
thf(fact_273_image__constant__conv,axiom,
! [C_31: hoare_1167836817_state,A_74: pname > $o] :
( ( ( A_74 = bot_bot_pname_o )
=> ( ( image_575578384_state
@ ^ [X: pname] : C_31
@ A_74 )
= bot_bo70021908tate_o ) )
& ( ( A_74 != bot_bot_pname_o )
=> ( ( image_575578384_state
@ ^ [X: pname] : C_31
@ A_74 )
= ( insert2134838167_state @ C_31 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_274_image__constant__conv,axiom,
! [C_31: hoare_1775062406iple_a,A_74: pname > $o] :
( ( ( A_74 = bot_bot_pname_o )
=> ( ( image_2063119815iple_a
@ ^ [X: pname] : C_31
@ A_74 )
= bot_bo751897185le_a_o ) )
& ( ( A_74 != bot_bot_pname_o )
=> ( ( image_2063119815iple_a
@ ^ [X: pname] : C_31
@ A_74 )
= ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_275_image__constant,axiom,
! [C_30: hoare_1167836817_state,X_28: pname,A_73: pname > $o] :
( ( member_pname @ X_28 @ A_73 )
=> ( ( image_575578384_state
@ ^ [X: pname] : C_30
@ A_73 )
= ( insert2134838167_state @ C_30 @ bot_bo70021908tate_o ) ) ) ).
thf(fact_276_image__constant,axiom,
! [C_30: pname,X_28: pname,A_73: pname > $o] :
( ( member_pname @ X_28 @ A_73 )
=> ( ( image_pname_pname
@ ^ [X: pname] : C_30
@ A_73 )
= ( insert_pname @ C_30 @ bot_bot_pname_o ) ) ) ).
thf(fact_277_image__constant,axiom,
! [C_30: hoare_1775062406iple_a,X_28: pname,A_73: pname > $o] :
( ( member_pname @ X_28 @ A_73 )
=> ( ( image_2063119815iple_a
@ ^ [X: pname] : C_30
@ A_73 )
= ( insert1281456128iple_a @ C_30 @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_278_image__insert,axiom,
! [F_31: pname > hoare_1167836817_state,A_72: pname,B_42: pname > $o] :
( ( image_575578384_state @ F_31 @ ( insert_pname @ A_72 @ B_42 ) )
= ( insert2134838167_state @ ( F_31 @ A_72 ) @ ( image_575578384_state @ F_31 @ B_42 ) ) ) ).
thf(fact_279_image__insert,axiom,
! [F_31: pname > hoare_1775062406iple_a,A_72: pname,B_42: pname > $o] :
( ( image_2063119815iple_a @ F_31 @ ( insert_pname @ A_72 @ B_42 ) )
= ( insert1281456128iple_a @ ( F_31 @ A_72 ) @ ( image_2063119815iple_a @ F_31 @ B_42 ) ) ) ).
thf(fact_280_insert__image,axiom,
! [F_30: pname > hoare_1167836817_state,X_27: pname,A_71: pname > $o] :
( ( member_pname @ X_27 @ A_71 )
=> ( ( insert2134838167_state @ ( F_30 @ X_27 ) @ ( image_575578384_state @ F_30 @ A_71 ) )
= ( image_575578384_state @ F_30 @ A_71 ) ) ) ).
thf(fact_281_insert__image,axiom,
! [F_30: pname > pname,X_27: pname,A_71: pname > $o] :
( ( member_pname @ X_27 @ A_71 )
=> ( ( insert_pname @ ( F_30 @ X_27 ) @ ( image_pname_pname @ F_30 @ A_71 ) )
= ( image_pname_pname @ F_30 @ A_71 ) ) ) ).
thf(fact_282_insert__image,axiom,
! [F_30: pname > hoare_1775062406iple_a,X_27: pname,A_71: pname > $o] :
( ( member_pname @ X_27 @ A_71 )
=> ( ( insert1281456128iple_a @ ( F_30 @ X_27 ) @ ( image_2063119815iple_a @ F_30 @ A_71 ) )
= ( image_2063119815iple_a @ F_30 @ A_71 ) ) ) ).
thf(fact_283_Un__insert__right,axiom,
! [A_70: hoare_1167836817_state > $o,A_69: hoare_1167836817_state,B_41: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_70 @ ( insert2134838167_state @ A_69 @ B_41 ) )
= ( insert2134838167_state @ A_69 @ ( semila1172322802tate_o @ A_70 @ B_41 ) ) ) ).
thf(fact_284_Un__insert__right,axiom,
! [A_70: pname > $o,A_69: pname,B_41: pname > $o] :
( ( semila1780557381name_o @ A_70 @ ( insert_pname @ A_69 @ B_41 ) )
= ( insert_pname @ A_69 @ ( semila1780557381name_o @ A_70 @ B_41 ) ) ) ).
thf(fact_285_Un__insert__right,axiom,
! [A_70: hoare_1775062406iple_a > $o,A_69: hoare_1775062406iple_a,B_41: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_70 @ ( insert1281456128iple_a @ A_69 @ B_41 ) )
= ( insert1281456128iple_a @ A_69 @ ( semila13410563le_a_o @ A_70 @ B_41 ) ) ) ).
thf(fact_286_Un__insert__left,axiom,
! [A_68: hoare_1167836817_state,B_40: hoare_1167836817_state > $o,C_29: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( insert2134838167_state @ A_68 @ B_40 ) @ C_29 )
= ( insert2134838167_state @ A_68 @ ( semila1172322802tate_o @ B_40 @ C_29 ) ) ) ).
thf(fact_287_Un__insert__left,axiom,
! [A_68: pname,B_40: pname > $o,C_29: pname > $o] :
( ( semila1780557381name_o @ ( insert_pname @ A_68 @ B_40 ) @ C_29 )
= ( insert_pname @ A_68 @ ( semila1780557381name_o @ B_40 @ C_29 ) ) ) ).
thf(fact_288_Un__insert__left,axiom,
! [A_68: hoare_1775062406iple_a,B_40: hoare_1775062406iple_a > $o,C_29: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_68 @ B_40 ) @ C_29 )
= ( insert1281456128iple_a @ A_68 @ ( semila13410563le_a_o @ B_40 @ C_29 ) ) ) ).
thf(fact_289_empty__is__image,axiom,
! [F_29: pname > hoare_1167836817_state,A_67: pname > $o] :
( ( bot_bo70021908tate_o
= ( image_575578384_state @ F_29 @ A_67 ) )
<=> ( A_67 = bot_bot_pname_o ) ) ).
thf(fact_290_empty__is__image,axiom,
! [F_29: pname > hoare_1775062406iple_a,A_67: pname > $o] :
( ( bot_bo751897185le_a_o
= ( image_2063119815iple_a @ F_29 @ A_67 ) )
<=> ( A_67 = bot_bot_pname_o ) ) ).
thf(fact_291_image__empty,axiom,
! [F_28: pname > hoare_1167836817_state] :
( ( image_575578384_state @ F_28 @ bot_bot_pname_o )
= bot_bo70021908tate_o ) ).
thf(fact_292_image__empty,axiom,
! [F_28: pname > hoare_1775062406iple_a] :
( ( image_2063119815iple_a @ F_28 @ bot_bot_pname_o )
= bot_bo751897185le_a_o ) ).
thf(fact_293_image__is__empty,axiom,
! [F_27: pname > hoare_1167836817_state,A_66: pname > $o] :
( ( ( image_575578384_state @ F_27 @ A_66 )
= bot_bo70021908tate_o )
<=> ( A_66 = bot_bot_pname_o ) ) ).
thf(fact_294_image__is__empty,axiom,
! [F_27: pname > hoare_1775062406iple_a,A_66: pname > $o] :
( ( ( image_2063119815iple_a @ F_27 @ A_66 )
= bot_bo751897185le_a_o )
<=> ( A_66 = bot_bot_pname_o ) ) ).
thf(fact_295_ball__empty,axiom,
! [P_24: hoare_1167836817_state > $o,X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ bot_bo70021908tate_o )
=> ( P_24 @ X ) ) ).
thf(fact_296_ball__empty,axiom,
! [P_24: hoare_1775062406iple_a > $o,X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ bot_bo751897185le_a_o )
=> ( P_24 @ X ) ) ).
thf(fact_297_ball__empty,axiom,
! [P_24: pname > $o,X: pname] :
( ( member_pname @ X @ bot_bot_pname_o )
=> ( P_24 @ X ) ) ).
thf(fact_298_Un__empty__left,axiom,
! [B_39: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ bot_bo70021908tate_o @ B_39 )
= B_39 ) ).
thf(fact_299_Un__empty__left,axiom,
! [B_39: pname > $o] :
( ( semila1780557381name_o @ bot_bot_pname_o @ B_39 )
= B_39 ) ).
thf(fact_300_Un__empty__left,axiom,
! [B_39: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ B_39 )
= B_39 ) ).
thf(fact_301_Un__empty__right,axiom,
! [A_65: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_65 @ bot_bo70021908tate_o )
= A_65 ) ).
thf(fact_302_Un__empty__right,axiom,
! [A_65: pname > $o] :
( ( semila1780557381name_o @ A_65 @ bot_bot_pname_o )
= A_65 ) ).
thf(fact_303_Un__empty__right,axiom,
! [A_65: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_65 @ bot_bo751897185le_a_o )
= A_65 ) ).
thf(fact_304_Un__empty,axiom,
! [A_64: hoare_1167836817_state > $o,B_38: hoare_1167836817_state > $o] :
( ( ( semila1172322802tate_o @ A_64 @ B_38 )
= bot_bo70021908tate_o )
<=> ( ( A_64 = bot_bo70021908tate_o )
& ( B_38 = bot_bo70021908tate_o ) ) ) ).
thf(fact_305_Un__empty,axiom,
! [A_64: pname > $o,B_38: pname > $o] :
( ( ( semila1780557381name_o @ A_64 @ B_38 )
= bot_bot_pname_o )
<=> ( ( A_64 = bot_bot_pname_o )
& ( B_38 = bot_bot_pname_o ) ) ) ).
thf(fact_306_Un__empty,axiom,
! [A_64: hoare_1775062406iple_a > $o,B_38: hoare_1775062406iple_a > $o] :
( ( ( semila13410563le_a_o @ A_64 @ B_38 )
= bot_bo751897185le_a_o )
<=> ( ( A_64 = bot_bo751897185le_a_o )
& ( B_38 = bot_bo751897185le_a_o ) ) ) ).
thf(fact_307_constant,axiom,
! [G_14: hoare_1775062406iple_a > $o,P_23: x_a > state > $o,C_28: com,Q_15: x_a > state > $o,C_27: $o] :
( ( C_27
=> ( hoare_1508237396rivs_a @ G_14 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_23 @ C_28 @ Q_15 ) @ bot_bo751897185le_a_o ) ) )
=> ( hoare_1508237396rivs_a @ G_14
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Z_8: x_a,S_3: state] : ( (&) @ ( P_23 @ Z_8 @ S_3 ) @ C_27 )
@ C_28
@ Q_15 )
@ bot_bo751897185le_a_o ) ) ) ).
thf(fact_308_constant,axiom,
! [G_14: hoare_1167836817_state > $o,P_23: state > state > $o,C_28: com,Q_15: state > state > $o,C_27: $o] :
( ( C_27
=> ( hoare_123228589_state @ G_14 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_23 @ C_28 @ Q_15 ) @ bot_bo70021908tate_o ) ) )
=> ( hoare_123228589_state @ G_14
@ ( insert2134838167_state
@ ( hoare_908217195_state
@ ^ [Z_8: state,S_3: state] : ( (&) @ ( P_23 @ Z_8 @ S_3 ) @ C_27 )
@ C_28
@ Q_15 )
@ bot_bo70021908tate_o ) ) ) ).
thf(fact_309_empty,axiom,
! [G_13: hoare_1167836817_state > $o] : ( hoare_123228589_state @ G_13 @ bot_bo70021908tate_o ) ).
thf(fact_310_empty,axiom,
! [G_13: hoare_1775062406iple_a > $o] : ( hoare_1508237396rivs_a @ G_13 @ bot_bo751897185le_a_o ) ).
thf(fact_311_sup__bot__left,axiom,
! [X_26: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ bot_bo70021908tate_o @ X_26 )
= X_26 ) ).
thf(fact_312_sup__bot__left,axiom,
! [X_26: pname > $o] :
( ( semila1780557381name_o @ bot_bot_pname_o @ X_26 )
= X_26 ) ).
thf(fact_313_sup__bot__left,axiom,
! [X_26: $o] :
( ( semila10642723_sup_o @ bot_bot_o @ X_26 )
<=> X_26 ) ).
thf(fact_314_sup__bot__left,axiom,
! [X_26: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ X_26 )
= X_26 ) ).
thf(fact_315_sup__bot__right,axiom,
! [X_25: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_25 @ bot_bo70021908tate_o )
= X_25 ) ).
thf(fact_316_sup__bot__right,axiom,
! [X_25: pname > $o] :
( ( semila1780557381name_o @ X_25 @ bot_bot_pname_o )
= X_25 ) ).
thf(fact_317_sup__bot__right,axiom,
! [X_25: $o] :
( ( semila10642723_sup_o @ X_25 @ bot_bot_o )
<=> X_25 ) ).
thf(fact_318_sup__bot__right,axiom,
! [X_25: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_25 @ bot_bo751897185le_a_o )
= X_25 ) ).
thf(fact_319_sup__eq__bot__iff,axiom,
! [X_24: hoare_1167836817_state > $o,Y_10: hoare_1167836817_state > $o] :
( ( ( semila1172322802tate_o @ X_24 @ Y_10 )
= bot_bo70021908tate_o )
<=> ( ( X_24 = bot_bo70021908tate_o )
& ( Y_10 = bot_bo70021908tate_o ) ) ) ).
thf(fact_320_sup__eq__bot__iff,axiom,
! [X_24: pname > $o,Y_10: pname > $o] :
( ( ( semila1780557381name_o @ X_24 @ Y_10 )
= bot_bot_pname_o )
<=> ( ( X_24 = bot_bot_pname_o )
& ( Y_10 = bot_bot_pname_o ) ) ) ).
thf(fact_321_sup__eq__bot__iff,axiom,
! [X_24: $o,Y_10: $o] :
( ( ( semila10642723_sup_o @ X_24 @ Y_10 )
<=> bot_bot_o )
<=> ( ( X_24
<=> bot_bot_o )
& ( Y_10
<=> bot_bot_o ) ) ) ).
thf(fact_322_sup__eq__bot__iff,axiom,
! [X_24: hoare_1775062406iple_a > $o,Y_10: hoare_1775062406iple_a > $o] :
( ( ( semila13410563le_a_o @ X_24 @ Y_10 )
= bot_bo751897185le_a_o )
<=> ( ( X_24 = bot_bo751897185le_a_o )
& ( Y_10 = bot_bo751897185le_a_o ) ) ) ).
thf(fact_323_triple__valid__Suc,axiom,
! [N_6: nat,T_1: hoare_1167836817_state] :
( ( hoare_56934129_state @ ( suc @ N_6 ) @ T_1 )
=> ( hoare_56934129_state @ N_6 @ T_1 ) ) ).
thf(fact_324_triple__valid__Suc,axiom,
! [N_6: nat,T_1: hoare_1775062406iple_a] :
( ( hoare_1462269968alid_a @ ( suc @ N_6 ) @ T_1 )
=> ( hoare_1462269968alid_a @ N_6 @ T_1 ) ) ).
thf(fact_325_insert__def,axiom,
! [A_63: hoare_1167836817_state,B_37: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_63 @ B_37 )
= ( semila1172322802tate_o
@ ( collec1027672124_state
@ ^ [X: hoare_1167836817_state] : X = A_63 )
@ B_37 ) ) ).
thf(fact_326_insert__def,axiom,
! [A_63: pname,B_37: pname > $o] :
( ( insert_pname @ A_63 @ B_37 )
= ( semila1780557381name_o
@ ( collect_pname
@ ^ [X: pname] : X = A_63 )
@ B_37 ) ) ).
thf(fact_327_insert__def,axiom,
! [A_63: hoare_1775062406iple_a,B_37: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_63 @ B_37 )
= ( semila13410563le_a_o
@ ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : X = A_63 )
@ B_37 ) ) ).
thf(fact_328_weak__Body,axiom,
! [G_12: hoare_1775062406iple_a > $o,P_22: x_a > state > $o,Pn_3: pname,Q_14: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_22 @ ( the_com @ ( body_1 @ Pn_3 ) ) @ Q_14 ) @ bot_bo751897185le_a_o ) )
=> ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_22 @ ( body @ Pn_3 ) @ Q_14 ) @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_329_weak__Body,axiom,
! [G_12: hoare_1167836817_state > $o,P_22: state > state > $o,Pn_3: pname,Q_14: state > state > $o] :
( ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_22 @ ( the_com @ ( body_1 @ Pn_3 ) ) @ Q_14 ) @ bot_bo70021908tate_o ) )
=> ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_22 @ ( body @ Pn_3 ) @ Q_14 ) @ bot_bo70021908tate_o ) ) ) ).
thf(fact_330_BodyN,axiom,
! [P_21: x_a > state > $o,Pn_2: pname,Q_13: x_a > state > $o,G_11: hoare_1775062406iple_a > $o] :
( ( hoare_1508237396rivs_a @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_21 @ ( body @ Pn_2 ) @ Q_13 ) @ G_11 ) @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_21 @ ( the_com @ ( body_1 @ Pn_2 ) ) @ Q_13 ) @ bot_bo751897185le_a_o ) )
=> ( hoare_1508237396rivs_a @ G_11 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_21 @ ( body @ Pn_2 ) @ Q_13 ) @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_331_BodyN,axiom,
! [P_21: state > state > $o,Pn_2: pname,Q_13: state > state > $o,G_11: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ ( body @ Pn_2 ) @ Q_13 ) @ G_11 ) @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ ( the_com @ ( body_1 @ Pn_2 ) ) @ Q_13 ) @ bot_bo70021908tate_o ) )
=> ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ ( body @ Pn_2 ) @ Q_13 ) @ bot_bo70021908tate_o ) ) ) ).
thf(fact_332_triples__valid__Suc,axiom,
! [N_5: nat,Ts: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ Ts )
=> ( hoare_56934129_state @ ( suc @ N_5 ) @ X ) )
=> ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ Ts )
=> ( hoare_56934129_state @ N_5 @ X ) ) ) ).
thf(fact_333_triples__valid__Suc,axiom,
! [N_5: nat,Ts: hoare_1775062406iple_a > $o] :
( ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ Ts )
=> ( hoare_1462269968alid_a @ ( suc @ N_5 ) @ X ) )
=> ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ Ts )
=> ( hoare_1462269968alid_a @ N_5 @ X ) ) ) ).
thf(fact_334_escape,axiom,
! [G_10: hoare_1775062406iple_a > $o,C_26: com,Q_12: x_a > state > $o,P_20: x_a > state > $o] :
( ! [Z_8: x_a,S_3: state] :
( ( P_20 @ Z_8 @ S_3 )
=> ( hoare_1508237396rivs_a @ G_10
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Za: x_a,S_4: state] : S_4 = S_3
@ C_26
@ ^ [Z_9: x_a] : ( Q_12 @ Z_8 ) )
@ bot_bo751897185le_a_o ) ) )
=> ( hoare_1508237396rivs_a @ G_10 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_20 @ C_26 @ Q_12 ) @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_335_escape,axiom,
! [G_10: hoare_1167836817_state > $o,C_26: com,Q_12: state > state > $o,P_20: state > state > $o] :
( ! [Z_8: state,S_3: state] :
( ( P_20 @ Z_8 @ S_3 )
=> ( hoare_123228589_state @ G_10
@ ( insert2134838167_state
@ ( hoare_908217195_state
@ ^ [Za: state,S_4: state] : S_4 = S_3
@ C_26
@ ^ [Z_9: state] : ( Q_12 @ Z_8 ) )
@ bot_bo70021908tate_o ) ) )
=> ( hoare_123228589_state @ G_10 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ C_26 @ Q_12 ) @ bot_bo70021908tate_o ) ) ) ).
thf(fact_336_conseq1,axiom,
! [P_19: x_a > state > $o,G_9: hoare_1775062406iple_a > $o,P_18: x_a > state > $o,C_25: com,Q_11: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_18 @ C_25 @ Q_11 ) @ bot_bo751897185le_a_o ) )
=> ( ! [Z_8: x_a,S_3: state] :
( ( P_19 @ Z_8 @ S_3 )
=> ( P_18 @ Z_8 @ S_3 ) )
=> ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_19 @ C_25 @ Q_11 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_337_conseq1,axiom,
! [P_19: state > state > $o,G_9: hoare_1167836817_state > $o,P_18: state > state > $o,C_25: com,Q_11: state > state > $o] :
( ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_18 @ C_25 @ Q_11 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z_8: state,S_3: state] :
( ( P_19 @ Z_8 @ S_3 )
=> ( P_18 @ Z_8 @ S_3 ) )
=> ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_19 @ C_25 @ Q_11 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_338_conseq2,axiom,
! [Q_10: x_a > state > $o,G_8: hoare_1775062406iple_a > $o,P_17: x_a > state > $o,C_24: com,Q_9: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_17 @ C_24 @ Q_9 ) @ bot_bo751897185le_a_o ) )
=> ( ! [Z_8: x_a,S_3: state] :
( ( Q_9 @ Z_8 @ S_3 )
=> ( Q_10 @ Z_8 @ S_3 ) )
=> ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_17 @ C_24 @ Q_10 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_339_conseq2,axiom,
! [Q_10: state > state > $o,G_8: hoare_1167836817_state > $o,P_17: state > state > $o,C_24: com,Q_9: state > state > $o] :
( ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_17 @ C_24 @ Q_9 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z_8: state,S_3: state] :
( ( Q_9 @ Z_8 @ S_3 )
=> ( Q_10 @ Z_8 @ S_3 ) )
=> ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_17 @ C_24 @ Q_10 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_340_triple_Osize_I1_J,axiom,
! [Fa: state > nat,Fun1_1: state > state > $o,Com_3: com,Fun2_1: state > state > $o] :
( ( hoare_545207370_state @ Fa @ ( hoare_908217195_state @ Fun1_1 @ Com_3 @ Fun2_1 ) )
= zero_zero_nat ) ).
thf(fact_341_triple_Osize_I1_J,axiom,
! [Fa: x_a > nat,Fun1_1: x_a > state > $o,Com_3: com,Fun2_1: x_a > state > $o] :
( ( hoare_1118907895size_a @ Fa @ ( hoare_1766022166iple_a @ Fun1_1 @ Com_3 @ Fun2_1 ) )
= zero_zero_nat ) ).
thf(fact_342_MGT__def,axiom,
! [C_19: com] :
( ( hoare_Mirabelle_MGT @ C_19 )
= ( hoare_908217195_state @ fequal_state @ C_19 @ ( evalc @ C_19 ) ) ) ).
thf(fact_343_triple_Osize_I2_J,axiom,
! [Fun1: state > state > $o,Com_2: com,Fun2: state > state > $o] :
( ( size_s645941755_state @ ( hoare_908217195_state @ Fun1 @ Com_2 @ Fun2 ) )
= zero_zero_nat ) ).
thf(fact_344_triple_Osize_I2_J,axiom,
! [Fun1: x_a > state > $o,Com_2: com,Fun2: x_a > state > $o] :
( ( size_s724313756iple_a @ ( hoare_1766022166iple_a @ Fun1 @ Com_2 @ Fun2 ) )
= zero_zero_nat ) ).
thf(fact_345_conseq12,axiom,
! [Q_8: state > state > $o,P_16: state > state > $o,G_7: hoare_1167836817_state > $o,P_15: state > state > $o,C_23: com,Q_7: state > state > $o] :
( ( hoare_123228589_state @ G_7 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_15 @ C_23 @ Q_7 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z_8: state,S_3: state] :
( ( P_16 @ Z_8 @ S_3 )
=> ! [S_4: state] :
( ! [Z_9: state] :
( ( P_15 @ Z_9 @ S_3 )
=> ( Q_7 @ Z_9 @ S_4 ) )
=> ( Q_8 @ Z_8 @ S_4 ) ) )
=> ( hoare_123228589_state @ G_7 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_16 @ C_23 @ Q_8 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_346_conseq12,axiom,
! [Q_8: x_a > state > $o,P_16: x_a > state > $o,G_7: hoare_1775062406iple_a > $o,P_15: x_a > state > $o,C_23: com,Q_7: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_7 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_15 @ C_23 @ Q_7 ) @ bot_bo751897185le_a_o ) )
=> ( ! [Z_8: x_a,S_3: state] :
( ( P_16 @ Z_8 @ S_3 )
=> ! [S_4: state] :
( ! [Z_9: x_a] :
( ( P_15 @ Z_9 @ S_3 )
=> ( Q_7 @ Z_9 @ S_4 ) )
=> ( Q_8 @ Z_8 @ S_4 ) ) )
=> ( hoare_1508237396rivs_a @ G_7 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_16 @ C_23 @ Q_8 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_347_the__elem__eq,axiom,
! [X_23: hoare_1167836817_state] :
( ( the_el323660082_state @ ( insert2134838167_state @ X_23 @ bot_bo70021908tate_o ) )
= X_23 ) ).
thf(fact_348_the__elem__eq,axiom,
! [X_23: hoare_1775062406iple_a] :
( ( the_el1844711461iple_a @ ( insert1281456128iple_a @ X_23 @ bot_bo751897185le_a_o ) )
= X_23 ) ).
thf(fact_349_the__elem__eq,axiom,
! [X_23: pname] :
( ( the_elem_pname @ ( insert_pname @ X_23 @ bot_bot_pname_o ) )
= X_23 ) ).
thf(fact_350_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_351_nat_Osimps_I2_J,axiom,
! [Nat_1: nat] :
( zero_zero_nat
!= ( suc @ Nat_1 ) ) ).
thf(fact_352_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_353_nat_Osimps_I3_J,axiom,
! [Nat_3: nat] :
( ( suc @ Nat_3 )
!= zero_zero_nat ) ).
thf(fact_354_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_355_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_356_bot__fun__def,axiom,
! [X: pname] :
( ( bot_bot_pname_o @ X )
<=> bot_bot_o ) ).
thf(fact_357_bot__fun__def,axiom,
! [X: hoare_1775062406iple_a] :
( ( bot_bo751897185le_a_o @ X )
<=> bot_bot_o ) ).
thf(fact_358_bot__fun__def,axiom,
! [X: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X )
<=> bot_bot_o ) ).
thf(fact_359_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
thf(fact_360_Suc__inject,axiom,
! [X_1: nat,Y: nat] :
( ( ( suc @ X_1 )
= ( suc @ Y ) )
=> ( X_1 = Y ) ) ).
thf(fact_361_nat_Oinject,axiom,
! [Nat_2: nat,Nat_1: nat] :
( ( ( suc @ Nat_2 )
= ( suc @ Nat_1 ) )
<=> ( Nat_2 = Nat_1 ) ) ).
thf(fact_362_Suc__n__not__n,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
!= N_1 ) ).
thf(fact_363_n__not__Suc__n,axiom,
! [N_1: nat] :
( N_1
!= ( suc @ N_1 ) ) ).
thf(fact_364_bot__apply,axiom,
! [X_22: pname] :
( ( bot_bot_pname_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_365_bot__apply,axiom,
! [X_22: hoare_1775062406iple_a] :
( ( bot_bo751897185le_a_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_366_bot__apply,axiom,
! [X_22: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_367_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
thf(fact_368_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_369_nat__induct,axiom,
! [N_1: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N_1 ) ) ) ).
thf(fact_370_not0__implies__Suc,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
=> ? [M_1: nat] :
( N_1
= ( suc @ M_1 ) ) ) ).
thf(fact_371_evaln_OBody,axiom,
! [Pn_1: pname,S0: state,N_1: nat,S1: state] :
( ( evaln @ ( the_com @ ( body_1 @ Pn_1 ) ) @ S0 @ N_1 @ S1 )
=> ( evaln @ ( body @ Pn_1 ) @ S0 @ ( suc @ N_1 ) @ S1 ) ) ).
thf(fact_372_hoare__derivs_OSkip,axiom,
! [G_6: hoare_1167836817_state > $o,P_14: state > state > $o] : ( hoare_123228589_state @ G_6 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_14 @ skip @ P_14 ) @ bot_bo70021908tate_o ) ) ).
thf(fact_373_hoare__derivs_OSkip,axiom,
! [G_6: hoare_1775062406iple_a > $o,P_14: x_a > state > $o] : ( hoare_1508237396rivs_a @ G_6 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_14 @ skip @ P_14 ) @ bot_bo751897185le_a_o ) ) ).
thf(fact_374_evaln__elim__cases_I1_J,axiom,
! [S: state,N_1: nat,T: state] :
( ( evaln @ skip @ S @ N_1 @ T )
=> ( T = S ) ) ).
thf(fact_375_evaln_OSkip,axiom,
! [S: state,N_1: nat] : ( evaln @ skip @ S @ N_1 @ S ) ).
thf(fact_376_evalc_OSkip,axiom,
! [S: state] : ( evalc @ skip @ S @ S ) ).
thf(fact_377_evalc__elim__cases_I1_J,axiom,
! [S: state,T: state] :
( ( evalc @ skip @ S @ T )
=> ( T = S ) ) ).
thf(fact_378_evaln__Suc,axiom,
! [C_19: com,S: state,N_1: nat,S_5: state] :
( ( evaln @ C_19 @ S @ N_1 @ S_5 )
=> ( evaln @ C_19 @ S @ ( suc @ N_1 ) @ S_5 ) ) ).
thf(fact_379_eval__eq,axiom,
! [C_19: com,S: state,T: state] :
( ( evalc @ C_19 @ S @ T )
<=> ? [N: nat] : ( evaln @ C_19 @ S @ N @ T ) ) ).
thf(fact_380_evaln__evalc,axiom,
! [C_19: com,S: state,N_1: nat,T: state] :
( ( evaln @ C_19 @ S @ N_1 @ T )
=> ( evalc @ C_19 @ S @ T ) ) ).
thf(fact_381_com_Osimps_I19_J,axiom,
! [Pname_1: pname] :
( ( body @ Pname_1 )
!= skip ) ).
thf(fact_382_com_Osimps_I18_J,axiom,
! [Pname_1: pname] :
( skip
!= ( body @ Pname_1 ) ) ).
thf(fact_383_triple__valid__def2,axiom,
! [N_4: nat,P_13: state > state > $o,C_22: com,Q_6: state > state > $o] :
( ( hoare_56934129_state @ N_4 @ ( hoare_908217195_state @ P_13 @ C_22 @ Q_6 ) )
<=> ! [Z_8: state,S_3: state] :
( ( P_13 @ Z_8 @ S_3 )
=> ! [S_4: state] :
( ( evaln @ C_22 @ S_3 @ N_4 @ S_4 )
=> ( Q_6 @ Z_8 @ S_4 ) ) ) ) ).
thf(fact_384_triple__valid__def2,axiom,
! [N_4: nat,P_13: x_a > state > $o,C_22: com,Q_6: x_a > state > $o] :
( ( hoare_1462269968alid_a @ N_4 @ ( hoare_1766022166iple_a @ P_13 @ C_22 @ Q_6 ) )
<=> ! [Z_8: x_a,S_3: state] :
( ( P_13 @ Z_8 @ S_3 )
=> ! [S_4: state] :
( ( evaln @ C_22 @ S_3 @ N_4 @ S_4 )
=> ( Q_6 @ Z_8 @ S_4 ) ) ) ) ).
thf(fact_385_evaln__elim__cases_I6_J,axiom,
! [P: pname,S: state,N_1: nat,S1: state] :
( ( evaln @ ( body @ P ) @ S @ N_1 @ S1 )
=> ~ ! [N: nat] :
( ( N_1
= ( suc @ N ) )
=> ~ ( evaln @ ( the_com @ ( body_1 @ P ) ) @ S @ N @ S1 ) ) ) ).
thf(fact_386_evalc__evaln,axiom,
! [C_19: com,S: state,T: state] :
( ( evalc @ C_19 @ S @ T )
=> ? [N: nat] : ( evaln @ C_19 @ S @ N @ T ) ) ).
thf(fact_387_LoopF,axiom,
! [G_5: hoare_1167836817_state > $o,P_12: state > state > $o,B_36: state > $o,C_21: com] :
( hoare_123228589_state @ G_5
@ ( insert2134838167_state
@ ( hoare_908217195_state
@ ^ [Z_8: state,S_3: state] : ( (&) @ ( P_12 @ Z_8 @ S_3 ) @ ( (~) @ ( B_36 @ S_3 ) ) )
@ ( while @ B_36 @ C_21 )
@ P_12 )
@ bot_bo70021908tate_o ) ) ).
thf(fact_388_LoopF,axiom,
! [G_5: hoare_1775062406iple_a > $o,P_12: x_a > state > $o,B_36: state > $o,C_21: com] :
( hoare_1508237396rivs_a @ G_5
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Z_8: x_a,S_3: state] : ( (&) @ ( P_12 @ Z_8 @ S_3 ) @ ( (~) @ ( B_36 @ S_3 ) ) )
@ ( while @ B_36 @ C_21 )
@ P_12 )
@ bot_bo751897185le_a_o ) ) ).
thf(fact_389_Comp,axiom,
! [D: com,R_1: state > state > $o,G_4: hoare_1167836817_state > $o,P_11: state > state > $o,C_20: com,Q_5: state > state > $o] :
( ( hoare_123228589_state @ G_4 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_11 @ C_20 @ Q_5 ) @ bot_bo70021908tate_o ) )
=> ( ( hoare_123228589_state @ G_4 @ ( insert2134838167_state @ ( hoare_908217195_state @ Q_5 @ D @ R_1 ) @ bot_bo70021908tate_o ) )
=> ( hoare_123228589_state @ G_4 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_11 @ ( semi @ C_20 @ D ) @ R_1 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_390_Comp,axiom,
! [D: com,R_1: x_a > state > $o,G_4: hoare_1775062406iple_a > $o,P_11: x_a > state > $o,C_20: com,Q_5: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_4 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_11 @ C_20 @ Q_5 ) @ bot_bo751897185le_a_o ) )
=> ( ( hoare_1508237396rivs_a @ G_4 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ Q_5 @ D @ R_1 ) @ bot_bo751897185le_a_o ) )
=> ( hoare_1508237396rivs_a @ G_4 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_11 @ ( semi @ C_20 @ D ) @ R_1 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_391_the__elem__def,axiom,
! [X_21: hoare_1167836817_state > $o] :
( ( the_el323660082_state @ X_21 )
= ( the_Ho310147232_state
@ ^ [X: hoare_1167836817_state] :
( X_21
= ( insert2134838167_state @ X @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_392_the__elem__def,axiom,
! [X_21: hoare_1775062406iple_a > $o] :
( ( the_el1844711461iple_a @ X_21 )
= ( the_Ho1155011127iple_a
@ ^ [X: hoare_1775062406iple_a] :
( X_21
= ( insert1281456128iple_a @ X @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_393_the__elem__def,axiom,
! [X_21: pname > $o] :
( ( the_elem_pname @ X_21 )
= ( the_pname
@ ^ [X: pname] :
( X_21
= ( insert_pname @ X @ bot_bot_pname_o ) ) ) ) ).
thf(fact_394_finite__pointwise,axiom,
! [P_9: pname > state > state > $o,Q_4: pname > state > state > $o,G_3: hoare_1167836817_state > $o,P_8: pname > state > state > $o,C0_1: pname > com,Q_3: pname > state > state > $o,U: pname > $o] :
( ( finite_finite_pname @ U )
=> ( ! [P_10: pname] :
( ( hoare_123228589_state @ G_3 @ ( insert2134838167_state @ ( hoare_908217195_state @ ( P_8 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_3 @ P_10 ) ) @ bot_bo70021908tate_o ) )
=> ( hoare_123228589_state @ G_3 @ ( insert2134838167_state @ ( hoare_908217195_state @ ( P_9 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_4 @ P_10 ) ) @ bot_bo70021908tate_o ) ) )
=> ( ( hoare_123228589_state @ G_3
@ ( image_575578384_state
@ ^ [P_10: pname] : ( hoare_908217195_state @ ( P_8 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_3 @ P_10 ) )
@ U ) )
=> ( hoare_123228589_state @ G_3
@ ( image_575578384_state
@ ^ [P_10: pname] : ( hoare_908217195_state @ ( P_9 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_4 @ P_10 ) )
@ U ) ) ) ) ) ).
thf(fact_395_finite__pointwise,axiom,
! [P_9: pname > x_a > state > $o,Q_4: pname > x_a > state > $o,G_3: hoare_1775062406iple_a > $o,P_8: pname > x_a > state > $o,C0_1: pname > com,Q_3: pname > x_a > state > $o,U: pname > $o] :
( ( finite_finite_pname @ U )
=> ( ! [P_10: pname] :
( ( hoare_1508237396rivs_a @ G_3 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ ( P_8 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_3 @ P_10 ) ) @ bot_bo751897185le_a_o ) )
=> ( hoare_1508237396rivs_a @ G_3 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ ( P_9 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_4 @ P_10 ) ) @ bot_bo751897185le_a_o ) ) )
=> ( ( hoare_1508237396rivs_a @ G_3
@ ( image_2063119815iple_a
@ ^ [P_10: pname] : ( hoare_1766022166iple_a @ ( P_8 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_3 @ P_10 ) )
@ U ) )
=> ( hoare_1508237396rivs_a @ G_3
@ ( image_2063119815iple_a
@ ^ [P_10: pname] : ( hoare_1766022166iple_a @ ( P_9 @ P_10 ) @ ( C0_1 @ P_10 ) @ ( Q_4 @ P_10 ) )
@ U ) ) ) ) ) ).
thf(fact_396_evaln_OWhileFalse,axiom,
! [C_19: com,N_1: nat,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( evaln @ ( while @ B @ C_19 ) @ S @ N_1 @ S ) ) ).
thf(fact_397_evaln_OWhileTrue,axiom,
! [S2: state,C_19: com,N_1: nat,S1: state,B: state > $o,S0: state] :
( ( B @ S0 )
=> ( ( evaln @ C_19 @ S0 @ N_1 @ S1 )
=> ( ( evaln @ ( while @ B @ C_19 ) @ S1 @ N_1 @ S2 )
=> ( evaln @ ( while @ B @ C_19 ) @ S0 @ N_1 @ S2 ) ) ) ) ).
thf(fact_398_evalc_OWhileTrue,axiom,
! [S2: state,C_19: com,S1: state,B: state > $o,S0: state] :
( ( B @ S0 )
=> ( ( evalc @ C_19 @ S0 @ S1 )
=> ( ( evalc @ ( while @ B @ C_19 ) @ S1 @ S2 )
=> ( evalc @ ( while @ B @ C_19 ) @ S0 @ S2 ) ) ) ) ).
thf(fact_399_evalc_OWhileFalse,axiom,
! [C_19: com,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( evalc @ ( while @ B @ C_19 ) @ S @ S ) ) ).
thf(fact_400_evaln_OSemi,axiom,
! [C1: com,S2: state,C0: com,S0: state,N_1: nat,S1: state] :
( ( evaln @ C0 @ S0 @ N_1 @ S1 )
=> ( ( evaln @ C1 @ S1 @ N_1 @ S2 )
=> ( evaln @ ( semi @ C0 @ C1 ) @ S0 @ N_1 @ S2 ) ) ) ).
thf(fact_401_evalc_OSemi,axiom,
! [C1: com,S2: state,C0: com,S0: state,S1: state] :
( ( evalc @ C0 @ S0 @ S1 )
=> ( ( evalc @ C1 @ S1 @ S2 )
=> ( evalc @ ( semi @ C0 @ C1 ) @ S0 @ S2 ) ) ) ).
thf(fact_402_com_Osimps_I46_J,axiom,
! [Com1: com,Com2: com,Fun_1: state > $o,Com_1: com] :
( ( semi @ Com1 @ Com2 )
!= ( while @ Fun_1 @ Com_1 ) ) ).
thf(fact_403_com_Osimps_I47_J,axiom,
! [Fun_1: state > $o,Com_1: com,Com1: com,Com2: com] :
( ( while @ Fun_1 @ Com_1 )
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_404_com_Osimps_I3_J,axiom,
! [Com1: com,Com2: com,Com1_1: com,Com2_1: com] :
( ( ( semi @ Com1 @ Com2 )
= ( semi @ Com1_1 @ Com2_1 ) )
<=> ( ( Com1 = Com1_1 )
& ( Com2 = Com2_1 ) ) ) ).
thf(fact_405_com_Osimps_I5_J,axiom,
! [Fun: state > $o,Com: com,Fun_1: state > $o,Com_1: com] :
( ( ( while @ Fun @ Com )
= ( while @ Fun_1 @ Com_1 ) )
<=> ( ( Fun = Fun_1 )
& ( Com = Com_1 ) ) ) ).
thf(fact_406_com_Osimps_I59_J,axiom,
! [Pname_1: pname,Fun: state > $o,Com: com] :
( ( body @ Pname_1 )
!= ( while @ Fun @ Com ) ) ).
thf(fact_407_com_Osimps_I58_J,axiom,
! [Fun: state > $o,Com: com,Pname_1: pname] :
( ( while @ Fun @ Com )
!= ( body @ Pname_1 ) ) ).
thf(fact_408_com_Osimps_I16_J,axiom,
! [Fun_1: state > $o,Com_1: com] :
( skip
!= ( while @ Fun_1 @ Com_1 ) ) ).
thf(fact_409_com_Osimps_I17_J,axiom,
! [Fun_1: state > $o,Com_1: com] :
( ( while @ Fun_1 @ Com_1 )
!= skip ) ).
thf(fact_410_com_Osimps_I49_J,axiom,
! [Pname_1: pname,Com1: com,Com2: com] :
( ( body @ Pname_1 )
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_411_com_Osimps_I48_J,axiom,
! [Com1: com,Com2: com,Pname_1: pname] :
( ( semi @ Com1 @ Com2 )
!= ( body @ Pname_1 ) ) ).
thf(fact_412_com_Osimps_I12_J,axiom,
! [Com1_1: com,Com2_1: com] :
( skip
!= ( semi @ Com1_1 @ Com2_1 ) ) ).
thf(fact_413_com_Osimps_I13_J,axiom,
! [Com1_1: com,Com2_1: com] :
( ( semi @ Com1_1 @ Com2_1 )
!= skip ) ).
thf(fact_414_evalc__elim__cases_I4_J,axiom,
! [C1: com,C2: com,S: state,T: state] :
( ( evalc @ ( semi @ C1 @ C2 ) @ S @ T )
=> ~ ! [S1_1: state] :
( ( evalc @ C1 @ S @ S1_1 )
=> ~ ( evalc @ C2 @ S1_1 @ T ) ) ) ).
thf(fact_415_evaln__elim__cases_I4_J,axiom,
! [C1: com,C2: com,S: state,N_1: nat,T: state] :
( ( evaln @ ( semi @ C1 @ C2 ) @ S @ N_1 @ T )
=> ~ ! [S1_1: state] :
( ( evaln @ C1 @ S @ N_1 @ S1_1 )
=> ~ ( evaln @ C2 @ S1_1 @ N_1 @ T ) ) ) ).
thf(fact_416_finite__imageI,axiom,
! [H_2: pname > hoare_1167836817_state,F_26: pname > $o] :
( ( finite_finite_pname @ F_26 )
=> ( finite1084549118_state @ ( image_575578384_state @ H_2 @ F_26 ) ) ) ).
thf(fact_417_finite__imageI,axiom,
! [H_2: pname > hoare_1775062406iple_a,F_26: pname > $o] :
( ( finite_finite_pname @ F_26 )
=> ( finite2063573081iple_a @ ( image_2063119815iple_a @ H_2 @ F_26 ) ) ) ).
thf(fact_418_finite_OinsertI,axiom,
! [A_62: hoare_1167836817_state,A_61: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_61 )
=> ( finite1084549118_state @ ( insert2134838167_state @ A_62 @ A_61 ) ) ) ).
thf(fact_419_finite_OinsertI,axiom,
! [A_62: hoare_1775062406iple_a,A_61: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_61 )
=> ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_62 @ A_61 ) ) ) ).
thf(fact_420_finite_OinsertI,axiom,
! [A_62: pname,A_61: pname > $o] :
( ( finite_finite_pname @ A_61 )
=> ( finite_finite_pname @ ( insert_pname @ A_62 @ A_61 ) ) ) ).
thf(fact_421_finite_OemptyI,axiom,
finite_finite_pname @ bot_bot_pname_o ).
thf(fact_422_finite_OemptyI,axiom,
finite2063573081iple_a @ bot_bo751897185le_a_o ).
thf(fact_423_finite_OemptyI,axiom,
finite1084549118_state @ bot_bo70021908tate_o ).
thf(fact_424_finite__Collect__conjI,axiom,
! [Q_2: hoare_1775062406iple_a > $o,P_7: hoare_1775062406iple_a > $o] :
( ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_7 ) )
| ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_2 ) ) )
=> ( finite2063573081iple_a
@ ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (&) @ ( P_7 @ X ) @ ( Q_2 @ X ) ) ) ) ) ).
thf(fact_425_finite__Collect__conjI,axiom,
! [Q_2: pname > $o,P_7: pname > $o] :
( ( ( finite_finite_pname @ ( collect_pname @ P_7 ) )
| ( finite_finite_pname @ ( collect_pname @ Q_2 ) ) )
=> ( finite_finite_pname
@ ( collect_pname
@ ^ [X: pname] : ( (&) @ ( P_7 @ X ) @ ( Q_2 @ X ) ) ) ) ) ).
thf(fact_426_finite__Collect__disjI,axiom,
! [P_6: hoare_1775062406iple_a > $o,Q_1: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a
@ ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (|) @ ( P_6 @ X ) @ ( Q_1 @ X ) ) ) )
<=> ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_6 ) )
& ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_1 ) ) ) ) ).
thf(fact_427_finite__Collect__disjI,axiom,
! [P_6: pname > $o,Q_1: pname > $o] :
( ( finite_finite_pname
@ ( collect_pname
@ ^ [X: pname] : ( (|) @ ( P_6 @ X ) @ ( Q_1 @ X ) ) ) )
<=> ( ( finite_finite_pname @ ( collect_pname @ P_6 ) )
& ( finite_finite_pname @ ( collect_pname @ Q_1 ) ) ) ) ).
thf(fact_428_finite__insert,axiom,
! [A_60: hoare_1167836817_state,A_59: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ ( insert2134838167_state @ A_60 @ A_59 ) )
<=> ( finite1084549118_state @ A_59 ) ) ).
thf(fact_429_finite__insert,axiom,
! [A_60: hoare_1775062406iple_a,A_59: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_60 @ A_59 ) )
<=> ( finite2063573081iple_a @ A_59 ) ) ).
thf(fact_430_finite__insert,axiom,
! [A_60: pname,A_59: pname > $o] :
( ( finite_finite_pname @ ( insert_pname @ A_60 @ A_59 ) )
<=> ( finite_finite_pname @ A_59 ) ) ).
thf(fact_431_finite__Un,axiom,
! [F_25: pname > $o,G_2: pname > $o] :
( ( finite_finite_pname @ ( semila1780557381name_o @ F_25 @ G_2 ) )
<=> ( ( finite_finite_pname @ F_25 )
& ( finite_finite_pname @ G_2 ) ) ) ).
thf(fact_432_finite__Un,axiom,
! [F_25: hoare_1167836817_state > $o,G_2: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ ( semila1172322802tate_o @ F_25 @ G_2 ) )
<=> ( ( finite1084549118_state @ F_25 )
& ( finite1084549118_state @ G_2 ) ) ) ).
thf(fact_433_finite__Un,axiom,
! [F_25: hoare_1775062406iple_a > $o,G_2: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ ( semila13410563le_a_o @ F_25 @ G_2 ) )
<=> ( ( finite2063573081iple_a @ F_25 )
& ( finite2063573081iple_a @ G_2 ) ) ) ).
thf(fact_434_finite__UnI,axiom,
! [G_1: pname > $o,F_24: pname > $o] :
( ( finite_finite_pname @ F_24 )
=> ( ( finite_finite_pname @ G_1 )
=> ( finite_finite_pname @ ( semila1780557381name_o @ F_24 @ G_1 ) ) ) ) ).
thf(fact_435_finite__UnI,axiom,
! [G_1: hoare_1167836817_state > $o,F_24: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_24 )
=> ( ( finite1084549118_state @ G_1 )
=> ( finite1084549118_state @ ( semila1172322802tate_o @ F_24 @ G_1 ) ) ) ) ).
thf(fact_436_finite__UnI,axiom,
! [G_1: hoare_1775062406iple_a > $o,F_24: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_24 )
=> ( ( finite2063573081iple_a @ G_1 )
=> ( finite2063573081iple_a @ ( semila13410563le_a_o @ F_24 @ G_1 ) ) ) ) ).
thf(fact_437_finite_Osimps,axiom,
! [A_57: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_57 )
<=> ( ( A_57 = bot_bo70021908tate_o )
| ? [A_58: hoare_1167836817_state > $o,A_56: hoare_1167836817_state] :
( ( A_57
= ( insert2134838167_state @ A_56 @ A_58 ) )
& ( finite1084549118_state @ A_58 ) ) ) ) ).
thf(fact_438_finite_Osimps,axiom,
! [A_57: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_57 )
<=> ( ( A_57 = bot_bo751897185le_a_o )
| ? [A_58: hoare_1775062406iple_a > $o,A_56: hoare_1775062406iple_a] :
( ( A_57
= ( insert1281456128iple_a @ A_56 @ A_58 ) )
& ( finite2063573081iple_a @ A_58 ) ) ) ) ).
thf(fact_439_finite_Osimps,axiom,
! [A_57: pname > $o] :
( ( finite_finite_pname @ A_57 )
<=> ( ( A_57 = bot_bot_pname_o )
| ? [A_58: pname > $o,A_56: pname] :
( ( A_57
= ( insert_pname @ A_56 @ A_58 ) )
& ( finite_finite_pname @ A_58 ) ) ) ) ).
thf(fact_440_finite__induct,axiom,
! [P_5: ( hoare_1167836817_state > $o ) > $o,F_23: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_23 )
=> ( ( P_5 @ bot_bo70021908tate_o )
=> ( ! [X: hoare_1167836817_state,F_16: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_16 )
=> ( ~ ( member2058392318_state @ X @ F_16 )
=> ( ( P_5 @ F_16 )
=> ( P_5 @ ( insert2134838167_state @ X @ F_16 ) ) ) ) )
=> ( P_5 @ F_23 ) ) ) ) ).
thf(fact_441_finite__induct,axiom,
! [P_5: ( hoare_1775062406iple_a > $o ) > $o,F_23: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_23 )
=> ( ( P_5 @ bot_bo751897185le_a_o )
=> ( ! [X: hoare_1775062406iple_a,F_16: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_16 )
=> ( ~ ( member2122167641iple_a @ X @ F_16 )
=> ( ( P_5 @ F_16 )
=> ( P_5 @ ( insert1281456128iple_a @ X @ F_16 ) ) ) ) )
=> ( P_5 @ F_23 ) ) ) ) ).
thf(fact_442_finite__induct,axiom,
! [P_5: ( pname > $o ) > $o,F_23: pname > $o] :
( ( finite_finite_pname @ F_23 )
=> ( ( P_5 @ bot_bot_pname_o )
=> ( ! [X: pname,F_16: pname > $o] :
( ( finite_finite_pname @ F_16 )
=> ( ~ ( member_pname @ X @ F_16 )
=> ( ( P_5 @ F_16 )
=> ( P_5 @ ( insert_pname @ X @ F_16 ) ) ) ) )
=> ( P_5 @ F_23 ) ) ) ) ).
thf(fact_443_pigeonhole__infinite,axiom,
! [F_22: pname > hoare_1167836817_state,A_55: pname > $o] :
( ~ ( finite_finite_pname @ A_55 )
=> ( ( finite1084549118_state @ ( image_575578384_state @ F_22 @ A_55 ) )
=> ? [X: pname] :
( ( member_pname @ X @ A_55 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_56: pname] :
( (&) @ ( member_pname @ A_56 @ A_55 )
@ ( ( F_22 @ A_56 )
= ( F_22 @ X ) ) ) ) ) ) ) ) ).
thf(fact_444_pigeonhole__infinite,axiom,
! [F_22: pname > hoare_1775062406iple_a,A_55: pname > $o] :
( ~ ( finite_finite_pname @ A_55 )
=> ( ( finite2063573081iple_a @ ( image_2063119815iple_a @ F_22 @ A_55 ) )
=> ? [X: pname] :
( ( member_pname @ X @ A_55 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_56: pname] :
( (&) @ ( member_pname @ A_56 @ A_55 )
@ ( ( F_22 @ A_56 )
= ( F_22 @ X ) ) ) ) ) ) ) ) ).
thf(fact_445_evalc__WHILE__case,axiom,
! [B: state > $o,C_19: com,S: state,T: state] :
( ( evalc @ ( while @ B @ C_19 ) @ S @ T )
=> ( ( ( T = S )
=> ( B @ S ) )
=> ~ ( ( B @ S )
=> ! [S1_1: state] :
( ( evalc @ C_19 @ S @ S1_1 )
=> ~ ( evalc @ ( while @ B @ C_19 ) @ S1_1 @ T ) ) ) ) ) ).
thf(fact_446_evaln__WHILE__case,axiom,
! [B: state > $o,C_19: com,S: state,N_1: nat,T: state] :
( ( evaln @ ( while @ B @ C_19 ) @ S @ N_1 @ T )
=> ( ( ( T = S )
=> ( B @ S ) )
=> ~ ( ( B @ S )
=> ! [S1_1: state] :
( ( evaln @ C_19 @ S @ N_1 @ S1_1 )
=> ~ ( evaln @ ( while @ B @ C_19 ) @ S1_1 @ N_1 @ T ) ) ) ) ) ).
thf(fact_447_nonempty__iff,axiom,
! [A_54: hoare_1167836817_state > $o] :
( ( A_54 != bot_bo70021908tate_o )
<=> ? [X: hoare_1167836817_state,B_34: hoare_1167836817_state > $o] :
( ( A_54
= ( insert2134838167_state @ X @ B_34 ) )
& ~ ( member2058392318_state @ X @ B_34 ) ) ) ).
thf(fact_448_nonempty__iff,axiom,
! [A_54: hoare_1775062406iple_a > $o] :
( ( A_54 != bot_bo751897185le_a_o )
<=> ? [X: hoare_1775062406iple_a,B_34: hoare_1775062406iple_a > $o] :
( ( A_54
= ( insert1281456128iple_a @ X @ B_34 ) )
& ~ ( member2122167641iple_a @ X @ B_34 ) ) ) ).
thf(fact_449_nonempty__iff,axiom,
! [A_54: pname > $o] :
( ( A_54 != bot_bot_pname_o )
<=> ? [X: pname,B_34: pname > $o] :
( ( A_54
= ( insert_pname @ X @ B_34 ) )
& ~ ( member_pname @ X @ B_34 ) ) ) ).
thf(fact_450_folding__one__idem_Ounion__idem,axiom,
! [B_35: pname > $o,A_53: pname > $o,F_21: pname > pname > pname,F_20: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_21 @ F_20 )
=> ( ( finite_finite_pname @ A_53 )
=> ( ( A_53 != bot_bot_pname_o )
=> ( ( finite_finite_pname @ B_35 )
=> ( ( B_35 != bot_bot_pname_o )
=> ( ( F_20 @ ( semila1780557381name_o @ A_53 @ B_35 ) )
= ( F_21 @ ( F_20 @ A_53 ) @ ( F_20 @ B_35 ) ) ) ) ) ) ) ) ).
thf(fact_451_folding__one__idem_Ounion__idem,axiom,
! [B_35: hoare_1775062406iple_a > $o,A_53: hoare_1775062406iple_a > $o,F_21: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_20: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_21 @ F_20 )
=> ( ( finite2063573081iple_a @ A_53 )
=> ( ( A_53 != bot_bo751897185le_a_o )
=> ( ( finite2063573081iple_a @ B_35 )
=> ( ( B_35 != bot_bo751897185le_a_o )
=> ( ( F_20 @ ( semila13410563le_a_o @ A_53 @ B_35 ) )
= ( F_21 @ ( F_20 @ A_53 ) @ ( F_20 @ B_35 ) ) ) ) ) ) ) ) ).
thf(fact_452_folding__one__idem_Ounion__idem,axiom,
! [B_35: hoare_1167836817_state > $o,A_53: hoare_1167836817_state > $o,F_21: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_20: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_21 @ F_20 )
=> ( ( finite1084549118_state @ A_53 )
=> ( ( A_53 != bot_bo70021908tate_o )
=> ( ( finite1084549118_state @ B_35 )
=> ( ( B_35 != bot_bo70021908tate_o )
=> ( ( F_20 @ ( semila1172322802tate_o @ A_53 @ B_35 ) )
= ( F_21 @ ( F_20 @ A_53 ) @ ( F_20 @ B_35 ) ) ) ) ) ) ) ) ).
thf(fact_453_folding__one__idem_Oinsert__idem,axiom,
! [X_20: hoare_1167836817_state,A_52: hoare_1167836817_state > $o,F_19: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_18: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_19 @ F_18 )
=> ( ( finite1084549118_state @ A_52 )
=> ( ( A_52 != bot_bo70021908tate_o )
=> ( ( F_18 @ ( insert2134838167_state @ X_20 @ A_52 ) )
= ( F_19 @ X_20 @ ( F_18 @ A_52 ) ) ) ) ) ) ).
thf(fact_454_folding__one__idem_Oinsert__idem,axiom,
! [X_20: hoare_1775062406iple_a,A_52: hoare_1775062406iple_a > $o,F_19: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_18: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_19 @ F_18 )
=> ( ( finite2063573081iple_a @ A_52 )
=> ( ( A_52 != bot_bo751897185le_a_o )
=> ( ( F_18 @ ( insert1281456128iple_a @ X_20 @ A_52 ) )
= ( F_19 @ X_20 @ ( F_18 @ A_52 ) ) ) ) ) ) ).
thf(fact_455_folding__one__idem_Oinsert__idem,axiom,
! [X_20: pname,A_52: pname > $o,F_19: pname > pname > pname,F_18: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_19 @ F_18 )
=> ( ( finite_finite_pname @ A_52 )
=> ( ( A_52 != bot_bot_pname_o )
=> ( ( F_18 @ ( insert_pname @ X_20 @ A_52 ) )
= ( F_19 @ X_20 @ ( F_18 @ A_52 ) ) ) ) ) ) ).
thf(fact_456_image__eq__fold__image,axiom,
! [F_17: pname > hoare_1167836817_state,A_51: pname > $o] :
( ( finite_finite_pname @ A_51 )
=> ( ( image_575578384_state @ F_17 @ A_51 )
= ( finite1068437657_pname @ semila1172322802tate_o
@ ^ [X: pname] : ( insert2134838167_state @ ( F_17 @ X ) @ bot_bo70021908tate_o )
@ bot_bo70021908tate_o
@ A_51 ) ) ) ).
thf(fact_457_image__eq__fold__image,axiom,
! [F_17: pname > hoare_1775062406iple_a,A_51: pname > $o] :
( ( finite_finite_pname @ A_51 )
=> ( ( image_2063119815iple_a @ F_17 @ A_51 )
= ( finite1805141964_pname @ semila13410563le_a_o
@ ^ [X: pname] : ( insert1281456128iple_a @ ( F_17 @ X ) @ bot_bo751897185le_a_o )
@ bot_bo751897185le_a_o
@ A_51 ) ) ) ).
thf(fact_458_finite__ne__induct,axiom,
! [P_4: ( hoare_1167836817_state > $o ) > $o,F_15: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_15 )
=> ( ( F_15 != bot_bo70021908tate_o )
=> ( ! [X: hoare_1167836817_state] : ( P_4 @ ( insert2134838167_state @ X @ bot_bo70021908tate_o ) )
=> ( ! [X: hoare_1167836817_state,F_16: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_16 )
=> ( ( F_16 != bot_bo70021908tate_o )
=> ( ~ ( member2058392318_state @ X @ F_16 )
=> ( ( P_4 @ F_16 )
=> ( P_4 @ ( insert2134838167_state @ X @ F_16 ) ) ) ) ) )
=> ( P_4 @ F_15 ) ) ) ) ) ).
thf(fact_459_finite__ne__induct,axiom,
! [P_4: ( hoare_1775062406iple_a > $o ) > $o,F_15: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_15 )
=> ( ( F_15 != bot_bo751897185le_a_o )
=> ( ! [X: hoare_1775062406iple_a] : ( P_4 @ ( insert1281456128iple_a @ X @ bot_bo751897185le_a_o ) )
=> ( ! [X: hoare_1775062406iple_a,F_16: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_16 )
=> ( ( F_16 != bot_bo751897185le_a_o )
=> ( ~ ( member2122167641iple_a @ X @ F_16 )
=> ( ( P_4 @ F_16 )
=> ( P_4 @ ( insert1281456128iple_a @ X @ F_16 ) ) ) ) ) )
=> ( P_4 @ F_15 ) ) ) ) ) ).
thf(fact_460_finite__ne__induct,axiom,
! [P_4: ( pname > $o ) > $o,F_15: pname > $o] :
( ( finite_finite_pname @ F_15 )
=> ( ( F_15 != bot_bot_pname_o )
=> ( ! [X: pname] : ( P_4 @ ( insert_pname @ X @ bot_bot_pname_o ) )
=> ( ! [X: pname,F_16: pname > $o] :
( ( finite_finite_pname @ F_16 )
=> ( ( F_16 != bot_bot_pname_o )
=> ( ~ ( member_pname @ X @ F_16 )
=> ( ( P_4 @ F_16 )
=> ( P_4 @ ( insert_pname @ X @ F_16 ) ) ) ) ) )
=> ( P_4 @ F_15 ) ) ) ) ) ).
thf(fact_461_folding__one__idem_Oin__idem,axiom,
! [X_19: hoare_1775062406iple_a,A_50: hoare_1775062406iple_a > $o,F_14: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_13: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_14 @ F_13 )
=> ( ( finite2063573081iple_a @ A_50 )
=> ( ( member2122167641iple_a @ X_19 @ A_50 )
=> ( ( F_14 @ X_19 @ ( F_13 @ A_50 ) )
= ( F_13 @ A_50 ) ) ) ) ) ).
thf(fact_462_folding__one__idem_Oin__idem,axiom,
! [X_19: pname,A_50: pname > $o,F_14: pname > pname > pname,F_13: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_14 @ F_13 )
=> ( ( finite_finite_pname @ A_50 )
=> ( ( member_pname @ X_19 @ A_50 )
=> ( ( F_14 @ X_19 @ ( F_13 @ A_50 ) )
= ( F_13 @ A_50 ) ) ) ) ) ).
thf(fact_463_folding__one__idem_Ohom__commute,axiom,
! [N_3: pname > $o,H_1: pname > pname,F_12: pname > pname > pname,F_11: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_12 @ F_11 )
=> ( ! [X: pname,Y_2: pname] :
( ( H_1 @ ( F_12 @ X @ Y_2 ) )
= ( F_12 @ ( H_1 @ X ) @ ( H_1 @ Y_2 ) ) )
=> ( ( finite_finite_pname @ N_3 )
=> ( ( N_3 != bot_bot_pname_o )
=> ( ( H_1 @ ( F_11 @ N_3 ) )
= ( F_11 @ ( image_pname_pname @ H_1 @ N_3 ) ) ) ) ) ) ) ).
thf(fact_464_folding__one__idem_Ohom__commute,axiom,
! [N_3: hoare_1775062406iple_a > $o,H_1: hoare_1775062406iple_a > hoare_1775062406iple_a,F_12: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_11: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_12 @ F_11 )
=> ( ! [X: hoare_1775062406iple_a,Y_2: hoare_1775062406iple_a] :
( ( H_1 @ ( F_12 @ X @ Y_2 ) )
= ( F_12 @ ( H_1 @ X ) @ ( H_1 @ Y_2 ) ) )
=> ( ( finite2063573081iple_a @ N_3 )
=> ( ( N_3 != bot_bo751897185le_a_o )
=> ( ( H_1 @ ( F_11 @ N_3 ) )
= ( F_11 @ ( image_1170193413iple_a @ H_1 @ N_3 ) ) ) ) ) ) ) ).
thf(fact_465_folding__one__idem_Ohom__commute,axiom,
! [N_3: hoare_1167836817_state > $o,H_1: hoare_1167836817_state > hoare_1167836817_state,F_12: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_11: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_12 @ F_11 )
=> ( ! [X: hoare_1167836817_state,Y_2: hoare_1167836817_state] :
( ( H_1 @ ( F_12 @ X @ Y_2 ) )
= ( F_12 @ ( H_1 @ X ) @ ( H_1 @ Y_2 ) ) )
=> ( ( finite1084549118_state @ N_3 )
=> ( ( N_3 != bot_bo70021908tate_o )
=> ( ( H_1 @ ( F_11 @ N_3 ) )
= ( F_11 @ ( image_31595733_state @ H_1 @ N_3 ) ) ) ) ) ) ) ).
thf(fact_466_folding__one_Oinsert,axiom,
! [X_18: hoare_1167836817_state,A_49: hoare_1167836817_state > $o,F_10: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_9: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_10 @ F_9 )
=> ( ( finite1084549118_state @ A_49 )
=> ( ~ ( member2058392318_state @ X_18 @ A_49 )
=> ( ( A_49 != bot_bo70021908tate_o )
=> ( ( F_9 @ ( insert2134838167_state @ X_18 @ A_49 ) )
= ( F_10 @ X_18 @ ( F_9 @ A_49 ) ) ) ) ) ) ) ).
thf(fact_467_folding__one_Oinsert,axiom,
! [X_18: hoare_1775062406iple_a,A_49: hoare_1775062406iple_a > $o,F_10: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_9: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_10 @ F_9 )
=> ( ( finite2063573081iple_a @ A_49 )
=> ( ~ ( member2122167641iple_a @ X_18 @ A_49 )
=> ( ( A_49 != bot_bo751897185le_a_o )
=> ( ( F_9 @ ( insert1281456128iple_a @ X_18 @ A_49 ) )
= ( F_10 @ X_18 @ ( F_9 @ A_49 ) ) ) ) ) ) ) ).
thf(fact_468_folding__one_Oinsert,axiom,
! [X_18: pname,A_49: pname > $o,F_10: pname > pname > pname,F_9: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_10 @ F_9 )
=> ( ( finite_finite_pname @ A_49 )
=> ( ~ ( member_pname @ X_18 @ A_49 )
=> ( ( A_49 != bot_bot_pname_o )
=> ( ( F_9 @ ( insert_pname @ X_18 @ A_49 ) )
= ( F_10 @ X_18 @ ( F_9 @ A_49 ) ) ) ) ) ) ) ).
thf(fact_469_folding__one_Osingleton,axiom,
! [X_17: hoare_1167836817_state,F_8: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_7: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_8 @ F_7 )
=> ( ( F_7 @ ( insert2134838167_state @ X_17 @ bot_bo70021908tate_o ) )
= X_17 ) ) ).
thf(fact_470_folding__one_Osingleton,axiom,
! [X_17: hoare_1775062406iple_a,F_8: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_7: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_8 @ F_7 )
=> ( ( F_7 @ ( insert1281456128iple_a @ X_17 @ bot_bo751897185le_a_o ) )
= X_17 ) ) ).
thf(fact_471_folding__one_Osingleton,axiom,
! [X_17: pname,F_8: pname > pname > pname,F_7: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_8 @ F_7 )
=> ( ( F_7 @ ( insert_pname @ X_17 @ bot_bot_pname_o ) )
= X_17 ) ) ).
thf(fact_472_folding__one_Oclosed,axiom,
! [A_48: hoare_1167836817_state > $o,F_6: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_5: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_6 @ F_5 )
=> ( ( finite1084549118_state @ A_48 )
=> ( ( A_48 != bot_bo70021908tate_o )
=> ( ! [X: hoare_1167836817_state,Y_2: hoare_1167836817_state] : ( member2058392318_state @ ( F_6 @ X @ Y_2 ) @ ( insert2134838167_state @ X @ ( insert2134838167_state @ Y_2 @ bot_bo70021908tate_o ) ) )
=> ( member2058392318_state @ ( F_5 @ A_48 ) @ A_48 ) ) ) ) ) ).
thf(fact_473_folding__one_Oclosed,axiom,
! [A_48: hoare_1775062406iple_a > $o,F_6: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_5: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_6 @ F_5 )
=> ( ( finite2063573081iple_a @ A_48 )
=> ( ( A_48 != bot_bo751897185le_a_o )
=> ( ! [X: hoare_1775062406iple_a,Y_2: hoare_1775062406iple_a] : ( member2122167641iple_a @ ( F_6 @ X @ Y_2 ) @ ( insert1281456128iple_a @ X @ ( insert1281456128iple_a @ Y_2 @ bot_bo751897185le_a_o ) ) )
=> ( member2122167641iple_a @ ( F_5 @ A_48 ) @ A_48 ) ) ) ) ) ).
thf(fact_474_folding__one_Oclosed,axiom,
! [A_48: pname > $o,F_6: pname > pname > pname,F_5: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_6 @ F_5 )
=> ( ( finite_finite_pname @ A_48 )
=> ( ( A_48 != bot_bot_pname_o )
=> ( ! [X: pname,Y_2: pname] : ( member_pname @ ( F_6 @ X @ Y_2 ) @ ( insert_pname @ X @ ( insert_pname @ Y_2 @ bot_bot_pname_o ) ) )
=> ( member_pname @ ( F_5 @ A_48 ) @ A_48 ) ) ) ) ) ).
thf(fact_475_evaln__max2,axiom,
! [C2: com,S2: state,N2: nat,T2: state,C1: com,S1: state,N1: nat,T1: state] :
( ( evaln @ C1 @ S1 @ N1 @ T1 )
=> ( ( evaln @ C2 @ S2 @ N2 @ T2 )
=> ? [N: nat] :
( ( evaln @ C1 @ S1 @ N @ T1 )
& ( evaln @ C2 @ S2 @ N @ T2 ) ) ) ) ).
thf(fact_476_mk__disjoint__insert,axiom,
! [A_47: hoare_1167836817_state,A_46: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_47 @ A_46 )
=> ? [B_34: hoare_1167836817_state > $o] :
( ( A_46
= ( insert2134838167_state @ A_47 @ B_34 ) )
& ~ ( member2058392318_state @ A_47 @ B_34 ) ) ) ).
thf(fact_477_mk__disjoint__insert,axiom,
! [A_47: hoare_1775062406iple_a,A_46: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_47 @ A_46 )
=> ? [B_34: hoare_1775062406iple_a > $o] :
( ( A_46
= ( insert1281456128iple_a @ A_47 @ B_34 ) )
& ~ ( member2122167641iple_a @ A_47 @ B_34 ) ) ) ).
thf(fact_478_mk__disjoint__insert,axiom,
! [A_47: pname,A_46: pname > $o] :
( ( member_pname @ A_47 @ A_46 )
=> ? [B_34: pname > $o] :
( ( A_46
= ( insert_pname @ A_47 @ B_34 ) )
& ~ ( member_pname @ A_47 @ B_34 ) ) ) ).
thf(fact_479_Set_Oset__insert,axiom,
! [X_16: hoare_1167836817_state,A_45: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_16 @ A_45 )
=> ~ ! [B_34: hoare_1167836817_state > $o] :
( ( A_45
= ( insert2134838167_state @ X_16 @ B_34 ) )
=> ( member2058392318_state @ X_16 @ B_34 ) ) ) ).
thf(fact_480_Set_Oset__insert,axiom,
! [X_16: hoare_1775062406iple_a,A_45: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_16 @ A_45 )
=> ~ ! [B_34: hoare_1775062406iple_a > $o] :
( ( A_45
= ( insert1281456128iple_a @ X_16 @ B_34 ) )
=> ( member2122167641iple_a @ X_16 @ B_34 ) ) ) ).
thf(fact_481_Set_Oset__insert,axiom,
! [X_16: pname,A_45: pname > $o] :
( ( member_pname @ X_16 @ A_45 )
=> ~ ! [B_34: pname > $o] :
( ( A_45
= ( insert_pname @ X_16 @ B_34 ) )
=> ( member_pname @ X_16 @ B_34 ) ) ) ).
thf(fact_482_equals0I,axiom,
! [A_44: hoare_1775062406iple_a > $o] :
( ! [Y_2: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ Y_2 @ A_44 )
=> ( A_44 = bot_bo751897185le_a_o ) ) ).
thf(fact_483_equals0I,axiom,
! [A_44: pname > $o] :
( ! [Y_2: pname] :
~ ( member_pname @ Y_2 @ A_44 )
=> ( A_44 = bot_bot_pname_o ) ) ).
thf(fact_484_equals0I,axiom,
! [A_44: hoare_1167836817_state > $o] :
( ! [Y_2: hoare_1167836817_state] :
~ ( member2058392318_state @ Y_2 @ A_44 )
=> ( A_44 = bot_bo70021908tate_o ) ) ).
thf(fact_485_Sup__fin_Ounion__idem,axiom,
! [B_33: $o > $o,A_43: $o > $o] :
( ( finite_finite_o @ A_43 )
=> ( ( A_43 != bot_bot_o_o )
=> ( ( finite_finite_o @ B_33 )
=> ( ( B_33 != bot_bot_o_o )
=> ( ( big_la727467310_fin_o @ ( semila2062604954up_o_o @ A_43 @ B_33 ) )
<=> ( semila10642723_sup_o @ ( big_la727467310_fin_o @ A_43 ) @ ( big_la727467310_fin_o @ B_33 ) ) ) ) ) ) ) ).
thf(fact_486_Sup__fin_Ounion__idem,axiom,
! [B_33: ( pname > $o ) > $o,A_43: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_43 )
=> ( ( A_43 != bot_bot_pname_o_o )
=> ( ( finite297249702name_o @ B_33 )
=> ( ( B_33 != bot_bot_pname_o_o )
=> ( ( big_la1286884090name_o @ ( semila181081674me_o_o @ A_43 @ B_33 ) )
= ( semila1780557381name_o @ ( big_la1286884090name_o @ A_43 ) @ ( big_la1286884090name_o @ B_33 ) ) ) ) ) ) ) ).
thf(fact_487_Sup__fin_Ounion__idem,axiom,
! [B_33: ( hoare_1167836817_state > $o ) > $o,A_43: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_43 )
=> ( ( A_43 != bot_bo691907561te_o_o )
=> ( ( finite1380128977tate_o @ B_33 )
=> ( ( B_33 != bot_bo691907561te_o_o )
=> ( ( big_la1138507389tate_o @ ( semila866907787te_o_o @ A_43 @ B_33 ) )
= ( semila1172322802tate_o @ ( big_la1138507389tate_o @ A_43 ) @ ( big_la1138507389tate_o @ B_33 ) ) ) ) ) ) ) ).
thf(fact_488_Sup__fin_Ounion__idem,axiom,
! [B_33: ( hoare_1775062406iple_a > $o ) > $o,A_43: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_43 )
=> ( ( A_43 != bot_bo1976773294_a_o_o )
=> ( ( finite789576932le_a_o @ B_33 )
=> ( ( B_33 != bot_bo1976773294_a_o_o )
=> ( ( big_la1843772984le_a_o @ ( semila2069193356_a_o_o @ A_43 @ B_33 ) )
= ( semila13410563le_a_o @ ( big_la1843772984le_a_o @ A_43 ) @ ( big_la1843772984le_a_o @ B_33 ) ) ) ) ) ) ) ).
thf(fact_489_Sup__fin_Oin__idem,axiom,
! [X_15: $o,A_42: $o > $o] :
( ( finite_finite_o @ A_42 )
=> ( ( member_o @ X_15 @ A_42 )
=> ( ( semila10642723_sup_o @ X_15 @ ( big_la727467310_fin_o @ A_42 ) )
<=> ( big_la727467310_fin_o @ A_42 ) ) ) ) ).
thf(fact_490_Sup__fin_Oin__idem,axiom,
! [X_15: pname > $o,A_42: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_42 )
=> ( ( member_pname_o @ X_15 @ A_42 )
=> ( ( semila1780557381name_o @ X_15 @ ( big_la1286884090name_o @ A_42 ) )
= ( big_la1286884090name_o @ A_42 ) ) ) ) ).
thf(fact_491_Sup__fin_Oin__idem,axiom,
! [X_15: hoare_1167836817_state > $o,A_42: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_42 )
=> ( ( member864234961tate_o @ X_15 @ A_42 )
=> ( ( semila1172322802tate_o @ X_15 @ ( big_la1138507389tate_o @ A_42 ) )
= ( big_la1138507389tate_o @ A_42 ) ) ) ) ).
thf(fact_492_Sup__fin_Oin__idem,axiom,
! [X_15: hoare_1775062406iple_a > $o,A_42: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_42 )
=> ( ( member1207314404le_a_o @ X_15 @ A_42 )
=> ( ( semila13410563le_a_o @ X_15 @ ( big_la1843772984le_a_o @ A_42 ) )
= ( big_la1843772984le_a_o @ A_42 ) ) ) ) ).
thf(fact_493_Sup__fin_Oinsert,axiom,
! [X_14: $o,A_41: $o > $o] :
( ( finite_finite_o @ A_41 )
=> ( ~ ( member_o @ X_14 @ A_41 )
=> ( ( A_41 != bot_bot_o_o )
=> ( ( big_la727467310_fin_o @ ( insert_o @ X_14 @ A_41 ) )
<=> ( semila10642723_sup_o @ X_14 @ ( big_la727467310_fin_o @ A_41 ) ) ) ) ) ) ).
thf(fact_494_Sup__fin_Oinsert,axiom,
! [X_14: pname > $o,A_41: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_41 )
=> ( ~ ( member_pname_o @ X_14 @ A_41 )
=> ( ( A_41 != bot_bot_pname_o_o )
=> ( ( big_la1286884090name_o @ ( insert_pname_o @ X_14 @ A_41 ) )
= ( semila1780557381name_o @ X_14 @ ( big_la1286884090name_o @ A_41 ) ) ) ) ) ) ).
thf(fact_495_Sup__fin_Oinsert,axiom,
! [X_14: hoare_1167836817_state > $o,A_41: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_41 )
=> ( ~ ( member864234961tate_o @ X_14 @ A_41 )
=> ( ( A_41 != bot_bo691907561te_o_o )
=> ( ( big_la1138507389tate_o @ ( insert999278200tate_o @ X_14 @ A_41 ) )
= ( semila1172322802tate_o @ X_14 @ ( big_la1138507389tate_o @ A_41 ) ) ) ) ) ) ).
thf(fact_496_Sup__fin_Oinsert,axiom,
! [X_14: hoare_1775062406iple_a > $o,A_41: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_41 )
=> ( ~ ( member1207314404le_a_o @ X_14 @ A_41 )
=> ( ( A_41 != bot_bo1976773294_a_o_o )
=> ( ( big_la1843772984le_a_o @ ( insert1210049533le_a_o @ X_14 @ A_41 ) )
= ( semila13410563le_a_o @ X_14 @ ( big_la1843772984le_a_o @ A_41 ) ) ) ) ) ) ).
thf(fact_497_Sup__fin_Oinsert__idem,axiom,
! [X_13: $o,A_40: $o > $o] :
( ( finite_finite_o @ A_40 )
=> ( ( A_40 != bot_bot_o_o )
=> ( ( big_la727467310_fin_o @ ( insert_o @ X_13 @ A_40 ) )
<=> ( semila10642723_sup_o @ X_13 @ ( big_la727467310_fin_o @ A_40 ) ) ) ) ) ).
thf(fact_498_Sup__fin_Oinsert__idem,axiom,
! [X_13: pname > $o,A_40: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_40 )
=> ( ( A_40 != bot_bot_pname_o_o )
=> ( ( big_la1286884090name_o @ ( insert_pname_o @ X_13 @ A_40 ) )
= ( semila1780557381name_o @ X_13 @ ( big_la1286884090name_o @ A_40 ) ) ) ) ) ).
thf(fact_499_Sup__fin_Oinsert__idem,axiom,
! [X_13: hoare_1167836817_state > $o,A_40: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_40 )
=> ( ( A_40 != bot_bo691907561te_o_o )
=> ( ( big_la1138507389tate_o @ ( insert999278200tate_o @ X_13 @ A_40 ) )
= ( semila1172322802tate_o @ X_13 @ ( big_la1138507389tate_o @ A_40 ) ) ) ) ) ).
thf(fact_500_Sup__fin_Oinsert__idem,axiom,
! [X_13: hoare_1775062406iple_a > $o,A_40: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_40 )
=> ( ( A_40 != bot_bo1976773294_a_o_o )
=> ( ( big_la1843772984le_a_o @ ( insert1210049533le_a_o @ X_13 @ A_40 ) )
= ( semila13410563le_a_o @ X_13 @ ( big_la1843772984le_a_o @ A_40 ) ) ) ) ) ).
thf(fact_501_Sup__fin_Ohom__commute,axiom,
! [N_2: $o > $o,H: $o > $o] :
( ! [X: $o,Y_2: $o] :
( ( H @ ( semila10642723_sup_o @ X @ Y_2 ) )
<=> ( semila10642723_sup_o @ ( H @ X ) @ ( H @ Y_2 ) ) )
=> ( ( finite_finite_o @ N_2 )
=> ( ( N_2 != bot_bot_o_o )
=> ( ( H @ ( big_la727467310_fin_o @ N_2 ) )
<=> ( big_la727467310_fin_o @ ( image_o_o @ H @ N_2 ) ) ) ) ) ) ).
thf(fact_502_Sup__fin_Ohom__commute,axiom,
! [N_2: ( pname > $o ) > $o,H: ( pname > $o ) > pname > $o] :
( ! [X: pname > $o,Y_2: pname > $o] :
( ( H @ ( semila1780557381name_o @ X @ Y_2 ) )
= ( semila1780557381name_o @ ( H @ X ) @ ( H @ Y_2 ) ) )
=> ( ( finite297249702name_o @ N_2 )
=> ( ( N_2 != bot_bot_pname_o_o )
=> ( ( H @ ( big_la1286884090name_o @ N_2 ) )
= ( big_la1286884090name_o @ ( image_1085733413name_o @ H @ N_2 ) ) ) ) ) ) ).
thf(fact_503_Sup__fin_Ohom__commute,axiom,
! [N_2: ( hoare_1167836817_state > $o ) > $o,H: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o] :
( ( H @ ( semila1172322802tate_o @ X @ Y_2 ) )
= ( semila1172322802tate_o @ ( H @ X ) @ ( H @ Y_2 ) ) )
=> ( ( finite1380128977tate_o @ N_2 )
=> ( ( N_2 != bot_bo691907561te_o_o )
=> ( ( H @ ( big_la1138507389tate_o @ N_2 ) )
= ( big_la1138507389tate_o @ ( image_1488525317tate_o @ H @ N_2 ) ) ) ) ) ) ).
thf(fact_504_Sup__fin_Ohom__commute,axiom,
! [N_2: ( hoare_1775062406iple_a > $o ) > $o,H: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o] :
( ! [X: hoare_1775062406iple_a > $o,Y_2: hoare_1775062406iple_a > $o] :
( ( H @ ( semila13410563le_a_o @ X @ Y_2 ) )
= ( semila13410563le_a_o @ ( H @ X ) @ ( H @ Y_2 ) ) )
=> ( ( finite789576932le_a_o @ N_2 )
=> ( ( N_2 != bot_bo1976773294_a_o_o )
=> ( ( H @ ( big_la1843772984le_a_o @ N_2 ) )
= ( big_la1843772984le_a_o @ ( image_2014247585le_a_o @ H @ N_2 ) ) ) ) ) ) ).
thf(fact_505_Sup__fin_Oclosed,axiom,
! [A_39: $o > $o] :
( ( finite_finite_o @ A_39 )
=> ( ( A_39 != bot_bot_o_o )
=> ( ! [X: $o,Y_2: $o] : ( member_o @ ( semila10642723_sup_o @ X @ Y_2 ) @ ( insert_o @ X @ ( insert_o @ Y_2 @ bot_bot_o_o ) ) )
=> ( member_o @ ( big_la727467310_fin_o @ A_39 ) @ A_39 ) ) ) ) ).
thf(fact_506_Sup__fin_Oclosed,axiom,
! [A_39: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_39 )
=> ( ( A_39 != bot_bot_pname_o_o )
=> ( ! [X: pname > $o,Y_2: pname > $o] : ( member_pname_o @ ( semila1780557381name_o @ X @ Y_2 ) @ ( insert_pname_o @ X @ ( insert_pname_o @ Y_2 @ bot_bot_pname_o_o ) ) )
=> ( member_pname_o @ ( big_la1286884090name_o @ A_39 ) @ A_39 ) ) ) ) ).
thf(fact_507_Sup__fin_Oclosed,axiom,
! [A_39: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_39 )
=> ( ( A_39 != bot_bo691907561te_o_o )
=> ( ! [X: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o] : ( member864234961tate_o @ ( semila1172322802tate_o @ X @ Y_2 ) @ ( insert999278200tate_o @ X @ ( insert999278200tate_o @ Y_2 @ bot_bo691907561te_o_o ) ) )
=> ( member864234961tate_o @ ( big_la1138507389tate_o @ A_39 ) @ A_39 ) ) ) ) ).
thf(fact_508_Sup__fin_Oclosed,axiom,
! [A_39: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_39 )
=> ( ( A_39 != bot_bo1976773294_a_o_o )
=> ( ! [X: hoare_1775062406iple_a > $o,Y_2: hoare_1775062406iple_a > $o] : ( member1207314404le_a_o @ ( semila13410563le_a_o @ X @ Y_2 ) @ ( insert1210049533le_a_o @ X @ ( insert1210049533le_a_o @ Y_2 @ bot_bo1976773294_a_o_o ) ) )
=> ( member1207314404le_a_o @ ( big_la1843772984le_a_o @ A_39 ) @ A_39 ) ) ) ) ).
thf(fact_509_Sup__fin_Ounion__inter,axiom,
! [B_32: $o > $o,A_38: $o > $o] :
( ( finite_finite_o @ A_38 )
=> ( ( finite_finite_o @ B_32 )
=> ( ( ( semila232696320nf_o_o @ A_38 @ B_32 )
!= bot_bot_o_o )
=> ( ( semila10642723_sup_o @ ( big_la727467310_fin_o @ ( semila2062604954up_o_o @ A_38 @ B_32 ) ) @ ( big_la727467310_fin_o @ ( semila232696320nf_o_o @ A_38 @ B_32 ) ) )
<=> ( semila10642723_sup_o @ ( big_la727467310_fin_o @ A_38 ) @ ( big_la727467310_fin_o @ B_32 ) ) ) ) ) ) ).
thf(fact_510_Sup__fin_Ounion__inter,axiom,
! [B_32: ( pname > $o ) > $o,A_38: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_38 )
=> ( ( finite297249702name_o @ B_32 )
=> ( ( ( semila2013987940me_o_o @ A_38 @ B_32 )
!= bot_bot_pname_o_o )
=> ( ( semila1780557381name_o @ ( big_la1286884090name_o @ ( semila181081674me_o_o @ A_38 @ B_32 ) ) @ ( big_la1286884090name_o @ ( semila2013987940me_o_o @ A_38 @ B_32 ) ) )
= ( semila1780557381name_o @ ( big_la1286884090name_o @ A_38 ) @ ( big_la1286884090name_o @ B_32 ) ) ) ) ) ) ).
thf(fact_511_Sup__fin_Ounion__inter,axiom,
! [B_32: ( hoare_1167836817_state > $o ) > $o,A_38: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_38 )
=> ( ( finite1380128977tate_o @ B_32 )
=> ( ( ( semila1758709489te_o_o @ A_38 @ B_32 )
!= bot_bo691907561te_o_o )
=> ( ( semila1172322802tate_o @ ( big_la1138507389tate_o @ ( semila866907787te_o_o @ A_38 @ B_32 ) ) @ ( big_la1138507389tate_o @ ( semila1758709489te_o_o @ A_38 @ B_32 ) ) )
= ( semila1172322802tate_o @ ( big_la1138507389tate_o @ A_38 ) @ ( big_la1138507389tate_o @ B_32 ) ) ) ) ) ) ).
thf(fact_512_Sup__fin_Ounion__inter,axiom,
! [B_32: ( hoare_1775062406iple_a > $o ) > $o,A_38: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_38 )
=> ( ( finite789576932le_a_o @ B_32 )
=> ( ( ( semila1691990438_a_o_o @ A_38 @ B_32 )
!= bot_bo1976773294_a_o_o )
=> ( ( semila13410563le_a_o @ ( big_la1843772984le_a_o @ ( semila2069193356_a_o_o @ A_38 @ B_32 ) ) @ ( big_la1843772984le_a_o @ ( semila1691990438_a_o_o @ A_38 @ B_32 ) ) )
= ( semila13410563le_a_o @ ( big_la1843772984le_a_o @ A_38 ) @ ( big_la1843772984le_a_o @ B_32 ) ) ) ) ) ) ).
thf(fact_513_Sup__fin_Ounion__disjoint,axiom,
! [B_31: $o > $o,A_37: $o > $o] :
( ( finite_finite_o @ A_37 )
=> ( ( A_37 != bot_bot_o_o )
=> ( ( finite_finite_o @ B_31 )
=> ( ( B_31 != bot_bot_o_o )
=> ( ( ( semila232696320nf_o_o @ A_37 @ B_31 )
= bot_bot_o_o )
=> ( ( big_la727467310_fin_o @ ( semila2062604954up_o_o @ A_37 @ B_31 ) )
<=> ( semila10642723_sup_o @ ( big_la727467310_fin_o @ A_37 ) @ ( big_la727467310_fin_o @ B_31 ) ) ) ) ) ) ) ) ).
thf(fact_514_Sup__fin_Ounion__disjoint,axiom,
! [B_31: ( pname > $o ) > $o,A_37: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_37 )
=> ( ( A_37 != bot_bot_pname_o_o )
=> ( ( finite297249702name_o @ B_31 )
=> ( ( B_31 != bot_bot_pname_o_o )
=> ( ( ( semila2013987940me_o_o @ A_37 @ B_31 )
= bot_bot_pname_o_o )
=> ( ( big_la1286884090name_o @ ( semila181081674me_o_o @ A_37 @ B_31 ) )
= ( semila1780557381name_o @ ( big_la1286884090name_o @ A_37 ) @ ( big_la1286884090name_o @ B_31 ) ) ) ) ) ) ) ) ).
thf(fact_515_Sup__fin_Ounion__disjoint,axiom,
! [B_31: ( hoare_1167836817_state > $o ) > $o,A_37: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_37 )
=> ( ( A_37 != bot_bo691907561te_o_o )
=> ( ( finite1380128977tate_o @ B_31 )
=> ( ( B_31 != bot_bo691907561te_o_o )
=> ( ( ( semila1758709489te_o_o @ A_37 @ B_31 )
= bot_bo691907561te_o_o )
=> ( ( big_la1138507389tate_o @ ( semila866907787te_o_o @ A_37 @ B_31 ) )
= ( semila1172322802tate_o @ ( big_la1138507389tate_o @ A_37 ) @ ( big_la1138507389tate_o @ B_31 ) ) ) ) ) ) ) ) ).
thf(fact_516_Sup__fin_Ounion__disjoint,axiom,
! [B_31: ( hoare_1775062406iple_a > $o ) > $o,A_37: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_37 )
=> ( ( A_37 != bot_bo1976773294_a_o_o )
=> ( ( finite789576932le_a_o @ B_31 )
=> ( ( B_31 != bot_bo1976773294_a_o_o )
=> ( ( ( semila1691990438_a_o_o @ A_37 @ B_31 )
= bot_bo1976773294_a_o_o )
=> ( ( big_la1843772984le_a_o @ ( semila2069193356_a_o_o @ A_37 @ B_31 ) )
= ( semila13410563le_a_o @ ( big_la1843772984le_a_o @ A_37 ) @ ( big_la1843772984le_a_o @ B_31 ) ) ) ) ) ) ) ) ).
thf(fact_517_IntI,axiom,
! [B_30: hoare_1775062406iple_a > $o,C_18: hoare_1775062406iple_a,A_36: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_18 @ A_36 )
=> ( ( member2122167641iple_a @ C_18 @ B_30 )
=> ( member2122167641iple_a @ C_18 @ ( semila966743401le_a_o @ A_36 @ B_30 ) ) ) ) ).
thf(fact_518_IntI,axiom,
! [B_30: pname > $o,C_18: pname,A_36: pname > $o] :
( ( member_pname @ C_18 @ A_36 )
=> ( ( member_pname @ C_18 @ B_30 )
=> ( member_pname @ C_18 @ ( semila1673364395name_o @ A_36 @ B_30 ) ) ) ) ).
thf(fact_519_IntE,axiom,
! [C_17: hoare_1775062406iple_a,A_35: hoare_1775062406iple_a > $o,B_29: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_17 @ ( semila966743401le_a_o @ A_35 @ B_29 ) )
=> ~ ( ( member2122167641iple_a @ C_17 @ A_35 )
=> ~ ( member2122167641iple_a @ C_17 @ B_29 ) ) ) ).
thf(fact_520_IntE,axiom,
! [C_17: pname,A_35: pname > $o,B_29: pname > $o] :
( ( member_pname @ C_17 @ ( semila1673364395name_o @ A_35 @ B_29 ) )
=> ~ ( ( member_pname @ C_17 @ A_35 )
=> ~ ( member_pname @ C_17 @ B_29 ) ) ) ).
thf(fact_521_inf__bot__right,axiom,
! [X_12: pname > $o] :
( ( semila1673364395name_o @ X_12 @ bot_bot_pname_o )
= bot_bot_pname_o ) ).
thf(fact_522_inf__bot__right,axiom,
! [X_12: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ X_12 @ bot_bo751897185le_a_o )
= bot_bo751897185le_a_o ) ).
thf(fact_523_inf__bot__right,axiom,
! [X_12: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ X_12 @ bot_bo70021908tate_o )
= bot_bo70021908tate_o ) ).
thf(fact_524_inf__bot__left,axiom,
! [X_11: pname > $o] :
( ( semila1673364395name_o @ bot_bot_pname_o @ X_11 )
= bot_bot_pname_o ) ).
thf(fact_525_inf__bot__left,axiom,
! [X_11: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ X_11 )
= bot_bo751897185le_a_o ) ).
thf(fact_526_inf__bot__left,axiom,
! [X_11: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ bot_bo70021908tate_o @ X_11 )
= bot_bo70021908tate_o ) ).
thf(fact_527_sup__inf__distrib2,axiom,
! [Y_9: $o,Z_7: $o,X_10: $o] :
( ( semila10642723_sup_o @ ( semila854092349_inf_o @ Y_9 @ Z_7 ) @ X_10 )
<=> ( semila854092349_inf_o @ ( semila10642723_sup_o @ Y_9 @ X_10 ) @ ( semila10642723_sup_o @ Z_7 @ X_10 ) ) ) ).
thf(fact_528_sup__inf__distrib2,axiom,
! [Y_9: pname > $o,Z_7: pname > $o,X_10: pname > $o] :
( ( semila1780557381name_o @ ( semila1673364395name_o @ Y_9 @ Z_7 ) @ X_10 )
= ( semila1673364395name_o @ ( semila1780557381name_o @ Y_9 @ X_10 ) @ ( semila1780557381name_o @ Z_7 @ X_10 ) ) ) ).
thf(fact_529_sup__inf__distrib2,axiom,
! [Y_9: hoare_1167836817_state > $o,Z_7: hoare_1167836817_state > $o,X_10: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( semila179895820tate_o @ Y_9 @ Z_7 ) @ X_10 )
= ( semila179895820tate_o @ ( semila1172322802tate_o @ Y_9 @ X_10 ) @ ( semila1172322802tate_o @ Z_7 @ X_10 ) ) ) ).
thf(fact_530_sup__inf__distrib2,axiom,
! [Y_9: hoare_1775062406iple_a > $o,Z_7: hoare_1775062406iple_a > $o,X_10: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( semila966743401le_a_o @ Y_9 @ Z_7 ) @ X_10 )
= ( semila966743401le_a_o @ ( semila13410563le_a_o @ Y_9 @ X_10 ) @ ( semila13410563le_a_o @ Z_7 @ X_10 ) ) ) ).
thf(fact_531_inf__sup__distrib2,axiom,
! [Y_8: $o,Z_6: $o,X_9: $o] :
( ( semila854092349_inf_o @ ( semila10642723_sup_o @ Y_8 @ Z_6 ) @ X_9 )
<=> ( semila10642723_sup_o @ ( semila854092349_inf_o @ Y_8 @ X_9 ) @ ( semila854092349_inf_o @ Z_6 @ X_9 ) ) ) ).
thf(fact_532_inf__sup__distrib2,axiom,
! [Y_8: pname > $o,Z_6: pname > $o,X_9: pname > $o] :
( ( semila1673364395name_o @ ( semila1780557381name_o @ Y_8 @ Z_6 ) @ X_9 )
= ( semila1780557381name_o @ ( semila1673364395name_o @ Y_8 @ X_9 ) @ ( semila1673364395name_o @ Z_6 @ X_9 ) ) ) ).
thf(fact_533_inf__sup__distrib2,axiom,
! [Y_8: hoare_1167836817_state > $o,Z_6: hoare_1167836817_state > $o,X_9: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ ( semila1172322802tate_o @ Y_8 @ Z_6 ) @ X_9 )
= ( semila1172322802tate_o @ ( semila179895820tate_o @ Y_8 @ X_9 ) @ ( semila179895820tate_o @ Z_6 @ X_9 ) ) ) ).
thf(fact_534_inf__sup__distrib2,axiom,
! [Y_8: hoare_1775062406iple_a > $o,Z_6: hoare_1775062406iple_a > $o,X_9: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ ( semila13410563le_a_o @ Y_8 @ Z_6 ) @ X_9 )
= ( semila13410563le_a_o @ ( semila966743401le_a_o @ Y_8 @ X_9 ) @ ( semila966743401le_a_o @ Z_6 @ X_9 ) ) ) ).
thf(fact_535_sup__inf__distrib1,axiom,
! [X_8: $o,Y_7: $o,Z_5: $o] :
( ( semila10642723_sup_o @ X_8 @ ( semila854092349_inf_o @ Y_7 @ Z_5 ) )
<=> ( semila854092349_inf_o @ ( semila10642723_sup_o @ X_8 @ Y_7 ) @ ( semila10642723_sup_o @ X_8 @ Z_5 ) ) ) ).
thf(fact_536_sup__inf__distrib1,axiom,
! [X_8: pname > $o,Y_7: pname > $o,Z_5: pname > $o] :
( ( semila1780557381name_o @ X_8 @ ( semila1673364395name_o @ Y_7 @ Z_5 ) )
= ( semila1673364395name_o @ ( semila1780557381name_o @ X_8 @ Y_7 ) @ ( semila1780557381name_o @ X_8 @ Z_5 ) ) ) ).
thf(fact_537_sup__inf__distrib1,axiom,
! [X_8: hoare_1167836817_state > $o,Y_7: hoare_1167836817_state > $o,Z_5: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_8 @ ( semila179895820tate_o @ Y_7 @ Z_5 ) )
= ( semila179895820tate_o @ ( semila1172322802tate_o @ X_8 @ Y_7 ) @ ( semila1172322802tate_o @ X_8 @ Z_5 ) ) ) ).
thf(fact_538_sup__inf__distrib1,axiom,
! [X_8: hoare_1775062406iple_a > $o,Y_7: hoare_1775062406iple_a > $o,Z_5: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_8 @ ( semila966743401le_a_o @ Y_7 @ Z_5 ) )
= ( semila966743401le_a_o @ ( semila13410563le_a_o @ X_8 @ Y_7 ) @ ( semila13410563le_a_o @ X_8 @ Z_5 ) ) ) ).
thf(fact_539_inf__sup__distrib1,axiom,
! [X_7: $o,Y_6: $o,Z_4: $o] :
( ( semila854092349_inf_o @ X_7 @ ( semila10642723_sup_o @ Y_6 @ Z_4 ) )
<=> ( semila10642723_sup_o @ ( semila854092349_inf_o @ X_7 @ Y_6 ) @ ( semila854092349_inf_o @ X_7 @ Z_4 ) ) ) ).
thf(fact_540_inf__sup__distrib1,axiom,
! [X_7: pname > $o,Y_6: pname > $o,Z_4: pname > $o] :
( ( semila1673364395name_o @ X_7 @ ( semila1780557381name_o @ Y_6 @ Z_4 ) )
= ( semila1780557381name_o @ ( semila1673364395name_o @ X_7 @ Y_6 ) @ ( semila1673364395name_o @ X_7 @ Z_4 ) ) ) ).
thf(fact_541_inf__sup__distrib1,axiom,
! [X_7: hoare_1167836817_state > $o,Y_6: hoare_1167836817_state > $o,Z_4: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ X_7 @ ( semila1172322802tate_o @ Y_6 @ Z_4 ) )
= ( semila1172322802tate_o @ ( semila179895820tate_o @ X_7 @ Y_6 ) @ ( semila179895820tate_o @ X_7 @ Z_4 ) ) ) ).
thf(fact_542_inf__sup__distrib1,axiom,
! [X_7: hoare_1775062406iple_a > $o,Y_6: hoare_1775062406iple_a > $o,Z_4: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ X_7 @ ( semila13410563le_a_o @ Y_6 @ Z_4 ) )
= ( semila13410563le_a_o @ ( semila966743401le_a_o @ X_7 @ Y_6 ) @ ( semila966743401le_a_o @ X_7 @ Z_4 ) ) ) ).
thf(fact_543_sup__inf__absorb,axiom,
! [X_6: $o,Y_5: $o] :
( ( semila10642723_sup_o @ X_6 @ ( semila854092349_inf_o @ X_6 @ Y_5 ) )
<=> X_6 ) ).
thf(fact_544_sup__inf__absorb,axiom,
! [X_6: pname > $o,Y_5: pname > $o] :
( ( semila1780557381name_o @ X_6 @ ( semila1673364395name_o @ X_6 @ Y_5 ) )
= X_6 ) ).
thf(fact_545_sup__inf__absorb,axiom,
! [X_6: hoare_1167836817_state > $o,Y_5: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_6 @ ( semila179895820tate_o @ X_6 @ Y_5 ) )
= X_6 ) ).
thf(fact_546_sup__inf__absorb,axiom,
! [X_6: hoare_1775062406iple_a > $o,Y_5: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_6 @ ( semila966743401le_a_o @ X_6 @ Y_5 ) )
= X_6 ) ).
thf(fact_547_inf__sup__absorb,axiom,
! [X_5: $o,Y_4: $o] :
( ( semila854092349_inf_o @ X_5 @ ( semila10642723_sup_o @ X_5 @ Y_4 ) )
<=> X_5 ) ).
thf(fact_548_inf__sup__absorb,axiom,
! [X_5: pname > $o,Y_4: pname > $o] :
( ( semila1673364395name_o @ X_5 @ ( semila1780557381name_o @ X_5 @ Y_4 ) )
= X_5 ) ).
thf(fact_549_inf__sup__absorb,axiom,
! [X_5: hoare_1167836817_state > $o,Y_4: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ X_5 @ ( semila1172322802tate_o @ X_5 @ Y_4 ) )
= X_5 ) ).
thf(fact_550_inf__sup__absorb,axiom,
! [X_5: hoare_1775062406iple_a > $o,Y_4: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ X_5 @ ( semila13410563le_a_o @ X_5 @ Y_4 ) )
= X_5 ) ).
thf(fact_551_disjoint__iff__not__equal,axiom,
! [A_34: pname > $o,B_28: pname > $o] :
( ( ( semila1673364395name_o @ A_34 @ B_28 )
= bot_bot_pname_o )
<=> ! [X: pname] :
( ( member_pname @ X @ A_34 )
=> ! [Xa: pname] :
( ( member_pname @ Xa @ B_28 )
=> ( X != Xa ) ) ) ) ).
thf(fact_552_disjoint__iff__not__equal,axiom,
! [A_34: hoare_1775062406iple_a > $o,B_28: hoare_1775062406iple_a > $o] :
( ( ( semila966743401le_a_o @ A_34 @ B_28 )
= bot_bo751897185le_a_o )
<=> ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ A_34 )
=> ! [Xa: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ Xa @ B_28 )
=> ( X != Xa ) ) ) ) ).
thf(fact_553_disjoint__iff__not__equal,axiom,
! [A_34: hoare_1167836817_state > $o,B_28: hoare_1167836817_state > $o] :
( ( ( semila179895820tate_o @ A_34 @ B_28 )
= bot_bo70021908tate_o )
<=> ! [X: hoare_1167836817_state] :
( ( member2058392318_state @ X @ A_34 )
=> ! [Xa: hoare_1167836817_state] :
( ( member2058392318_state @ Xa @ B_28 )
=> ( X != Xa ) ) ) ) ).
thf(fact_554_Int__empty__right,axiom,
! [A_33: pname > $o] :
( ( semila1673364395name_o @ A_33 @ bot_bot_pname_o )
= bot_bot_pname_o ) ).
thf(fact_555_Int__empty__right,axiom,
! [A_33: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ A_33 @ bot_bo751897185le_a_o )
= bot_bo751897185le_a_o ) ).
thf(fact_556_Int__empty__right,axiom,
! [A_33: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ A_33 @ bot_bo70021908tate_o )
= bot_bo70021908tate_o ) ).
thf(fact_557_Int__empty__left,axiom,
! [B_27: pname > $o] :
( ( semila1673364395name_o @ bot_bot_pname_o @ B_27 )
= bot_bot_pname_o ) ).
thf(fact_558_Int__empty__left,axiom,
! [B_27: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ B_27 )
= bot_bo751897185le_a_o ) ).
thf(fact_559_Int__empty__left,axiom,
! [B_27: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ bot_bo70021908tate_o @ B_27 )
= bot_bo70021908tate_o ) ).
thf(fact_560_Int__def,axiom,
! [A_32: hoare_1775062406iple_a > $o,B_26: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ A_32 @ B_26 )
= ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (&) @ ( member2122167641iple_a @ X @ A_32 ) @ ( member2122167641iple_a @ X @ B_26 ) ) ) ) ).
thf(fact_561_Int__def,axiom,
! [A_32: pname > $o,B_26: pname > $o] :
( ( semila1673364395name_o @ A_32 @ B_26 )
= ( collect_pname
@ ^ [X: pname] : ( (&) @ ( member_pname @ X @ A_32 ) @ ( member_pname @ X @ B_26 ) ) ) ) ).
thf(fact_562_Int__iff,axiom,
! [C_16: hoare_1775062406iple_a,A_31: hoare_1775062406iple_a > $o,B_25: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_16 @ ( semila966743401le_a_o @ A_31 @ B_25 ) )
<=> ( ( member2122167641iple_a @ C_16 @ A_31 )
& ( member2122167641iple_a @ C_16 @ B_25 ) ) ) ).
thf(fact_563_Int__iff,axiom,
! [C_16: pname,A_31: pname > $o,B_25: pname > $o] :
( ( member_pname @ C_16 @ ( semila1673364395name_o @ A_31 @ B_25 ) )
<=> ( ( member_pname @ C_16 @ A_31 )
& ( member_pname @ C_16 @ B_25 ) ) ) ).
thf(fact_564_IntD1,axiom,
! [C_15: hoare_1775062406iple_a,A_30: hoare_1775062406iple_a > $o,B_24: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_15 @ ( semila966743401le_a_o @ A_30 @ B_24 ) )
=> ( member2122167641iple_a @ C_15 @ A_30 ) ) ).
thf(fact_565_IntD1,axiom,
! [C_15: pname,A_30: pname > $o,B_24: pname > $o] :
( ( member_pname @ C_15 @ ( semila1673364395name_o @ A_30 @ B_24 ) )
=> ( member_pname @ C_15 @ A_30 ) ) ).
thf(fact_566_IntD2,axiom,
! [C_14: hoare_1775062406iple_a,A_29: hoare_1775062406iple_a > $o,B_23: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_14 @ ( semila966743401le_a_o @ A_29 @ B_23 ) )
=> ( member2122167641iple_a @ C_14 @ B_23 ) ) ).
thf(fact_567_IntD2,axiom,
! [C_14: pname,A_29: pname > $o,B_23: pname > $o] :
( ( member_pname @ C_14 @ ( semila1673364395name_o @ A_29 @ B_23 ) )
=> ( member_pname @ C_14 @ B_23 ) ) ).
thf(fact_568_Collect__conj__eq,axiom,
! [P_3: hoare_1775062406iple_a > $o,Q: hoare_1775062406iple_a > $o] :
( ( collec676402587iple_a
@ ^ [X: hoare_1775062406iple_a] : ( (&) @ ( P_3 @ X ) @ ( Q @ X ) ) )
= ( semila966743401le_a_o @ ( collec676402587iple_a @ P_3 ) @ ( collec676402587iple_a @ Q ) ) ) ).
thf(fact_569_Collect__conj__eq,axiom,
! [P_3: pname > $o,Q: pname > $o] :
( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( P_3 @ X ) @ ( Q @ X ) ) )
= ( semila1673364395name_o @ ( collect_pname @ P_3 ) @ ( collect_pname @ Q ) ) ) ).
thf(fact_570_Int__Collect,axiom,
! [X_4: hoare_1775062406iple_a,A_28: hoare_1775062406iple_a > $o,P_2: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_4 @ ( semila966743401le_a_o @ A_28 @ ( collec676402587iple_a @ P_2 ) ) )
<=> ( ( member2122167641iple_a @ X_4 @ A_28 )
& ( P_2 @ X_4 ) ) ) ).
thf(fact_571_Int__Collect,axiom,
! [X_4: pname,A_28: pname > $o,P_2: pname > $o] :
( ( member_pname @ X_4 @ ( semila1673364395name_o @ A_28 @ ( collect_pname @ P_2 ) ) )
<=> ( ( member_pname @ X_4 @ A_28 )
& ( P_2 @ X_4 ) ) ) ).
thf(fact_572_inf__Int__eq,axiom,
! [R: hoare_1775062406iple_a > $o,S_2: hoare_1775062406iple_a > $o,X: hoare_1775062406iple_a] :
( ( semila966743401le_a_o
@ ^ [Y_2: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_2 @ R )
@ ^ [Y_2: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_2 @ S_2 )
@ X )
<=> ( member2122167641iple_a @ X @ ( semila966743401le_a_o @ R @ S_2 ) ) ) ).
thf(fact_573_inf__Int__eq,axiom,
! [R: pname > $o,S_2: pname > $o,X: pname] :
( ( semila1673364395name_o
@ ^ [Y_2: pname] : ( member_pname @ Y_2 @ R )
@ ^ [Y_2: pname] : ( member_pname @ Y_2 @ S_2 )
@ X )
<=> ( member_pname @ X @ ( semila1673364395name_o @ R @ S_2 ) ) ) ).
thf(fact_574_Un__Int__crazy,axiom,
! [A_27: pname > $o,B_22: pname > $o,C_13: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ ( semila1673364395name_o @ A_27 @ B_22 ) @ ( semila1673364395name_o @ B_22 @ C_13 ) ) @ ( semila1673364395name_o @ C_13 @ A_27 ) )
= ( semila1673364395name_o @ ( semila1673364395name_o @ ( semila1780557381name_o @ A_27 @ B_22 ) @ ( semila1780557381name_o @ B_22 @ C_13 ) ) @ ( semila1780557381name_o @ C_13 @ A_27 ) ) ) ).
thf(fact_575_Un__Int__crazy,axiom,
! [A_27: hoare_1167836817_state > $o,B_22: hoare_1167836817_state > $o,C_13: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( semila1172322802tate_o @ ( semila179895820tate_o @ A_27 @ B_22 ) @ ( semila179895820tate_o @ B_22 @ C_13 ) ) @ ( semila179895820tate_o @ C_13 @ A_27 ) )
= ( semila179895820tate_o @ ( semila179895820tate_o @ ( semila1172322802tate_o @ A_27 @ B_22 ) @ ( semila1172322802tate_o @ B_22 @ C_13 ) ) @ ( semila1172322802tate_o @ C_13 @ A_27 ) ) ) ).
thf(fact_576_Un__Int__crazy,axiom,
! [A_27: hoare_1775062406iple_a > $o,B_22: hoare_1775062406iple_a > $o,C_13: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( semila13410563le_a_o @ ( semila966743401le_a_o @ A_27 @ B_22 ) @ ( semila966743401le_a_o @ B_22 @ C_13 ) ) @ ( semila966743401le_a_o @ C_13 @ A_27 ) )
= ( semila966743401le_a_o @ ( semila966743401le_a_o @ ( semila13410563le_a_o @ A_27 @ B_22 ) @ ( semila13410563le_a_o @ B_22 @ C_13 ) ) @ ( semila13410563le_a_o @ C_13 @ A_27 ) ) ) ).
thf(fact_577_Un__Int__distrib2,axiom,
! [B_21: pname > $o,C_12: pname > $o,A_26: pname > $o] :
( ( semila1780557381name_o @ ( semila1673364395name_o @ B_21 @ C_12 ) @ A_26 )
= ( semila1673364395name_o @ ( semila1780557381name_o @ B_21 @ A_26 ) @ ( semila1780557381name_o @ C_12 @ A_26 ) ) ) ).
thf(fact_578_Un__Int__distrib2,axiom,
! [B_21: hoare_1167836817_state > $o,C_12: hoare_1167836817_state > $o,A_26: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( semila179895820tate_o @ B_21 @ C_12 ) @ A_26 )
= ( semila179895820tate_o @ ( semila1172322802tate_o @ B_21 @ A_26 ) @ ( semila1172322802tate_o @ C_12 @ A_26 ) ) ) ).
thf(fact_579_Un__Int__distrib2,axiom,
! [B_21: hoare_1775062406iple_a > $o,C_12: hoare_1775062406iple_a > $o,A_26: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( semila966743401le_a_o @ B_21 @ C_12 ) @ A_26 )
= ( semila966743401le_a_o @ ( semila13410563le_a_o @ B_21 @ A_26 ) @ ( semila13410563le_a_o @ C_12 @ A_26 ) ) ) ).
thf(fact_580_Int__Un__distrib2,axiom,
! [B_20: pname > $o,C_11: pname > $o,A_25: pname > $o] :
( ( semila1673364395name_o @ ( semila1780557381name_o @ B_20 @ C_11 ) @ A_25 )
= ( semila1780557381name_o @ ( semila1673364395name_o @ B_20 @ A_25 ) @ ( semila1673364395name_o @ C_11 @ A_25 ) ) ) ).
thf(fact_581_Int__Un__distrib2,axiom,
! [B_20: hoare_1167836817_state > $o,C_11: hoare_1167836817_state > $o,A_25: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ ( semila1172322802tate_o @ B_20 @ C_11 ) @ A_25 )
= ( semila1172322802tate_o @ ( semila179895820tate_o @ B_20 @ A_25 ) @ ( semila179895820tate_o @ C_11 @ A_25 ) ) ) ).
thf(fact_582_Int__Un__distrib2,axiom,
! [B_20: hoare_1775062406iple_a > $o,C_11: hoare_1775062406iple_a > $o,A_25: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ ( semila13410563le_a_o @ B_20 @ C_11 ) @ A_25 )
= ( semila13410563le_a_o @ ( semila966743401le_a_o @ B_20 @ A_25 ) @ ( semila966743401le_a_o @ C_11 @ A_25 ) ) ) ).
thf(fact_583_Un__Int__distrib,axiom,
! [A_24: pname > $o,B_19: pname > $o,C_10: pname > $o] :
( ( semila1780557381name_o @ A_24 @ ( semila1673364395name_o @ B_19 @ C_10 ) )
= ( semila1673364395name_o @ ( semila1780557381name_o @ A_24 @ B_19 ) @ ( semila1780557381name_o @ A_24 @ C_10 ) ) ) ).
thf(fact_584_Un__Int__distrib,axiom,
! [A_24: hoare_1167836817_state > $o,B_19: hoare_1167836817_state > $o,C_10: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_24 @ ( semila179895820tate_o @ B_19 @ C_10 ) )
= ( semila179895820tate_o @ ( semila1172322802tate_o @ A_24 @ B_19 ) @ ( semila1172322802tate_o @ A_24 @ C_10 ) ) ) ).
thf(fact_585_Un__Int__distrib,axiom,
! [A_24: hoare_1775062406iple_a > $o,B_19: hoare_1775062406iple_a > $o,C_10: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_24 @ ( semila966743401le_a_o @ B_19 @ C_10 ) )
= ( semila966743401le_a_o @ ( semila13410563le_a_o @ A_24 @ B_19 ) @ ( semila13410563le_a_o @ A_24 @ C_10 ) ) ) ).
thf(fact_586_Int__Un__distrib,axiom,
! [A_23: pname > $o,B_18: pname > $o,C_9: pname > $o] :
( ( semila1673364395name_o @ A_23 @ ( semila1780557381name_o @ B_18 @ C_9 ) )
= ( semila1780557381name_o @ ( semila1673364395name_o @ A_23 @ B_18 ) @ ( semila1673364395name_o @ A_23 @ C_9 ) ) ) ).
thf(fact_587_Int__Un__distrib,axiom,
! [A_23: hoare_1167836817_state > $o,B_18: hoare_1167836817_state > $o,C_9: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ A_23 @ ( semila1172322802tate_o @ B_18 @ C_9 ) )
= ( semila1172322802tate_o @ ( semila179895820tate_o @ A_23 @ B_18 ) @ ( semila179895820tate_o @ A_23 @ C_9 ) ) ) ).
thf(fact_588_Int__Un__distrib,axiom,
! [A_23: hoare_1775062406iple_a > $o,B_18: hoare_1775062406iple_a > $o,C_9: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ A_23 @ ( semila13410563le_a_o @ B_18 @ C_9 ) )
= ( semila13410563le_a_o @ ( semila966743401le_a_o @ A_23 @ B_18 ) @ ( semila966743401le_a_o @ A_23 @ C_9 ) ) ) ).
thf(fact_589_Int__insert__left__if1,axiom,
! [B_17: hoare_1167836817_state > $o,A_22: hoare_1167836817_state,C_8: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_22 @ C_8 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_22 @ B_17 ) @ C_8 )
= ( insert2134838167_state @ A_22 @ ( semila179895820tate_o @ B_17 @ C_8 ) ) ) ) ).
thf(fact_590_Int__insert__left__if1,axiom,
! [B_17: hoare_1775062406iple_a > $o,A_22: hoare_1775062406iple_a,C_8: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_22 @ C_8 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_22 @ B_17 ) @ C_8 )
= ( insert1281456128iple_a @ A_22 @ ( semila966743401le_a_o @ B_17 @ C_8 ) ) ) ) ).
thf(fact_591_Int__insert__left__if1,axiom,
! [B_17: pname > $o,A_22: pname,C_8: pname > $o] :
( ( member_pname @ A_22 @ C_8 )
=> ( ( semila1673364395name_o @ ( insert_pname @ A_22 @ B_17 ) @ C_8 )
= ( insert_pname @ A_22 @ ( semila1673364395name_o @ B_17 @ C_8 ) ) ) ) ).
thf(fact_592_Int__insert__right__if1,axiom,
! [B_16: hoare_1167836817_state > $o,A_21: hoare_1167836817_state,A_20: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_21 @ A_20 )
=> ( ( semila179895820tate_o @ A_20 @ ( insert2134838167_state @ A_21 @ B_16 ) )
= ( insert2134838167_state @ A_21 @ ( semila179895820tate_o @ A_20 @ B_16 ) ) ) ) ).
thf(fact_593_Int__insert__right__if1,axiom,
! [B_16: hoare_1775062406iple_a > $o,A_21: hoare_1775062406iple_a,A_20: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_21 @ A_20 )
=> ( ( semila966743401le_a_o @ A_20 @ ( insert1281456128iple_a @ A_21 @ B_16 ) )
= ( insert1281456128iple_a @ A_21 @ ( semila966743401le_a_o @ A_20 @ B_16 ) ) ) ) ).
thf(fact_594_Int__insert__right__if1,axiom,
! [B_16: pname > $o,A_21: pname,A_20: pname > $o] :
( ( member_pname @ A_21 @ A_20 )
=> ( ( semila1673364395name_o @ A_20 @ ( insert_pname @ A_21 @ B_16 ) )
= ( insert_pname @ A_21 @ ( semila1673364395name_o @ A_20 @ B_16 ) ) ) ) ).
thf(fact_595_Int__insert__left__if0,axiom,
! [B_15: hoare_1167836817_state > $o,A_19: hoare_1167836817_state,C_7: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ A_19 @ C_7 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_19 @ B_15 ) @ C_7 )
= ( semila179895820tate_o @ B_15 @ C_7 ) ) ) ).
thf(fact_596_Int__insert__left__if0,axiom,
! [B_15: hoare_1775062406iple_a > $o,A_19: hoare_1775062406iple_a,C_7: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ A_19 @ C_7 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_19 @ B_15 ) @ C_7 )
= ( semila966743401le_a_o @ B_15 @ C_7 ) ) ) ).
thf(fact_597_Int__insert__left__if0,axiom,
! [B_15: pname > $o,A_19: pname,C_7: pname > $o] :
( ~ ( member_pname @ A_19 @ C_7 )
=> ( ( semila1673364395name_o @ ( insert_pname @ A_19 @ B_15 ) @ C_7 )
= ( semila1673364395name_o @ B_15 @ C_7 ) ) ) ).
thf(fact_598_Int__insert__right__if0,axiom,
! [B_14: hoare_1167836817_state > $o,A_18: hoare_1167836817_state,A_17: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ A_18 @ A_17 )
=> ( ( semila179895820tate_o @ A_17 @ ( insert2134838167_state @ A_18 @ B_14 ) )
= ( semila179895820tate_o @ A_17 @ B_14 ) ) ) ).
thf(fact_599_Int__insert__right__if0,axiom,
! [B_14: hoare_1775062406iple_a > $o,A_18: hoare_1775062406iple_a,A_17: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ A_18 @ A_17 )
=> ( ( semila966743401le_a_o @ A_17 @ ( insert1281456128iple_a @ A_18 @ B_14 ) )
= ( semila966743401le_a_o @ A_17 @ B_14 ) ) ) ).
thf(fact_600_Int__insert__right__if0,axiom,
! [B_14: pname > $o,A_18: pname,A_17: pname > $o] :
( ~ ( member_pname @ A_18 @ A_17 )
=> ( ( semila1673364395name_o @ A_17 @ ( insert_pname @ A_18 @ B_14 ) )
= ( semila1673364395name_o @ A_17 @ B_14 ) ) ) ).
thf(fact_601_insert__inter__insert,axiom,
! [A_16: hoare_1167836817_state,A_15: hoare_1167836817_state > $o,B_13: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ ( insert2134838167_state @ A_16 @ A_15 ) @ ( insert2134838167_state @ A_16 @ B_13 ) )
= ( insert2134838167_state @ A_16 @ ( semila179895820tate_o @ A_15 @ B_13 ) ) ) ).
thf(fact_602_insert__inter__insert,axiom,
! [A_16: hoare_1775062406iple_a,A_15: hoare_1775062406iple_a > $o,B_13: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_16 @ A_15 ) @ ( insert1281456128iple_a @ A_16 @ B_13 ) )
= ( insert1281456128iple_a @ A_16 @ ( semila966743401le_a_o @ A_15 @ B_13 ) ) ) ).
thf(fact_603_insert__inter__insert,axiom,
! [A_16: pname,A_15: pname > $o,B_13: pname > $o] :
( ( semila1673364395name_o @ ( insert_pname @ A_16 @ A_15 ) @ ( insert_pname @ A_16 @ B_13 ) )
= ( insert_pname @ A_16 @ ( semila1673364395name_o @ A_15 @ B_13 ) ) ) ).
thf(fact_604_Int__insert__left,axiom,
! [B_12: hoare_1167836817_state > $o,A_14: hoare_1167836817_state,C_6: hoare_1167836817_state > $o] :
( ( ( member2058392318_state @ A_14 @ C_6 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_14 @ B_12 ) @ C_6 )
= ( insert2134838167_state @ A_14 @ ( semila179895820tate_o @ B_12 @ C_6 ) ) ) )
& ( ~ ( member2058392318_state @ A_14 @ C_6 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_14 @ B_12 ) @ C_6 )
= ( semila179895820tate_o @ B_12 @ C_6 ) ) ) ) ).
thf(fact_605_Int__insert__left,axiom,
! [B_12: hoare_1775062406iple_a > $o,A_14: hoare_1775062406iple_a,C_6: hoare_1775062406iple_a > $o] :
( ( ( member2122167641iple_a @ A_14 @ C_6 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_14 @ B_12 ) @ C_6 )
= ( insert1281456128iple_a @ A_14 @ ( semila966743401le_a_o @ B_12 @ C_6 ) ) ) )
& ( ~ ( member2122167641iple_a @ A_14 @ C_6 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_14 @ B_12 ) @ C_6 )
= ( semila966743401le_a_o @ B_12 @ C_6 ) ) ) ) ).
thf(fact_606_Int__insert__left,axiom,
! [B_12: pname > $o,A_14: pname,C_6: pname > $o] :
( ( ( member_pname @ A_14 @ C_6 )
=> ( ( semila1673364395name_o @ ( insert_pname @ A_14 @ B_12 ) @ C_6 )
= ( insert_pname @ A_14 @ ( semila1673364395name_o @ B_12 @ C_6 ) ) ) )
& ( ~ ( member_pname @ A_14 @ C_6 )
=> ( ( semila1673364395name_o @ ( insert_pname @ A_14 @ B_12 ) @ C_6 )
= ( semila1673364395name_o @ B_12 @ C_6 ) ) ) ) ).
thf(fact_607_Int__insert__right,axiom,
! [B_11: hoare_1167836817_state > $o,A_13: hoare_1167836817_state,A_12: hoare_1167836817_state > $o] :
( ( ( member2058392318_state @ A_13 @ A_12 )
=> ( ( semila179895820tate_o @ A_12 @ ( insert2134838167_state @ A_13 @ B_11 ) )
= ( insert2134838167_state @ A_13 @ ( semila179895820tate_o @ A_12 @ B_11 ) ) ) )
& ( ~ ( member2058392318_state @ A_13 @ A_12 )
=> ( ( semila179895820tate_o @ A_12 @ ( insert2134838167_state @ A_13 @ B_11 ) )
= ( semila179895820tate_o @ A_12 @ B_11 ) ) ) ) ).
thf(fact_608_Int__insert__right,axiom,
! [B_11: hoare_1775062406iple_a > $o,A_13: hoare_1775062406iple_a,A_12: hoare_1775062406iple_a > $o] :
( ( ( member2122167641iple_a @ A_13 @ A_12 )
=> ( ( semila966743401le_a_o @ A_12 @ ( insert1281456128iple_a @ A_13 @ B_11 ) )
= ( insert1281456128iple_a @ A_13 @ ( semila966743401le_a_o @ A_12 @ B_11 ) ) ) )
& ( ~ ( member2122167641iple_a @ A_13 @ A_12 )
=> ( ( semila966743401le_a_o @ A_12 @ ( insert1281456128iple_a @ A_13 @ B_11 ) )
= ( semila966743401le_a_o @ A_12 @ B_11 ) ) ) ) ).
thf(fact_609_Int__insert__right,axiom,
! [B_11: pname > $o,A_13: pname,A_12: pname > $o] :
( ( ( member_pname @ A_13 @ A_12 )
=> ( ( semila1673364395name_o @ A_12 @ ( insert_pname @ A_13 @ B_11 ) )
= ( insert_pname @ A_13 @ ( semila1673364395name_o @ A_12 @ B_11 ) ) ) )
& ( ~ ( member_pname @ A_13 @ A_12 )
=> ( ( semila1673364395name_o @ A_12 @ ( insert_pname @ A_13 @ B_11 ) )
= ( semila1673364395name_o @ A_12 @ B_11 ) ) ) ) ).
thf(fact_610_if__image__distrib,axiom,
! [P_1: pname > $o,F_4: pname > hoare_1167836817_state,G: pname > hoare_1167836817_state,S_1: pname > $o] :
( ( image_575578384_state
@ ^ [X: pname] : ( if_Hoa833675553_state @ ( P_1 @ X ) @ ( F_4 @ X ) @ ( G @ X ) )
@ S_1 )
= ( semila1172322802tate_o @ ( image_575578384_state @ F_4 @ ( semila1673364395name_o @ S_1 @ ( collect_pname @ P_1 ) ) )
@ ( image_575578384_state @ G
@ ( semila1673364395name_o @ S_1
@ ( collect_pname
@ ^ [X: pname] : ( (~) @ ( P_1 @ X ) ) ) ) ) ) ) ).
thf(fact_611_if__image__distrib,axiom,
! [P_1: pname > $o,F_4: pname > hoare_1775062406iple_a,G: pname > hoare_1775062406iple_a,S_1: pname > $o] :
( ( image_2063119815iple_a
@ ^ [X: pname] : ( if_Hoa1047340790iple_a @ ( P_1 @ X ) @ ( F_4 @ X ) @ ( G @ X ) )
@ S_1 )
= ( semila13410563le_a_o @ ( image_2063119815iple_a @ F_4 @ ( semila1673364395name_o @ S_1 @ ( collect_pname @ P_1 ) ) )
@ ( image_2063119815iple_a @ G
@ ( semila1673364395name_o @ S_1
@ ( collect_pname
@ ^ [X: pname] : ( (~) @ ( P_1 @ X ) ) ) ) ) ) ) ).
thf(fact_612_folding__one_Ounion__inter,axiom,
! [B_10: pname > $o,A_11: pname > $o,F_3: pname > pname > pname,F_2: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_3 @ F_2 )
=> ( ( finite_finite_pname @ A_11 )
=> ( ( finite_finite_pname @ B_10 )
=> ( ( ( semila1673364395name_o @ A_11 @ B_10 )
!= bot_bot_pname_o )
=> ( ( F_3 @ ( F_2 @ ( semila1780557381name_o @ A_11 @ B_10 ) ) @ ( F_2 @ ( semila1673364395name_o @ A_11 @ B_10 ) ) )
= ( F_3 @ ( F_2 @ A_11 ) @ ( F_2 @ B_10 ) ) ) ) ) ) ) ).
thf(fact_613_folding__one_Ounion__inter,axiom,
! [B_10: hoare_1775062406iple_a > $o,A_11: hoare_1775062406iple_a > $o,F_3: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_2: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_3 @ F_2 )
=> ( ( finite2063573081iple_a @ A_11 )
=> ( ( finite2063573081iple_a @ B_10 )
=> ( ( ( semila966743401le_a_o @ A_11 @ B_10 )
!= bot_bo751897185le_a_o )
=> ( ( F_3 @ ( F_2 @ ( semila13410563le_a_o @ A_11 @ B_10 ) ) @ ( F_2 @ ( semila966743401le_a_o @ A_11 @ B_10 ) ) )
= ( F_3 @ ( F_2 @ A_11 ) @ ( F_2 @ B_10 ) ) ) ) ) ) ) ).
thf(fact_614_folding__one_Ounion__inter,axiom,
! [B_10: hoare_1167836817_state > $o,A_11: hoare_1167836817_state > $o,F_3: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_2: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_3 @ F_2 )
=> ( ( finite1084549118_state @ A_11 )
=> ( ( finite1084549118_state @ B_10 )
=> ( ( ( semila179895820tate_o @ A_11 @ B_10 )
!= bot_bo70021908tate_o )
=> ( ( F_3 @ ( F_2 @ ( semila1172322802tate_o @ A_11 @ B_10 ) ) @ ( F_2 @ ( semila179895820tate_o @ A_11 @ B_10 ) ) )
= ( F_3 @ ( F_2 @ A_11 ) @ ( F_2 @ B_10 ) ) ) ) ) ) ) ).
thf(fact_615_folding__one_Ounion__disjoint,axiom,
! [B_9: pname > $o,A_10: pname > $o,F_1: pname > pname > pname,F: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_1 @ F )
=> ( ( finite_finite_pname @ A_10 )
=> ( ( A_10 != bot_bot_pname_o )
=> ( ( finite_finite_pname @ B_9 )
=> ( ( B_9 != bot_bot_pname_o )
=> ( ( ( semila1673364395name_o @ A_10 @ B_9 )
= bot_bot_pname_o )
=> ( ( F @ ( semila1780557381name_o @ A_10 @ B_9 ) )
= ( F_1 @ ( F @ A_10 ) @ ( F @ B_9 ) ) ) ) ) ) ) ) ) ).
thf(fact_616_folding__one_Ounion__disjoint,axiom,
! [B_9: hoare_1775062406iple_a > $o,A_10: hoare_1775062406iple_a > $o,F_1: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_1 @ F )
=> ( ( finite2063573081iple_a @ A_10 )
=> ( ( A_10 != bot_bo751897185le_a_o )
=> ( ( finite2063573081iple_a @ B_9 )
=> ( ( B_9 != bot_bo751897185le_a_o )
=> ( ( ( semila966743401le_a_o @ A_10 @ B_9 )
= bot_bo751897185le_a_o )
=> ( ( F @ ( semila13410563le_a_o @ A_10 @ B_9 ) )
= ( F_1 @ ( F @ A_10 ) @ ( F @ B_9 ) ) ) ) ) ) ) ) ) ).
thf(fact_617_folding__one_Ounion__disjoint,axiom,
! [B_9: hoare_1167836817_state > $o,A_10: hoare_1167836817_state > $o,F_1: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_1 @ F )
=> ( ( finite1084549118_state @ A_10 )
=> ( ( A_10 != bot_bo70021908tate_o )
=> ( ( finite1084549118_state @ B_9 )
=> ( ( B_9 != bot_bo70021908tate_o )
=> ( ( ( semila179895820tate_o @ A_10 @ B_9 )
= bot_bo70021908tate_o )
=> ( ( F @ ( semila1172322802tate_o @ A_10 @ B_9 ) )
= ( F_1 @ ( F @ A_10 ) @ ( F @ B_9 ) ) ) ) ) ) ) ) ) ).
thf(fact_618_distrib__imp2,axiom,
! [X_3: $o,Y_3: $o,Z_3: $o] :
( ! [X: $o,Y_2: $o,Z_2: $o] :
( ( semila10642723_sup_o @ X @ ( semila854092349_inf_o @ Y_2 @ Z_2 ) )
<=> ( semila854092349_inf_o @ ( semila10642723_sup_o @ X @ Y_2 ) @ ( semila10642723_sup_o @ X @ Z_2 ) ) )
=> ( ( semila854092349_inf_o @ X_3 @ ( semila10642723_sup_o @ Y_3 @ Z_3 ) )
<=> ( semila10642723_sup_o @ ( semila854092349_inf_o @ X_3 @ Y_3 ) @ ( semila854092349_inf_o @ X_3 @ Z_3 ) ) ) ) ).
thf(fact_619_distrib__imp2,axiom,
! [X_3: pname > $o,Y_3: pname > $o,Z_3: pname > $o] :
( ! [X: pname > $o,Y_2: pname > $o,Z_2: pname > $o] :
( ( semila1780557381name_o @ X @ ( semila1673364395name_o @ Y_2 @ Z_2 ) )
= ( semila1673364395name_o @ ( semila1780557381name_o @ X @ Y_2 ) @ ( semila1780557381name_o @ X @ Z_2 ) ) )
=> ( ( semila1673364395name_o @ X_3 @ ( semila1780557381name_o @ Y_3 @ Z_3 ) )
= ( semila1780557381name_o @ ( semila1673364395name_o @ X_3 @ Y_3 ) @ ( semila1673364395name_o @ X_3 @ Z_3 ) ) ) ) ).
thf(fact_620_distrib__imp2,axiom,
! [X_3: hoare_1167836817_state > $o,Y_3: hoare_1167836817_state > $o,Z_3: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o,Z_2: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X @ ( semila179895820tate_o @ Y_2 @ Z_2 ) )
= ( semila179895820tate_o @ ( semila1172322802tate_o @ X @ Y_2 ) @ ( semila1172322802tate_o @ X @ Z_2 ) ) )
=> ( ( semila179895820tate_o @ X_3 @ ( semila1172322802tate_o @ Y_3 @ Z_3 ) )
= ( semila1172322802tate_o @ ( semila179895820tate_o @ X_3 @ Y_3 ) @ ( semila179895820tate_o @ X_3 @ Z_3 ) ) ) ) ).
thf(fact_621_distrib__imp2,axiom,
! [X_3: hoare_1775062406iple_a > $o,Y_3: hoare_1775062406iple_a > $o,Z_3: hoare_1775062406iple_a > $o] :
( ! [X: hoare_1775062406iple_a > $o,Y_2: hoare_1775062406iple_a > $o,Z_2: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X @ ( semila966743401le_a_o @ Y_2 @ Z_2 ) )
= ( semila966743401le_a_o @ ( semila13410563le_a_o @ X @ Y_2 ) @ ( semila13410563le_a_o @ X @ Z_2 ) ) )
=> ( ( semila966743401le_a_o @ X_3 @ ( semila13410563le_a_o @ Y_3 @ Z_3 ) )
= ( semila13410563le_a_o @ ( semila966743401le_a_o @ X_3 @ Y_3 ) @ ( semila966743401le_a_o @ X_3 @ Z_3 ) ) ) ) ).
thf(fact_622_distrib__imp1,axiom,
! [X_2: $o,Y_1: $o,Z_1: $o] :
( ! [X: $o,Y_2: $o,Z_2: $o] :
( ( semila854092349_inf_o @ X @ ( semila10642723_sup_o @ Y_2 @ Z_2 ) )
<=> ( semila10642723_sup_o @ ( semila854092349_inf_o @ X @ Y_2 ) @ ( semila854092349_inf_o @ X @ Z_2 ) ) )
=> ( ( semila10642723_sup_o @ X_2 @ ( semila854092349_inf_o @ Y_1 @ Z_1 ) )
<=> ( semila854092349_inf_o @ ( semila10642723_sup_o @ X_2 @ Y_1 ) @ ( semila10642723_sup_o @ X_2 @ Z_1 ) ) ) ) ).
thf(fact_623_distrib__imp1,axiom,
! [X_2: pname > $o,Y_1: pname > $o,Z_1: pname > $o] :
( ! [X: pname > $o,Y_2: pname > $o,Z_2: pname > $o] :
( ( semila1673364395name_o @ X @ ( semila1780557381name_o @ Y_2 @ Z_2 ) )
= ( semila1780557381name_o @ ( semila1673364395name_o @ X @ Y_2 ) @ ( semila1673364395name_o @ X @ Z_2 ) ) )
=> ( ( semila1780557381name_o @ X_2 @ ( semila1673364395name_o @ Y_1 @ Z_1 ) )
= ( semila1673364395name_o @ ( semila1780557381name_o @ X_2 @ Y_1 ) @ ( semila1780557381name_o @ X_2 @ Z_1 ) ) ) ) ).
thf(fact_624_distrib__imp1,axiom,
! [X_2: hoare_1167836817_state > $o,Y_1: hoare_1167836817_state > $o,Z_1: hoare_1167836817_state > $o] :
( ! [X: hoare_1167836817_state > $o,Y_2: hoare_1167836817_state > $o,Z_2: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ X @ ( semila1172322802tate_o @ Y_2 @ Z_2 ) )
= ( semila1172322802tate_o @ ( semila179895820tate_o @ X @ Y_2 ) @ ( semila179895820tate_o @ X @ Z_2 ) ) )
=> ( ( semila1172322802tate_o @ X_2 @ ( semila179895820tate_o @ Y_1 @ Z_1 ) )
= ( semila179895820tate_o @ ( semila1172322802tate_o @ X_2 @ Y_1 ) @ ( semila1172322802tate_o @ X_2 @ Z_1 ) ) ) ) ).
thf(fact_625_distrib__imp1,axiom,
! [X_2: hoare_1775062406iple_a > $o,Y_1: hoare_1775062406iple_a > $o,Z_1: hoare_1775062406iple_a > $o] :
( ! [X: hoare_1775062406iple_a > $o,Y_2: hoare_1775062406iple_a > $o,Z_2: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ X @ ( semila13410563le_a_o @ Y_2 @ Z_2 ) )
= ( semila13410563le_a_o @ ( semila966743401le_a_o @ X @ Y_2 ) @ ( semila966743401le_a_o @ X @ Z_2 ) ) )
=> ( ( semila13410563le_a_o @ X_2 @ ( semila966743401le_a_o @ Y_1 @ Z_1 ) )
= ( semila966743401le_a_o @ ( semila13410563le_a_o @ X_2 @ Y_1 ) @ ( semila13410563le_a_o @ X_2 @ Z_1 ) ) ) ) ).
thf(fact_626_sup__Inf__absorb,axiom,
! [A_9: $o,A_8: $o > $o] :
( ( finite_finite_o @ A_8 )
=> ( ( member_o @ A_9 @ A_8 )
=> ( ( semila10642723_sup_o @ A_9 @ ( big_la1690136417_fin_o @ A_8 ) )
<=> A_9 ) ) ) ).
thf(fact_627_sup__Inf__absorb,axiom,
! [A_9: pname > $o,A_8: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_8 )
=> ( ( member_pname_o @ A_9 @ A_8 )
=> ( ( semila1780557381name_o @ A_9 @ ( big_la1126801287name_o @ A_8 ) )
= A_9 ) ) ) ).
thf(fact_628_sup__Inf__absorb,axiom,
! [A_9: hoare_1167836817_state > $o,A_8: ( hoare_1167836817_state > $o ) > $o] :
( ( finite1380128977tate_o @ A_8 )
=> ( ( member864234961tate_o @ A_9 @ A_8 )
=> ( ( semila1172322802tate_o @ A_9 @ ( big_la831793456tate_o @ A_8 ) )
= A_9 ) ) ) ).
thf(fact_629_sup__Inf__absorb,axiom,
! [A_9: hoare_1775062406iple_a > $o,A_8: ( hoare_1775062406iple_a > $o ) > $o] :
( ( finite789576932le_a_o @ A_8 )
=> ( ( member1207314404le_a_o @ A_9 @ A_8 )
=> ( ( semila13410563le_a_o @ A_9 @ ( big_la447547205le_a_o @ A_8 ) )
= A_9 ) ) ) ).
thf(fact_630_DiffE,axiom,
! [C_5: hoare_1775062406iple_a,A_7: hoare_1775062406iple_a > $o,B_8: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_5 @ ( minus_1944206118le_a_o @ A_7 @ B_8 ) )
=> ~ ( ( member2122167641iple_a @ C_5 @ A_7 )
=> ( member2122167641iple_a @ C_5 @ B_8 ) ) ) ).
thf(fact_631_DiffE,axiom,
! [C_5: pname,A_7: pname > $o,B_8: pname > $o] :
( ( member_pname @ C_5 @ ( minus_minus_pname_o @ A_7 @ B_8 ) )
=> ~ ( ( member_pname @ C_5 @ A_7 )
=> ( member_pname @ C_5 @ B_8 ) ) ) ).
thf(fact_632_DiffI,axiom,
! [B_7: hoare_1775062406iple_a > $o,C_4: hoare_1775062406iple_a,A_6: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_4 @ A_6 )
=> ( ~ ( member2122167641iple_a @ C_4 @ B_7 )
=> ( member2122167641iple_a @ C_4 @ ( minus_1944206118le_a_o @ A_6 @ B_7 ) ) ) ) ).
thf(fact_633_DiffI,axiom,
! [B_7: pname > $o,C_4: pname,A_6: pname > $o] :
( ( member_pname @ C_4 @ A_6 )
=> ( ~ ( member_pname @ C_4 @ B_7 )
=> ( member_pname @ C_4 @ ( minus_minus_pname_o @ A_6 @ B_7 ) ) ) ) ).
thf(fact_634_Un__Diff__cancel,axiom,
! [A_5: pname > $o,B_6: pname > $o] :
( ( semila1780557381name_o @ A_5 @ ( minus_minus_pname_o @ B_6 @ A_5 ) )
= ( semila1780557381name_o @ A_5 @ B_6 ) ) ).
thf(fact_635_Un__Diff__cancel,axiom,
! [A_5: hoare_1167836817_state > $o,B_6: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_5 @ ( minus_2107060239tate_o @ B_6 @ A_5 ) )
= ( semila1172322802tate_o @ A_5 @ B_6 ) ) ).
thf(fact_636_Un__Diff__cancel,axiom,
! [A_5: hoare_1775062406iple_a > $o,B_6: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_5 @ ( minus_1944206118le_a_o @ B_6 @ A_5 ) )
= ( semila13410563le_a_o @ A_5 @ B_6 ) ) ).
thf(fact_637_Un__Diff__cancel2,axiom,
! [B_5: pname > $o,A_4: pname > $o] :
( ( semila1780557381name_o @ ( minus_minus_pname_o @ B_5 @ A_4 ) @ A_4 )
= ( semila1780557381name_o @ B_5 @ A_4 ) ) ).
thf(fact_638_Un__Diff__cancel2,axiom,
! [B_5: hoare_1167836817_state > $o,A_4: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( minus_2107060239tate_o @ B_5 @ A_4 ) @ A_4 )
= ( semila1172322802tate_o @ B_5 @ A_4 ) ) ).
thf(fact_639_Un__Diff__cancel2,axiom,
! [B_5: hoare_1775062406iple_a > $o,A_4: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( minus_1944206118le_a_o @ B_5 @ A_4 ) @ A_4 )
= ( semila13410563le_a_o @ B_5 @ A_4 ) ) ).
thf(fact_640_Un__Diff,axiom,
! [A_3: pname > $o,B_4: pname > $o,C_3: pname > $o] :
( ( minus_minus_pname_o @ ( semila1780557381name_o @ A_3 @ B_4 ) @ C_3 )
= ( semila1780557381name_o @ ( minus_minus_pname_o @ A_3 @ C_3 ) @ ( minus_minus_pname_o @ B_4 @ C_3 ) ) ) ).
thf(fact_641_Un__Diff,axiom,
! [A_3: hoare_1167836817_state > $o,B_4: hoare_1167836817_state > $o,C_3: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ ( semila1172322802tate_o @ A_3 @ B_4 ) @ C_3 )
= ( semila1172322802tate_o @ ( minus_2107060239tate_o @ A_3 @ C_3 ) @ ( minus_2107060239tate_o @ B_4 @ C_3 ) ) ) ).
thf(fact_642_Un__Diff,axiom,
! [A_3: hoare_1775062406iple_a > $o,B_4: hoare_1775062406iple_a > $o,C_3: hoare_1775062406iple_a > $o] :
( ( minus_1944206118le_a_o @ ( semila13410563le_a_o @ A_3 @ B_4 ) @ C_3 )
= ( semila13410563le_a_o @ ( minus_1944206118le_a_o @ A_3 @ C_3 ) @ ( minus_1944206118le_a_o @ B_4 @ C_3 ) ) ) ).
thf(fact_643_DiffD2,axiom,
! [C_2: hoare_1775062406iple_a,A_2: hoare_1775062406iple_a > $o,B_3: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_2 @ ( minus_1944206118le_a_o @ A_2 @ B_3 ) )
=> ~ ( member2122167641iple_a @ C_2 @ B_3 ) ) ).
thf(fact_644_DiffD2,axiom,
! [C_2: pname,A_2: pname > $o,B_3: pname > $o] :
( ( member_pname @ C_2 @ ( minus_minus_pname_o @ A_2 @ B_3 ) )
=> ~ ( member_pname @ C_2 @ B_3 ) ) ).
thf(fact_645_DiffD1,axiom,
! [C_1: hoare_1775062406iple_a,A_1: hoare_1775062406iple_a > $o,B_2: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_1 @ ( minus_1944206118le_a_o @ A_1 @ B_2 ) )
=> ( member2122167641iple_a @ C_1 @ A_1 ) ) ).
thf(fact_646_DiffD1,axiom,
! [C_1: pname,A_1: pname > $o,B_2: pname > $o] :
( ( member_pname @ C_1 @ ( minus_minus_pname_o @ A_1 @ B_2 ) )
=> ( member_pname @ C_1 @ A_1 ) ) ).
thf(fact_647_Diff__iff,axiom,
! [C: pname,A: pname > $o,B_1: pname > $o] :
( ( member_pname @ C @ ( minus_minus_pname_o @ A @ B_1 ) )
<=> ( ( member_pname @ C @ A )
& ~ ( member_pname @ C @ B_1 ) ) ) ).
thf(fact_648_diff__Suc__Suc,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_649_Suc__diff__diff,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N_1 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N_1 ) @ K ) ) ).
thf(fact_650_diff__0__eq__0,axiom,
! [N_1: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N_1 )
= zero_zero_nat ) ).
thf(fact_651_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_652_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_653_diffs0__imp__equal,axiom,
! [M: nat,N_1: nat] :
( ( ( minus_minus_nat @ M @ N_1 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N_1 @ M )
= zero_zero_nat )
=> ( M = N_1 ) ) ) ).
thf(fact_654_zero__induct__lemma,axiom,
! [I: nat,P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
thf(fact_655_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
thf(fact_656_diff__Suc,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ M @ ( suc @ N_1 ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [K_1: nat] : K_1
@ ( minus_minus_nat @ M @ N_1 ) ) ) ).
thf(fact_657_diff__Suc__1,axiom,
! [N_1: nat] :
( ( minus_minus_nat @ ( suc @ N_1 ) @ one_one_nat )
= N_1 ) ).
thf(fact_658_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ M @ ( suc @ N_1 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ).
thf(fact_659_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_660_Suc__eq__plus1,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
= ( plus_plus_nat @ N_1 @ one_one_nat ) ) ).
thf(fact_661_Suc__eq__plus1__left,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
= ( plus_plus_nat @ one_one_nat @ N_1 ) ) ).
thf(fact_662_diff__cancel2,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N_1 @ K ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_663_diff__cancel,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_664_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_665_diff__add__inverse,axiom,
! [N_1: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N_1 @ M ) @ N_1 )
= M ) ).
thf(fact_666_diff__add__inverse2,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N_1 ) @ N_1 )
= M ) ).
thf(fact_667_diff__add__0,axiom,
! [N_1: nat,M: nat] :
( ( minus_minus_nat @ N_1 @ ( plus_plus_nat @ N_1 @ M ) )
= zero_zero_nat ) ).
thf(fact_668_nat__add__commute,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ M @ N_1 )
= ( plus_plus_nat @ N_1 @ M ) ) ).
thf(fact_669_nat__add__left__commute,axiom,
! [X_1: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X_1 @ ( plus_plus_nat @ Y @ Z ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X_1 @ Z ) ) ) ).
thf(fact_670_nat__add__assoc,axiom,
! [M: nat,N_1: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N_1 ) @ K )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N_1 @ K ) ) ) ).
thf(fact_671_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N_1: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N_1 ) )
<=> ( M = N_1 ) ) ).
thf(fact_672_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N_1 @ K ) )
<=> ( M = N_1 ) ) ).
thf(fact_673_add__Suc__shift,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_1 )
= ( plus_plus_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_674_add__Suc,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_1 )
= ( suc @ ( plus_plus_nat @ M @ N_1 ) ) ) ).
thf(fact_675_add__Suc__right,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ M @ ( suc @ N_1 ) )
= ( suc @ ( plus_plus_nat @ M @ N_1 ) ) ) ).
thf(fact_676_one__is__add,axiom,
! [M: nat,N_1: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N_1 ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_677_add__is__1,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_678_plus__nat_Oadd__0,axiom,
! [N_1: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N_1 )
= N_1 ) ).
thf(fact_679_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_680_add__is__0,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N_1 = zero_zero_nat ) ) ) ).
thf(fact_681_add__eq__self__zero,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= M )
=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_682_add__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_1 )
= N_1 ) )
& ( ( M != zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_1 )
= ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ) ) ) ).
thf(fact_683_com_Osize_I4_J,axiom,
! [Com1: com,Com2: com] :
( ( com_size @ ( semi @ Com1 @ Com2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( com_size @ Com1 ) @ ( com_size @ Com2 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_684_com_Osize_I7_J,axiom,
! [Pname: pname] :
( ( com_size @ ( body @ Pname ) )
= zero_zero_nat ) ).
thf(fact_685_com_Osize_I1_J,axiom,
( ( com_size @ skip )
= zero_zero_nat ) ).
thf(fact_686_com_Osize_I6_J,axiom,
! [Fun: state > $o,Com: com] :
( ( com_size @ ( while @ Fun @ Com ) )
= ( plus_plus_nat @ ( com_size @ Com ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_687_com_Osize_I12_J,axiom,
! [Com1: com,Com2: com] :
( ( size_size_com @ ( semi @ Com1 @ Com2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ Com1 ) @ ( size_size_com @ Com2 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_688_com_Osize_I15_J,axiom,
! [Pname: pname] :
( ( size_size_com @ ( body @ Pname ) )
= zero_zero_nat ) ).
thf(fact_689_com_Osize_I9_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
thf(fact_690_com_Osize_I14_J,axiom,
! [Fun: state > $o,Com: com] :
( ( size_size_com @ ( while @ Fun @ Com ) )
= ( plus_plus_nat @ ( size_size_com @ Com ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_691_com_Osize_I13_J,axiom,
! [Fun: state > $o,Com1: com,Com2: com] :
( ( size_size_com @ ( cond @ Fun @ Com1 @ Com2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ Com1 ) @ ( size_size_com @ Com2 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_692_evaln_OIfFalse,axiom,
! [C0: com,C1: com,N_1: nat,S1: state,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( ( evaln @ C1 @ S @ N_1 @ S1 )
=> ( evaln @ ( cond @ B @ C0 @ C1 ) @ S @ N_1 @ S1 ) ) ) ).
thf(fact_693_evaln_OIfTrue,axiom,
! [C1: com,C0: com,N_1: nat,S1: state,B: state > $o,S: state] :
( ( B @ S )
=> ( ( evaln @ C0 @ S @ N_1 @ S1 )
=> ( evaln @ ( cond @ B @ C0 @ C1 ) @ S @ N_1 @ S1 ) ) ) ).
thf(fact_694_evaln__elim__cases_I5_J,axiom,
! [B: state > $o,C1: com,C2: com,S: state,N_1: nat,T: state] :
( ( evaln @ ( cond @ B @ C1 @ C2 ) @ S @ N_1 @ T )
=> ( ( ( B @ S )
=> ~ ( evaln @ C1 @ S @ N_1 @ T ) )
=> ~ ( ~ ( B @ S )
=> ~ ( evaln @ C2 @ S @ N_1 @ T ) ) ) ) ).
thf(fact_695_evalc__elim__cases_I5_J,axiom,
! [B: state > $o,C1: com,C2: com,S: state,T: state] :
( ( evalc @ ( cond @ B @ C1 @ C2 ) @ S @ T )
=> ( ( ( B @ S )
=> ~ ( evalc @ C1 @ S @ T ) )
=> ~ ( ~ ( B @ S )
=> ~ ( evalc @ C2 @ S @ T ) ) ) ) ).
thf(fact_696_evalc_OIfTrue,axiom,
! [C1: com,C0: com,S1: state,B: state > $o,S: state] :
( ( B @ S )
=> ( ( evalc @ C0 @ S @ S1 )
=> ( evalc @ ( cond @ B @ C0 @ C1 ) @ S @ S1 ) ) ) ).
thf(fact_697_evalc_OIfFalse,axiom,
! [C0: com,C1: com,S1: state,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( ( evalc @ C1 @ S @ S1 )
=> ( evalc @ ( cond @ B @ C0 @ C1 ) @ S @ S1 ) ) ) ).
%----Helper facts (14)
thf(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ~ ( fequal_pname @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ( X_1 != Y )
| ( fequal_pname @ X_1 @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
! [X_1: state,Y: state] :
( ~ ( fequal_state @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
! [X_1: state,Y: state] :
( ( X_1 != Y )
| ( fequal_state @ X_1 @ Y ) ) ).
thf(help_If_1_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_T,axiom,
! [X_1: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
( ( if_Hoa1047340790iple_a @ $true @ X_1 @ Y )
= X_1 ) ).
thf(help_If_2_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_T,axiom,
! [X_1: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
( ( if_Hoa1047340790iple_a @ $false @ X_1 @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
! [X_1: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
( ~ ( fequal1288209029iple_a @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
! [X_1: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
( ( X_1 != Y )
| ( fequal1288209029iple_a @ X_1 @ Y ) ) ).
thf(help_If_1_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate,axiom,
! [X_1: hoare_1167836817_state,Y: hoare_1167836817_state] :
( ( if_Hoa833675553_state @ $true @ X_1 @ Y )
= X_1 ) ).
thf(help_If_2_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate,axiom,
! [X_1: hoare_1167836817_state,Y: hoare_1167836817_state] :
( ( if_Hoa833675553_state @ $false @ X_1 @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
! [X_1: hoare_1167836817_state,Y: hoare_1167836817_state] :
( ~ ( fequal1831255762_state @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
! [X_1: hoare_1167836817_state,Y: hoare_1167836817_state] :
( ( X_1 != Y )
| ( fequal1831255762_state @ X_1 @ Y ) ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [N: nat] :
( ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X
@ ( semila13410563le_a_o @ g
@ ( image_2063119815iple_a
@ ^ [Pn: pname] : ( hoare_1766022166iple_a @ ( p @ Pn ) @ ( body @ Pn ) @ ( q @ Pn ) )
@ procs ) ) )
=> ( hoare_1462269968alid_a @ N @ X ) )
=> ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X
@ ( image_2063119815iple_a
@ ^ [Pn: pname] : ( hoare_1766022166iple_a @ ( p @ Pn ) @ ( the_com @ ( body_1 @ Pn ) ) @ ( q @ Pn ) )
@ procs ) )
=> ( hoare_1462269968alid_a @ N @ X ) ) ) ).
thf(conj_1,conjecture,
( ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X @ g )
=> ( hoare_1462269968alid_a @ n @ X ) )
=> ! [X: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X
@ ( image_2063119815iple_a
@ ^ [Pn: pname] : ( hoare_1766022166iple_a @ ( p @ Pn ) @ ( body @ Pn ) @ ( q @ Pn ) )
@ procs ) )
=> ( hoare_1462269968alid_a @ n @ X ) ) ) ).
%------------------------------------------------------------------------------