TPTP Problem File: SWW470^2.p
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%------------------------------------------------------------------------------
% File : SWW470^2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 200, 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_l200 [Bla11]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0, 1.00 v7.2.0, 0.75 v7.1.0, 0.00 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.80 v5.3.0
% Syntax : Number of formulae : 832 ( 178 unt; 121 typ; 0 def)
% Number of atoms : 3211 ( 690 equ; 62 cnn)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 6257 ( 261 ~; 46 |; 141 &;4902 @)
% ( 145 <=>; 758 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 1667 (1667 >; 0 *; 0 +; 0 <<)
% Number of symbols : 119 ( 112 usr; 10 con; 0-4 aty)
% ( 0 !!; 4 ??; 0 @@+; 0 @@-)
% Number of variables : 2068 ( 140 ^;1880 !; 48 ?;2068 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:17:23
%------------------------------------------------------------------------------
%----Should-be-implicit typings (9)
thf(ty_ty_t__a,type,
x_a: $tType ).
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Oglb,type,
glb: $tType ).
thf(ty_ty_tc__Com__Oloc,type,
loc: $tType ).
thf(ty_ty_tc__Com__Ostate,type,
state: $tType ).
thf(ty_ty_tc__Com__Ovname,type,
vname: $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 ).
%----Explicit typings (115)
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000tc__Nat__Onat,type,
big_la43341705in_nat: ( nat > $o ) > nat ).
thf(sy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat,type,
big_se275732192ig_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(sy_c_Com_Ocom_OAss,type,
ass: vname > ( state > nat ) > com ).
thf(sy_c_Com_Ocom_OLocal,type,
local: loc > ( state > nat ) > 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_Ovname_OGlb,type,
glb_1: glb > vname ).
thf(sy_c_Com_Ovname_OLoc,type,
loc_1: loc > vname ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
finite2064891473iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip,type,
finite2120172977le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip_001,type,
finite856902323tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
finite389864113_nat_o: ( nat > ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Nat__Onat_M_Eo_J,type,
finite_finite_nat_o: ( ( nat > $o ) > $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_Ofinite_000tc__Nat__Onat,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__,type,
finite1946188886iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc_,type,
finite309220289_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Nat__Onat,type,
finite_fold1Set_nat: ( nat > nat > nat ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
finite1790765286iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Co,type,
finite1646097201_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_Finite__Set_Ofold1_000tc__Nat__Onat,type,
finite_fold1_nat: ( nat > nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,type,
finite1544171829le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J__002,type,
finite1842721992iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,type,
finite291020855tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com_003,type,
finite1731015960_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
finite326637109_nat_o: ( nat > ( nat > $o ) > nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____srushsumbx__Otriple_I,type,
finite727644230iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____srushsumbx__Otriple_I_004,type,
finite1316643734_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__Nat__Onat,type,
finite929467206at_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat > $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_000tc__Nat__Onat,type,
finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $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_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $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_000_062_Itc__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
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_HOL_OThe_000tc__Nat__Onat,type,
the_nat: ( nat > $o ) > nat ).
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_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__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_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_008,type,
semila179895820tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Nat__Onat_M_Eo_J,type,
semila1947288293_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_Eo,type,
semila854092349_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__Nat__Onat,type,
semila80283416nf_nat: nat > nat > nat ).
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_009,type,
semila1172322802tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Nat__Onat_M_Eo_J,type,
semila848761471_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_Eo,type,
semila10642723_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__Nat__Onat,type,
semila972727038up_nat: nat > nat > 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_Natural_Ogetlocs,type,
getlocs: state > loc > nat ).
thf(sy_c_Natural_Oupdate,type,
update: state > vname > nat > state ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
bot_bot_nat_o_o: ( nat > $o ) > $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_010,type,
bot_bo70021908tate_o: hoare_1167836817_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $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_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum,type,
ord_le1143225901le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum_011,type,
ord_le827224136tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____srushsumbx__Otrip,type,
partia126998524iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____srushsumbx__Otrip_012,type,
partia715677851_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(sy_c_Partial__Function_Oflat__lub_000tc__Nat__Onat,type,
partial_flat_lub_nat: nat > ( nat > $o ) > nat ).
thf(sy_c_Set_OCollect_000_062_Itc__Nat__Onat_M_Eo_J,type,
collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $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_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc__,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___013,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_It__a_J_000tc___014,type,
image_1806517641_a_nat: ( hoare_1775062406iple_a > nat ) > ( hoare_1775062406iple_a > $o ) > nat > $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_015,type,
image_31595733_state: ( hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_016,type,
image_1476618182te_nat: ( hoare_1167836817_state > nat ) > ( hoare_1167836817_state > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____srushsumbx__Otripl,type,
image_43014529iple_a: ( nat > hoare_1775062406iple_a ) > ( nat > $o ) > hoare_1775062406iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____srushsumbx__Otripl_017,type,
image_2121260246_state: ( nat > hoare_1167836817_state ) > ( nat > $o ) > hoare_1167836817_state > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Nat__Onat_M_Eo_J,type,
insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $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_Oinsert_000tc__Nat__Onat,type,
insert_nat: nat > ( nat > $o ) > nat > $o ).
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_Set_Othe__elem_000tc__Nat__Onat,type,
the_elem_nat: ( nat > $o ) > nat ).
thf(sy_c_fequal_000_062_Itc__Nat__Onat_M_Eo_J,type,
fequal_nat_o: ( nat > $o ) > ( nat > $o ) > $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_fequal_000tc__Nat__Onat,type,
fequal_nat: nat > nat > $o ).
thf(sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $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_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_v_G,type,
g: hoare_1775062406iple_a > $o ).
thf(sy_v_P,type,
p: x_a > state > $o ).
thf(sy_v_b,type,
b: state > $o ).
thf(sy_v_c,type,
c: com ).
%----Relevant facts (700)
thf(fact_0_empty,axiom,
! [G_19: hoare_1775062406iple_a > $o] : ( hoare_1508237396rivs_a @ G_19 @ bot_bo751897185le_a_o ) ).
thf(fact_1_empty,axiom,
! [G_19: hoare_1167836817_state > $o] : ( hoare_123228589_state @ G_19 @ bot_bo70021908tate_o ) ).
thf(fact_2_triple_Oinject,axiom,
! [Fun1_2: state > state > $o,Com_4: com,Fun2_2: state > state > $o,Fun1_1: state > state > $o,Com_3: com,Fun2_1: state > state > $o] :
( ( ( hoare_908217195_state @ Fun1_2 @ Com_4 @ Fun2_2 )
= ( hoare_908217195_state @ Fun1_1 @ Com_3 @ Fun2_1 ) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_4 = Com_3 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
thf(fact_3_triple_Oinject,axiom,
! [Fun1_2: x_a > state > $o,Com_4: com,Fun2_2: x_a > state > $o,Fun1_1: x_a > state > $o,Com_3: com,Fun2_1: x_a > state > $o] :
( ( ( hoare_1766022166iple_a @ Fun1_2 @ Com_4 @ Fun2_2 )
= ( hoare_1766022166iple_a @ Fun1_1 @ Com_3 @ Fun2_1 ) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_4 = Com_3 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
thf(fact_4_cut,axiom,
! [G_18: hoare_1775062406iple_a > $o,G_17: hoare_1775062406iple_a > $o,Ts_4: hoare_1775062406iple_a > $o] :
( ( hoare_1508237396rivs_a @ G_17 @ Ts_4 )
=> ( ( hoare_1508237396rivs_a @ G_18 @ G_17 )
=> ( hoare_1508237396rivs_a @ G_18 @ Ts_4 ) ) ) ).
thf(fact_5_cut,axiom,
! [G_18: hoare_1167836817_state > $o,G_17: hoare_1167836817_state > $o,Ts_4: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_17 @ Ts_4 )
=> ( ( hoare_123228589_state @ G_18 @ G_17 )
=> ( hoare_123228589_state @ G_18 @ Ts_4 ) ) ) ).
thf(fact_6_hoare__derivs_Oinsert,axiom,
! [Ts_3: hoare_1775062406iple_a > $o,G_16: hoare_1775062406iple_a > $o,T_1: hoare_1775062406iple_a] :
( ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_1 @ bot_bo751897185le_a_o ) )
=> ( ( hoare_1508237396rivs_a @ G_16 @ Ts_3 )
=> ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_1 @ Ts_3 ) ) ) ) ).
thf(fact_7_hoare__derivs_Oinsert,axiom,
! [Ts_3: hoare_1167836817_state > $o,G_16: hoare_1167836817_state > $o,T_1: hoare_1167836817_state] :
( ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_1 @ bot_bo70021908tate_o ) )
=> ( ( hoare_123228589_state @ G_16 @ Ts_3 )
=> ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_1 @ Ts_3 ) ) ) ) ).
thf(fact_8_constant,axiom,
! [G_15: hoare_1775062406iple_a > $o,P_33: x_a > state > $o,C_42: com,Q_15: x_a > state > $o,C_41: $o] :
( ( C_41
=> ( hoare_1508237396rivs_a @ G_15 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_33 @ C_42 @ Q_15 ) @ bot_bo751897185le_a_o ) ) )
=> ( hoare_1508237396rivs_a @ G_15
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Z: x_a,S: state] : ( (&) @ ( P_33 @ Z @ S ) @ C_41 )
@ C_42
@ Q_15 )
@ bot_bo751897185le_a_o ) ) ) ).
thf(fact_9_constant,axiom,
! [G_15: hoare_1167836817_state > $o,P_33: state > state > $o,C_42: com,Q_15: state > state > $o,C_41: $o] :
( ( C_41
=> ( hoare_123228589_state @ G_15 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_33 @ C_42 @ Q_15 ) @ bot_bo70021908tate_o ) ) )
=> ( hoare_123228589_state @ G_15
@ ( insert2134838167_state
@ ( hoare_908217195_state
@ ^ [Z: state,S: state] : ( (&) @ ( P_33 @ Z @ S ) @ C_41 )
@ C_42
@ Q_15 )
@ bot_bo70021908tate_o ) ) ) ).
thf(fact_10_escape,axiom,
! [G_14: hoare_1775062406iple_a > $o,C_40: com,Q_14: x_a > state > $o,P_32: x_a > state > $o] :
( ! [Z: x_a,S: state] :
( ( P_32 @ Z @ S )
=> ( hoare_1508237396rivs_a @ G_14
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Za: x_a,S_5: state] : S_5 = S
@ C_40
@ ^ [Z_23: x_a] : ( Q_14 @ Z ) )
@ bot_bo751897185le_a_o ) ) )
=> ( hoare_1508237396rivs_a @ G_14 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_32 @ C_40 @ Q_14 ) @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_11_escape,axiom,
! [G_14: hoare_1167836817_state > $o,C_40: com,Q_14: state > state > $o,P_32: state > state > $o] :
( ! [Z: state,S: state] :
( ( P_32 @ Z @ S )
=> ( hoare_123228589_state @ G_14
@ ( insert2134838167_state
@ ( hoare_908217195_state
@ ^ [Za: state,S_5: state] : S_5 = S
@ C_40
@ ^ [Z_23: state] : ( Q_14 @ Z ) )
@ bot_bo70021908tate_o ) ) )
=> ( hoare_123228589_state @ G_14 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_32 @ C_40 @ Q_14 ) @ bot_bo70021908tate_o ) ) ) ).
thf(fact_12_conseq2,axiom,
! [Q_13: x_a > state > $o,G_13: hoare_1775062406iple_a > $o,P_31: x_a > state > $o,C_39: com,Q_12: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_13 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_31 @ C_39 @ Q_12 ) @ bot_bo751897185le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( Q_12 @ Z @ S )
=> ( Q_13 @ Z @ S ) )
=> ( hoare_1508237396rivs_a @ G_13 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_31 @ C_39 @ Q_13 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_13_conseq2,axiom,
! [Q_13: state > state > $o,G_13: hoare_1167836817_state > $o,P_31: state > state > $o,C_39: com,Q_12: state > state > $o] :
( ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_31 @ C_39 @ Q_12 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z: state,S: state] :
( ( Q_12 @ Z @ S )
=> ( Q_13 @ Z @ S ) )
=> ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_31 @ C_39 @ Q_13 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_14_conseq1,axiom,
! [P_30: x_a > state > $o,G_12: hoare_1775062406iple_a > $o,P_29: x_a > state > $o,C_38: com,Q_11: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_29 @ C_38 @ Q_11 ) @ bot_bo751897185le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( P_30 @ Z @ S )
=> ( P_29 @ Z @ S ) )
=> ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_30 @ C_38 @ Q_11 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_15_conseq1,axiom,
! [P_30: state > state > $o,G_12: hoare_1167836817_state > $o,P_29: state > state > $o,C_38: com,Q_11: state > state > $o] :
( ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_29 @ C_38 @ Q_11 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z: state,S: state] :
( ( P_30 @ Z @ S )
=> ( P_29 @ Z @ S ) )
=> ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_30 @ C_38 @ Q_11 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_16_conseq12,axiom,
! [Q_10: x_a > state > $o,P_28: x_a > state > $o,G_11: hoare_1775062406iple_a > $o,P_27: x_a > state > $o,C_37: com,Q_9: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_11 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_27 @ C_37 @ Q_9 ) @ bot_bo751897185le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( P_28 @ Z @ S )
=> ! [S_5: state] :
( ! [Z_23: x_a] :
( ( P_27 @ Z_23 @ S )
=> ( Q_9 @ Z_23 @ S_5 ) )
=> ( Q_10 @ Z @ S_5 ) ) )
=> ( hoare_1508237396rivs_a @ G_11 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_28 @ C_37 @ Q_10 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_17_conseq12,axiom,
! [Q_10: state > state > $o,P_28: state > state > $o,G_11: hoare_1167836817_state > $o,P_27: state > state > $o,C_37: com,Q_9: state > state > $o] :
( ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_27 @ C_37 @ Q_9 ) @ bot_bo70021908tate_o ) )
=> ( ! [Z: state,S: state] :
( ( P_28 @ Z @ S )
=> ! [S_5: state] :
( ! [Z_23: state] :
( ( P_27 @ Z_23 @ S )
=> ( Q_9 @ Z_23 @ S_5 ) )
=> ( Q_10 @ Z @ S_5 ) ) )
=> ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_28 @ C_37 @ Q_10 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_18_insertE,axiom,
! [A_217: hoare_1775062406iple_a,B_99: hoare_1775062406iple_a,A_216: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_217 @ ( insert1281456128iple_a @ B_99 @ A_216 ) )
=> ( ( A_217 != B_99 )
=> ( member2122167641iple_a @ A_217 @ A_216 ) ) ) ).
thf(fact_19_insertE,axiom,
! [A_217: nat,B_99: nat,A_216: nat > $o] :
( ( member_nat @ A_217 @ ( insert_nat @ B_99 @ A_216 ) )
=> ( ( A_217 != B_99 )
=> ( member_nat @ A_217 @ A_216 ) ) ) ).
thf(fact_20_insertE,axiom,
! [A_217: hoare_1167836817_state,B_99: hoare_1167836817_state,A_216: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_217 @ ( insert2134838167_state @ B_99 @ A_216 ) )
=> ( ( A_217 != B_99 )
=> ( member2058392318_state @ A_217 @ A_216 ) ) ) ).
thf(fact_21_insertCI,axiom,
! [B_98: hoare_1775062406iple_a,A_215: hoare_1775062406iple_a,B_97: hoare_1775062406iple_a > $o] :
( ( ~ ( member2122167641iple_a @ A_215 @ B_97 )
=> ( A_215 = B_98 ) )
=> ( member2122167641iple_a @ A_215 @ ( insert1281456128iple_a @ B_98 @ B_97 ) ) ) ).
thf(fact_22_insertCI,axiom,
! [B_98: nat,A_215: nat,B_97: nat > $o] :
( ( ~ ( member_nat @ A_215 @ B_97 )
=> ( A_215 = B_98 ) )
=> ( member_nat @ A_215 @ ( insert_nat @ B_98 @ B_97 ) ) ) ).
thf(fact_23_insertCI,axiom,
! [B_98: hoare_1167836817_state,A_215: hoare_1167836817_state,B_97: hoare_1167836817_state > $o] :
( ( ~ ( member2058392318_state @ A_215 @ B_97 )
=> ( A_215 = B_98 ) )
=> ( member2058392318_state @ A_215 @ ( insert2134838167_state @ B_98 @ B_97 ) ) ) ).
thf(fact_24_emptyE,axiom,
! [A_214: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ A_214 @ bot_bo751897185le_a_o ) ).
thf(fact_25_emptyE,axiom,
! [A_214: hoare_1167836817_state] :
~ ( member2058392318_state @ A_214 @ bot_bo70021908tate_o ) ).
thf(fact_26_emptyE,axiom,
! [A_214: nat] :
~ ( member_nat @ A_214 @ bot_bot_nat_o ) ).
thf(fact_27_singleton__conv2,axiom,
! [A_213: hoare_1775062406iple_a] :
( ( collec676402587iple_a @ ( fequal1288209029iple_a @ A_213 ) )
= ( insert1281456128iple_a @ A_213 @ bot_bo751897185le_a_o ) ) ).
thf(fact_28_singleton__conv2,axiom,
! [A_213: nat] :
( ( collect_nat @ ( fequal_nat @ A_213 ) )
= ( insert_nat @ A_213 @ bot_bot_nat_o ) ) ).
thf(fact_29_singleton__conv2,axiom,
! [A_213: nat > $o] :
( ( collect_nat_o @ ( fequal_nat_o @ A_213 ) )
= ( insert_nat_o @ A_213 @ bot_bot_nat_o_o ) ) ).
thf(fact_30_singleton__conv2,axiom,
! [A_213: hoare_1167836817_state] :
( ( collec1027672124_state @ ( fequal1831255762_state @ A_213 ) )
= ( insert2134838167_state @ A_213 @ bot_bo70021908tate_o ) ) ).
thf(fact_31_singleton__conv,axiom,
! [A_212: hoare_1775062406iple_a] :
( ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : X_3 = A_212 )
= ( insert1281456128iple_a @ A_212 @ bot_bo751897185le_a_o ) ) ).
thf(fact_32_singleton__conv,axiom,
! [A_212: nat] :
( ( collect_nat
@ ^ [X_3: nat] : X_3 = A_212 )
= ( insert_nat @ A_212 @ bot_bot_nat_o ) ) ).
thf(fact_33_singleton__conv,axiom,
! [A_212: nat > $o] :
( ( collect_nat_o
@ ^ [X_3: nat > $o] : X_3 = A_212 )
= ( insert_nat_o @ A_212 @ bot_bot_nat_o_o ) ) ).
thf(fact_34_singleton__conv,axiom,
! [A_212: hoare_1167836817_state] :
( ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : X_3 = A_212 )
= ( insert2134838167_state @ A_212 @ bot_bo70021908tate_o ) ) ).
thf(fact_35_Collect__conv__if2,axiom,
! [P_26: hoare_1775062406iple_a > $o,A_211: hoare_1775062406iple_a] :
( ( ( P_26 @ A_211 )
=> ( ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= ( insert1281456128iple_a @ A_211 @ bot_bo751897185le_a_o ) ) )
& ( ~ ( P_26 @ A_211 )
=> ( ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= bot_bo751897185le_a_o ) ) ) ).
thf(fact_36_Collect__conv__if2,axiom,
! [P_26: nat > $o,A_211: nat] :
( ( ( P_26 @ A_211 )
=> ( ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= ( insert_nat @ A_211 @ bot_bot_nat_o ) ) )
& ( ~ ( P_26 @ A_211 )
=> ( ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_37_Collect__conv__if2,axiom,
! [P_26: ( nat > $o ) > $o,A_211: nat > $o] :
( ( ( P_26 @ A_211 )
=> ( ( collect_nat_o
@ ^ [X_3: nat > $o] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= ( insert_nat_o @ A_211 @ bot_bot_nat_o_o ) ) )
& ( ~ ( P_26 @ A_211 )
=> ( ( collect_nat_o
@ ^ [X_3: nat > $o] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= bot_bot_nat_o_o ) ) ) ).
thf(fact_38_Collect__conv__if2,axiom,
! [P_26: hoare_1167836817_state > $o,A_211: hoare_1167836817_state] :
( ( ( P_26 @ A_211 )
=> ( ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= ( insert2134838167_state @ A_211 @ bot_bo70021908tate_o ) ) )
& ( ~ ( P_26 @ A_211 )
=> ( ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
= bot_bo70021908tate_o ) ) ) ).
thf(fact_39_Collect__conv__if,axiom,
! [P_25: hoare_1775062406iple_a > $o,A_210: hoare_1775062406iple_a] :
( ( ( P_25 @ A_210 )
=> ( ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= ( insert1281456128iple_a @ A_210 @ bot_bo751897185le_a_o ) ) )
& ( ~ ( P_25 @ A_210 )
=> ( ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= bot_bo751897185le_a_o ) ) ) ).
thf(fact_40_Collect__conv__if,axiom,
! [P_25: nat > $o,A_210: nat] :
( ( ( P_25 @ A_210 )
=> ( ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= ( insert_nat @ A_210 @ bot_bot_nat_o ) ) )
& ( ~ ( P_25 @ A_210 )
=> ( ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_41_Collect__conv__if,axiom,
! [P_25: ( nat > $o ) > $o,A_210: nat > $o] :
( ( ( P_25 @ A_210 )
=> ( ( collect_nat_o
@ ^ [X_3: nat > $o] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= ( insert_nat_o @ A_210 @ bot_bot_nat_o_o ) ) )
& ( ~ ( P_25 @ A_210 )
=> ( ( collect_nat_o
@ ^ [X_3: nat > $o] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= bot_bot_nat_o_o ) ) ) ).
thf(fact_42_Collect__conv__if,axiom,
! [P_25: hoare_1167836817_state > $o,A_210: hoare_1167836817_state] :
( ( ( P_25 @ A_210 )
=> ( ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= ( insert2134838167_state @ A_210 @ bot_bo70021908tate_o ) ) )
& ( ~ ( P_25 @ A_210 )
=> ( ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
= bot_bo70021908tate_o ) ) ) ).
thf(fact_43_equals0D,axiom,
! [A_209: hoare_1775062406iple_a,A_208: hoare_1775062406iple_a > $o] :
( ( A_208 = bot_bo751897185le_a_o )
=> ~ ( member2122167641iple_a @ A_209 @ A_208 ) ) ).
thf(fact_44_equals0D,axiom,
! [A_209: hoare_1167836817_state,A_208: hoare_1167836817_state > $o] :
( ( A_208 = bot_bo70021908tate_o )
=> ~ ( member2058392318_state @ A_209 @ A_208 ) ) ).
thf(fact_45_equals0D,axiom,
! [A_209: nat,A_208: nat > $o] :
( ( A_208 = bot_bot_nat_o )
=> ~ ( member_nat @ A_209 @ A_208 ) ) ).
thf(fact_46_Collect__empty__eq,axiom,
! [P_24: nat > $o] :
( ( ( collect_nat @ P_24 )
= bot_bot_nat_o )
<=> ! [X_3: nat] :
~ ( P_24 @ X_3 ) ) ).
thf(fact_47_Collect__empty__eq,axiom,
! [P_24: hoare_1775062406iple_a > $o] :
( ( ( collec676402587iple_a @ P_24 )
= bot_bo751897185le_a_o )
<=> ! [X_3: hoare_1775062406iple_a] :
~ ( P_24 @ X_3 ) ) ).
thf(fact_48_Collect__empty__eq,axiom,
! [P_24: ( nat > $o ) > $o] :
( ( ( collect_nat_o @ P_24 )
= bot_bot_nat_o_o )
<=> ! [X_3: nat > $o] :
~ ( P_24 @ X_3 ) ) ).
thf(fact_49_Collect__empty__eq,axiom,
! [P_24: hoare_1167836817_state > $o] :
( ( ( collec1027672124_state @ P_24 )
= bot_bo70021908tate_o )
<=> ! [X_3: hoare_1167836817_state] :
~ ( P_24 @ X_3 ) ) ).
thf(fact_50_empty__iff,axiom,
! [C_36: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ C_36 @ bot_bo751897185le_a_o ) ).
thf(fact_51_empty__iff,axiom,
! [C_36: hoare_1167836817_state] :
~ ( member2058392318_state @ C_36 @ bot_bo70021908tate_o ) ).
thf(fact_52_empty__iff,axiom,
! [C_36: nat] :
~ ( member_nat @ C_36 @ bot_bot_nat_o ) ).
thf(fact_53_empty__Collect__eq,axiom,
! [P_23: nat > $o] :
( ( bot_bot_nat_o
= ( collect_nat @ P_23 ) )
<=> ! [X_3: nat] :
~ ( P_23 @ X_3 ) ) ).
thf(fact_54_empty__Collect__eq,axiom,
! [P_23: hoare_1775062406iple_a > $o] :
( ( bot_bo751897185le_a_o
= ( collec676402587iple_a @ P_23 ) )
<=> ! [X_3: hoare_1775062406iple_a] :
~ ( P_23 @ X_3 ) ) ).
thf(fact_55_empty__Collect__eq,axiom,
! [P_23: ( nat > $o ) > $o] :
( ( bot_bot_nat_o_o
= ( collect_nat_o @ P_23 ) )
<=> ! [X_3: nat > $o] :
~ ( P_23 @ X_3 ) ) ).
thf(fact_56_empty__Collect__eq,axiom,
! [P_23: hoare_1167836817_state > $o] :
( ( bot_bo70021908tate_o
= ( collec1027672124_state @ P_23 ) )
<=> ! [X_3: hoare_1167836817_state] :
~ ( P_23 @ X_3 ) ) ).
thf(fact_57_ex__in__conv,axiom,
! [A_207: hoare_1775062406iple_a > $o] :
( ? [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ A_207 )
<=> ( A_207 != bot_bo751897185le_a_o ) ) ).
thf(fact_58_ex__in__conv,axiom,
! [A_207: hoare_1167836817_state > $o] :
( ? [X_3: hoare_1167836817_state] : ( member2058392318_state @ X_3 @ A_207 )
<=> ( A_207 != bot_bo70021908tate_o ) ) ).
thf(fact_59_ex__in__conv,axiom,
! [A_207: nat > $o] :
( ? [X_3: nat] : ( member_nat @ X_3 @ A_207 )
<=> ( A_207 != bot_bot_nat_o ) ) ).
thf(fact_60_all__not__in__conv,axiom,
! [A_206: hoare_1775062406iple_a > $o] :
( ! [X_3: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ X_3 @ A_206 )
<=> ( A_206 = bot_bo751897185le_a_o ) ) ).
thf(fact_61_all__not__in__conv,axiom,
! [A_206: hoare_1167836817_state > $o] :
( ! [X_3: hoare_1167836817_state] :
~ ( member2058392318_state @ X_3 @ A_206 )
<=> ( A_206 = bot_bo70021908tate_o ) ) ).
thf(fact_62_all__not__in__conv,axiom,
! [A_206: nat > $o] :
( ! [X_3: nat] :
~ ( member_nat @ X_3 @ A_206 )
<=> ( A_206 = bot_bot_nat_o ) ) ).
thf(fact_63_empty__def,axiom,
( bot_bot_nat_o
= ( collect_nat
@ ^ [X_3: nat] : $false ) ) ).
thf(fact_64_empty__def,axiom,
( bot_bo751897185le_a_o
= ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : $false ) ) ).
thf(fact_65_empty__def,axiom,
( bot_bot_nat_o_o
= ( collect_nat_o
@ ^ [X_3: nat > $o] : $false ) ) ).
thf(fact_66_empty__def,axiom,
( bot_bo70021908tate_o
= ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : $false ) ) ).
thf(fact_67_insert__absorb,axiom,
! [A_205: hoare_1775062406iple_a,A_204: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_205 @ A_204 )
=> ( ( insert1281456128iple_a @ A_205 @ A_204 )
= A_204 ) ) ).
thf(fact_68_insert__absorb,axiom,
! [A_205: nat,A_204: nat > $o] :
( ( member_nat @ A_205 @ A_204 )
=> ( ( insert_nat @ A_205 @ A_204 )
= A_204 ) ) ).
thf(fact_69_insert__absorb,axiom,
! [A_205: hoare_1167836817_state,A_204: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_205 @ A_204 )
=> ( ( insert2134838167_state @ A_205 @ A_204 )
= A_204 ) ) ).
thf(fact_70_insertI2,axiom,
! [B_96: hoare_1775062406iple_a,A_203: hoare_1775062406iple_a,B_95: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_203 @ B_95 )
=> ( member2122167641iple_a @ A_203 @ ( insert1281456128iple_a @ B_96 @ B_95 ) ) ) ).
thf(fact_71_insertI2,axiom,
! [B_96: nat,A_203: nat,B_95: nat > $o] :
( ( member_nat @ A_203 @ B_95 )
=> ( member_nat @ A_203 @ ( insert_nat @ B_96 @ B_95 ) ) ) ).
thf(fact_72_insertI2,axiom,
! [B_96: hoare_1167836817_state,A_203: hoare_1167836817_state,B_95: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_203 @ B_95 )
=> ( member2058392318_state @ A_203 @ ( insert2134838167_state @ B_96 @ B_95 ) ) ) ).
thf(fact_73_insert__ident,axiom,
! [B_94: hoare_1775062406iple_a > $o,X_100: hoare_1775062406iple_a,A_202: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ X_100 @ A_202 )
=> ( ~ ( member2122167641iple_a @ X_100 @ B_94 )
=> ( ( ( insert1281456128iple_a @ X_100 @ A_202 )
= ( insert1281456128iple_a @ X_100 @ B_94 ) )
<=> ( A_202 = B_94 ) ) ) ) ).
thf(fact_74_insert__ident,axiom,
! [B_94: nat > $o,X_100: nat,A_202: nat > $o] :
( ~ ( member_nat @ X_100 @ A_202 )
=> ( ~ ( member_nat @ X_100 @ B_94 )
=> ( ( ( insert_nat @ X_100 @ A_202 )
= ( insert_nat @ X_100 @ B_94 ) )
<=> ( A_202 = B_94 ) ) ) ) ).
thf(fact_75_insert__ident,axiom,
! [B_94: hoare_1167836817_state > $o,X_100: hoare_1167836817_state,A_202: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_100 @ A_202 )
=> ( ~ ( member2058392318_state @ X_100 @ B_94 )
=> ( ( ( insert2134838167_state @ X_100 @ A_202 )
= ( insert2134838167_state @ X_100 @ B_94 ) )
<=> ( A_202 = B_94 ) ) ) ) ).
thf(fact_76_insert__code,axiom,
! [Y_43: hoare_1775062406iple_a,A_201: hoare_1775062406iple_a > $o,X_99: hoare_1775062406iple_a] :
( ( insert1281456128iple_a @ Y_43 @ A_201 @ X_99 )
<=> ( ( Y_43 = X_99 )
| ( A_201 @ X_99 ) ) ) ).
thf(fact_77_insert__code,axiom,
! [Y_43: nat,A_201: nat > $o,X_99: nat] :
( ( insert_nat @ Y_43 @ A_201 @ X_99 )
<=> ( ( Y_43 = X_99 )
| ( A_201 @ X_99 ) ) ) ).
thf(fact_78_insert__code,axiom,
! [Y_43: hoare_1167836817_state,A_201: hoare_1167836817_state > $o,X_99: hoare_1167836817_state] :
( ( insert2134838167_state @ Y_43 @ A_201 @ X_99 )
<=> ( ( Y_43 = X_99 )
| ( A_201 @ X_99 ) ) ) ).
thf(fact_79_insert__iff,axiom,
! [A_200: hoare_1775062406iple_a,B_93: hoare_1775062406iple_a,A_199: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_200 @ ( insert1281456128iple_a @ B_93 @ A_199 ) )
<=> ( ( A_200 = B_93 )
| ( member2122167641iple_a @ A_200 @ A_199 ) ) ) ).
thf(fact_80_insert__iff,axiom,
! [A_200: nat,B_93: nat,A_199: nat > $o] :
( ( member_nat @ A_200 @ ( insert_nat @ B_93 @ A_199 ) )
<=> ( ( A_200 = B_93 )
| ( member_nat @ A_200 @ A_199 ) ) ) ).
thf(fact_81_insert__iff,axiom,
! [A_200: hoare_1167836817_state,B_93: hoare_1167836817_state,A_199: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_200 @ ( insert2134838167_state @ B_93 @ A_199 ) )
<=> ( ( A_200 = B_93 )
| ( member2058392318_state @ A_200 @ A_199 ) ) ) ).
thf(fact_82_insert__commute,axiom,
! [X_98: hoare_1775062406iple_a,Y_42: hoare_1775062406iple_a,A_198: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ X_98 @ ( insert1281456128iple_a @ Y_42 @ A_198 ) )
= ( insert1281456128iple_a @ Y_42 @ ( insert1281456128iple_a @ X_98 @ A_198 ) ) ) ).
thf(fact_83_insert__commute,axiom,
! [X_98: nat,Y_42: nat,A_198: nat > $o] :
( ( insert_nat @ X_98 @ ( insert_nat @ Y_42 @ A_198 ) )
= ( insert_nat @ Y_42 @ ( insert_nat @ X_98 @ A_198 ) ) ) ).
thf(fact_84_insert__commute,axiom,
! [X_98: hoare_1167836817_state,Y_42: hoare_1167836817_state,A_198: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X_98 @ ( insert2134838167_state @ Y_42 @ A_198 ) )
= ( insert2134838167_state @ Y_42 @ ( insert2134838167_state @ X_98 @ A_198 ) ) ) ).
thf(fact_85_insert__absorb2,axiom,
! [X_97: hoare_1775062406iple_a,A_197: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ X_97 @ ( insert1281456128iple_a @ X_97 @ A_197 ) )
= ( insert1281456128iple_a @ X_97 @ A_197 ) ) ).
thf(fact_86_insert__absorb2,axiom,
! [X_97: nat,A_197: nat > $o] :
( ( insert_nat @ X_97 @ ( insert_nat @ X_97 @ A_197 ) )
= ( insert_nat @ X_97 @ A_197 ) ) ).
thf(fact_87_insert__absorb2,axiom,
! [X_97: hoare_1167836817_state,A_197: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X_97 @ ( insert2134838167_state @ X_97 @ A_197 ) )
= ( insert2134838167_state @ X_97 @ A_197 ) ) ).
thf(fact_88_insert__Collect,axiom,
! [A_196: hoare_1775062406iple_a,P_22: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_196 @ ( collec676402587iple_a @ P_22 ) )
= ( collec676402587iple_a
@ ^ [U_1: hoare_1775062406iple_a] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
thf(fact_89_insert__Collect,axiom,
! [A_196: nat,P_22: nat > $o] :
( ( insert_nat @ A_196 @ ( collect_nat @ P_22 ) )
= ( collect_nat
@ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
thf(fact_90_insert__Collect,axiom,
! [A_196: nat > $o,P_22: ( nat > $o ) > $o] :
( ( insert_nat_o @ A_196 @ ( collect_nat_o @ P_22 ) )
= ( collect_nat_o
@ ^ [U_1: nat > $o] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
thf(fact_91_insert__Collect,axiom,
! [A_196: hoare_1167836817_state,P_22: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_196 @ ( collec1027672124_state @ P_22 ) )
= ( collec1027672124_state
@ ^ [U_1: hoare_1167836817_state] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
thf(fact_92_insert__compr,axiom,
! [A_195: hoare_1775062406iple_a,B_92: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_195 @ B_92 )
= ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( X_3 = A_195 ) @ ( member2122167641iple_a @ X_3 @ B_92 ) ) ) ) ).
thf(fact_93_insert__compr,axiom,
! [A_195: nat,B_92: nat > $o] :
( ( insert_nat @ A_195 @ B_92 )
= ( collect_nat
@ ^ [X_3: nat] : ( (|) @ ( X_3 = A_195 ) @ ( member_nat @ X_3 @ B_92 ) ) ) ) ).
thf(fact_94_insert__compr,axiom,
! [A_195: nat > $o,B_92: ( nat > $o ) > $o] :
( ( insert_nat_o @ A_195 @ B_92 )
= ( collect_nat_o
@ ^ [X_3: nat > $o] : ( (|) @ ( X_3 = A_195 ) @ ( member_nat_o @ X_3 @ B_92 ) ) ) ) ).
thf(fact_95_insert__compr,axiom,
! [A_195: hoare_1167836817_state,B_92: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_195 @ B_92 )
= ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : ( (|) @ ( X_3 = A_195 ) @ ( member2058392318_state @ X_3 @ B_92 ) ) ) ) ).
thf(fact_96_insertI1,axiom,
! [A_194: hoare_1775062406iple_a,B_91: hoare_1775062406iple_a > $o] : ( member2122167641iple_a @ A_194 @ ( insert1281456128iple_a @ A_194 @ B_91 ) ) ).
thf(fact_97_insertI1,axiom,
! [A_194: nat,B_91: nat > $o] : ( member_nat @ A_194 @ ( insert_nat @ A_194 @ B_91 ) ) ).
thf(fact_98_insertI1,axiom,
! [A_194: hoare_1167836817_state,B_91: hoare_1167836817_state > $o] : ( member2058392318_state @ A_194 @ ( insert2134838167_state @ A_194 @ B_91 ) ) ).
thf(fact_99_insert__compr__raw,axiom,
! [X_3: hoare_1775062406iple_a,Xa: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ X_3 @ Xa )
= ( collec676402587iple_a
@ ^ [Y_1: hoare_1775062406iple_a] : ( (|) @ ( Y_1 = X_3 ) @ ( member2122167641iple_a @ Y_1 @ Xa ) ) ) ) ).
thf(fact_100_insert__compr__raw,axiom,
! [X_3: nat,Xa: nat > $o] :
( ( insert_nat @ X_3 @ Xa )
= ( collect_nat
@ ^ [Y_1: nat] : ( (|) @ ( Y_1 = X_3 ) @ ( member_nat @ Y_1 @ Xa ) ) ) ) ).
thf(fact_101_insert__compr__raw,axiom,
! [X_3: nat > $o,Xa: ( nat > $o ) > $o] :
( ( insert_nat_o @ X_3 @ Xa )
= ( collect_nat_o
@ ^ [Y_1: nat > $o] : ( (|) @ ( Y_1 = X_3 ) @ ( member_nat_o @ Y_1 @ Xa ) ) ) ) ).
thf(fact_102_insert__compr__raw,axiom,
! [X_3: hoare_1167836817_state,Xa: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ X_3 @ Xa )
= ( collec1027672124_state
@ ^ [Y_1: hoare_1167836817_state] : ( (|) @ ( Y_1 = X_3 ) @ ( member2058392318_state @ Y_1 @ Xa ) ) ) ) ).
thf(fact_103_singleton__inject,axiom,
! [A_193: hoare_1775062406iple_a,B_90: hoare_1775062406iple_a] :
( ( ( insert1281456128iple_a @ A_193 @ bot_bo751897185le_a_o )
= ( insert1281456128iple_a @ B_90 @ bot_bo751897185le_a_o ) )
=> ( A_193 = B_90 ) ) ).
thf(fact_104_singleton__inject,axiom,
! [A_193: nat,B_90: nat] :
( ( ( insert_nat @ A_193 @ bot_bot_nat_o )
= ( insert_nat @ B_90 @ bot_bot_nat_o ) )
=> ( A_193 = B_90 ) ) ).
thf(fact_105_singleton__inject,axiom,
! [A_193: hoare_1167836817_state,B_90: hoare_1167836817_state] :
( ( ( insert2134838167_state @ A_193 @ bot_bo70021908tate_o )
= ( insert2134838167_state @ B_90 @ bot_bo70021908tate_o ) )
=> ( A_193 = B_90 ) ) ).
thf(fact_106_singletonE,axiom,
! [B_89: hoare_1775062406iple_a,A_192: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ B_89 @ ( insert1281456128iple_a @ A_192 @ bot_bo751897185le_a_o ) )
=> ( B_89 = A_192 ) ) ).
thf(fact_107_singletonE,axiom,
! [B_89: nat,A_192: nat] :
( ( member_nat @ B_89 @ ( insert_nat @ A_192 @ bot_bot_nat_o ) )
=> ( B_89 = A_192 ) ) ).
thf(fact_108_singletonE,axiom,
! [B_89: hoare_1167836817_state,A_192: hoare_1167836817_state] :
( ( member2058392318_state @ B_89 @ ( insert2134838167_state @ A_192 @ bot_bo70021908tate_o ) )
=> ( B_89 = A_192 ) ) ).
thf(fact_109_doubleton__eq__iff,axiom,
! [A_191: hoare_1775062406iple_a,B_88: hoare_1775062406iple_a,C_35: hoare_1775062406iple_a,D_4: hoare_1775062406iple_a] :
( ( ( insert1281456128iple_a @ A_191 @ ( insert1281456128iple_a @ B_88 @ bot_bo751897185le_a_o ) )
= ( insert1281456128iple_a @ C_35 @ ( insert1281456128iple_a @ D_4 @ bot_bo751897185le_a_o ) ) )
<=> ( ( ( A_191 = C_35 )
& ( B_88 = D_4 ) )
| ( ( A_191 = D_4 )
& ( B_88 = C_35 ) ) ) ) ).
thf(fact_110_doubleton__eq__iff,axiom,
! [A_191: nat,B_88: nat,C_35: nat,D_4: nat] :
( ( ( insert_nat @ A_191 @ ( insert_nat @ B_88 @ bot_bot_nat_o ) )
= ( insert_nat @ C_35 @ ( insert_nat @ D_4 @ bot_bot_nat_o ) ) )
<=> ( ( ( A_191 = C_35 )
& ( B_88 = D_4 ) )
| ( ( A_191 = D_4 )
& ( B_88 = C_35 ) ) ) ) ).
thf(fact_111_doubleton__eq__iff,axiom,
! [A_191: hoare_1167836817_state,B_88: hoare_1167836817_state,C_35: hoare_1167836817_state,D_4: hoare_1167836817_state] :
( ( ( insert2134838167_state @ A_191 @ ( insert2134838167_state @ B_88 @ bot_bo70021908tate_o ) )
= ( insert2134838167_state @ C_35 @ ( insert2134838167_state @ D_4 @ bot_bo70021908tate_o ) ) )
<=> ( ( ( A_191 = C_35 )
& ( B_88 = D_4 ) )
| ( ( A_191 = D_4 )
& ( B_88 = C_35 ) ) ) ) ).
thf(fact_112_singleton__iff,axiom,
! [B_87: hoare_1775062406iple_a,A_190: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ B_87 @ ( insert1281456128iple_a @ A_190 @ bot_bo751897185le_a_o ) )
<=> ( B_87 = A_190 ) ) ).
thf(fact_113_singleton__iff,axiom,
! [B_87: nat,A_190: nat] :
( ( member_nat @ B_87 @ ( insert_nat @ A_190 @ bot_bot_nat_o ) )
<=> ( B_87 = A_190 ) ) ).
thf(fact_114_singleton__iff,axiom,
! [B_87: hoare_1167836817_state,A_190: hoare_1167836817_state] :
( ( member2058392318_state @ B_87 @ ( insert2134838167_state @ A_190 @ bot_bo70021908tate_o ) )
<=> ( B_87 = A_190 ) ) ).
thf(fact_115_insert__not__empty,axiom,
! [A_189: hoare_1775062406iple_a,A_188: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_189 @ A_188 )
!= bot_bo751897185le_a_o ) ).
thf(fact_116_insert__not__empty,axiom,
! [A_189: nat,A_188: nat > $o] :
( ( insert_nat @ A_189 @ A_188 )
!= bot_bot_nat_o ) ).
thf(fact_117_insert__not__empty,axiom,
! [A_189: hoare_1167836817_state,A_188: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_189 @ A_188 )
!= bot_bo70021908tate_o ) ).
thf(fact_118_empty__not__insert,axiom,
! [A_187: hoare_1775062406iple_a,A_186: hoare_1775062406iple_a > $o] :
( bot_bo751897185le_a_o
!= ( insert1281456128iple_a @ A_187 @ A_186 ) ) ).
thf(fact_119_empty__not__insert,axiom,
! [A_187: nat,A_186: nat > $o] :
( bot_bot_nat_o
!= ( insert_nat @ A_187 @ A_186 ) ) ).
thf(fact_120_empty__not__insert,axiom,
! [A_187: hoare_1167836817_state,A_186: hoare_1167836817_state > $o] :
( bot_bo70021908tate_o
!= ( insert2134838167_state @ A_187 @ A_186 ) ) ).
thf(fact_121_the__elem__eq,axiom,
! [X_96: hoare_1775062406iple_a] :
( ( the_el1844711461iple_a @ ( insert1281456128iple_a @ X_96 @ bot_bo751897185le_a_o ) )
= X_96 ) ).
thf(fact_122_the__elem__eq,axiom,
! [X_96: nat] :
( ( the_elem_nat @ ( insert_nat @ X_96 @ bot_bot_nat_o ) )
= X_96 ) ).
thf(fact_123_the__elem__eq,axiom,
! [X_96: hoare_1167836817_state] :
( ( the_el323660082_state @ ( insert2134838167_state @ X_96 @ bot_bo70021908tate_o ) )
= X_96 ) ).
thf(fact_124_bot__apply,axiom,
! [X_95: hoare_1775062406iple_a] :
( ( bot_bo751897185le_a_o @ X_95 )
<=> bot_bot_o ) ).
thf(fact_125_bot__apply,axiom,
! [X_95: nat] :
( ( bot_bot_nat_o @ X_95 )
<=> bot_bot_o ) ).
thf(fact_126_bot__apply,axiom,
! [X_95: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X_95 )
<=> bot_bot_o ) ).
thf(fact_127_bot__fun__def,axiom,
! [X_3: hoare_1775062406iple_a] :
( ( bot_bo751897185le_a_o @ X_3 )
<=> bot_bot_o ) ).
thf(fact_128_bot__fun__def,axiom,
! [X_3: nat] :
( ( bot_bot_nat_o @ X_3 )
<=> bot_bot_o ) ).
thf(fact_129_bot__fun__def,axiom,
! [X_3: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X_3 )
<=> bot_bot_o ) ).
thf(fact_130_hoare__derivs_OSkip,axiom,
! [G_10: hoare_1775062406iple_a > $o,P_21: x_a > state > $o] : ( hoare_1508237396rivs_a @ G_10 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_21 @ skip @ P_21 ) @ bot_bo751897185le_a_o ) ) ).
thf(fact_131_hoare__derivs_OSkip,axiom,
! [G_10: hoare_1167836817_state > $o,P_21: state > state > $o] : ( hoare_123228589_state @ G_10 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ skip @ P_21 ) @ bot_bo70021908tate_o ) ) ).
thf(fact_132_Comp,axiom,
! [D_3: com,R_3: x_a > state > $o,G_9: hoare_1775062406iple_a > $o,P_20: x_a > state > $o,C_34: com,Q_8: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_20 @ C_34 @ Q_8 ) @ bot_bo751897185le_a_o ) )
=> ( ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ Q_8 @ D_3 @ R_3 ) @ bot_bo751897185le_a_o ) )
=> ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_20 @ ( semi @ C_34 @ D_3 ) @ R_3 ) @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_133_Comp,axiom,
! [D_3: com,R_3: state > state > $o,G_9: hoare_1167836817_state > $o,P_20: state > state > $o,C_34: com,Q_8: state > state > $o] :
( ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ C_34 @ Q_8 ) @ bot_bo70021908tate_o ) )
=> ( ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ Q_8 @ D_3 @ R_3 ) @ bot_bo70021908tate_o ) )
=> ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ ( semi @ C_34 @ D_3 ) @ R_3 ) @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_134_triple_Oexhaust,axiom,
! [Y_41: hoare_1167836817_state] :
~ ! [Fun1: state > state > $o,Com_2: com,Fun2: state > state > $o] :
( Y_41
!= ( hoare_908217195_state @ Fun1 @ Com_2 @ Fun2 ) ) ).
thf(fact_135_triple_Oexhaust,axiom,
! [Y_41: hoare_1775062406iple_a] :
~ ! [Fun1: x_a > state > $o,Com_2: com,Fun2: x_a > state > $o] :
( Y_41
!= ( hoare_1766022166iple_a @ Fun1 @ Com_2 @ Fun2 ) ) ).
thf(fact_136_Set_Oset__insert,axiom,
! [X_94: hoare_1775062406iple_a,A_185: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_94 @ A_185 )
=> ~ ! [B_62: hoare_1775062406iple_a > $o] :
( ( A_185
= ( insert1281456128iple_a @ X_94 @ B_62 ) )
=> ( member2122167641iple_a @ X_94 @ B_62 ) ) ) ).
thf(fact_137_Set_Oset__insert,axiom,
! [X_94: nat,A_185: nat > $o] :
( ( member_nat @ X_94 @ A_185 )
=> ~ ! [B_62: nat > $o] :
( ( A_185
= ( insert_nat @ X_94 @ B_62 ) )
=> ( member_nat @ X_94 @ B_62 ) ) ) ).
thf(fact_138_Set_Oset__insert,axiom,
! [X_94: hoare_1167836817_state,A_185: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_94 @ A_185 )
=> ~ ! [B_62: hoare_1167836817_state > $o] :
( ( A_185
= ( insert2134838167_state @ X_94 @ B_62 ) )
=> ( member2058392318_state @ X_94 @ B_62 ) ) ) ).
thf(fact_139_mk__disjoint__insert,axiom,
! [A_184: hoare_1775062406iple_a,A_183: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_184 @ A_183 )
=> ? [B_62: hoare_1775062406iple_a > $o] :
( ( A_183
= ( insert1281456128iple_a @ A_184 @ B_62 ) )
& ~ ( member2122167641iple_a @ A_184 @ B_62 ) ) ) ).
thf(fact_140_mk__disjoint__insert,axiom,
! [A_184: nat,A_183: nat > $o] :
( ( member_nat @ A_184 @ A_183 )
=> ? [B_62: nat > $o] :
( ( A_183
= ( insert_nat @ A_184 @ B_62 ) )
& ~ ( member_nat @ A_184 @ B_62 ) ) ) ).
thf(fact_141_mk__disjoint__insert,axiom,
! [A_184: hoare_1167836817_state,A_183: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_184 @ A_183 )
=> ? [B_62: hoare_1167836817_state > $o] :
( ( A_183
= ( insert2134838167_state @ A_184 @ B_62 ) )
& ~ ( member2058392318_state @ A_184 @ B_62 ) ) ) ).
thf(fact_142_equals0I,axiom,
! [A_182: hoare_1775062406iple_a > $o] :
( ! [Y_1: hoare_1775062406iple_a] :
~ ( member2122167641iple_a @ Y_1 @ A_182 )
=> ( A_182 = bot_bo751897185le_a_o ) ) ).
thf(fact_143_equals0I,axiom,
! [A_182: hoare_1167836817_state > $o] :
( ! [Y_1: hoare_1167836817_state] :
~ ( member2058392318_state @ Y_1 @ A_182 )
=> ( A_182 = bot_bo70021908tate_o ) ) ).
thf(fact_144_equals0I,axiom,
! [A_182: nat > $o] :
( ! [Y_1: nat] :
~ ( member_nat @ Y_1 @ A_182 )
=> ( A_182 = bot_bot_nat_o ) ) ).
thf(fact_145_conseq,axiom,
! [Q_6: x_a > state > $o,G_8: hoare_1775062406iple_a > $o,C_33: com,P_18: x_a > state > $o] :
( ! [Z: x_a,S: state] :
( ( P_18 @ Z @ S )
=> ? [P_19: x_a > state > $o,Q_7: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_19 @ C_33 @ Q_7 ) @ bot_bo751897185le_a_o ) )
& ! [S_5: state] :
( ! [Z_23: x_a] :
( ( P_19 @ Z_23 @ S )
=> ( Q_7 @ Z_23 @ S_5 ) )
=> ( Q_6 @ Z @ S_5 ) ) ) )
=> ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_18 @ C_33 @ Q_6 ) @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_146_conseq,axiom,
! [Q_6: state > state > $o,G_8: hoare_1167836817_state > $o,C_33: com,P_18: state > state > $o] :
( ! [Z: state,S: state] :
( ( P_18 @ Z @ S )
=> ? [P_19: state > state > $o,Q_7: state > state > $o] :
( ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_19 @ C_33 @ Q_7 ) @ bot_bo70021908tate_o ) )
& ! [S_5: state] :
( ! [Z_23: state] :
( ( P_19 @ Z_23 @ S )
=> ( Q_7 @ Z_23 @ S_5 ) )
=> ( Q_6 @ Z @ S_5 ) ) ) )
=> ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_18 @ C_33 @ Q_6 ) @ bot_bo70021908tate_o ) ) ) ).
thf(fact_147_com_Osimps_I13_J,axiom,
! [Com1: com,Com2: com] :
( ( semi @ Com1 @ Com2 )
!= skip ) ).
thf(fact_148_com_Osimps_I12_J,axiom,
! [Com1: com,Com2: com] :
( skip
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_149_the__elem__def,axiom,
! [X_93: hoare_1775062406iple_a > $o] :
( ( the_el1844711461iple_a @ X_93 )
= ( the_Ho1155011127iple_a
@ ^ [X_3: hoare_1775062406iple_a] :
( X_93
= ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_150_the__elem__def,axiom,
! [X_93: nat > $o] :
( ( the_elem_nat @ X_93 )
= ( the_nat
@ ^ [X_3: nat] :
( X_93
= ( insert_nat @ X_3 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_151_the__elem__def,axiom,
! [X_93: hoare_1167836817_state > $o] :
( ( the_el323660082_state @ X_93 )
= ( the_Ho310147232_state
@ ^ [X_3: hoare_1167836817_state] :
( X_93
= ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_152_com_Osimps_I3_J,axiom,
! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
( ( ( semi @ Com1_1 @ Com2_1 )
= ( semi @ Com1 @ Com2 ) )
<=> ( ( Com1_1 = Com1 )
& ( Com2_1 = Com2 ) ) ) ).
thf(fact_153_nonempty__iff,axiom,
! [A_181: hoare_1775062406iple_a > $o] :
( ( A_181 != bot_bo751897185le_a_o )
<=> ? [X_3: hoare_1775062406iple_a,B_62: hoare_1775062406iple_a > $o] :
( ( A_181
= ( insert1281456128iple_a @ X_3 @ B_62 ) )
& ~ ( member2122167641iple_a @ X_3 @ B_62 ) ) ) ).
thf(fact_154_nonempty__iff,axiom,
! [A_181: nat > $o] :
( ( A_181 != bot_bot_nat_o )
<=> ? [X_3: nat,B_62: nat > $o] :
( ( A_181
= ( insert_nat @ X_3 @ B_62 ) )
& ~ ( member_nat @ X_3 @ B_62 ) ) ) ).
thf(fact_155_nonempty__iff,axiom,
! [A_181: hoare_1167836817_state > $o] :
( ( A_181 != bot_bo70021908tate_o )
<=> ? [X_3: hoare_1167836817_state,B_62: hoare_1167836817_state > $o] :
( ( A_181
= ( insert2134838167_state @ X_3 @ B_62 ) )
& ~ ( member2058392318_state @ X_3 @ B_62 ) ) ) ).
thf(fact_156_bot__empty__eq,axiom,
! [X_3: hoare_1775062406iple_a] :
( ( bot_bo751897185le_a_o @ X_3 )
<=> ( member2122167641iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ).
thf(fact_157_bot__empty__eq,axiom,
! [X_3: hoare_1167836817_state] :
( ( bot_bo70021908tate_o @ X_3 )
<=> ( member2058392318_state @ X_3 @ bot_bo70021908tate_o ) ) ).
thf(fact_158_bot__empty__eq,axiom,
! [X_3: nat] :
( ( bot_bot_nat_o @ X_3 )
<=> ( member_nat @ X_3 @ bot_bot_nat_o ) ) ).
thf(fact_159_Ass,axiom,
! [G_7: hoare_1775062406iple_a > $o,P_17: x_a > state > $o,X_92: vname,A_180: state > nat] :
( hoare_1508237396rivs_a @ G_7
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Z: x_a,S: state] : ( P_17 @ Z @ ( update @ S @ X_92 @ ( A_180 @ S ) ) )
@ ( ass @ X_92 @ A_180 )
@ P_17 )
@ bot_bo751897185le_a_o ) ) ).
thf(fact_160_Ass,axiom,
! [G_7: hoare_1167836817_state > $o,P_17: state > state > $o,X_92: vname,A_180: state > nat] :
( hoare_123228589_state @ G_7
@ ( insert2134838167_state
@ ( hoare_908217195_state
@ ^ [Z: state,S: state] : ( P_17 @ Z @ ( update @ S @ X_92 @ ( A_180 @ S ) ) )
@ ( ass @ X_92 @ A_180 )
@ P_17 )
@ bot_bo70021908tate_o ) ) ).
thf(fact_161_image__constant__conv,axiom,
! [C_32: hoare_1775062406iple_a,A_179: nat > $o] :
( ( ( A_179 = bot_bot_nat_o )
=> ( ( image_43014529iple_a
@ ^ [X_3: nat] : C_32
@ A_179 )
= bot_bo751897185le_a_o ) )
& ( ( A_179 != bot_bot_nat_o )
=> ( ( image_43014529iple_a
@ ^ [X_3: nat] : C_32
@ A_179 )
= ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_162_image__constant__conv,axiom,
! [C_32: hoare_1775062406iple_a,A_179: hoare_1167836817_state > $o] :
( ( ( A_179 = bot_bo70021908tate_o )
=> ( ( image_1802845250iple_a
@ ^ [X_3: hoare_1167836817_state] : C_32
@ A_179 )
= bot_bo751897185le_a_o ) )
& ( ( A_179 != bot_bo70021908tate_o )
=> ( ( image_1802845250iple_a
@ ^ [X_3: hoare_1167836817_state] : C_32
@ A_179 )
= ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_163_image__constant__conv,axiom,
! [C_32: nat,A_179: hoare_1775062406iple_a > $o] :
( ( ( A_179 = bot_bo751897185le_a_o )
=> ( ( image_1806517641_a_nat
@ ^ [X_3: hoare_1775062406iple_a] : C_32
@ A_179 )
= bot_bot_nat_o ) )
& ( ( A_179 != bot_bo751897185le_a_o )
=> ( ( image_1806517641_a_nat
@ ^ [X_3: hoare_1775062406iple_a] : C_32
@ A_179 )
= ( insert_nat @ C_32 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_164_image__constant__conv,axiom,
! [C_32: hoare_1167836817_state,A_179: hoare_1775062406iple_a > $o] :
( ( ( A_179 = bot_bo751897185le_a_o )
=> ( ( image_1021683026_state
@ ^ [X_3: hoare_1775062406iple_a] : C_32
@ A_179 )
= bot_bo70021908tate_o ) )
& ( ( A_179 != bot_bo751897185le_a_o )
=> ( ( image_1021683026_state
@ ^ [X_3: hoare_1775062406iple_a] : C_32
@ A_179 )
= ( insert2134838167_state @ C_32 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_165_image__constant__conv,axiom,
! [C_32: nat,A_179: nat > $o] :
( ( ( A_179 = bot_bot_nat_o )
=> ( ( image_nat_nat
@ ^ [X_3: nat] : C_32
@ A_179 )
= bot_bot_nat_o ) )
& ( ( A_179 != bot_bot_nat_o )
=> ( ( image_nat_nat
@ ^ [X_3: nat] : C_32
@ A_179 )
= ( insert_nat @ C_32 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_166_image__constant__conv,axiom,
! [C_32: hoare_1167836817_state,A_179: hoare_1167836817_state > $o] :
( ( ( A_179 = bot_bo70021908tate_o )
=> ( ( image_31595733_state
@ ^ [X_3: hoare_1167836817_state] : C_32
@ A_179 )
= bot_bo70021908tate_o ) )
& ( ( A_179 != bot_bo70021908tate_o )
=> ( ( image_31595733_state
@ ^ [X_3: hoare_1167836817_state] : C_32
@ A_179 )
= ( insert2134838167_state @ C_32 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_167_image__constant__conv,axiom,
! [C_32: hoare_1775062406iple_a,A_179: hoare_1775062406iple_a > $o] :
( ( ( A_179 = bot_bo751897185le_a_o )
=> ( ( image_1170193413iple_a
@ ^ [X_3: hoare_1775062406iple_a] : C_32
@ A_179 )
= bot_bo751897185le_a_o ) )
& ( ( A_179 != bot_bo751897185le_a_o )
=> ( ( image_1170193413iple_a
@ ^ [X_3: hoare_1775062406iple_a] : C_32
@ A_179 )
= ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_168_image__constant,axiom,
! [C_31: hoare_1775062406iple_a,X_91: nat,A_178: nat > $o] :
( ( member_nat @ X_91 @ A_178 )
=> ( ( image_43014529iple_a
@ ^ [X_3: nat] : C_31
@ A_178 )
= ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_169_image__constant,axiom,
! [C_31: hoare_1775062406iple_a,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_91 @ A_178 )
=> ( ( image_1170193413iple_a
@ ^ [X_3: hoare_1775062406iple_a] : C_31
@ A_178 )
= ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_170_image__constant,axiom,
! [C_31: hoare_1775062406iple_a,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_91 @ A_178 )
=> ( ( image_1802845250iple_a
@ ^ [X_3: hoare_1167836817_state] : C_31
@ A_178 )
= ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_171_image__constant,axiom,
! [C_31: nat,X_91: nat,A_178: nat > $o] :
( ( member_nat @ X_91 @ A_178 )
=> ( ( image_nat_nat
@ ^ [X_3: nat] : C_31
@ A_178 )
= ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
thf(fact_172_image__constant,axiom,
! [C_31: nat,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_91 @ A_178 )
=> ( ( image_1806517641_a_nat
@ ^ [X_3: hoare_1775062406iple_a] : C_31
@ A_178 )
= ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
thf(fact_173_image__constant,axiom,
! [C_31: nat,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_91 @ A_178 )
=> ( ( image_1476618182te_nat
@ ^ [X_3: hoare_1167836817_state] : C_31
@ A_178 )
= ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
thf(fact_174_image__constant,axiom,
! [C_31: hoare_1167836817_state,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_91 @ A_178 )
=> ( ( image_31595733_state
@ ^ [X_3: hoare_1167836817_state] : C_31
@ A_178 )
= ( insert2134838167_state @ C_31 @ bot_bo70021908tate_o ) ) ) ).
thf(fact_175_image__constant,axiom,
! [C_31: hoare_1167836817_state,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_91 @ A_178 )
=> ( ( image_1021683026_state
@ ^ [X_3: hoare_1775062406iple_a] : C_31
@ A_178 )
= ( insert2134838167_state @ C_31 @ bot_bo70021908tate_o ) ) ) ).
thf(fact_176_image__eqI,axiom,
! [A_177: hoare_1167836817_state > $o,B_86: hoare_1167836817_state,F_65: hoare_1167836817_state > hoare_1167836817_state,X_90: hoare_1167836817_state] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member2058392318_state @ X_90 @ A_177 )
=> ( member2058392318_state @ B_86 @ ( image_31595733_state @ F_65 @ A_177 ) ) ) ) ).
thf(fact_177_image__eqI,axiom,
! [A_177: nat > $o,B_86: hoare_1167836817_state,F_65: nat > hoare_1167836817_state,X_90: nat] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member_nat @ X_90 @ A_177 )
=> ( member2058392318_state @ B_86 @ ( image_2121260246_state @ F_65 @ A_177 ) ) ) ) ).
thf(fact_178_image__eqI,axiom,
! [A_177: hoare_1775062406iple_a > $o,B_86: hoare_1167836817_state,F_65: hoare_1775062406iple_a > hoare_1167836817_state,X_90: hoare_1775062406iple_a] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member2122167641iple_a @ X_90 @ A_177 )
=> ( member2058392318_state @ B_86 @ ( image_1021683026_state @ F_65 @ A_177 ) ) ) ) ).
thf(fact_179_image__eqI,axiom,
! [A_177: nat > $o,B_86: nat,F_65: nat > nat,X_90: nat] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member_nat @ X_90 @ A_177 )
=> ( member_nat @ B_86 @ ( image_nat_nat @ F_65 @ A_177 ) ) ) ) ).
thf(fact_180_image__eqI,axiom,
! [A_177: hoare_1775062406iple_a > $o,B_86: nat,F_65: hoare_1775062406iple_a > nat,X_90: hoare_1775062406iple_a] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member2122167641iple_a @ X_90 @ A_177 )
=> ( member_nat @ B_86 @ ( image_1806517641_a_nat @ F_65 @ A_177 ) ) ) ) ).
thf(fact_181_image__eqI,axiom,
! [A_177: hoare_1167836817_state > $o,B_86: nat,F_65: hoare_1167836817_state > nat,X_90: hoare_1167836817_state] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member2058392318_state @ X_90 @ A_177 )
=> ( member_nat @ B_86 @ ( image_1476618182te_nat @ F_65 @ A_177 ) ) ) ) ).
thf(fact_182_image__eqI,axiom,
! [A_177: hoare_1775062406iple_a > $o,B_86: hoare_1775062406iple_a,F_65: hoare_1775062406iple_a > hoare_1775062406iple_a,X_90: hoare_1775062406iple_a] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member2122167641iple_a @ X_90 @ A_177 )
=> ( member2122167641iple_a @ B_86 @ ( image_1170193413iple_a @ F_65 @ A_177 ) ) ) ) ).
thf(fact_183_image__eqI,axiom,
! [A_177: hoare_1167836817_state > $o,B_86: hoare_1775062406iple_a,F_65: hoare_1167836817_state > hoare_1775062406iple_a,X_90: hoare_1167836817_state] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member2058392318_state @ X_90 @ A_177 )
=> ( member2122167641iple_a @ B_86 @ ( image_1802845250iple_a @ F_65 @ A_177 ) ) ) ) ).
thf(fact_184_image__eqI,axiom,
! [A_177: nat > $o,B_86: hoare_1775062406iple_a,F_65: nat > hoare_1775062406iple_a,X_90: nat] :
( ( B_86
= ( F_65 @ X_90 ) )
=> ( ( member_nat @ X_90 @ A_177 )
=> ( member2122167641iple_a @ B_86 @ ( image_43014529iple_a @ F_65 @ A_177 ) ) ) ) ).
thf(fact_185_image__ident,axiom,
! [Y_40: nat > $o] :
( ( image_nat_nat
@ ^ [X_3: nat] : X_3
@ Y_40 )
= Y_40 ) ).
thf(fact_186_image__ident,axiom,
! [Y_40: hoare_1167836817_state > $o] :
( ( image_31595733_state
@ ^ [X_3: hoare_1167836817_state] : X_3
@ Y_40 )
= Y_40 ) ).
thf(fact_187_image__ident,axiom,
! [Y_40: hoare_1775062406iple_a > $o] :
( ( image_1170193413iple_a
@ ^ [X_3: hoare_1775062406iple_a] : X_3
@ Y_40 )
= Y_40 ) ).
thf(fact_188_com_Osimps_I1_J,axiom,
! [Vname: vname,Fun_1: state > nat,Vname_1: vname,Fun: state > nat] :
( ( ( ass @ Vname @ Fun_1 )
= ( ass @ Vname_1 @ Fun ) )
<=> ( ( Vname = Vname_1 )
& ( Fun_1 = Fun ) ) ) ).
thf(fact_189_rev__image__eqI,axiom,
! [B_85: hoare_1167836817_state,F_64: hoare_1167836817_state > hoare_1167836817_state,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member2058392318_state @ B_85 @ ( image_31595733_state @ F_64 @ A_176 ) ) ) ) ).
thf(fact_190_rev__image__eqI,axiom,
! [B_85: hoare_1167836817_state,F_64: nat > hoare_1167836817_state,X_89: nat,A_176: nat > $o] :
( ( member_nat @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member2058392318_state @ B_85 @ ( image_2121260246_state @ F_64 @ A_176 ) ) ) ) ).
thf(fact_191_rev__image__eqI,axiom,
! [B_85: hoare_1167836817_state,F_64: hoare_1775062406iple_a > hoare_1167836817_state,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member2058392318_state @ B_85 @ ( image_1021683026_state @ F_64 @ A_176 ) ) ) ) ).
thf(fact_192_rev__image__eqI,axiom,
! [B_85: nat,F_64: nat > nat,X_89: nat,A_176: nat > $o] :
( ( member_nat @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member_nat @ B_85 @ ( image_nat_nat @ F_64 @ A_176 ) ) ) ) ).
thf(fact_193_rev__image__eqI,axiom,
! [B_85: nat,F_64: hoare_1775062406iple_a > nat,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member_nat @ B_85 @ ( image_1806517641_a_nat @ F_64 @ A_176 ) ) ) ) ).
thf(fact_194_rev__image__eqI,axiom,
! [B_85: nat,F_64: hoare_1167836817_state > nat,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member_nat @ B_85 @ ( image_1476618182te_nat @ F_64 @ A_176 ) ) ) ) ).
thf(fact_195_rev__image__eqI,axiom,
! [B_85: hoare_1775062406iple_a,F_64: hoare_1775062406iple_a > hoare_1775062406iple_a,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member2122167641iple_a @ B_85 @ ( image_1170193413iple_a @ F_64 @ A_176 ) ) ) ) ).
thf(fact_196_rev__image__eqI,axiom,
! [B_85: hoare_1775062406iple_a,F_64: hoare_1167836817_state > hoare_1775062406iple_a,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member2122167641iple_a @ B_85 @ ( image_1802845250iple_a @ F_64 @ A_176 ) ) ) ) ).
thf(fact_197_rev__image__eqI,axiom,
! [B_85: hoare_1775062406iple_a,F_64: nat > hoare_1775062406iple_a,X_89: nat,A_176: nat > $o] :
( ( member_nat @ X_89 @ A_176 )
=> ( ( B_85
= ( F_64 @ X_89 ) )
=> ( member2122167641iple_a @ B_85 @ ( image_43014529iple_a @ F_64 @ A_176 ) ) ) ) ).
thf(fact_198_imageI,axiom,
! [F_63: hoare_1167836817_state > hoare_1167836817_state,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_88 @ A_175 )
=> ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_31595733_state @ F_63 @ A_175 ) ) ) ).
thf(fact_199_imageI,axiom,
! [F_63: nat > hoare_1167836817_state,X_88: nat,A_175: nat > $o] :
( ( member_nat @ X_88 @ A_175 )
=> ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_2121260246_state @ F_63 @ A_175 ) ) ) ).
thf(fact_200_imageI,axiom,
! [F_63: hoare_1775062406iple_a > hoare_1167836817_state,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_88 @ A_175 )
=> ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_1021683026_state @ F_63 @ A_175 ) ) ) ).
thf(fact_201_imageI,axiom,
! [F_63: nat > nat,X_88: nat,A_175: nat > $o] :
( ( member_nat @ X_88 @ A_175 )
=> ( member_nat @ ( F_63 @ X_88 ) @ ( image_nat_nat @ F_63 @ A_175 ) ) ) ).
thf(fact_202_imageI,axiom,
! [F_63: hoare_1775062406iple_a > nat,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_88 @ A_175 )
=> ( member_nat @ ( F_63 @ X_88 ) @ ( image_1806517641_a_nat @ F_63 @ A_175 ) ) ) ).
thf(fact_203_imageI,axiom,
! [F_63: hoare_1167836817_state > nat,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_88 @ A_175 )
=> ( member_nat @ ( F_63 @ X_88 ) @ ( image_1476618182te_nat @ F_63 @ A_175 ) ) ) ).
thf(fact_204_imageI,axiom,
! [F_63: hoare_1775062406iple_a > hoare_1775062406iple_a,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_88 @ A_175 )
=> ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_1170193413iple_a @ F_63 @ A_175 ) ) ) ).
thf(fact_205_imageI,axiom,
! [F_63: hoare_1167836817_state > hoare_1775062406iple_a,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_88 @ A_175 )
=> ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_1802845250iple_a @ F_63 @ A_175 ) ) ) ).
thf(fact_206_imageI,axiom,
! [F_63: nat > hoare_1775062406iple_a,X_88: nat,A_175: nat > $o] :
( ( member_nat @ X_88 @ A_175 )
=> ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_43014529iple_a @ F_63 @ A_175 ) ) ) ).
thf(fact_207_image__iff,axiom,
! [Z_22: hoare_1167836817_state,F_62: hoare_1167836817_state > hoare_1167836817_state,A_174: hoare_1167836817_state > $o] :
( ( member2058392318_state @ Z_22 @ ( image_31595733_state @ F_62 @ A_174 ) )
<=> ? [X_3: hoare_1167836817_state] :
( ( member2058392318_state @ X_3 @ A_174 )
& ( Z_22
= ( F_62 @ X_3 ) ) ) ) ).
thf(fact_208_image__iff,axiom,
! [Z_22: hoare_1167836817_state,F_62: hoare_1775062406iple_a > hoare_1167836817_state,A_174: hoare_1775062406iple_a > $o] :
( ( member2058392318_state @ Z_22 @ ( image_1021683026_state @ F_62 @ A_174 ) )
<=> ? [X_3: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X_3 @ A_174 )
& ( Z_22
= ( F_62 @ X_3 ) ) ) ) ).
thf(fact_209_image__iff,axiom,
! [Z_22: nat,F_62: nat > nat,A_174: nat > $o] :
( ( member_nat @ Z_22 @ ( image_nat_nat @ F_62 @ A_174 ) )
<=> ? [X_3: nat] :
( ( member_nat @ X_3 @ A_174 )
& ( Z_22
= ( F_62 @ X_3 ) ) ) ) ).
thf(fact_210_image__iff,axiom,
! [Z_22: nat,F_62: hoare_1775062406iple_a > nat,A_174: hoare_1775062406iple_a > $o] :
( ( member_nat @ Z_22 @ ( image_1806517641_a_nat @ F_62 @ A_174 ) )
<=> ? [X_3: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X_3 @ A_174 )
& ( Z_22
= ( F_62 @ X_3 ) ) ) ) ).
thf(fact_211_image__iff,axiom,
! [Z_22: hoare_1775062406iple_a,F_62: hoare_1775062406iple_a > hoare_1775062406iple_a,A_174: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ Z_22 @ ( image_1170193413iple_a @ F_62 @ A_174 ) )
<=> ? [X_3: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X_3 @ A_174 )
& ( Z_22
= ( F_62 @ X_3 ) ) ) ) ).
thf(fact_212_image__iff,axiom,
! [Z_22: hoare_1775062406iple_a,F_62: hoare_1167836817_state > hoare_1775062406iple_a,A_174: hoare_1167836817_state > $o] :
( ( member2122167641iple_a @ Z_22 @ ( image_1802845250iple_a @ F_62 @ A_174 ) )
<=> ? [X_3: hoare_1167836817_state] :
( ( member2058392318_state @ X_3 @ A_174 )
& ( Z_22
= ( F_62 @ X_3 ) ) ) ) ).
thf(fact_213_image__iff,axiom,
! [Z_22: hoare_1775062406iple_a,F_62: nat > hoare_1775062406iple_a,A_174: nat > $o] :
( ( member2122167641iple_a @ Z_22 @ ( image_43014529iple_a @ F_62 @ A_174 ) )
<=> ? [X_3: nat] :
( ( member_nat @ X_3 @ A_174 )
& ( Z_22
= ( F_62 @ X_3 ) ) ) ) ).
thf(fact_214_com_Osimps_I24_J,axiom,
! [Vname: vname,Fun_1: state > nat,Com1: com,Com2: com] :
( ( ass @ Vname @ Fun_1 )
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_215_com_Osimps_I25_J,axiom,
! [Com1: com,Com2: com,Vname: vname,Fun_1: state > nat] :
( ( semi @ Com1 @ Com2 )
!= ( ass @ Vname @ Fun_1 ) ) ).
thf(fact_216_com_Osimps_I8_J,axiom,
! [Vname_1: vname,Fun: state > nat] :
( skip
!= ( ass @ Vname_1 @ Fun ) ) ).
thf(fact_217_com_Osimps_I9_J,axiom,
! [Vname_1: vname,Fun: state > nat] :
( ( ass @ Vname_1 @ Fun )
!= skip ) ).
thf(fact_218_image__is__empty,axiom,
! [F_61: hoare_1167836817_state > hoare_1775062406iple_a,A_173: hoare_1167836817_state > $o] :
( ( ( image_1802845250iple_a @ F_61 @ A_173 )
= bot_bo751897185le_a_o )
<=> ( A_173 = bot_bo70021908tate_o ) ) ).
thf(fact_219_image__is__empty,axiom,
! [F_61: nat > hoare_1775062406iple_a,A_173: nat > $o] :
( ( ( image_43014529iple_a @ F_61 @ A_173 )
= bot_bo751897185le_a_o )
<=> ( A_173 = bot_bot_nat_o ) ) ).
thf(fact_220_image__is__empty,axiom,
! [F_61: hoare_1775062406iple_a > hoare_1167836817_state,A_173: hoare_1775062406iple_a > $o] :
( ( ( image_1021683026_state @ F_61 @ A_173 )
= bot_bo70021908tate_o )
<=> ( A_173 = bot_bo751897185le_a_o ) ) ).
thf(fact_221_image__is__empty,axiom,
! [F_61: hoare_1775062406iple_a > nat,A_173: hoare_1775062406iple_a > $o] :
( ( ( image_1806517641_a_nat @ F_61 @ A_173 )
= bot_bot_nat_o )
<=> ( A_173 = bot_bo751897185le_a_o ) ) ).
thf(fact_222_image__empty,axiom,
! [F_60: hoare_1775062406iple_a > hoare_1167836817_state] :
( ( image_1021683026_state @ F_60 @ bot_bo751897185le_a_o )
= bot_bo70021908tate_o ) ).
thf(fact_223_image__empty,axiom,
! [F_60: hoare_1775062406iple_a > nat] :
( ( image_1806517641_a_nat @ F_60 @ bot_bo751897185le_a_o )
= bot_bot_nat_o ) ).
thf(fact_224_image__empty,axiom,
! [F_60: hoare_1167836817_state > hoare_1775062406iple_a] :
( ( image_1802845250iple_a @ F_60 @ bot_bo70021908tate_o )
= bot_bo751897185le_a_o ) ).
thf(fact_225_image__empty,axiom,
! [F_60: nat > hoare_1775062406iple_a] :
( ( image_43014529iple_a @ F_60 @ bot_bot_nat_o )
= bot_bo751897185le_a_o ) ).
thf(fact_226_empty__is__image,axiom,
! [F_59: hoare_1167836817_state > hoare_1775062406iple_a,A_172: hoare_1167836817_state > $o] :
( ( bot_bo751897185le_a_o
= ( image_1802845250iple_a @ F_59 @ A_172 ) )
<=> ( A_172 = bot_bo70021908tate_o ) ) ).
thf(fact_227_empty__is__image,axiom,
! [F_59: nat > hoare_1775062406iple_a,A_172: nat > $o] :
( ( bot_bo751897185le_a_o
= ( image_43014529iple_a @ F_59 @ A_172 ) )
<=> ( A_172 = bot_bot_nat_o ) ) ).
thf(fact_228_empty__is__image,axiom,
! [F_59: hoare_1775062406iple_a > hoare_1167836817_state,A_172: hoare_1775062406iple_a > $o] :
( ( bot_bo70021908tate_o
= ( image_1021683026_state @ F_59 @ A_172 ) )
<=> ( A_172 = bot_bo751897185le_a_o ) ) ).
thf(fact_229_empty__is__image,axiom,
! [F_59: hoare_1775062406iple_a > nat,A_172: hoare_1775062406iple_a > $o] :
( ( bot_bot_nat_o
= ( image_1806517641_a_nat @ F_59 @ A_172 ) )
<=> ( A_172 = bot_bo751897185le_a_o ) ) ).
thf(fact_230_mem__def,axiom,
! [X_87: hoare_1775062406iple_a,A_171: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_87 @ A_171 )
<=> ( A_171 @ X_87 ) ) ).
thf(fact_231_mem__def,axiom,
! [X_87: nat,A_171: nat > $o] :
( ( member_nat @ X_87 @ A_171 )
<=> ( A_171 @ X_87 ) ) ).
thf(fact_232_Collect__def,axiom,
! [P_16: hoare_1775062406iple_a > $o] :
( ( collec676402587iple_a @ P_16 )
= P_16 ) ).
thf(fact_233_Collect__def,axiom,
! [P_16: nat > $o] :
( ( collect_nat @ P_16 )
= P_16 ) ).
thf(fact_234_insert__image,axiom,
! [F_58: hoare_1775062406iple_a > hoare_1775062406iple_a,X_86: hoare_1775062406iple_a,A_170: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_86 @ A_170 )
=> ( ( insert1281456128iple_a @ ( F_58 @ X_86 ) @ ( image_1170193413iple_a @ F_58 @ A_170 ) )
= ( image_1170193413iple_a @ F_58 @ A_170 ) ) ) ).
thf(fact_235_insert__image,axiom,
! [F_58: nat > hoare_1775062406iple_a,X_86: nat,A_170: nat > $o] :
( ( member_nat @ X_86 @ A_170 )
=> ( ( insert1281456128iple_a @ ( F_58 @ X_86 ) @ ( image_43014529iple_a @ F_58 @ A_170 ) )
= ( image_43014529iple_a @ F_58 @ A_170 ) ) ) ).
thf(fact_236_image__insert,axiom,
! [F_57: hoare_1775062406iple_a > hoare_1167836817_state,A_169: hoare_1775062406iple_a,B_84: hoare_1775062406iple_a > $o] :
( ( image_1021683026_state @ F_57 @ ( insert1281456128iple_a @ A_169 @ B_84 ) )
= ( insert2134838167_state @ ( F_57 @ A_169 ) @ ( image_1021683026_state @ F_57 @ B_84 ) ) ) ).
thf(fact_237_image__insert,axiom,
! [F_57: hoare_1775062406iple_a > nat,A_169: hoare_1775062406iple_a,B_84: hoare_1775062406iple_a > $o] :
( ( image_1806517641_a_nat @ F_57 @ ( insert1281456128iple_a @ A_169 @ B_84 ) )
= ( insert_nat @ ( F_57 @ A_169 ) @ ( image_1806517641_a_nat @ F_57 @ B_84 ) ) ) ).
thf(fact_238_image__insert,axiom,
! [F_57: hoare_1167836817_state > hoare_1775062406iple_a,A_169: hoare_1167836817_state,B_84: hoare_1167836817_state > $o] :
( ( image_1802845250iple_a @ F_57 @ ( insert2134838167_state @ A_169 @ B_84 ) )
= ( insert1281456128iple_a @ ( F_57 @ A_169 ) @ ( image_1802845250iple_a @ F_57 @ B_84 ) ) ) ).
thf(fact_239_image__insert,axiom,
! [F_57: nat > hoare_1775062406iple_a,A_169: nat,B_84: nat > $o] :
( ( image_43014529iple_a @ F_57 @ ( insert_nat @ A_169 @ B_84 ) )
= ( insert1281456128iple_a @ ( F_57 @ A_169 ) @ ( image_43014529iple_a @ F_57 @ B_84 ) ) ) ).
thf(fact_240_the__sym__eq__trivial,axiom,
! [X_85: hoare_1775062406iple_a] :
( ( the_Ho1155011127iple_a @ ( fequal1288209029iple_a @ X_85 ) )
= X_85 ) ).
thf(fact_241_the__eq__trivial,axiom,
! [A_168: hoare_1775062406iple_a] :
( ( the_Ho1155011127iple_a
@ ^ [X_3: hoare_1775062406iple_a] : X_3 = A_168 )
= A_168 ) ).
thf(fact_242_If__def,axiom,
! [X_84: hoare_1775062406iple_a,Y_39: hoare_1775062406iple_a,P_15: $o] :
( ( P_15
=> ( X_84
= ( the_Ho1155011127iple_a
@ ^ [Z_21: hoare_1775062406iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_21 = X_84 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_21 = Y_39 ) ) ) ) ) )
& ( ~ P_15
=> ( Y_39
= ( the_Ho1155011127iple_a
@ ^ [Z_21: hoare_1775062406iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_21 = X_84 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_21 = Y_39 ) ) ) ) ) ) ) ).
thf(fact_243_fold1Set__sing,axiom,
! [F_56: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_167: hoare_1167836817_state,B_83: hoare_1167836817_state] :
( ( finite309220289_state @ F_56 @ ( insert2134838167_state @ A_167 @ bot_bo70021908tate_o ) @ B_83 )
<=> ( A_167 = B_83 ) ) ).
thf(fact_244_fold1Set__sing,axiom,
! [F_56: nat > nat > nat,A_167: nat,B_83: nat] :
( ( finite_fold1Set_nat @ F_56 @ ( insert_nat @ A_167 @ bot_bot_nat_o ) @ B_83 )
<=> ( A_167 = B_83 ) ) ).
thf(fact_245_fold1Set__sing,axiom,
! [F_56: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_167: hoare_1775062406iple_a,B_83: hoare_1775062406iple_a] :
( ( finite1946188886iple_a @ F_56 @ ( insert1281456128iple_a @ A_167 @ bot_bo751897185le_a_o ) @ B_83 )
<=> ( A_167 = B_83 ) ) ).
thf(fact_246_the__equality,axiom,
! [P_14: hoare_1775062406iple_a > $o,A_166: hoare_1775062406iple_a] :
( ( P_14 @ A_166 )
=> ( ! [X_3: hoare_1775062406iple_a] :
( ( P_14 @ X_3 )
=> ( X_3 = A_166 ) )
=> ( ( the_Ho1155011127iple_a @ P_14 )
= A_166 ) ) ) ).
thf(fact_247_folding__one_Osingleton,axiom,
! [X_83: hoare_1167836817_state,F_55: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_54: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_55 @ F_54 )
=> ( ( F_54 @ ( insert2134838167_state @ X_83 @ bot_bo70021908tate_o ) )
= X_83 ) ) ).
thf(fact_248_folding__one_Osingleton,axiom,
! [X_83: nat,F_55: nat > nat > nat,F_54: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_55 @ F_54 )
=> ( ( F_54 @ ( insert_nat @ X_83 @ bot_bot_nat_o ) )
= X_83 ) ) ).
thf(fact_249_folding__one_Osingleton,axiom,
! [X_83: hoare_1775062406iple_a,F_55: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_54: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_55 @ F_54 )
=> ( ( F_54 @ ( insert1281456128iple_a @ X_83 @ bot_bo751897185le_a_o ) )
= X_83 ) ) ).
thf(fact_250_hoare__derivs_OLocal,axiom,
! [A_165: state > nat,G_6: hoare_1167836817_state > $o,P_13: state > state > $o,C_30: com,Q_5: state > state > $o,X_82: loc,S_6: state] :
( ( hoare_123228589_state @ G_6
@ ( insert2134838167_state
@ ( hoare_908217195_state @ P_13 @ C_30
@ ^ [Z: state,S: state] : ( Q_5 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( getlocs @ S_6 @ X_82 ) ) ) )
@ bot_bo70021908tate_o ) )
=> ( hoare_123228589_state @ G_6
@ ( insert2134838167_state
@ ( hoare_908217195_state
@ ^ [Z: state,S: state] : ( (&) @ ( S_6 = S ) @ ( P_13 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( A_165 @ S ) ) ) )
@ ( local @ X_82 @ A_165 @ C_30 )
@ Q_5 )
@ bot_bo70021908tate_o ) ) ) ).
thf(fact_251_hoare__derivs_OLocal,axiom,
! [A_165: state > nat,G_6: hoare_1775062406iple_a > $o,P_13: x_a > state > $o,C_30: com,Q_5: x_a > state > $o,X_82: loc,S_6: state] :
( ( hoare_1508237396rivs_a @ G_6
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ P_13 @ C_30
@ ^ [Z: x_a,S: state] : ( Q_5 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( getlocs @ S_6 @ X_82 ) ) ) )
@ bot_bo751897185le_a_o ) )
=> ( hoare_1508237396rivs_a @ G_6
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Z: x_a,S: state] : ( (&) @ ( S_6 = S ) @ ( P_13 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( A_165 @ S ) ) ) )
@ ( local @ X_82 @ A_165 @ C_30 )
@ Q_5 )
@ bot_bo751897185le_a_o ) ) ) ).
thf(fact_252_vname_Osimps_I2_J,axiom,
! [Loc_2: loc,Loc_1: loc] :
( ( ( loc_1 @ Loc_2 )
= ( loc_1 @ Loc_1 ) )
<=> ( Loc_2 = Loc_1 ) ) ).
thf(fact_253_com_Osimps_I2_J,axiom,
! [Loc_2: loc,Fun_1: state > nat,Com_1: com,Loc_1: loc,Fun: state > nat,Com: com] :
( ( ( local @ Loc_2 @ Fun_1 @ Com_1 )
= ( local @ Loc_1 @ Fun @ Com ) )
<=> ( ( Loc_2 = Loc_1 )
& ( Fun_1 = Fun )
& ( Com_1 = Com ) ) ) ).
thf(fact_254_com_Osimps_I34_J,axiom,
! [Loc_2: loc,Fun_1: state > nat,Com_1: com,Com1: com,Com2: com] :
( ( local @ Loc_2 @ Fun_1 @ Com_1 )
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_255_com_Osimps_I35_J,axiom,
! [Com1: com,Com2: com,Loc_2: loc,Fun_1: state > nat,Com_1: com] :
( ( semi @ Com1 @ Com2 )
!= ( local @ Loc_2 @ Fun_1 @ Com_1 ) ) ).
thf(fact_256_com_Osimps_I23_J,axiom,
! [Loc_1: loc,Fun: state > nat,Com: com,Vname: vname,Fun_1: state > nat] :
( ( local @ Loc_1 @ Fun @ Com )
!= ( ass @ Vname @ Fun_1 ) ) ).
thf(fact_257_com_Osimps_I22_J,axiom,
! [Vname: vname,Fun_1: state > nat,Loc_1: loc,Fun: state > nat,Com: com] :
( ( ass @ Vname @ Fun_1 )
!= ( local @ Loc_1 @ Fun @ Com ) ) ).
thf(fact_258_com_Osimps_I11_J,axiom,
! [Loc_1: loc,Fun: state > nat,Com: com] :
( ( local @ Loc_1 @ Fun @ Com )
!= skip ) ).
thf(fact_259_com_Osimps_I10_J,axiom,
! [Loc_1: loc,Fun: state > nat,Com: com] :
( skip
!= ( local @ Loc_1 @ Fun @ Com ) ) ).
thf(fact_260_empty__fold1SetE,axiom,
! [F_53: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,X_81: hoare_1167836817_state] :
~ ( finite309220289_state @ F_53 @ bot_bo70021908tate_o @ X_81 ) ).
thf(fact_261_empty__fold1SetE,axiom,
! [F_53: nat > nat > nat,X_81: nat] :
~ ( finite_fold1Set_nat @ F_53 @ bot_bot_nat_o @ X_81 ) ).
thf(fact_262_empty__fold1SetE,axiom,
! [F_53: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,X_81: hoare_1775062406iple_a] :
~ ( finite1946188886iple_a @ F_53 @ bot_bo751897185le_a_o @ X_81 ) ).
thf(fact_263_fold1Set__nonempty,axiom,
! [F_52: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_164: hoare_1167836817_state > $o,X_80: hoare_1167836817_state] :
( ( finite309220289_state @ F_52 @ A_164 @ X_80 )
=> ( A_164 != bot_bo70021908tate_o ) ) ).
thf(fact_264_fold1Set__nonempty,axiom,
! [F_52: nat > nat > nat,A_164: nat > $o,X_80: nat] :
( ( finite_fold1Set_nat @ F_52 @ A_164 @ X_80 )
=> ( A_164 != bot_bot_nat_o ) ) ).
thf(fact_265_fold1Set__nonempty,axiom,
! [F_52: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_164: hoare_1775062406iple_a > $o,X_80: hoare_1775062406iple_a] :
( ( finite1946188886iple_a @ F_52 @ A_164 @ X_80 )
=> ( A_164 != bot_bo751897185le_a_o ) ) ).
thf(fact_266_theI,axiom,
! [P_12: hoare_1775062406iple_a > $o,A_163: hoare_1775062406iple_a] :
( ( P_12 @ A_163 )
=> ( ! [X_3: hoare_1775062406iple_a] :
( ( P_12 @ X_3 )
=> ( X_3 = A_163 ) )
=> ( P_12 @ ( the_Ho1155011127iple_a @ P_12 ) ) ) ) ).
thf(fact_267_the1__equality,axiom,
! [A_162: hoare_1775062406iple_a,P_11: hoare_1775062406iple_a > $o] :
( ? [X_3: hoare_1775062406iple_a] :
( ( P_11 @ X_3 )
& ! [Y_1: hoare_1775062406iple_a] :
( ( P_11 @ Y_1 )
=> ( Y_1 = X_3 ) ) )
=> ( ( P_11 @ A_162 )
=> ( ( the_Ho1155011127iple_a @ P_11 )
= A_162 ) ) ) ).
thf(fact_268_theI_H,axiom,
! [P_10: hoare_1775062406iple_a > $o] :
( ? [X_3: hoare_1775062406iple_a] :
( ( P_10 @ X_3 )
& ! [Y_1: hoare_1775062406iple_a] :
( ( P_10 @ Y_1 )
=> ( Y_1 = X_3 ) ) )
=> ( P_10 @ ( the_Ho1155011127iple_a @ P_10 ) ) ) ).
thf(fact_269_evalc_OLocal,axiom,
! [C_9: com,S0: state,Y_37: loc,A_158: state > nat,S1: state] :
( ( evalc @ C_9 @ ( update @ S0 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S0 ) ) @ S1 )
=> ( evalc @ ( local @ Y_37 @ A_158 @ C_9 ) @ S0 @ ( update @ S1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S0 @ Y_37 ) ) ) ) ).
thf(fact_270_evaln_OLocal,axiom,
! [C_9: com,S0: state,Y_37: loc,A_158: state > nat,N_3: nat,S1: state] :
( ( evaln @ C_9 @ ( update @ S0 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S0 ) ) @ N_3 @ S1 )
=> ( evaln @ ( local @ Y_37 @ A_158 @ C_9 ) @ S0 @ N_3 @ ( update @ S1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S0 @ Y_37 ) ) ) ) ).
thf(fact_271_fold1Set_Ointros,axiom,
! [F_51: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_161: hoare_1167836817_state,A_160: hoare_1167836817_state > $o,X_79: hoare_1167836817_state] :
( ( finite1316643734_state @ F_51 @ A_161 @ A_160 @ X_79 )
=> ( ~ ( member2058392318_state @ A_161 @ A_160 )
=> ( finite309220289_state @ F_51 @ ( insert2134838167_state @ A_161 @ A_160 ) @ X_79 ) ) ) ).
thf(fact_272_fold1Set_Ointros,axiom,
! [F_51: nat > nat > nat,A_161: nat,A_160: nat > $o,X_79: nat] :
( ( finite929467206at_nat @ F_51 @ A_161 @ A_160 @ X_79 )
=> ( ~ ( member_nat @ A_161 @ A_160 )
=> ( finite_fold1Set_nat @ F_51 @ ( insert_nat @ A_161 @ A_160 ) @ X_79 ) ) ) ).
thf(fact_273_fold1Set_Ointros,axiom,
! [F_51: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_161: hoare_1775062406iple_a,A_160: hoare_1775062406iple_a > $o,X_79: hoare_1775062406iple_a] :
( ( finite727644230iple_a @ F_51 @ A_161 @ A_160 @ X_79 )
=> ( ~ ( member2122167641iple_a @ A_161 @ A_160 )
=> ( finite1946188886iple_a @ F_51 @ ( insert1281456128iple_a @ A_161 @ A_160 ) @ X_79 ) ) ) ).
thf(fact_274_evaln_OSemi,axiom,
! [C1: com,S2: state,C0: com,S0: state,N_3: nat,S1: state] :
( ( evaln @ C0 @ S0 @ N_3 @ S1 )
=> ( ( evaln @ C1 @ S1 @ N_3 @ S2 )
=> ( evaln @ ( semi @ C0 @ C1 ) @ S0 @ N_3 @ S2 ) ) ) ).
thf(fact_275_evaln_OSkip,axiom,
! [S_2: state,N_3: nat] : ( evaln @ skip @ S_2 @ N_3 @ S_2 ) ).
thf(fact_276_evaln__elim__cases_I1_J,axiom,
! [S_2: state,N_3: nat,T: state] :
( ( evaln @ skip @ S_2 @ N_3 @ T )
=> ( T = S_2 ) ) ).
thf(fact_277_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_278_evalc_OSkip,axiom,
! [S_2: state] : ( evalc @ skip @ S_2 @ S_2 ) ).
thf(fact_279_evalc__elim__cases_I1_J,axiom,
! [S_2: state,T: state] :
( ( evalc @ skip @ S_2 @ T )
=> ( T = S_2 ) ) ).
thf(fact_280_evaln_OAssign,axiom,
! [X_78: vname,A_158: state > nat,S_2: state,N_3: nat] : ( evaln @ ( ass @ X_78 @ A_158 ) @ S_2 @ N_3 @ ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ).
thf(fact_281_evaln__elim__cases_I2_J,axiom,
! [X_78: vname,A_158: state > nat,S_2: state,N_3: nat,T: state] :
( ( evaln @ ( ass @ X_78 @ A_158 ) @ S_2 @ N_3 @ T )
=> ( T
= ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ) ).
thf(fact_282_evalc_OAssign,axiom,
! [X_78: vname,A_158: state > nat,S_2: state] : ( evalc @ ( ass @ X_78 @ A_158 ) @ S_2 @ ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ).
thf(fact_283_evalc__elim__cases_I2_J,axiom,
! [X_78: vname,A_158: state > nat,S_2: state,T: state] :
( ( evalc @ ( ass @ X_78 @ A_158 ) @ S_2 @ T )
=> ( T
= ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ) ).
thf(fact_284_eval__eq,axiom,
! [C_9: com,S_2: state,T: state] :
( ( evalc @ C_9 @ S_2 @ T )
<=> ? [N_1: nat] : ( evaln @ C_9 @ S_2 @ N_1 @ T ) ) ).
thf(fact_285_com__det,axiom,
! [U: state,C_9: com,S_2: state,T: state] :
( ( evalc @ C_9 @ S_2 @ T )
=> ( ( evalc @ C_9 @ S_2 @ U )
=> ( U = T ) ) ) ).
thf(fact_286_evaln__evalc,axiom,
! [C_9: com,S_2: state,N_3: nat,T: state] :
( ( evaln @ C_9 @ S_2 @ N_3 @ T )
=> ( evalc @ C_9 @ S_2 @ T ) ) ).
thf(fact_287_empty__fold__graphE,axiom,
! [F_50: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_20: hoare_1775062406iple_a,X_77: hoare_1775062406iple_a] :
( ( finite727644230iple_a @ F_50 @ Z_20 @ bot_bo751897185le_a_o @ X_77 )
=> ( X_77 = Z_20 ) ) ).
thf(fact_288_fold__graph_OemptyI,axiom,
! [F_49: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_19: hoare_1775062406iple_a] : ( finite727644230iple_a @ F_49 @ Z_19 @ bot_bo751897185le_a_o @ Z_19 ) ).
thf(fact_289_fold__graph_OinsertI,axiom,
! [F_48: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_18: hoare_1775062406iple_a,Y_38: hoare_1775062406iple_a,X_76: hoare_1775062406iple_a,A_159: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ X_76 @ A_159 )
=> ( ( finite727644230iple_a @ F_48 @ Z_18 @ A_159 @ Y_38 )
=> ( finite727644230iple_a @ F_48 @ Z_18 @ ( insert1281456128iple_a @ X_76 @ A_159 ) @ ( F_48 @ X_76 @ Y_38 ) ) ) ) ).
thf(fact_290_evalc__elim__cases_I3_J,axiom,
! [Y_37: loc,A_158: state > nat,C_9: com,S_2: state,T: state] :
( ( evalc @ ( local @ Y_37 @ A_158 @ C_9 ) @ S_2 @ T )
=> ~ ! [S1_1: state] :
( ( T
= ( update @ S1_1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S_2 @ Y_37 ) ) )
=> ~ ( evalc @ C_9 @ ( update @ S_2 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S_2 ) ) @ S1_1 ) ) ) ).
thf(fact_291_evaln__elim__cases_I3_J,axiom,
! [Y_37: loc,A_158: state > nat,C_9: com,S_2: state,N_3: nat,T: state] :
( ( evaln @ ( local @ Y_37 @ A_158 @ C_9 ) @ S_2 @ N_3 @ T )
=> ~ ! [S1_1: state] :
( ( T
= ( update @ S1_1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S_2 @ Y_37 ) ) )
=> ~ ( evaln @ C_9 @ ( update @ S_2 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S_2 ) ) @ N_3 @ S1_1 ) ) ) ).
thf(fact_292_evalc__elim__cases_I4_J,axiom,
! [C1: com,C2: com,S_2: state,T: state] :
( ( evalc @ ( semi @ C1 @ C2 ) @ S_2 @ T )
=> ~ ! [S1_1: state] :
( ( evalc @ C1 @ S_2 @ S1_1 )
=> ~ ( evalc @ C2 @ S1_1 @ T ) ) ) ).
thf(fact_293_evaln__elim__cases_I4_J,axiom,
! [C1: com,C2: com,S_2: state,N_3: nat,T: state] :
( ( evaln @ ( semi @ C1 @ C2 ) @ S_2 @ N_3 @ T )
=> ~ ! [S1_1: state] :
( ( evaln @ C1 @ S_2 @ N_3 @ S1_1 )
=> ~ ( evaln @ C2 @ S1_1 @ N_3 @ T ) ) ) ).
thf(fact_294_insert__fold1SetE,axiom,
! [F_47: nat > nat > nat,A_157: nat,X_75: nat > $o,X_74: nat] :
( ( finite_fold1Set_nat @ F_47 @ ( insert_nat @ A_157 @ X_75 ) @ X_74 )
=> ~ ! [A_43: nat,A_98: nat > $o] :
( ( ( insert_nat @ A_157 @ X_75 )
= ( insert_nat @ A_43 @ A_98 ) )
=> ( ( finite929467206at_nat @ F_47 @ A_43 @ A_98 @ X_74 )
=> ( member_nat @ A_43 @ A_98 ) ) ) ) ).
thf(fact_295_insert__fold1SetE,axiom,
! [F_47: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_157: hoare_1167836817_state,X_75: hoare_1167836817_state > $o,X_74: hoare_1167836817_state] :
( ( finite309220289_state @ F_47 @ ( insert2134838167_state @ A_157 @ X_75 ) @ X_74 )
=> ~ ! [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] :
( ( ( insert2134838167_state @ A_157 @ X_75 )
= ( insert2134838167_state @ A_43 @ A_98 ) )
=> ( ( finite1316643734_state @ F_47 @ A_43 @ A_98 @ X_74 )
=> ( member2058392318_state @ A_43 @ A_98 ) ) ) ) ).
thf(fact_296_insert__fold1SetE,axiom,
! [F_47: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_157: hoare_1775062406iple_a,X_75: hoare_1775062406iple_a > $o,X_74: hoare_1775062406iple_a] :
( ( finite1946188886iple_a @ F_47 @ ( insert1281456128iple_a @ A_157 @ X_75 ) @ X_74 )
=> ~ ! [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] :
( ( ( insert1281456128iple_a @ A_157 @ X_75 )
= ( insert1281456128iple_a @ A_43 @ A_98 ) )
=> ( ( finite727644230iple_a @ F_47 @ A_43 @ A_98 @ X_74 )
=> ( member2122167641iple_a @ A_43 @ A_98 ) ) ) ) ).
thf(fact_297_MGT__def,axiom,
! [C_9: com] :
( ( hoare_Mirabelle_MGT @ C_9 )
= ( hoare_908217195_state @ fequal_state @ C_9 @ ( evalc @ C_9 ) ) ) ).
thf(fact_298_evalc__evaln,axiom,
! [C_9: com,S_2: state,T: state] :
( ( evalc @ C_9 @ S_2 @ T )
=> ? [N_1: nat] : ( evaln @ C_9 @ S_2 @ N_1 @ T ) ) ).
thf(fact_299_fold1Set_Osimps,axiom,
! [F_46: nat > nat > nat,A1_1: nat > $o,A2_1: nat] :
( ( finite_fold1Set_nat @ F_46 @ A1_1 @ A2_1 )
<=> ? [A_43: nat,A_98: nat > $o,X_3: nat] :
( ( A1_1
= ( insert_nat @ A_43 @ A_98 ) )
& ( A2_1 = X_3 )
& ( finite929467206at_nat @ F_46 @ A_43 @ A_98 @ X_3 )
& ~ ( member_nat @ A_43 @ A_98 ) ) ) ).
thf(fact_300_fold1Set_Osimps,axiom,
! [F_46: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A1_1: hoare_1167836817_state > $o,A2_1: hoare_1167836817_state] :
( ( finite309220289_state @ F_46 @ A1_1 @ A2_1 )
<=> ? [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o,X_3: hoare_1167836817_state] :
( ( A1_1
= ( insert2134838167_state @ A_43 @ A_98 ) )
& ( A2_1 = X_3 )
& ( finite1316643734_state @ F_46 @ A_43 @ A_98 @ X_3 )
& ~ ( member2058392318_state @ A_43 @ A_98 ) ) ) ).
thf(fact_301_fold1Set_Osimps,axiom,
! [F_46: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A1_1: hoare_1775062406iple_a > $o,A2_1: hoare_1775062406iple_a] :
( ( finite1946188886iple_a @ F_46 @ A1_1 @ A2_1 )
<=> ? [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
( ( A1_1
= ( insert1281456128iple_a @ A_43 @ A_98 ) )
& ( A2_1 = X_3 )
& ( finite727644230iple_a @ F_46 @ A_43 @ A_98 @ X_3 )
& ~ ( member2122167641iple_a @ A_43 @ A_98 ) ) ) ).
thf(fact_302_fold__graph_Osimps,axiom,
! [F_45: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_17: hoare_1775062406iple_a,A1: hoare_1775062406iple_a > $o,A2: hoare_1775062406iple_a] :
( ( finite727644230iple_a @ F_45 @ Z_17 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bo751897185le_a_o )
& ( A2 = Z_17 ) )
| ? [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o,Y_1: hoare_1775062406iple_a] :
( ( A1
= ( insert1281456128iple_a @ X_3 @ A_98 ) )
& ( A2
= ( F_45 @ X_3 @ Y_1 ) )
& ~ ( member2122167641iple_a @ X_3 @ A_98 )
& ( finite727644230iple_a @ F_45 @ Z_17 @ A_98 @ Y_1 ) ) ) ) ).
thf(fact_303_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_1: nat] :
( ( evaln @ C1 @ S1 @ N_1 @ T1 )
& ( evaln @ C2 @ S2 @ N_1 @ T2 ) ) ) ) ).
thf(fact_304_triple__valid__def2,axiom,
! [N_5: nat,P_9: state > state > $o,C_29: com,Q_4: state > state > $o] :
( ( hoare_56934129_state @ N_5 @ ( hoare_908217195_state @ P_9 @ C_29 @ Q_4 ) )
<=> ! [Z: state,S: state] :
( ( P_9 @ Z @ S )
=> ! [S_5: state] :
( ( evaln @ C_29 @ S @ N_5 @ S_5 )
=> ( Q_4 @ Z @ S_5 ) ) ) ) ).
thf(fact_305_triple__valid__def2,axiom,
! [N_5: nat,P_9: x_a > state > $o,C_29: com,Q_4: x_a > state > $o] :
( ( hoare_1462269968alid_a @ N_5 @ ( hoare_1766022166iple_a @ P_9 @ C_29 @ Q_4 ) )
<=> ! [Z: x_a,S: state] :
( ( P_9 @ Z @ S )
=> ! [S_5: state] :
( ( evaln @ C_29 @ S @ N_5 @ S_5 )
=> ( Q_4 @ Z @ S_5 ) ) ) ) ).
thf(fact_306_folding__one_Oinsert,axiom,
! [X_73: nat,A_156: nat > $o,F_44: nat > nat > nat,F_43: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_44 @ F_43 )
=> ( ( finite_finite_nat @ A_156 )
=> ( ~ ( member_nat @ X_73 @ A_156 )
=> ( ( A_156 != bot_bot_nat_o )
=> ( ( F_43 @ ( insert_nat @ X_73 @ A_156 ) )
= ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
thf(fact_307_folding__one_Oinsert,axiom,
! [X_73: hoare_1167836817_state,A_156: hoare_1167836817_state > $o,F_44: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_43: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_44 @ F_43 )
=> ( ( finite1084549118_state @ A_156 )
=> ( ~ ( member2058392318_state @ X_73 @ A_156 )
=> ( ( A_156 != bot_bo70021908tate_o )
=> ( ( F_43 @ ( insert2134838167_state @ X_73 @ A_156 ) )
= ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
thf(fact_308_folding__one_Oinsert,axiom,
! [X_73: hoare_1775062406iple_a,A_156: hoare_1775062406iple_a > $o,F_44: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_43: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_44 @ F_43 )
=> ( ( finite2063573081iple_a @ A_156 )
=> ( ~ ( member2122167641iple_a @ X_73 @ A_156 )
=> ( ( A_156 != bot_bo751897185le_a_o )
=> ( ( F_43 @ ( insert1281456128iple_a @ X_73 @ A_156 ) )
= ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
thf(fact_309_finite__Collect__conjI,axiom,
! [Q_3: hoare_1775062406iple_a > $o,P_8: hoare_1775062406iple_a > $o] :
( ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_8 ) )
| ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_3 ) ) )
=> ( finite2063573081iple_a
@ ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( P_8 @ X_3 ) @ ( Q_3 @ X_3 ) ) ) ) ) ).
thf(fact_310_finite__Collect__conjI,axiom,
! [Q_3: nat > $o,P_8: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P_8 ) )
| ( finite_finite_nat @ ( collect_nat @ Q_3 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( P_8 @ X_3 ) @ ( Q_3 @ X_3 ) ) ) ) ) ).
thf(fact_311_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_nat_o ).
thf(fact_312_finite_OemptyI,axiom,
finite1084549118_state @ bot_bo70021908tate_o ).
thf(fact_313_finite_OemptyI,axiom,
finite2063573081iple_a @ bot_bo751897185le_a_o ).
thf(fact_314_finite_OinsertI,axiom,
! [A_155: nat,A_154: nat > $o] :
( ( finite_finite_nat @ A_154 )
=> ( finite_finite_nat @ ( insert_nat @ A_155 @ A_154 ) ) ) ).
thf(fact_315_finite_OinsertI,axiom,
! [A_155: hoare_1167836817_state,A_154: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_154 )
=> ( finite1084549118_state @ ( insert2134838167_state @ A_155 @ A_154 ) ) ) ).
thf(fact_316_finite_OinsertI,axiom,
! [A_155: hoare_1775062406iple_a,A_154: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_154 )
=> ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_155 @ A_154 ) ) ) ).
thf(fact_317_finite__Collect__disjI,axiom,
! [P_7: hoare_1775062406iple_a > $o,Q_2: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a
@ ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( P_7 @ X_3 ) @ ( Q_2 @ X_3 ) ) ) )
<=> ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_7 ) )
& ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_2 ) ) ) ) ).
thf(fact_318_finite__Collect__disjI,axiom,
! [P_7: nat > $o,Q_2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X_3: nat] : ( (|) @ ( P_7 @ X_3 ) @ ( Q_2 @ X_3 ) ) ) )
<=> ( ( finite_finite_nat @ ( collect_nat @ P_7 ) )
& ( finite_finite_nat @ ( collect_nat @ Q_2 ) ) ) ) ).
thf(fact_319_vname_Osimps_I1_J,axiom,
! [Glb_1: glb,Glb_2: glb] :
( ( ( glb_1 @ Glb_1 )
= ( glb_1 @ Glb_2 ) )
<=> ( Glb_1 = Glb_2 ) ) ).
thf(fact_320_finite__insert,axiom,
! [A_153: nat,A_152: nat > $o] :
( ( finite_finite_nat @ ( insert_nat @ A_153 @ A_152 ) )
<=> ( finite_finite_nat @ A_152 ) ) ).
thf(fact_321_finite__insert,axiom,
! [A_153: hoare_1167836817_state,A_152: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ ( insert2134838167_state @ A_153 @ A_152 ) )
<=> ( finite1084549118_state @ A_152 ) ) ).
thf(fact_322_finite__insert,axiom,
! [A_153: hoare_1775062406iple_a,A_152: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_153 @ A_152 ) )
<=> ( finite2063573081iple_a @ A_152 ) ) ).
thf(fact_323_vname_Osimps_I4_J,axiom,
! [Loc_1: loc,Glb_1: glb] :
( ( loc_1 @ Loc_1 )
!= ( glb_1 @ Glb_1 ) ) ).
thf(fact_324_vname_Osimps_I3_J,axiom,
! [Glb_1: glb,Loc_1: loc] :
( ( glb_1 @ Glb_1 )
!= ( loc_1 @ Loc_1 ) ) ).
thf(fact_325_folding__one_Oclosed,axiom,
! [A_151: nat > $o,F_42: nat > nat > nat,F_41: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_42 @ F_41 )
=> ( ( finite_finite_nat @ A_151 )
=> ( ( A_151 != bot_bot_nat_o )
=> ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( F_42 @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
thf(fact_326_folding__one_Oclosed,axiom,
! [A_151: hoare_1167836817_state > $o,F_42: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_41: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_42 @ F_41 )
=> ( ( finite1084549118_state @ A_151 )
=> ( ( A_151 != bot_bo70021908tate_o )
=> ( ! [X_3: hoare_1167836817_state,Y_1: hoare_1167836817_state] : ( member2058392318_state @ ( F_42 @ X_3 @ Y_1 ) @ ( insert2134838167_state @ X_3 @ ( insert2134838167_state @ Y_1 @ bot_bo70021908tate_o ) ) )
=> ( member2058392318_state @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
thf(fact_327_folding__one_Oclosed,axiom,
! [A_151: hoare_1775062406iple_a > $o,F_42: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_41: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_42 @ F_41 )
=> ( ( finite2063573081iple_a @ A_151 )
=> ( ( A_151 != bot_bo751897185le_a_o )
=> ( ! [X_3: hoare_1775062406iple_a,Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ ( F_42 @ X_3 @ Y_1 ) @ ( insert1281456128iple_a @ X_3 @ ( insert1281456128iple_a @ Y_1 @ bot_bo751897185le_a_o ) ) )
=> ( member2122167641iple_a @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
thf(fact_328_finite__nonempty__imp__fold1Set,axiom,
! [F_40: nat > nat > nat,A_150: nat > $o] :
( ( finite_finite_nat @ A_150 )
=> ( ( A_150 != bot_bot_nat_o )
=> ( ?? @ nat @ ( finite_fold1Set_nat @ F_40 @ A_150 ) ) ) ) ).
thf(fact_329_finite__nonempty__imp__fold1Set,axiom,
! [F_40: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_150: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_150 )
=> ( ( A_150 != bot_bo751897185le_a_o )
=> ( ?? @ hoare_1775062406iple_a @ ( finite1946188886iple_a @ F_40 @ A_150 ) ) ) ) ).
thf(fact_330_finite__nonempty__imp__fold1Set,axiom,
! [F_40: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_150: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_150 )
=> ( ( A_150 != bot_bo70021908tate_o )
=> ( ?? @ hoare_1167836817_state @ ( finite309220289_state @ F_40 @ A_150 ) ) ) ) ).
thf(fact_331_finite__induct,axiom,
! [P_6: ( nat > $o ) > $o,F_39: nat > $o] :
( ( finite_finite_nat @ F_39 )
=> ( ( P_6 @ bot_bot_nat_o )
=> ( ! [X_3: nat,F_3: nat > $o] :
( ( finite_finite_nat @ F_3 )
=> ( ~ ( member_nat @ X_3 @ F_3 )
=> ( ( P_6 @ F_3 )
=> ( P_6 @ ( insert_nat @ X_3 @ F_3 ) ) ) ) )
=> ( P_6 @ F_39 ) ) ) ) ).
thf(fact_332_finite__induct,axiom,
! [P_6: ( hoare_1167836817_state > $o ) > $o,F_39: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_39 )
=> ( ( P_6 @ bot_bo70021908tate_o )
=> ( ! [X_3: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_3 )
=> ( ~ ( member2058392318_state @ X_3 @ F_3 )
=> ( ( P_6 @ F_3 )
=> ( P_6 @ ( insert2134838167_state @ X_3 @ F_3 ) ) ) ) )
=> ( P_6 @ F_39 ) ) ) ) ).
thf(fact_333_finite__induct,axiom,
! [P_6: ( hoare_1775062406iple_a > $o ) > $o,F_39: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_39 )
=> ( ( P_6 @ bot_bo751897185le_a_o )
=> ( ! [X_3: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_3 )
=> ( ~ ( member2122167641iple_a @ X_3 @ F_3 )
=> ( ( P_6 @ F_3 )
=> ( P_6 @ ( insert1281456128iple_a @ X_3 @ F_3 ) ) ) ) )
=> ( P_6 @ F_39 ) ) ) ) ).
thf(fact_334_finite_Osimps,axiom,
! [A_149: nat > $o] :
( ( finite_finite_nat @ A_149 )
<=> ( ( A_149 = bot_bot_nat_o )
| ? [A_98: nat > $o,A_43: nat] :
( ( A_149
= ( insert_nat @ A_43 @ A_98 ) )
& ( finite_finite_nat @ A_98 ) ) ) ) ).
thf(fact_335_finite_Osimps,axiom,
! [A_149: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_149 )
<=> ( ( A_149 = bot_bo70021908tate_o )
| ? [A_98: hoare_1167836817_state > $o,A_43: hoare_1167836817_state] :
( ( A_149
= ( insert2134838167_state @ A_43 @ A_98 ) )
& ( finite1084549118_state @ A_98 ) ) ) ) ).
thf(fact_336_finite_Osimps,axiom,
! [A_149: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_149 )
<=> ( ( A_149 = bot_bo751897185le_a_o )
| ? [A_98: hoare_1775062406iple_a > $o,A_43: hoare_1775062406iple_a] :
( ( A_149
= ( insert1281456128iple_a @ A_43 @ A_98 ) )
& ( finite2063573081iple_a @ A_98 ) ) ) ) ).
thf(fact_337_finite__imp__fold__graph,axiom,
! [F_38: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_16: hoare_1775062406iple_a,A_148: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_148 )
=> ( ?? @ hoare_1775062406iple_a @ ( finite727644230iple_a @ F_38 @ Z_16 @ A_148 ) ) ) ).
thf(fact_338_folding__one__idem_Oinsert__idem,axiom,
! [X_72: nat,A_147: nat > $o,F_37: nat > nat > nat,F_36: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_37 @ F_36 )
=> ( ( finite_finite_nat @ A_147 )
=> ( ( A_147 != bot_bot_nat_o )
=> ( ( F_36 @ ( insert_nat @ X_72 @ A_147 ) )
= ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
thf(fact_339_folding__one__idem_Oinsert__idem,axiom,
! [X_72: hoare_1167836817_state,A_147: hoare_1167836817_state > $o,F_37: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_36: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_37 @ F_36 )
=> ( ( finite1084549118_state @ A_147 )
=> ( ( A_147 != bot_bo70021908tate_o )
=> ( ( F_36 @ ( insert2134838167_state @ X_72 @ A_147 ) )
= ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
thf(fact_340_folding__one__idem_Oinsert__idem,axiom,
! [X_72: hoare_1775062406iple_a,A_147: hoare_1775062406iple_a > $o,F_37: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_36: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_37 @ F_36 )
=> ( ( finite2063573081iple_a @ A_147 )
=> ( ( A_147 != bot_bo751897185le_a_o )
=> ( ( F_36 @ ( insert1281456128iple_a @ X_72 @ A_147 ) )
= ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
thf(fact_341_finite__ne__induct,axiom,
! [P_5: ( nat > $o ) > $o,F_35: nat > $o] :
( ( finite_finite_nat @ F_35 )
=> ( ( F_35 != bot_bot_nat_o )
=> ( ! [X_3: nat] : ( P_5 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) )
=> ( ! [X_3: nat,F_3: nat > $o] :
( ( finite_finite_nat @ F_3 )
=> ( ( F_3 != bot_bot_nat_o )
=> ( ~ ( member_nat @ X_3 @ F_3 )
=> ( ( P_5 @ F_3 )
=> ( P_5 @ ( insert_nat @ X_3 @ F_3 ) ) ) ) ) )
=> ( P_5 @ F_35 ) ) ) ) ) ).
thf(fact_342_finite__ne__induct,axiom,
! [P_5: ( hoare_1167836817_state > $o ) > $o,F_35: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_35 )
=> ( ( F_35 != bot_bo70021908tate_o )
=> ( ! [X_3: hoare_1167836817_state] : ( P_5 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
=> ( ! [X_3: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_3 )
=> ( ( F_3 != bot_bo70021908tate_o )
=> ( ~ ( member2058392318_state @ X_3 @ F_3 )
=> ( ( P_5 @ F_3 )
=> ( P_5 @ ( insert2134838167_state @ X_3 @ F_3 ) ) ) ) ) )
=> ( P_5 @ F_35 ) ) ) ) ) ).
thf(fact_343_finite__ne__induct,axiom,
! [P_5: ( hoare_1775062406iple_a > $o ) > $o,F_35: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_35 )
=> ( ( F_35 != bot_bo751897185le_a_o )
=> ( ! [X_3: hoare_1775062406iple_a] : ( P_5 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) )
=> ( ! [X_3: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_3 )
=> ( ( F_3 != bot_bo751897185le_a_o )
=> ( ~ ( member2122167641iple_a @ X_3 @ F_3 )
=> ( ( P_5 @ F_3 )
=> ( P_5 @ ( insert1281456128iple_a @ X_3 @ F_3 ) ) ) ) ) )
=> ( P_5 @ F_35 ) ) ) ) ) ).
thf(fact_344_vname_Oexhaust,axiom,
! [Y: vname] :
( ! [Glb: glb] :
( Y
!= ( glb_1 @ Glb ) )
=> ~ ! [Loc: loc] :
( Y
!= ( loc_1 @ Loc ) ) ) ).
thf(fact_345_folding__one_Oremove,axiom,
! [X_71: nat,A_146: nat > $o,F_34: nat > nat > nat,F_33: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_34 @ F_33 )
=> ( ( finite_finite_nat @ A_146 )
=> ( ( member_nat @ X_71 @ A_146 )
=> ( ( ( ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_33 @ A_146 )
= X_71 ) )
& ( ( ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_33 @ A_146 )
= ( F_34 @ X_71 @ ( F_33 @ ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_346_folding__one_Oremove,axiom,
! [X_71: hoare_1167836817_state,A_146: hoare_1167836817_state > $o,F_34: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_33: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_34 @ F_33 )
=> ( ( finite1084549118_state @ A_146 )
=> ( ( member2058392318_state @ X_71 @ A_146 )
=> ( ( ( ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) )
= bot_bo70021908tate_o )
=> ( ( F_33 @ A_146 )
= X_71 ) )
& ( ( ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) )
!= bot_bo70021908tate_o )
=> ( ( F_33 @ A_146 )
= ( F_34 @ X_71 @ ( F_33 @ ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_347_folding__one_Oremove,axiom,
! [X_71: hoare_1775062406iple_a,A_146: hoare_1775062406iple_a > $o,F_34: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_33: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_34 @ F_33 )
=> ( ( finite2063573081iple_a @ A_146 )
=> ( ( member2122167641iple_a @ X_71 @ A_146 )
=> ( ( ( ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) )
= bot_bo751897185le_a_o )
=> ( ( F_33 @ A_146 )
= X_71 ) )
& ( ( ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) )
!= bot_bo751897185le_a_o )
=> ( ( F_33 @ A_146 )
= ( F_34 @ X_71 @ ( F_33 @ ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_348_folding__one_Oinsert__remove,axiom,
! [X_70: nat,A_145: nat > $o,F_32: nat > nat > nat,F_31: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_32 @ F_31 )
=> ( ( finite_finite_nat @ A_145 )
=> ( ( ( ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_31 @ ( insert_nat @ X_70 @ A_145 ) )
= X_70 ) )
& ( ( ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_31 @ ( insert_nat @ X_70 @ A_145 ) )
= ( F_32 @ X_70 @ ( F_31 @ ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
thf(fact_349_folding__one_Oinsert__remove,axiom,
! [X_70: hoare_1167836817_state,A_145: hoare_1167836817_state > $o,F_32: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_31: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_32 @ F_31 )
=> ( ( finite1084549118_state @ A_145 )
=> ( ( ( ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) )
= bot_bo70021908tate_o )
=> ( ( F_31 @ ( insert2134838167_state @ X_70 @ A_145 ) )
= X_70 ) )
& ( ( ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) )
!= bot_bo70021908tate_o )
=> ( ( F_31 @ ( insert2134838167_state @ X_70 @ A_145 ) )
= ( F_32 @ X_70 @ ( F_31 @ ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ).
thf(fact_350_folding__one_Oinsert__remove,axiom,
! [X_70: hoare_1775062406iple_a,A_145: hoare_1775062406iple_a > $o,F_32: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_31: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_32 @ F_31 )
=> ( ( finite2063573081iple_a @ A_145 )
=> ( ( ( ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) )
= bot_bo751897185le_a_o )
=> ( ( F_31 @ ( insert1281456128iple_a @ X_70 @ A_145 ) )
= X_70 ) )
& ( ( ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) )
!= bot_bo751897185le_a_o )
=> ( ( F_31 @ ( insert1281456128iple_a @ X_70 @ A_145 ) )
= ( F_32 @ X_70 @ ( F_31 @ ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ) ) ).
thf(fact_351_DiffE,axiom,
! [C_28: hoare_1775062406iple_a,A_144: hoare_1775062406iple_a > $o,B_82: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_28 @ ( minus_1944206118le_a_o @ A_144 @ B_82 ) )
=> ~ ( ( member2122167641iple_a @ C_28 @ A_144 )
=> ( member2122167641iple_a @ C_28 @ B_82 ) ) ) ).
thf(fact_352_DiffE,axiom,
! [C_28: nat,A_144: nat > $o,B_82: nat > $o] :
( ( member_nat @ C_28 @ ( minus_minus_nat_o @ A_144 @ B_82 ) )
=> ~ ( ( member_nat @ C_28 @ A_144 )
=> ( member_nat @ C_28 @ B_82 ) ) ) ).
thf(fact_353_DiffI,axiom,
! [B_81: hoare_1775062406iple_a > $o,C_27: hoare_1775062406iple_a,A_143: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_27 @ A_143 )
=> ( ~ ( member2122167641iple_a @ C_27 @ B_81 )
=> ( member2122167641iple_a @ C_27 @ ( minus_1944206118le_a_o @ A_143 @ B_81 ) ) ) ) ).
thf(fact_354_DiffI,axiom,
! [B_81: nat > $o,C_27: nat,A_143: nat > $o] :
( ( member_nat @ C_27 @ A_143 )
=> ( ~ ( member_nat @ C_27 @ B_81 )
=> ( member_nat @ C_27 @ ( minus_minus_nat_o @ A_143 @ B_81 ) ) ) ) ).
thf(fact_355_finite__Diff,axiom,
! [B_80: nat > $o,A_142: nat > $o] :
( ( finite_finite_nat @ A_142 )
=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_142 @ B_80 ) ) ) ).
thf(fact_356_DiffD2,axiom,
! [C_26: hoare_1775062406iple_a,A_141: hoare_1775062406iple_a > $o,B_79: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_26 @ ( minus_1944206118le_a_o @ A_141 @ B_79 ) )
=> ~ ( member2122167641iple_a @ C_26 @ B_79 ) ) ).
thf(fact_357_DiffD2,axiom,
! [C_26: nat,A_141: nat > $o,B_79: nat > $o] :
( ( member_nat @ C_26 @ ( minus_minus_nat_o @ A_141 @ B_79 ) )
=> ~ ( member_nat @ C_26 @ B_79 ) ) ).
thf(fact_358_DiffD1,axiom,
! [C_25: hoare_1775062406iple_a,A_140: hoare_1775062406iple_a > $o,B_78: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_25 @ ( minus_1944206118le_a_o @ A_140 @ B_78 ) )
=> ( member2122167641iple_a @ C_25 @ A_140 ) ) ).
thf(fact_359_DiffD1,axiom,
! [C_25: nat,A_140: nat > $o,B_78: nat > $o] :
( ( member_nat @ C_25 @ ( minus_minus_nat_o @ A_140 @ B_78 ) )
=> ( member_nat @ C_25 @ A_140 ) ) ).
thf(fact_360_Diff__iff,axiom,
! [C_24: hoare_1775062406iple_a,A_139: hoare_1775062406iple_a > $o,B_77: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_24 @ ( minus_1944206118le_a_o @ A_139 @ B_77 ) )
<=> ( ( member2122167641iple_a @ C_24 @ A_139 )
& ~ ( member2122167641iple_a @ C_24 @ B_77 ) ) ) ).
thf(fact_361_Diff__iff,axiom,
! [C_24: nat,A_139: nat > $o,B_77: nat > $o] :
( ( member_nat @ C_24 @ ( minus_minus_nat_o @ A_139 @ B_77 ) )
<=> ( ( member_nat @ C_24 @ A_139 )
& ~ ( member_nat @ C_24 @ B_77 ) ) ) ).
thf(fact_362_set__diff__eq,axiom,
! [A_138: hoare_1775062406iple_a > $o,B_76: hoare_1775062406iple_a > $o] :
( ( minus_1944206118le_a_o @ A_138 @ B_76 )
= ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( member2122167641iple_a @ X_3 @ A_138 ) @ ( (~) @ ( member2122167641iple_a @ X_3 @ B_76 ) ) ) ) ) ).
thf(fact_363_set__diff__eq,axiom,
! [A_138: nat > $o,B_76: nat > $o] :
( ( minus_minus_nat_o @ A_138 @ B_76 )
= ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( member_nat @ X_3 @ A_138 ) @ ( (~) @ ( member_nat @ X_3 @ B_76 ) ) ) ) ) ).
thf(fact_364_Diff__cancel,axiom,
! [A_137: nat > $o] :
( ( minus_minus_nat_o @ A_137 @ A_137 )
= bot_bot_nat_o ) ).
thf(fact_365_Diff__cancel,axiom,
! [A_137: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_137 @ A_137 )
= bot_bo70021908tate_o ) ).
thf(fact_366_Diff__cancel,axiom,
! [A_137: hoare_1775062406iple_a > $o] :
( ( minus_1944206118le_a_o @ A_137 @ A_137 )
= bot_bo751897185le_a_o ) ).
thf(fact_367_Diff__empty,axiom,
! [A_136: nat > $o] :
( ( minus_minus_nat_o @ A_136 @ bot_bot_nat_o )
= A_136 ) ).
thf(fact_368_Diff__empty,axiom,
! [A_136: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_136 @ bot_bo70021908tate_o )
= A_136 ) ).
thf(fact_369_Diff__empty,axiom,
! [A_136: hoare_1775062406iple_a > $o] :
( ( minus_1944206118le_a_o @ A_136 @ bot_bo751897185le_a_o )
= A_136 ) ).
thf(fact_370_empty__Diff,axiom,
! [A_135: nat > $o] :
( ( minus_minus_nat_o @ bot_bot_nat_o @ A_135 )
= bot_bot_nat_o ) ).
thf(fact_371_empty__Diff,axiom,
! [A_135: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ bot_bo70021908tate_o @ A_135 )
= bot_bo70021908tate_o ) ).
thf(fact_372_empty__Diff,axiom,
! [A_135: hoare_1775062406iple_a > $o] :
( ( minus_1944206118le_a_o @ bot_bo751897185le_a_o @ A_135 )
= bot_bo751897185le_a_o ) ).
thf(fact_373_finite__Diff2,axiom,
! [A_134: nat > $o,B_75: nat > $o] :
( ( finite_finite_nat @ B_75 )
=> ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_134 @ B_75 ) )
<=> ( finite_finite_nat @ A_134 ) ) ) ).
thf(fact_374_insert__Diff1,axiom,
! [A_133: nat > $o,X_69: nat,B_74: nat > $o] :
( ( member_nat @ X_69 @ B_74 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_69 @ A_133 ) @ B_74 )
= ( minus_minus_nat_o @ A_133 @ B_74 ) ) ) ).
thf(fact_375_insert__Diff1,axiom,
! [A_133: hoare_1167836817_state > $o,X_69: hoare_1167836817_state,B_74: hoare_1167836817_state > $o] :
( ( member2058392318_state @ X_69 @ B_74 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_69 @ A_133 ) @ B_74 )
= ( minus_2107060239tate_o @ A_133 @ B_74 ) ) ) ).
thf(fact_376_insert__Diff1,axiom,
! [A_133: hoare_1775062406iple_a > $o,X_69: hoare_1775062406iple_a,B_74: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_69 @ B_74 )
=> ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_69 @ A_133 ) @ B_74 )
= ( minus_1944206118le_a_o @ A_133 @ B_74 ) ) ) ).
thf(fact_377_insert__Diff__if,axiom,
! [A_132: nat > $o,X_68: nat,B_73: nat > $o] :
( ( ( member_nat @ X_68 @ B_73 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_132 ) @ B_73 )
= ( minus_minus_nat_o @ A_132 @ B_73 ) ) )
& ( ~ ( member_nat @ X_68 @ B_73 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_132 ) @ B_73 )
= ( insert_nat @ X_68 @ ( minus_minus_nat_o @ A_132 @ B_73 ) ) ) ) ) ).
thf(fact_378_insert__Diff__if,axiom,
! [A_132: hoare_1167836817_state > $o,X_68: hoare_1167836817_state,B_73: hoare_1167836817_state > $o] :
( ( ( member2058392318_state @ X_68 @ B_73 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_68 @ A_132 ) @ B_73 )
= ( minus_2107060239tate_o @ A_132 @ B_73 ) ) )
& ( ~ ( member2058392318_state @ X_68 @ B_73 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_68 @ A_132 ) @ B_73 )
= ( insert2134838167_state @ X_68 @ ( minus_2107060239tate_o @ A_132 @ B_73 ) ) ) ) ) ).
thf(fact_379_insert__Diff__if,axiom,
! [A_132: hoare_1775062406iple_a > $o,X_68: hoare_1775062406iple_a,B_73: hoare_1775062406iple_a > $o] :
( ( ( member2122167641iple_a @ X_68 @ B_73 )
=> ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_68 @ A_132 ) @ B_73 )
= ( minus_1944206118le_a_o @ A_132 @ B_73 ) ) )
& ( ~ ( member2122167641iple_a @ X_68 @ B_73 )
=> ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_68 @ A_132 ) @ B_73 )
= ( insert1281456128iple_a @ X_68 @ ( minus_1944206118le_a_o @ A_132 @ B_73 ) ) ) ) ) ).
thf(fact_380_insert__Diff,axiom,
! [A_131: nat,A_130: nat > $o] :
( ( member_nat @ A_131 @ A_130 )
=> ( ( insert_nat @ A_131 @ ( minus_minus_nat_o @ A_130 @ ( insert_nat @ A_131 @ bot_bot_nat_o ) ) )
= A_130 ) ) ).
thf(fact_381_insert__Diff,axiom,
! [A_131: hoare_1167836817_state,A_130: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_131 @ A_130 )
=> ( ( insert2134838167_state @ A_131 @ ( minus_2107060239tate_o @ A_130 @ ( insert2134838167_state @ A_131 @ bot_bo70021908tate_o ) ) )
= A_130 ) ) ).
thf(fact_382_insert__Diff,axiom,
! [A_131: hoare_1775062406iple_a,A_130: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_131 @ A_130 )
=> ( ( insert1281456128iple_a @ A_131 @ ( minus_1944206118le_a_o @ A_130 @ ( insert1281456128iple_a @ A_131 @ bot_bo751897185le_a_o ) ) )
= A_130 ) ) ).
thf(fact_383_Diff__insert__absorb,axiom,
! [X_67: nat,A_129: nat > $o] :
( ~ ( member_nat @ X_67 @ A_129 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_67 @ A_129 ) @ ( insert_nat @ X_67 @ bot_bot_nat_o ) )
= A_129 ) ) ).
thf(fact_384_Diff__insert__absorb,axiom,
! [X_67: hoare_1167836817_state,A_129: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_67 @ A_129 )
=> ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_67 @ A_129 ) @ ( insert2134838167_state @ X_67 @ bot_bo70021908tate_o ) )
= A_129 ) ) ).
thf(fact_385_Diff__insert__absorb,axiom,
! [X_67: hoare_1775062406iple_a,A_129: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ X_67 @ A_129 )
=> ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_67 @ A_129 ) @ ( insert1281456128iple_a @ X_67 @ bot_bo751897185le_a_o ) )
= A_129 ) ) ).
thf(fact_386_insert__Diff__single,axiom,
! [A_128: nat,A_127: nat > $o] :
( ( insert_nat @ A_128 @ ( minus_minus_nat_o @ A_127 @ ( insert_nat @ A_128 @ bot_bot_nat_o ) ) )
= ( insert_nat @ A_128 @ A_127 ) ) ).
thf(fact_387_insert__Diff__single,axiom,
! [A_128: hoare_1167836817_state,A_127: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_128 @ ( minus_2107060239tate_o @ A_127 @ ( insert2134838167_state @ A_128 @ bot_bo70021908tate_o ) ) )
= ( insert2134838167_state @ A_128 @ A_127 ) ) ).
thf(fact_388_insert__Diff__single,axiom,
! [A_128: hoare_1775062406iple_a,A_127: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_128 @ ( minus_1944206118le_a_o @ A_127 @ ( insert1281456128iple_a @ A_128 @ bot_bo751897185le_a_o ) ) )
= ( insert1281456128iple_a @ A_128 @ A_127 ) ) ).
thf(fact_389_Diff__insert2,axiom,
! [A_126: nat > $o,A_125: nat,B_72: nat > $o] :
( ( minus_minus_nat_o @ A_126 @ ( insert_nat @ A_125 @ B_72 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_126 @ ( insert_nat @ A_125 @ bot_bot_nat_o ) ) @ B_72 ) ) ).
thf(fact_390_Diff__insert2,axiom,
! [A_126: hoare_1167836817_state > $o,A_125: hoare_1167836817_state,B_72: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_126 @ ( insert2134838167_state @ A_125 @ B_72 ) )
= ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_126 @ ( insert2134838167_state @ A_125 @ bot_bo70021908tate_o ) ) @ B_72 ) ) ).
thf(fact_391_Diff__insert2,axiom,
! [A_126: hoare_1775062406iple_a > $o,A_125: hoare_1775062406iple_a,B_72: hoare_1775062406iple_a > $o] :
( ( minus_1944206118le_a_o @ A_126 @ ( insert1281456128iple_a @ A_125 @ B_72 ) )
= ( minus_1944206118le_a_o @ ( minus_1944206118le_a_o @ A_126 @ ( insert1281456128iple_a @ A_125 @ bot_bo751897185le_a_o ) ) @ B_72 ) ) ).
thf(fact_392_Diff__insert,axiom,
! [A_124: nat > $o,A_123: nat,B_71: nat > $o] :
( ( minus_minus_nat_o @ A_124 @ ( insert_nat @ A_123 @ B_71 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_124 @ B_71 ) @ ( insert_nat @ A_123 @ bot_bot_nat_o ) ) ) ).
thf(fact_393_Diff__insert,axiom,
! [A_124: hoare_1167836817_state > $o,A_123: hoare_1167836817_state,B_71: hoare_1167836817_state > $o] :
( ( minus_2107060239tate_o @ A_124 @ ( insert2134838167_state @ A_123 @ B_71 ) )
= ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_124 @ B_71 ) @ ( insert2134838167_state @ A_123 @ bot_bo70021908tate_o ) ) ) ).
thf(fact_394_Diff__insert,axiom,
! [A_124: hoare_1775062406iple_a > $o,A_123: hoare_1775062406iple_a,B_71: hoare_1775062406iple_a > $o] :
( ( minus_1944206118le_a_o @ A_124 @ ( insert1281456128iple_a @ A_123 @ B_71 ) )
= ( minus_1944206118le_a_o @ ( minus_1944206118le_a_o @ A_124 @ B_71 ) @ ( insert1281456128iple_a @ A_123 @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_395_finite__Diff__insert,axiom,
! [A_122: nat > $o,A_121: nat,B_70: nat > $o] :
( ( finite_finite_nat @ ( minus_minus_nat_o @ A_122 @ ( insert_nat @ A_121 @ B_70 ) ) )
<=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_122 @ B_70 ) ) ) ).
thf(fact_396_finite__Diff__insert,axiom,
! [A_122: hoare_1167836817_state > $o,A_121: hoare_1167836817_state,B_70: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ ( minus_2107060239tate_o @ A_122 @ ( insert2134838167_state @ A_121 @ B_70 ) ) )
<=> ( finite1084549118_state @ ( minus_2107060239tate_o @ A_122 @ B_70 ) ) ) ).
thf(fact_397_finite__Diff__insert,axiom,
! [A_122: hoare_1775062406iple_a > $o,A_121: hoare_1775062406iple_a,B_70: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ ( minus_1944206118le_a_o @ A_122 @ ( insert1281456128iple_a @ A_121 @ B_70 ) ) )
<=> ( finite2063573081iple_a @ ( minus_1944206118le_a_o @ A_122 @ B_70 ) ) ) ).
thf(fact_398_folding__one__idem_Oin__idem,axiom,
! [X_66: hoare_1775062406iple_a,A_120: hoare_1775062406iple_a > $o,F_30: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_29: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_30 @ F_29 )
=> ( ( finite2063573081iple_a @ A_120 )
=> ( ( member2122167641iple_a @ X_66 @ A_120 )
=> ( ( F_30 @ X_66 @ ( F_29 @ A_120 ) )
= ( F_29 @ A_120 ) ) ) ) ) ).
thf(fact_399_folding__one__idem_Oin__idem,axiom,
! [X_66: nat,A_120: nat > $o,F_30: nat > nat > nat,F_29: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_30 @ F_29 )
=> ( ( finite_finite_nat @ A_120 )
=> ( ( member_nat @ X_66 @ A_120 )
=> ( ( F_30 @ X_66 @ ( F_29 @ A_120 ) )
= ( F_29 @ A_120 ) ) ) ) ) ).
thf(fact_400_folding__one__idem_Ohom__commute,axiom,
! [N_4: nat > $o,H_1: nat > nat,F_28: nat > nat > nat,F_27: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_28 @ F_27 )
=> ( ! [X_3: nat,Y_1: nat] :
( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
= ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
=> ( ( finite_finite_nat @ N_4 )
=> ( ( N_4 != bot_bot_nat_o )
=> ( ( H_1 @ ( F_27 @ N_4 ) )
= ( F_27 @ ( image_nat_nat @ H_1 @ N_4 ) ) ) ) ) ) ) ).
thf(fact_401_folding__one__idem_Ohom__commute,axiom,
! [N_4: hoare_1167836817_state > $o,H_1: hoare_1167836817_state > hoare_1167836817_state,F_28: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_27: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_28 @ F_27 )
=> ( ! [X_3: hoare_1167836817_state,Y_1: hoare_1167836817_state] :
( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
= ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
=> ( ( finite1084549118_state @ N_4 )
=> ( ( N_4 != bot_bo70021908tate_o )
=> ( ( H_1 @ ( F_27 @ N_4 ) )
= ( F_27 @ ( image_31595733_state @ H_1 @ N_4 ) ) ) ) ) ) ) ).
thf(fact_402_folding__one__idem_Ohom__commute,axiom,
! [N_4: hoare_1775062406iple_a > $o,H_1: hoare_1775062406iple_a > hoare_1775062406iple_a,F_28: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_27: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_28 @ F_27 )
=> ( ! [X_3: hoare_1775062406iple_a,Y_1: hoare_1775062406iple_a] :
( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
= ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
=> ( ( finite2063573081iple_a @ N_4 )
=> ( ( N_4 != bot_bo751897185le_a_o )
=> ( ( H_1 @ ( F_27 @ N_4 ) )
= ( F_27 @ ( image_1170193413iple_a @ H_1 @ N_4 ) ) ) ) ) ) ) ).
thf(fact_403_finite__empty__induct,axiom,
! [P_4: ( nat > $o ) > $o,A_119: nat > $o] :
( ( finite_finite_nat @ A_119 )
=> ( ( P_4 @ A_119 )
=> ( ! [A_43: nat,A_98: nat > $o] :
( ( finite_finite_nat @ A_98 )
=> ( ( member_nat @ A_43 @ A_98 )
=> ( ( P_4 @ A_98 )
=> ( P_4 @ ( minus_minus_nat_o @ A_98 @ ( insert_nat @ A_43 @ bot_bot_nat_o ) ) ) ) ) )
=> ( P_4 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_404_finite__empty__induct,axiom,
! [P_4: ( hoare_1167836817_state > $o ) > $o,A_119: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_119 )
=> ( ( P_4 @ A_119 )
=> ( ! [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_98 )
=> ( ( member2058392318_state @ A_43 @ A_98 )
=> ( ( P_4 @ A_98 )
=> ( P_4 @ ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ A_43 @ bot_bo70021908tate_o ) ) ) ) ) )
=> ( P_4 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_405_finite__empty__induct,axiom,
! [P_4: ( hoare_1775062406iple_a > $o ) > $o,A_119: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_119 )
=> ( ( P_4 @ A_119 )
=> ( ! [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_98 )
=> ( ( member2122167641iple_a @ A_43 @ A_98 )
=> ( ( P_4 @ A_98 )
=> ( P_4 @ ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ A_43 @ bot_bo751897185le_a_o ) ) ) ) ) )
=> ( P_4 @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_406_comp__fun__idem__remove,axiom,
( finite389864113_nat_o
@ ^ [X_3: nat,A_98: nat > $o] : ( minus_minus_nat_o @ A_98 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) ) ) ).
thf(fact_407_comp__fun__idem__remove,axiom,
( finite856902323tate_o
@ ^ [X_3: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] : ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) ) ) ).
thf(fact_408_comp__fun__idem__remove,axiom,
( finite2120172977le_a_o
@ ^ [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] : ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ) ).
thf(fact_409_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_118: hoare_1775062406iple_a,Z_15: hoare_1775062406iple_a,A_117: hoare_1775062406iple_a > $o,Y_35: hoare_1775062406iple_a,F_26: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
( ( finite2064891473iple_a @ F_26 )
=> ( ( finite727644230iple_a @ F_26 @ Z_15 @ A_117 @ Y_35 )
=> ( ( member2122167641iple_a @ A_118 @ A_117 )
=> ? [Y_36: hoare_1775062406iple_a] :
( ( Y_35
= ( F_26 @ A_118 @ Y_36 ) )
& ( finite727644230iple_a @ F_26 @ Z_15 @ ( minus_1944206118le_a_o @ A_117 @ ( insert1281456128iple_a @ A_118 @ bot_bo751897185le_a_o ) ) @ Y_36 ) ) ) ) ) ).
thf(fact_410_fold__graph__permute__diff,axiom,
! [A_116: nat,B_69: nat,A_115: nat > $o,X_65: nat] :
( ( finite929467206at_nat @ times_times_nat @ B_69 @ A_115 @ X_65 )
=> ( ( member_nat @ A_116 @ A_115 )
=> ( ~ ( member_nat @ B_69 @ A_115 )
=> ( finite929467206at_nat @ times_times_nat @ A_116 @ ( insert_nat @ B_69 @ ( minus_minus_nat_o @ A_115 @ ( insert_nat @ A_116 @ bot_bot_nat_o ) ) ) @ X_65 ) ) ) ) ).
thf(fact_411_comp__fun__idem__insert,axiom,
finite389864113_nat_o @ insert_nat ).
thf(fact_412_comp__fun__idem__insert,axiom,
finite856902323tate_o @ insert2134838167_state ).
thf(fact_413_comp__fun__idem__insert,axiom,
finite2120172977le_a_o @ insert1281456128iple_a ).
thf(fact_414_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_34: hoare_1775062406iple_a,Z_14: hoare_1775062406iple_a,A_114: hoare_1775062406iple_a > $o,X_64: hoare_1775062406iple_a,F_25: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
( ( finite2064891473iple_a @ F_25 )
=> ( ( finite727644230iple_a @ F_25 @ Z_14 @ A_114 @ X_64 )
=> ( ( finite727644230iple_a @ F_25 @ Z_14 @ A_114 @ Y_34 )
=> ( Y_34 = X_64 ) ) ) ) ).
thf(fact_415_fold__graph__insert__swap,axiom,
! [Z_13: nat,B_68: nat,A_113: nat > $o,Y_33: nat] :
( ( finite929467206at_nat @ times_times_nat @ B_68 @ A_113 @ Y_33 )
=> ( ~ ( member_nat @ B_68 @ A_113 )
=> ( finite929467206at_nat @ times_times_nat @ Z_13 @ ( insert_nat @ B_68 @ A_113 ) @ ( times_times_nat @ Z_13 @ Y_33 ) ) ) ) ).
thf(fact_416_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_12: hoare_1775062406iple_a,X_63: hoare_1775062406iple_a,A_112: hoare_1775062406iple_a > $o,V: hoare_1775062406iple_a,F_24: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
( ( finite2064891473iple_a @ F_24 )
=> ( ( finite727644230iple_a @ F_24 @ Z_12 @ ( insert1281456128iple_a @ X_63 @ A_112 ) @ V )
=> ( ~ ( member2122167641iple_a @ X_63 @ A_112 )
=> ~ ! [Y_1: hoare_1775062406iple_a] :
( ( V
= ( F_24 @ X_63 @ Y_1 ) )
=> ~ ( finite727644230iple_a @ F_24 @ Z_12 @ A_112 @ Y_1 ) ) ) ) ) ).
thf(fact_417_fold1__insert,axiom,
! [X_62: nat,A_111: nat > $o] :
( ( A_111 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ A_111 )
=> ( ~ ( member_nat @ X_62 @ A_111 )
=> ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ X_62 @ A_111 ) )
= ( times_times_nat @ X_62 @ ( finite_fold1_nat @ times_times_nat @ A_111 ) ) ) ) ) ) ).
thf(fact_418_fold1__eq__fold,axiom,
! [A_110: nat,A_109: nat > $o] :
( ( finite_finite_nat @ A_109 )
=> ( ~ ( member_nat @ A_110 @ A_109 )
=> ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ A_110 @ A_109 ) )
= ( finite_fold_nat_nat @ times_times_nat @ A_110 @ A_109 ) ) ) ) ).
thf(fact_419_fold1__singleton,axiom,
! [F_23: nat > nat > nat,A_108: nat] :
( ( finite_fold1_nat @ F_23 @ ( insert_nat @ A_108 @ bot_bot_nat_o ) )
= A_108 ) ).
thf(fact_420_fold1__singleton,axiom,
! [F_23: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_108: hoare_1167836817_state] :
( ( finite1646097201_state @ F_23 @ ( insert2134838167_state @ A_108 @ bot_bo70021908tate_o ) )
= A_108 ) ).
thf(fact_421_fold1__singleton,axiom,
! [F_23: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_108: hoare_1775062406iple_a] :
( ( finite1790765286iple_a @ F_23 @ ( insert1281456128iple_a @ A_108 @ bot_bo751897185le_a_o ) )
= A_108 ) ).
thf(fact_422_fold1__singleton__def,axiom,
! [A_107: nat,G_5: ( nat > $o ) > nat,F_22: nat > nat > nat] :
( ( G_5
= ( finite_fold1_nat @ F_22 ) )
=> ( ( G_5 @ ( insert_nat @ A_107 @ bot_bot_nat_o ) )
= A_107 ) ) ).
thf(fact_423_fold1__singleton__def,axiom,
! [A_107: hoare_1167836817_state,G_5: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,F_22: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
( ( G_5
= ( finite1646097201_state @ F_22 ) )
=> ( ( G_5 @ ( insert2134838167_state @ A_107 @ bot_bo70021908tate_o ) )
= A_107 ) ) ).
thf(fact_424_fold1__singleton__def,axiom,
! [A_107: hoare_1775062406iple_a,G_5: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a,F_22: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
( ( G_5
= ( finite1790765286iple_a @ F_22 ) )
=> ( ( G_5 @ ( insert1281456128iple_a @ A_107 @ bot_bo751897185le_a_o ) )
= A_107 ) ) ).
thf(fact_425_comp__fun__commute_Ofold__equality,axiom,
! [Z_11: hoare_1775062406iple_a,A_106: hoare_1775062406iple_a > $o,Y_32: hoare_1775062406iple_a,F_21: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
( ( finite2064891473iple_a @ F_21 )
=> ( ( finite727644230iple_a @ F_21 @ Z_11 @ A_106 @ Y_32 )
=> ( ( finite1842721992iple_a @ F_21 @ Z_11 @ A_106 )
= Y_32 ) ) ) ).
thf(fact_426_fold__def,axiom,
! [F_20: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_10: hoare_1775062406iple_a,A_105: hoare_1775062406iple_a > $o] :
( ( finite1842721992iple_a @ F_20 @ Z_10 @ A_105 )
= ( the_Ho1155011127iple_a @ ( finite727644230iple_a @ F_20 @ Z_10 @ A_105 ) ) ) ).
thf(fact_427_folding__one_Oeq__fold,axiom,
! [A_104: nat > $o,F_19: nat > nat > nat,F_18: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_19 @ F_18 )
=> ( ( finite_finite_nat @ A_104 )
=> ( ( F_18 @ A_104 )
= ( finite_fold1_nat @ F_19 @ A_104 ) ) ) ) ).
thf(fact_428_folding__one_Oeq__fold,axiom,
! [A_104: hoare_1775062406iple_a > $o,F_19: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_18: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_19 @ F_18 )
=> ( ( finite2063573081iple_a @ A_104 )
=> ( ( F_18 @ A_104 )
= ( finite1790765286iple_a @ F_19 @ A_104 ) ) ) ) ).
thf(fact_429_folding__one_Oeq__fold_H,axiom,
! [X_61: nat,A_103: nat > $o,F_17: nat > nat > nat,F_16: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_17 @ F_16 )
=> ( ( finite_finite_nat @ A_103 )
=> ( ~ ( member_nat @ X_61 @ A_103 )
=> ( ( F_16 @ ( insert_nat @ X_61 @ A_103 ) )
= ( finite_fold_nat_nat @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
thf(fact_430_folding__one_Oeq__fold_H,axiom,
! [X_61: hoare_1167836817_state,A_103: hoare_1167836817_state > $o,F_17: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_16: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite1074406356_state @ F_17 @ F_16 )
=> ( ( finite1084549118_state @ A_103 )
=> ( ~ ( member2058392318_state @ X_61 @ A_103 )
=> ( ( F_16 @ ( insert2134838167_state @ X_61 @ A_103 ) )
= ( finite1731015960_state @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
thf(fact_431_folding__one_Oeq__fold_H,axiom,
! [X_61: hoare_1775062406iple_a,A_103: hoare_1775062406iple_a > $o,F_17: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_16: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite2078349315iple_a @ F_17 @ F_16 )
=> ( ( finite2063573081iple_a @ A_103 )
=> ( ~ ( member2122167641iple_a @ X_61 @ A_103 )
=> ( ( F_16 @ ( insert1281456128iple_a @ X_61 @ A_103 ) )
= ( finite1842721992iple_a @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
thf(fact_432_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_102: nat,A_101: nat > $o,F_15: nat > nat > nat,F_14: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_15 @ F_14 )
=> ( ( finite_finite_nat @ A_101 )
=> ( ( F_14 @ ( insert_nat @ A_102 @ A_101 ) )
= ( finite_fold_nat_nat @ F_15 @ A_102 @ A_101 ) ) ) ) ).
thf(fact_433_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_102: hoare_1167836817_state,A_101: hoare_1167836817_state > $o,F_15: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_14: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_15 @ F_14 )
=> ( ( finite1084549118_state @ A_101 )
=> ( ( F_14 @ ( insert2134838167_state @ A_102 @ A_101 ) )
= ( finite1731015960_state @ F_15 @ A_102 @ A_101 ) ) ) ) ).
thf(fact_434_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_102: hoare_1775062406iple_a,A_101: hoare_1775062406iple_a > $o,F_15: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_14: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_15 @ F_14 )
=> ( ( finite2063573081iple_a @ A_101 )
=> ( ( F_14 @ ( insert1281456128iple_a @ A_102 @ A_101 ) )
= ( finite1842721992iple_a @ F_15 @ A_102 @ A_101 ) ) ) ) ).
thf(fact_435_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_9: hoare_1775062406iple_a,A_100: hoare_1775062406iple_a > $o,F_13: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
( ( finite2064891473iple_a @ F_13 )
=> ( ( finite2063573081iple_a @ A_100 )
=> ( finite727644230iple_a @ F_13 @ Z_9 @ A_100 @ ( finite1842721992iple_a @ F_13 @ Z_9 @ A_100 ) ) ) ) ).
thf(fact_436_fold1__def,axiom,
! [F_12: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_99: hoare_1775062406iple_a > $o] :
( ( finite1790765286iple_a @ F_12 @ A_99 )
= ( the_Ho1155011127iple_a @ ( finite1946188886iple_a @ F_12 @ A_99 ) ) ) ).
thf(fact_437_minus__fold__remove,axiom,
! [B_67: nat > $o,A_97: nat > $o] :
( ( finite_finite_nat @ A_97 )
=> ( ( minus_minus_nat_o @ B_67 @ A_97 )
= ( finite326637109_nat_o
@ ^ [X_3: nat,A_98: nat > $o] : ( minus_minus_nat_o @ A_98 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) )
@ B_67
@ A_97 ) ) ) ).
thf(fact_438_minus__fold__remove,axiom,
! [B_67: hoare_1167836817_state > $o,A_97: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_97 )
=> ( ( minus_2107060239tate_o @ B_67 @ A_97 )
= ( finite291020855tate_o
@ ^ [X_3: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] : ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
@ B_67
@ A_97 ) ) ) ).
thf(fact_439_minus__fold__remove,axiom,
! [B_67: hoare_1775062406iple_a > $o,A_97: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_97 )
=> ( ( minus_1944206118le_a_o @ B_67 @ A_97 )
= ( finite1544171829le_a_o
@ ^ [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] : ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) )
@ B_67
@ A_97 ) ) ) ).
thf(fact_440_fold1__in,axiom,
! [A_96: nat > $o] :
( ( finite_finite_nat @ A_96 )
=> ( ( A_96 != bot_bot_nat_o )
=> ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( times_times_nat @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( finite_fold1_nat @ times_times_nat @ A_96 ) @ A_96 ) ) ) ) ).
thf(fact_441_semilattice__big_OF__eq,axiom,
! [A_95: nat > $o,F_11: nat > nat > nat,F_10: ( nat > $o ) > nat] :
( ( big_se275732192ig_nat @ F_11 @ F_10 )
=> ( ( finite_finite_nat @ A_95 )
=> ( ( F_10 @ A_95 )
= ( finite_fold1_nat @ F_11 @ A_95 ) ) ) ) ).
thf(fact_442_folding__one__idem_Osubset__idem,axiom,
! [B_66: nat > $o,A_94: nat > $o,F_9: nat > nat > nat,F_8: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_9 @ F_8 )
=> ( ( finite_finite_nat @ A_94 )
=> ( ( B_66 != bot_bot_nat_o )
=> ( ( ord_less_eq_nat_o @ B_66 @ A_94 )
=> ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
= ( F_8 @ A_94 ) ) ) ) ) ) ).
thf(fact_443_folding__one__idem_Osubset__idem,axiom,
! [B_66: hoare_1167836817_state > $o,A_94: hoare_1167836817_state > $o,F_9: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_8: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_9 @ F_8 )
=> ( ( finite1084549118_state @ A_94 )
=> ( ( B_66 != bot_bo70021908tate_o )
=> ( ( ord_le827224136tate_o @ B_66 @ A_94 )
=> ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
= ( F_8 @ A_94 ) ) ) ) ) ) ).
thf(fact_444_folding__one__idem_Osubset__idem,axiom,
! [B_66: hoare_1775062406iple_a > $o,A_94: hoare_1775062406iple_a > $o,F_9: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_8: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_9 @ F_8 )
=> ( ( finite2063573081iple_a @ A_94 )
=> ( ( B_66 != bot_bo751897185le_a_o )
=> ( ( ord_le1143225901le_a_o @ B_66 @ A_94 )
=> ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
= ( F_8 @ A_94 ) ) ) ) ) ) ).
thf(fact_445_order__refl,axiom,
! [X_60: nat] : ( ord_less_eq_nat @ X_60 @ X_60 ) ).
thf(fact_446_subsetD,axiom,
! [C_23: hoare_1775062406iple_a,A_93: hoare_1775062406iple_a > $o,B_65: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_93 @ B_65 )
=> ( ( member2122167641iple_a @ C_23 @ A_93 )
=> ( member2122167641iple_a @ C_23 @ B_65 ) ) ) ).
thf(fact_447_subsetD,axiom,
! [C_23: nat,A_93: nat > $o,B_65: nat > $o] :
( ( ord_less_eq_nat_o @ A_93 @ B_65 )
=> ( ( member_nat @ C_23 @ A_93 )
=> ( member_nat @ C_23 @ B_65 ) ) ) ).
thf(fact_448_UnCI,axiom,
! [A_92: hoare_1775062406iple_a > $o,C_22: hoare_1775062406iple_a,B_64: hoare_1775062406iple_a > $o] :
( ( ~ ( member2122167641iple_a @ C_22 @ B_64 )
=> ( member2122167641iple_a @ C_22 @ A_92 ) )
=> ( member2122167641iple_a @ C_22 @ ( semila13410563le_a_o @ A_92 @ B_64 ) ) ) ).
thf(fact_449_UnCI,axiom,
! [A_92: nat > $o,C_22: nat,B_64: nat > $o] :
( ( ~ ( member_nat @ C_22 @ B_64 )
=> ( member_nat @ C_22 @ A_92 ) )
=> ( member_nat @ C_22 @ ( semila848761471_nat_o @ A_92 @ B_64 ) ) ) ).
thf(fact_450_UnE,axiom,
! [C_21: hoare_1775062406iple_a,A_91: hoare_1775062406iple_a > $o,B_63: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_21 @ ( semila13410563le_a_o @ A_91 @ B_63 ) )
=> ( ~ ( member2122167641iple_a @ C_21 @ A_91 )
=> ( member2122167641iple_a @ C_21 @ B_63 ) ) ) ).
thf(fact_451_UnE,axiom,
! [C_21: nat,A_91: nat > $o,B_63: nat > $o] :
( ( member_nat @ C_21 @ ( semila848761471_nat_o @ A_91 @ B_63 ) )
=> ( ~ ( member_nat @ C_21 @ A_91 )
=> ( member_nat @ C_21 @ B_63 ) ) ) ).
thf(fact_452_empty__subsetI,axiom,
! [A_90: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_90 ) ).
thf(fact_453_empty__subsetI,axiom,
! [A_90: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_90 ) ).
thf(fact_454_empty__subsetI,axiom,
! [A_90: hoare_1775062406iple_a > $o] : ( ord_le1143225901le_a_o @ bot_bo751897185le_a_o @ A_90 ) ).
thf(fact_455_finite__Collect__subsets,axiom,
! [A_89: nat > $o] :
( ( finite_finite_nat @ A_89 )
=> ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [B_62: nat > $o] : ( ord_less_eq_nat_o @ B_62 @ A_89 ) ) ) ) ).
thf(fact_456_sup__le__fold__sup,axiom,
! [B_61: nat,A_88: nat,A_87: nat > $o] :
( ( finite_finite_nat @ A_87 )
=> ( ( member_nat @ A_88 @ A_87 )
=> ( ord_less_eq_nat @ ( semila972727038up_nat @ A_88 @ B_61 ) @ ( finite_fold_nat_nat @ semila972727038up_nat @ B_61 @ A_87 ) ) ) ) ).
thf(fact_457_subset__empty,axiom,
! [A_86: nat > $o] :
( ( ord_less_eq_nat_o @ A_86 @ bot_bot_nat_o )
<=> ( A_86 = bot_bot_nat_o ) ) ).
thf(fact_458_subset__empty,axiom,
! [A_86: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_86 @ bot_bo70021908tate_o )
<=> ( A_86 = bot_bo70021908tate_o ) ) ).
thf(fact_459_subset__empty,axiom,
! [A_86: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_86 @ bot_bo751897185le_a_o )
<=> ( A_86 = bot_bo751897185le_a_o ) ) ).
thf(fact_460_rev__finite__subset,axiom,
! [A_85: nat > $o,B_60: nat > $o] :
( ( finite_finite_nat @ B_60 )
=> ( ( ord_less_eq_nat_o @ A_85 @ B_60 )
=> ( finite_finite_nat @ A_85 ) ) ) ).
thf(fact_461_finite__subset,axiom,
! [A_84: nat > $o,B_59: nat > $o] :
( ( ord_less_eq_nat_o @ A_84 @ B_59 )
=> ( ( finite_finite_nat @ B_59 )
=> ( finite_finite_nat @ A_84 ) ) ) ).
thf(fact_462_subset__insertI,axiom,
! [B_58: nat > $o,A_83: nat] : ( ord_less_eq_nat_o @ B_58 @ ( insert_nat @ A_83 @ B_58 ) ) ).
thf(fact_463_subset__insertI,axiom,
! [B_58: hoare_1167836817_state > $o,A_83: hoare_1167836817_state] : ( ord_le827224136tate_o @ B_58 @ ( insert2134838167_state @ A_83 @ B_58 ) ) ).
thf(fact_464_subset__insertI,axiom,
! [B_58: hoare_1775062406iple_a > $o,A_83: hoare_1775062406iple_a] : ( ord_le1143225901le_a_o @ B_58 @ ( insert1281456128iple_a @ A_83 @ B_58 ) ) ).
thf(fact_465_insert__subset,axiom,
! [X_59: nat,A_82: nat > $o,B_57: nat > $o] :
( ( ord_less_eq_nat_o @ ( insert_nat @ X_59 @ A_82 ) @ B_57 )
<=> ( ( member_nat @ X_59 @ B_57 )
& ( ord_less_eq_nat_o @ A_82 @ B_57 ) ) ) ).
thf(fact_466_insert__subset,axiom,
! [X_59: hoare_1167836817_state,A_82: hoare_1167836817_state > $o,B_57: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ ( insert2134838167_state @ X_59 @ A_82 ) @ B_57 )
<=> ( ( member2058392318_state @ X_59 @ B_57 )
& ( ord_le827224136tate_o @ A_82 @ B_57 ) ) ) ).
thf(fact_467_insert__subset,axiom,
! [X_59: hoare_1775062406iple_a,A_82: hoare_1775062406iple_a > $o,B_57: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ ( insert1281456128iple_a @ X_59 @ A_82 ) @ B_57 )
<=> ( ( member2122167641iple_a @ X_59 @ B_57 )
& ( ord_le1143225901le_a_o @ A_82 @ B_57 ) ) ) ).
thf(fact_468_subset__insert,axiom,
! [B_56: nat > $o,X_58: nat,A_81: nat > $o] :
( ~ ( member_nat @ X_58 @ A_81 )
=> ( ( ord_less_eq_nat_o @ A_81 @ ( insert_nat @ X_58 @ B_56 ) )
<=> ( ord_less_eq_nat_o @ A_81 @ B_56 ) ) ) ).
thf(fact_469_subset__insert,axiom,
! [B_56: hoare_1167836817_state > $o,X_58: hoare_1167836817_state,A_81: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ X_58 @ A_81 )
=> ( ( ord_le827224136tate_o @ A_81 @ ( insert2134838167_state @ X_58 @ B_56 ) )
<=> ( ord_le827224136tate_o @ A_81 @ B_56 ) ) ) ).
thf(fact_470_subset__insert,axiom,
! [B_56: hoare_1775062406iple_a > $o,X_58: hoare_1775062406iple_a,A_81: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ X_58 @ A_81 )
=> ( ( ord_le1143225901le_a_o @ A_81 @ ( insert1281456128iple_a @ X_58 @ B_56 ) )
<=> ( ord_le1143225901le_a_o @ A_81 @ B_56 ) ) ) ).
thf(fact_471_subset__insertI2,axiom,
! [B_55: nat,A_80: nat > $o,B_54: nat > $o] :
( ( ord_less_eq_nat_o @ A_80 @ B_54 )
=> ( ord_less_eq_nat_o @ A_80 @ ( insert_nat @ B_55 @ B_54 ) ) ) ).
thf(fact_472_subset__insertI2,axiom,
! [B_55: hoare_1167836817_state,A_80: hoare_1167836817_state > $o,B_54: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_80 @ B_54 )
=> ( ord_le827224136tate_o @ A_80 @ ( insert2134838167_state @ B_55 @ B_54 ) ) ) ).
thf(fact_473_subset__insertI2,axiom,
! [B_55: hoare_1775062406iple_a,A_80: hoare_1775062406iple_a > $o,B_54: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_80 @ B_54 )
=> ( ord_le1143225901le_a_o @ A_80 @ ( insert1281456128iple_a @ B_55 @ B_54 ) ) ) ).
thf(fact_474_insert__mono,axiom,
! [A_79: nat,C_20: nat > $o,D_2: nat > $o] :
( ( ord_less_eq_nat_o @ C_20 @ D_2 )
=> ( ord_less_eq_nat_o @ ( insert_nat @ A_79 @ C_20 ) @ ( insert_nat @ A_79 @ D_2 ) ) ) ).
thf(fact_475_insert__mono,axiom,
! [A_79: hoare_1167836817_state,C_20: hoare_1167836817_state > $o,D_2: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ C_20 @ D_2 )
=> ( ord_le827224136tate_o @ ( insert2134838167_state @ A_79 @ C_20 ) @ ( insert2134838167_state @ A_79 @ D_2 ) ) ) ).
thf(fact_476_insert__mono,axiom,
! [A_79: hoare_1775062406iple_a,C_20: hoare_1775062406iple_a > $o,D_2: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ C_20 @ D_2 )
=> ( ord_le1143225901le_a_o @ ( insert1281456128iple_a @ A_79 @ C_20 ) @ ( insert1281456128iple_a @ A_79 @ D_2 ) ) ) ).
thf(fact_477_le__supE,axiom,
! [A_78: nat,B_53: nat,X_57: nat] :
( ( ord_less_eq_nat @ ( semila972727038up_nat @ A_78 @ B_53 ) @ X_57 )
=> ~ ( ( ord_less_eq_nat @ A_78 @ X_57 )
=> ~ ( ord_less_eq_nat @ B_53 @ X_57 ) ) ) ).
thf(fact_478_sup__mono,axiom,
! [B_52: nat,D_1: nat,A_77: nat,C_19: nat] :
( ( ord_less_eq_nat @ A_77 @ C_19 )
=> ( ( ord_less_eq_nat @ B_52 @ D_1 )
=> ( ord_less_eq_nat @ ( semila972727038up_nat @ A_77 @ B_52 ) @ ( semila972727038up_nat @ C_19 @ D_1 ) ) ) ) ).
thf(fact_479_sup__least,axiom,
! [Z_8: nat,Y_31: nat,X_56: nat] :
( ( ord_less_eq_nat @ Y_31 @ X_56 )
=> ( ( ord_less_eq_nat @ Z_8 @ X_56 )
=> ( ord_less_eq_nat @ ( semila972727038up_nat @ Y_31 @ Z_8 ) @ X_56 ) ) ) ).
thf(fact_480_le__supI,axiom,
! [B_51: nat,A_76: nat,X_55: nat] :
( ( ord_less_eq_nat @ A_76 @ X_55 )
=> ( ( ord_less_eq_nat @ B_51 @ X_55 )
=> ( ord_less_eq_nat @ ( semila972727038up_nat @ A_76 @ B_51 ) @ X_55 ) ) ) ).
thf(fact_481_sup__absorb1,axiom,
! [Y_30: nat,X_54: nat] :
( ( ord_less_eq_nat @ Y_30 @ X_54 )
=> ( ( semila972727038up_nat @ X_54 @ Y_30 )
= X_54 ) ) ).
thf(fact_482_sup__absorb2,axiom,
! [X_53: nat,Y_29: nat] :
( ( ord_less_eq_nat @ X_53 @ Y_29 )
=> ( ( semila972727038up_nat @ X_53 @ Y_29 )
= Y_29 ) ) ).
thf(fact_483_le__supI2,axiom,
! [A_75: nat,X_52: nat,B_50: nat] :
( ( ord_less_eq_nat @ X_52 @ B_50 )
=> ( ord_less_eq_nat @ X_52 @ ( semila972727038up_nat @ A_75 @ B_50 ) ) ) ).
thf(fact_484_le__supI1,axiom,
! [B_49: nat,X_51: nat,A_74: nat] :
( ( ord_less_eq_nat @ X_51 @ A_74 )
=> ( ord_less_eq_nat @ X_51 @ ( semila972727038up_nat @ A_74 @ B_49 ) ) ) ).
thf(fact_485_le__sup__iff,axiom,
! [X_50: nat,Y_28: nat,Z_7: nat] :
( ( ord_less_eq_nat @ ( semila972727038up_nat @ X_50 @ Y_28 ) @ Z_7 )
<=> ( ( ord_less_eq_nat @ X_50 @ Z_7 )
& ( ord_less_eq_nat @ Y_28 @ Z_7 ) ) ) ).
thf(fact_486_le__iff__sup,axiom,
! [X_49: nat,Y_27: nat] :
( ( ord_less_eq_nat @ X_49 @ Y_27 )
<=> ( ( semila972727038up_nat @ X_49 @ Y_27 )
= Y_27 ) ) ).
thf(fact_487_sup__ge2,axiom,
! [Y_26: nat,X_48: nat] : ( ord_less_eq_nat @ Y_26 @ ( semila972727038up_nat @ X_48 @ Y_26 ) ) ).
thf(fact_488_inf__sup__ord_I4_J,axiom,
! [Y_25: nat,X_47: nat] : ( ord_less_eq_nat @ Y_25 @ ( semila972727038up_nat @ X_47 @ Y_25 ) ) ).
thf(fact_489_sup__ge1,axiom,
! [X_46: nat,Y_24: nat] : ( ord_less_eq_nat @ X_46 @ ( semila972727038up_nat @ X_46 @ Y_24 ) ) ).
thf(fact_490_inf__sup__ord_I3_J,axiom,
! [X_45: nat,Y_23: nat] : ( ord_less_eq_nat @ X_45 @ ( semila972727038up_nat @ X_45 @ Y_23 ) ) ).
thf(fact_491_sup__eq__bot__iff,axiom,
! [X_44: $o,Y_22: $o] :
( ( ( semila10642723_sup_o @ X_44 @ Y_22 )
<=> bot_bot_o )
<=> ( ( X_44
<=> bot_bot_o )
& ( Y_22
<=> bot_bot_o ) ) ) ).
thf(fact_492_sup__eq__bot__iff,axiom,
! [X_44: nat > $o,Y_22: nat > $o] :
( ( ( semila848761471_nat_o @ X_44 @ Y_22 )
= bot_bot_nat_o )
<=> ( ( X_44 = bot_bot_nat_o )
& ( Y_22 = bot_bot_nat_o ) ) ) ).
thf(fact_493_sup__eq__bot__iff,axiom,
! [X_44: hoare_1167836817_state > $o,Y_22: hoare_1167836817_state > $o] :
( ( ( semila1172322802tate_o @ X_44 @ Y_22 )
= bot_bo70021908tate_o )
<=> ( ( X_44 = bot_bo70021908tate_o )
& ( Y_22 = bot_bo70021908tate_o ) ) ) ).
thf(fact_494_sup__eq__bot__iff,axiom,
! [X_44: hoare_1775062406iple_a > $o,Y_22: hoare_1775062406iple_a > $o] :
( ( ( semila13410563le_a_o @ X_44 @ Y_22 )
= bot_bo751897185le_a_o )
<=> ( ( X_44 = bot_bo751897185le_a_o )
& ( Y_22 = bot_bo751897185le_a_o ) ) ) ).
thf(fact_495_sup__bot__right,axiom,
! [X_43: $o] :
( ( semila10642723_sup_o @ X_43 @ bot_bot_o )
<=> X_43 ) ).
thf(fact_496_sup__bot__right,axiom,
! [X_43: nat > $o] :
( ( semila848761471_nat_o @ X_43 @ bot_bot_nat_o )
= X_43 ) ).
thf(fact_497_sup__bot__right,axiom,
! [X_43: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ X_43 @ bot_bo70021908tate_o )
= X_43 ) ).
thf(fact_498_sup__bot__right,axiom,
! [X_43: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ X_43 @ bot_bo751897185le_a_o )
= X_43 ) ).
thf(fact_499_sup__bot__left,axiom,
! [X_42: $o] :
( ( semila10642723_sup_o @ bot_bot_o @ X_42 )
<=> X_42 ) ).
thf(fact_500_sup__bot__left,axiom,
! [X_42: nat > $o] :
( ( semila848761471_nat_o @ bot_bot_nat_o @ X_42 )
= X_42 ) ).
thf(fact_501_sup__bot__left,axiom,
! [X_42: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ bot_bo70021908tate_o @ X_42 )
= X_42 ) ).
thf(fact_502_sup__bot__left,axiom,
! [X_42: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ X_42 )
= X_42 ) ).
thf(fact_503_Un__empty__left,axiom,
! [B_48: nat > $o] :
( ( semila848761471_nat_o @ bot_bot_nat_o @ B_48 )
= B_48 ) ).
thf(fact_504_Un__empty__left,axiom,
! [B_48: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ bot_bo70021908tate_o @ B_48 )
= B_48 ) ).
thf(fact_505_Un__empty__left,axiom,
! [B_48: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ B_48 )
= B_48 ) ).
thf(fact_506_Un__empty__right,axiom,
! [A_73: nat > $o] :
( ( semila848761471_nat_o @ A_73 @ bot_bot_nat_o )
= A_73 ) ).
thf(fact_507_Un__empty__right,axiom,
! [A_73: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_73 @ bot_bo70021908tate_o )
= A_73 ) ).
thf(fact_508_Un__empty__right,axiom,
! [A_73: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_73 @ bot_bo751897185le_a_o )
= A_73 ) ).
thf(fact_509_Un__empty,axiom,
! [A_72: nat > $o,B_47: nat > $o] :
( ( ( semila848761471_nat_o @ A_72 @ B_47 )
= bot_bot_nat_o )
<=> ( ( A_72 = bot_bot_nat_o )
& ( B_47 = bot_bot_nat_o ) ) ) ).
thf(fact_510_Un__empty,axiom,
! [A_72: hoare_1167836817_state > $o,B_47: hoare_1167836817_state > $o] :
( ( ( semila1172322802tate_o @ A_72 @ B_47 )
= bot_bo70021908tate_o )
<=> ( ( A_72 = bot_bo70021908tate_o )
& ( B_47 = bot_bo70021908tate_o ) ) ) ).
thf(fact_511_Un__empty,axiom,
! [A_72: hoare_1775062406iple_a > $o,B_47: hoare_1775062406iple_a > $o] :
( ( ( semila13410563le_a_o @ A_72 @ B_47 )
= bot_bo751897185le_a_o )
<=> ( ( A_72 = bot_bo751897185le_a_o )
& ( B_47 = bot_bo751897185le_a_o ) ) ) ).
thf(fact_512_finite__Un,axiom,
! [F_7: nat > $o,G_4: nat > $o] :
( ( finite_finite_nat @ ( semila848761471_nat_o @ F_7 @ G_4 ) )
<=> ( ( finite_finite_nat @ F_7 )
& ( finite_finite_nat @ G_4 ) ) ) ).
thf(fact_513_finite__UnI,axiom,
! [G_3: nat > $o,F_6: nat > $o] :
( ( finite_finite_nat @ F_6 )
=> ( ( finite_finite_nat @ G_3 )
=> ( finite_finite_nat @ ( semila848761471_nat_o @ F_6 @ G_3 ) ) ) ) ).
thf(fact_514_linorder__le__cases,axiom,
! [X_41: nat,Y_21: nat] :
( ~ ( ord_less_eq_nat @ X_41 @ Y_21 )
=> ( ord_less_eq_nat @ Y_21 @ X_41 ) ) ).
thf(fact_515_xt1_I6_J,axiom,
! [Z_6: nat,Y_20: nat,X_40: nat] :
( ( ord_less_eq_nat @ Y_20 @ X_40 )
=> ( ( ord_less_eq_nat @ Z_6 @ Y_20 )
=> ( ord_less_eq_nat @ Z_6 @ X_40 ) ) ) ).
thf(fact_516_xt1_I5_J,axiom,
! [Y_19: nat,X_39: nat] :
( ( ord_less_eq_nat @ Y_19 @ X_39 )
=> ( ( ord_less_eq_nat @ X_39 @ Y_19 )
=> ( X_39 = Y_19 ) ) ) ).
thf(fact_517_order__trans,axiom,
! [Z_5: nat,X_38: nat,Y_18: nat] :
( ( ord_less_eq_nat @ X_38 @ Y_18 )
=> ( ( ord_less_eq_nat @ Y_18 @ Z_5 )
=> ( ord_less_eq_nat @ X_38 @ Z_5 ) ) ) ).
thf(fact_518_order__antisym,axiom,
! [X_37: nat,Y_17: nat] :
( ( ord_less_eq_nat @ X_37 @ Y_17 )
=> ( ( ord_less_eq_nat @ Y_17 @ X_37 )
=> ( X_37 = Y_17 ) ) ) ).
thf(fact_519_xt1_I4_J,axiom,
! [C_18: nat,B_46: nat,A_71: nat] :
( ( ord_less_eq_nat @ B_46 @ A_71 )
=> ( ( B_46 = C_18 )
=> ( ord_less_eq_nat @ C_18 @ A_71 ) ) ) ).
thf(fact_520_ord__le__eq__trans,axiom,
! [C_17: nat,A_70: nat,B_45: nat] :
( ( ord_less_eq_nat @ A_70 @ B_45 )
=> ( ( B_45 = C_17 )
=> ( ord_less_eq_nat @ A_70 @ C_17 ) ) ) ).
thf(fact_521_xt1_I3_J,axiom,
! [C_16: nat,A_69: nat,B_44: nat] :
( ( A_69 = B_44 )
=> ( ( ord_less_eq_nat @ C_16 @ B_44 )
=> ( ord_less_eq_nat @ C_16 @ A_69 ) ) ) ).
thf(fact_522_ord__eq__le__trans,axiom,
! [C_15: nat,A_68: nat,B_43: nat] :
( ( A_68 = B_43 )
=> ( ( ord_less_eq_nat @ B_43 @ C_15 )
=> ( ord_less_eq_nat @ A_68 @ C_15 ) ) ) ).
thf(fact_523_order__antisym__conv,axiom,
! [Y_16: nat,X_36: nat] :
( ( ord_less_eq_nat @ Y_16 @ X_36 )
=> ( ( ord_less_eq_nat @ X_36 @ Y_16 )
<=> ( X_36 = Y_16 ) ) ) ).
thf(fact_524_order__eq__refl,axiom,
! [X_35: nat,Y_15: nat] :
( ( X_35 = Y_15 )
=> ( ord_less_eq_nat @ X_35 @ Y_15 ) ) ).
thf(fact_525_order__eq__iff,axiom,
! [X_34: nat,Y_14: nat] :
( ( X_34 = Y_14 )
<=> ( ( ord_less_eq_nat @ X_34 @ Y_14 )
& ( ord_less_eq_nat @ Y_14 @ X_34 ) ) ) ).
thf(fact_526_linorder__linear,axiom,
! [X_33: nat,Y_13: nat] :
( ( ord_less_eq_nat @ X_33 @ Y_13 )
| ( ord_less_eq_nat @ Y_13 @ X_33 ) ) ).
thf(fact_527_Collect__disj__eq,axiom,
! [P_3: hoare_1775062406iple_a > $o,Q_1: hoare_1775062406iple_a > $o] :
( ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( P_3 @ X_3 ) @ ( Q_1 @ X_3 ) ) )
= ( semila13410563le_a_o @ ( collec676402587iple_a @ P_3 ) @ ( collec676402587iple_a @ Q_1 ) ) ) ).
thf(fact_528_Collect__disj__eq,axiom,
! [P_3: nat > $o,Q_1: nat > $o] :
( ( collect_nat
@ ^ [X_3: nat] : ( (|) @ ( P_3 @ X_3 ) @ ( Q_1 @ X_3 ) ) )
= ( semila848761471_nat_o @ ( collect_nat @ P_3 ) @ ( collect_nat @ Q_1 ) ) ) ).
thf(fact_529_set__mp,axiom,
! [X_32: hoare_1775062406iple_a,A_67: hoare_1775062406iple_a > $o,B_42: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_67 @ B_42 )
=> ( ( member2122167641iple_a @ X_32 @ A_67 )
=> ( member2122167641iple_a @ X_32 @ B_42 ) ) ) ).
thf(fact_530_set__mp,axiom,
! [X_32: nat,A_67: nat > $o,B_42: nat > $o] :
( ( ord_less_eq_nat_o @ A_67 @ B_42 )
=> ( ( member_nat @ X_32 @ A_67 )
=> ( member_nat @ X_32 @ B_42 ) ) ) ).
thf(fact_531_set__rev__mp,axiom,
! [B_41: hoare_1775062406iple_a > $o,X_31: hoare_1775062406iple_a,A_66: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_31 @ A_66 )
=> ( ( ord_le1143225901le_a_o @ A_66 @ B_41 )
=> ( member2122167641iple_a @ X_31 @ B_41 ) ) ) ).
thf(fact_532_set__rev__mp,axiom,
! [B_41: nat > $o,X_31: nat,A_66: nat > $o] :
( ( member_nat @ X_31 @ A_66 )
=> ( ( ord_less_eq_nat_o @ A_66 @ B_41 )
=> ( member_nat @ X_31 @ B_41 ) ) ) ).
thf(fact_533_in__mono,axiom,
! [X_30: hoare_1775062406iple_a,A_65: hoare_1775062406iple_a > $o,B_40: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_65 @ B_40 )
=> ( ( member2122167641iple_a @ X_30 @ A_65 )
=> ( member2122167641iple_a @ X_30 @ B_40 ) ) ) ).
thf(fact_534_in__mono,axiom,
! [X_30: nat,A_65: nat > $o,B_40: nat > $o] :
( ( ord_less_eq_nat_o @ A_65 @ B_40 )
=> ( ( member_nat @ X_30 @ A_65 )
=> ( member_nat @ X_30 @ B_40 ) ) ) ).
thf(fact_535_UnI2,axiom,
! [A_64: hoare_1775062406iple_a > $o,C_14: hoare_1775062406iple_a,B_39: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_14 @ B_39 )
=> ( member2122167641iple_a @ C_14 @ ( semila13410563le_a_o @ A_64 @ B_39 ) ) ) ).
thf(fact_536_UnI2,axiom,
! [A_64: nat > $o,C_14: nat,B_39: nat > $o] :
( ( member_nat @ C_14 @ B_39 )
=> ( member_nat @ C_14 @ ( semila848761471_nat_o @ A_64 @ B_39 ) ) ) ).
thf(fact_537_UnI1,axiom,
! [B_38: hoare_1775062406iple_a > $o,C_13: hoare_1775062406iple_a,A_63: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_13 @ A_63 )
=> ( member2122167641iple_a @ C_13 @ ( semila13410563le_a_o @ A_63 @ B_38 ) ) ) ).
thf(fact_538_UnI1,axiom,
! [B_38: nat > $o,C_13: nat,A_63: nat > $o] :
( ( member_nat @ C_13 @ A_63 )
=> ( member_nat @ C_13 @ ( semila848761471_nat_o @ A_63 @ B_38 ) ) ) ).
thf(fact_539_Un__iff,axiom,
! [C_12: hoare_1775062406iple_a,A_62: hoare_1775062406iple_a > $o,B_37: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_12 @ ( semila13410563le_a_o @ A_62 @ B_37 ) )
<=> ( ( member2122167641iple_a @ C_12 @ A_62 )
| ( member2122167641iple_a @ C_12 @ B_37 ) ) ) ).
thf(fact_540_Un__iff,axiom,
! [C_12: nat,A_62: nat > $o,B_37: nat > $o] :
( ( member_nat @ C_12 @ ( semila848761471_nat_o @ A_62 @ B_37 ) )
<=> ( ( member_nat @ C_12 @ A_62 )
| ( member_nat @ C_12 @ B_37 ) ) ) ).
thf(fact_541_Un__def,axiom,
! [A_61: hoare_1775062406iple_a > $o,B_36: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_61 @ B_36 )
= ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( member2122167641iple_a @ X_3 @ A_61 ) @ ( member2122167641iple_a @ X_3 @ B_36 ) ) ) ) ).
thf(fact_542_Un__def,axiom,
! [A_61: nat > $o,B_36: nat > $o] :
( ( semila848761471_nat_o @ A_61 @ B_36 )
= ( collect_nat
@ ^ [X_3: nat] : ( (|) @ ( member_nat @ X_3 @ A_61 ) @ ( member_nat @ X_3 @ B_36 ) ) ) ) ).
thf(fact_543_pred__subset__eq,axiom,
! [R_2: hoare_1775062406iple_a > $o,S_4: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o
@ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ R_2 )
@ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ S_4 ) )
<=> ( ord_le1143225901le_a_o @ R_2 @ S_4 ) ) ).
thf(fact_544_pred__subset__eq,axiom,
! [R_2: nat > $o,S_4: nat > $o] :
( ( ord_less_eq_nat_o
@ ^ [X_3: nat] : ( member_nat @ X_3 @ R_2 )
@ ^ [X_3: nat] : ( member_nat @ X_3 @ S_4 ) )
<=> ( ord_less_eq_nat_o @ R_2 @ S_4 ) ) ).
thf(fact_545_sup__Un__eq,axiom,
! [R_1: hoare_1775062406iple_a > $o,S_3: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
( ( semila13410563le_a_o
@ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ R_1 )
@ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ S_3 )
@ X_3 )
<=> ( member2122167641iple_a @ X_3 @ ( semila13410563le_a_o @ R_1 @ S_3 ) ) ) ).
thf(fact_546_sup__Un__eq,axiom,
! [R_1: nat > $o,S_3: nat > $o,X_3: nat] :
( ( semila848761471_nat_o
@ ^ [Y_1: nat] : ( member_nat @ Y_1 @ R_1 )
@ ^ [Y_1: nat] : ( member_nat @ Y_1 @ S_3 )
@ X_3 )
<=> ( member_nat @ X_3 @ ( semila848761471_nat_o @ R_1 @ S_3 ) ) ) ).
thf(fact_547_bot__least,axiom,
! [A_60: $o] : ( ord_less_eq_o @ bot_bot_o @ A_60 ) ).
thf(fact_548_bot__least,axiom,
! [A_60: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_60 ) ).
thf(fact_549_bot__least,axiom,
! [A_60: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_60 ) ).
thf(fact_550_bot__least,axiom,
! [A_60: hoare_1775062406iple_a > $o] : ( ord_le1143225901le_a_o @ bot_bo751897185le_a_o @ A_60 ) ).
thf(fact_551_bot__least,axiom,
! [A_60: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_60 ) ).
thf(fact_552_bot__unique,axiom,
! [A_59: $o] :
( ( ord_less_eq_o @ A_59 @ bot_bot_o )
<=> ( A_59
<=> bot_bot_o ) ) ).
thf(fact_553_bot__unique,axiom,
! [A_59: nat > $o] :
( ( ord_less_eq_nat_o @ A_59 @ bot_bot_nat_o )
<=> ( A_59 = bot_bot_nat_o ) ) ).
thf(fact_554_bot__unique,axiom,
! [A_59: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_59 @ bot_bo70021908tate_o )
<=> ( A_59 = bot_bo70021908tate_o ) ) ).
thf(fact_555_bot__unique,axiom,
! [A_59: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_59 @ bot_bo751897185le_a_o )
<=> ( A_59 = bot_bo751897185le_a_o ) ) ).
thf(fact_556_bot__unique,axiom,
! [A_59: nat] :
( ( ord_less_eq_nat @ A_59 @ bot_bot_nat )
<=> ( A_59 = bot_bot_nat ) ) ).
thf(fact_557_le__bot,axiom,
! [A_58: $o] :
( ( ord_less_eq_o @ A_58 @ bot_bot_o )
=> ( A_58
<=> bot_bot_o ) ) ).
thf(fact_558_le__bot,axiom,
! [A_58: nat > $o] :
( ( ord_less_eq_nat_o @ A_58 @ bot_bot_nat_o )
=> ( A_58 = bot_bot_nat_o ) ) ).
thf(fact_559_le__bot,axiom,
! [A_58: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_58 @ bot_bo70021908tate_o )
=> ( A_58 = bot_bo70021908tate_o ) ) ).
thf(fact_560_le__bot,axiom,
! [A_58: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_58 @ bot_bo751897185le_a_o )
=> ( A_58 = bot_bo751897185le_a_o ) ) ).
thf(fact_561_le__bot,axiom,
! [A_58: nat] :
( ( ord_less_eq_nat @ A_58 @ bot_bot_nat )
=> ( A_58 = bot_bot_nat ) ) ).
thf(fact_562_Un__insert__right,axiom,
! [A_57: nat > $o,A_56: nat,B_35: nat > $o] :
( ( semila848761471_nat_o @ A_57 @ ( insert_nat @ A_56 @ B_35 ) )
= ( insert_nat @ A_56 @ ( semila848761471_nat_o @ A_57 @ B_35 ) ) ) ).
thf(fact_563_Un__insert__right,axiom,
! [A_57: hoare_1167836817_state > $o,A_56: hoare_1167836817_state,B_35: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ A_57 @ ( insert2134838167_state @ A_56 @ B_35 ) )
= ( insert2134838167_state @ A_56 @ ( semila1172322802tate_o @ A_57 @ B_35 ) ) ) ).
thf(fact_564_Un__insert__right,axiom,
! [A_57: hoare_1775062406iple_a > $o,A_56: hoare_1775062406iple_a,B_35: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ A_57 @ ( insert1281456128iple_a @ A_56 @ B_35 ) )
= ( insert1281456128iple_a @ A_56 @ ( semila13410563le_a_o @ A_57 @ B_35 ) ) ) ).
thf(fact_565_Un__insert__left,axiom,
! [A_55: nat,B_34: nat > $o,C_11: nat > $o] :
( ( semila848761471_nat_o @ ( insert_nat @ A_55 @ B_34 ) @ C_11 )
= ( insert_nat @ A_55 @ ( semila848761471_nat_o @ B_34 @ C_11 ) ) ) ).
thf(fact_566_Un__insert__left,axiom,
! [A_55: hoare_1167836817_state,B_34: hoare_1167836817_state > $o,C_11: hoare_1167836817_state > $o] :
( ( semila1172322802tate_o @ ( insert2134838167_state @ A_55 @ B_34 ) @ C_11 )
= ( insert2134838167_state @ A_55 @ ( semila1172322802tate_o @ B_34 @ C_11 ) ) ) ).
thf(fact_567_Un__insert__left,axiom,
! [A_55: hoare_1775062406iple_a,B_34: hoare_1775062406iple_a > $o,C_11: hoare_1775062406iple_a > $o] :
( ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_55 @ B_34 ) @ C_11 )
= ( insert1281456128iple_a @ A_55 @ ( semila13410563le_a_o @ B_34 @ C_11 ) ) ) ).
thf(fact_568_weaken,axiom,
! [Ts_2: hoare_1167836817_state > $o,G_2: hoare_1167836817_state > $o,Ts_1: hoare_1167836817_state > $o] :
( ( hoare_123228589_state @ G_2 @ Ts_1 )
=> ( ( ord_le827224136tate_o @ Ts_2 @ Ts_1 )
=> ( hoare_123228589_state @ G_2 @ Ts_2 ) ) ) ).
thf(fact_569_weaken,axiom,
! [Ts_2: hoare_1775062406iple_a > $o,G_2: hoare_1775062406iple_a > $o,Ts_1: hoare_1775062406iple_a > $o] :
( ( hoare_1508237396rivs_a @ G_2 @ Ts_1 )
=> ( ( ord_le1143225901le_a_o @ Ts_2 @ Ts_1 )
=> ( hoare_1508237396rivs_a @ G_2 @ Ts_2 ) ) ) ).
thf(fact_570_asm,axiom,
! [Ts: hoare_1167836817_state > $o,G_1: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ Ts @ G_1 )
=> ( hoare_123228589_state @ G_1 @ Ts ) ) ).
thf(fact_571_asm,axiom,
! [Ts: hoare_1775062406iple_a > $o,G_1: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ Ts @ G_1 )
=> ( hoare_1508237396rivs_a @ G_1 @ Ts ) ) ).
thf(fact_572_insert__def,axiom,
! [A_54: nat,B_33: nat > $o] :
( ( insert_nat @ A_54 @ B_33 )
= ( semila848761471_nat_o
@ ( collect_nat
@ ^ [X_3: nat] : X_3 = A_54 )
@ B_33 ) ) ).
thf(fact_573_insert__def,axiom,
! [A_54: hoare_1167836817_state,B_33: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_54 @ B_33 )
= ( semila1172322802tate_o
@ ( collec1027672124_state
@ ^ [X_3: hoare_1167836817_state] : X_3 = A_54 )
@ B_33 ) ) ).
thf(fact_574_insert__def,axiom,
! [A_54: hoare_1775062406iple_a,B_33: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_54 @ B_33 )
= ( semila13410563le_a_o
@ ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : X_3 = A_54 )
@ B_33 ) ) ).
thf(fact_575_insert__is__Un,axiom,
! [A_53: nat,A_52: nat > $o] :
( ( insert_nat @ A_53 @ A_52 )
= ( semila848761471_nat_o @ ( insert_nat @ A_53 @ bot_bot_nat_o ) @ A_52 ) ) ).
thf(fact_576_insert__is__Un,axiom,
! [A_53: hoare_1167836817_state,A_52: hoare_1167836817_state > $o] :
( ( insert2134838167_state @ A_53 @ A_52 )
= ( semila1172322802tate_o @ ( insert2134838167_state @ A_53 @ bot_bo70021908tate_o ) @ A_52 ) ) ).
thf(fact_577_insert__is__Un,axiom,
! [A_53: hoare_1775062406iple_a,A_52: hoare_1775062406iple_a > $o] :
( ( insert1281456128iple_a @ A_53 @ A_52 )
= ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_53 @ bot_bo751897185le_a_o ) @ A_52 ) ) ).
thf(fact_578_fold__sup__insert,axiom,
! [B_32: nat,A_51: nat,A_50: nat > $o] :
( ( finite_finite_nat @ A_50 )
=> ( ( finite_fold_nat_nat @ semila972727038up_nat @ B_32 @ ( insert_nat @ A_51 @ A_50 ) )
= ( semila972727038up_nat @ A_51 @ ( finite_fold_nat_nat @ semila972727038up_nat @ B_32 @ A_50 ) ) ) ) ).
thf(fact_579_union__fold__insert,axiom,
! [B_31: nat > $o,A_49: nat > $o] :
( ( finite_finite_nat @ A_49 )
=> ( ( semila848761471_nat_o @ A_49 @ B_31 )
= ( finite326637109_nat_o @ insert_nat @ B_31 @ A_49 ) ) ) ).
thf(fact_580_union__fold__insert,axiom,
! [B_31: hoare_1167836817_state > $o,A_49: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ A_49 )
=> ( ( semila1172322802tate_o @ A_49 @ B_31 )
= ( finite291020855tate_o @ insert2134838167_state @ B_31 @ A_49 ) ) ) ).
thf(fact_581_union__fold__insert,axiom,
! [B_31: hoare_1775062406iple_a > $o,A_49: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ A_49 )
=> ( ( semila13410563le_a_o @ A_49 @ B_31 )
= ( finite1544171829le_a_o @ insert1281456128iple_a @ B_31 @ A_49 ) ) ) ).
thf(fact_582_subset__singletonD,axiom,
! [A_48: nat > $o,X_29: nat] :
( ( ord_less_eq_nat_o @ A_48 @ ( insert_nat @ X_29 @ bot_bot_nat_o ) )
=> ( ( A_48 = bot_bot_nat_o )
| ( A_48
= ( insert_nat @ X_29 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_583_subset__singletonD,axiom,
! [A_48: hoare_1167836817_state > $o,X_29: hoare_1167836817_state] :
( ( ord_le827224136tate_o @ A_48 @ ( insert2134838167_state @ X_29 @ bot_bo70021908tate_o ) )
=> ( ( A_48 = bot_bo70021908tate_o )
| ( A_48
= ( insert2134838167_state @ X_29 @ bot_bo70021908tate_o ) ) ) ) ).
thf(fact_584_subset__singletonD,axiom,
! [A_48: hoare_1775062406iple_a > $o,X_29: hoare_1775062406iple_a] :
( ( ord_le1143225901le_a_o @ A_48 @ ( insert1281456128iple_a @ X_29 @ bot_bo751897185le_a_o ) )
=> ( ( A_48 = bot_bo751897185le_a_o )
| ( A_48
= ( insert1281456128iple_a @ X_29 @ bot_bo751897185le_a_o ) ) ) ) ).
thf(fact_585_diff__single__insert,axiom,
! [A_47: nat > $o,X_28: nat,B_30: nat > $o] :
( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_47 @ ( insert_nat @ X_28 @ bot_bot_nat_o ) ) @ B_30 )
=> ( ( member_nat @ X_28 @ A_47 )
=> ( ord_less_eq_nat_o @ A_47 @ ( insert_nat @ X_28 @ B_30 ) ) ) ) ).
thf(fact_586_diff__single__insert,axiom,
! [A_47: hoare_1167836817_state > $o,X_28: hoare_1167836817_state,B_30: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_47 @ ( insert2134838167_state @ X_28 @ bot_bo70021908tate_o ) ) @ B_30 )
=> ( ( member2058392318_state @ X_28 @ A_47 )
=> ( ord_le827224136tate_o @ A_47 @ ( insert2134838167_state @ X_28 @ B_30 ) ) ) ) ).
thf(fact_587_diff__single__insert,axiom,
! [A_47: hoare_1775062406iple_a > $o,X_28: hoare_1775062406iple_a,B_30: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ ( minus_1944206118le_a_o @ A_47 @ ( insert1281456128iple_a @ X_28 @ bot_bo751897185le_a_o ) ) @ B_30 )
=> ( ( member2122167641iple_a @ X_28 @ A_47 )
=> ( ord_le1143225901le_a_o @ A_47 @ ( insert1281456128iple_a @ X_28 @ B_30 ) ) ) ) ).
thf(fact_588_subset__insert__iff,axiom,
! [A_46: nat > $o,X_27: nat,B_29: nat > $o] :
( ( ord_less_eq_nat_o @ A_46 @ ( insert_nat @ X_27 @ B_29 ) )
<=> ( ( ( member_nat @ X_27 @ A_46 )
=> ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_46 @ ( insert_nat @ X_27 @ bot_bot_nat_o ) ) @ B_29 ) )
& ( ~ ( member_nat @ X_27 @ A_46 )
=> ( ord_less_eq_nat_o @ A_46 @ B_29 ) ) ) ) ).
thf(fact_589_subset__insert__iff,axiom,
! [A_46: hoare_1167836817_state > $o,X_27: hoare_1167836817_state,B_29: hoare_1167836817_state > $o] :
( ( ord_le827224136tate_o @ A_46 @ ( insert2134838167_state @ X_27 @ B_29 ) )
<=> ( ( ( member2058392318_state @ X_27 @ A_46 )
=> ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_46 @ ( insert2134838167_state @ X_27 @ bot_bo70021908tate_o ) ) @ B_29 ) )
& ( ~ ( member2058392318_state @ X_27 @ A_46 )
=> ( ord_le827224136tate_o @ A_46 @ B_29 ) ) ) ) ).
thf(fact_590_subset__insert__iff,axiom,
! [A_46: hoare_1775062406iple_a > $o,X_27: hoare_1775062406iple_a,B_29: hoare_1775062406iple_a > $o] :
( ( ord_le1143225901le_a_o @ A_46 @ ( insert1281456128iple_a @ X_27 @ B_29 ) )
<=> ( ( ( member2122167641iple_a @ X_27 @ A_46 )
=> ( ord_le1143225901le_a_o @ ( minus_1944206118le_a_o @ A_46 @ ( insert1281456128iple_a @ X_27 @ bot_bo751897185le_a_o ) ) @ B_29 ) )
& ( ~ ( member2122167641iple_a @ X_27 @ A_46 )
=> ( ord_le1143225901le_a_o @ A_46 @ B_29 ) ) ) ) ).
thf(fact_591_folding__one__idem_Ounion__idem,axiom,
! [B_28: nat > $o,A_45: nat > $o,F_5: nat > nat > nat,F_4: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_5 @ F_4 )
=> ( ( finite_finite_nat @ A_45 )
=> ( ( A_45 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ B_28 )
=> ( ( B_28 != bot_bot_nat_o )
=> ( ( F_4 @ ( semila848761471_nat_o @ A_45 @ B_28 ) )
= ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
thf(fact_592_folding__one__idem_Ounion__idem,axiom,
! [B_28: hoare_1167836817_state > $o,A_45: hoare_1167836817_state > $o,F_5: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_4: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
( ( finite806517911_state @ F_5 @ F_4 )
=> ( ( finite1084549118_state @ A_45 )
=> ( ( A_45 != bot_bo70021908tate_o )
=> ( ( finite1084549118_state @ B_28 )
=> ( ( B_28 != bot_bo70021908tate_o )
=> ( ( F_4 @ ( semila1172322802tate_o @ A_45 @ B_28 ) )
= ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
thf(fact_593_folding__one__idem_Ounion__idem,axiom,
! [B_28: hoare_1775062406iple_a > $o,A_45: hoare_1775062406iple_a > $o,F_5: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_4: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
( ( finite1358382848iple_a @ F_5 @ F_4 )
=> ( ( finite2063573081iple_a @ A_45 )
=> ( ( A_45 != bot_bo751897185le_a_o )
=> ( ( finite2063573081iple_a @ B_28 )
=> ( ( B_28 != bot_bo751897185le_a_o )
=> ( ( F_4 @ ( semila13410563le_a_o @ A_45 @ B_28 ) )
= ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
thf(fact_594_fold__sup__le__sup,axiom,
! [C_10: nat,B_27: nat,A_44: nat > $o] :
( ( finite_finite_nat @ A_44 )
=> ( ! [X_3: nat] :
( ( member_nat @ X_3 @ A_44 )
=> ( ord_less_eq_nat @ X_3 @ B_27 ) )
=> ( ord_less_eq_nat @ ( finite_fold_nat_nat @ semila972727038up_nat @ C_10 @ A_44 ) @ ( semila972727038up_nat @ B_27 @ C_10 ) ) ) ) ).
thf(fact_595_finite__subset__induct,axiom,
! [P_2: ( nat > $o ) > $o,A_42: nat > $o,F_2: nat > $o] :
( ( finite_finite_nat @ F_2 )
=> ( ( ord_less_eq_nat_o @ F_2 @ A_42 )
=> ( ( P_2 @ bot_bot_nat_o )
=> ( ! [A_43: nat,F_3: nat > $o] :
( ( finite_finite_nat @ F_3 )
=> ( ( member_nat @ A_43 @ A_42 )
=> ( ~ ( member_nat @ A_43 @ F_3 )
=> ( ( P_2 @ F_3 )
=> ( P_2 @ ( insert_nat @ A_43 @ F_3 ) ) ) ) ) )
=> ( P_2 @ F_2 ) ) ) ) ) ).
thf(fact_596_finite__subset__induct,axiom,
! [P_2: ( hoare_1167836817_state > $o ) > $o,A_42: hoare_1167836817_state > $o,F_2: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_2 )
=> ( ( ord_le827224136tate_o @ F_2 @ A_42 )
=> ( ( P_2 @ bot_bo70021908tate_o )
=> ( ! [A_43: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
( ( finite1084549118_state @ F_3 )
=> ( ( member2058392318_state @ A_43 @ A_42 )
=> ( ~ ( member2058392318_state @ A_43 @ F_3 )
=> ( ( P_2 @ F_3 )
=> ( P_2 @ ( insert2134838167_state @ A_43 @ F_3 ) ) ) ) ) )
=> ( P_2 @ F_2 ) ) ) ) ) ).
thf(fact_597_finite__subset__induct,axiom,
! [P_2: ( hoare_1775062406iple_a > $o ) > $o,A_42: hoare_1775062406iple_a > $o,F_2: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_2 )
=> ( ( ord_le1143225901le_a_o @ F_2 @ A_42 )
=> ( ( P_2 @ bot_bo751897185le_a_o )
=> ( ! [A_43: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
( ( finite2063573081iple_a @ F_3 )
=> ( ( member2122167641iple_a @ A_43 @ A_42 )
=> ( ~ ( member2122167641iple_a @ A_43 @ F_3 )
=> ( ( P_2 @ F_3 )
=> ( P_2 @ ( insert1281456128iple_a @ A_43 @ F_3 ) ) ) ) ) )
=> ( P_2 @ F_2 ) ) ) ) ) ).
thf(fact_598_subsetI,axiom,
! [B_26: hoare_1775062406iple_a > $o,A_41: hoare_1775062406iple_a > $o] :
( ! [X_3: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X_3 @ A_41 )
=> ( member2122167641iple_a @ X_3 @ B_26 ) )
=> ( ord_le1143225901le_a_o @ A_41 @ B_26 ) ) ).
thf(fact_599_subsetI,axiom,
! [B_26: nat > $o,A_41: nat > $o] :
( ! [X_3: nat] :
( ( member_nat @ X_3 @ A_41 )
=> ( member_nat @ X_3 @ B_26 ) )
=> ( ord_less_eq_nat_o @ A_41 @ B_26 ) ) ).
thf(fact_600_evaln__nonstrict,axiom,
! [M_1: nat,C_9: com,S_2: state,N_3: nat,T: state] :
( ( evaln @ C_9 @ S_2 @ N_3 @ T )
=> ( ( ord_less_eq_nat @ N_3 @ M_1 )
=> ( evaln @ C_9 @ S_2 @ M_1 @ T ) ) ) ).
thf(fact_601_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
thf(fact_602_flat__lub__def,axiom,
! [A_40: hoare_1775062406iple_a > $o,B_25: hoare_1775062406iple_a] :
( ( ( ord_le1143225901le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) )
=> ( ( partia126998524iple_a @ B_25 @ A_40 )
= B_25 ) )
& ( ~ ( ord_le1143225901le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) )
=> ( ( partia126998524iple_a @ B_25 @ A_40 )
= ( the_Ho1155011127iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ ( minus_1944206118le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ).
thf(fact_603_flat__lub__def,axiom,
! [A_40: nat > $o,B_25: nat] :
( ( ( ord_less_eq_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) )
=> ( ( partial_flat_lub_nat @ B_25 @ A_40 )
= B_25 ) )
& ( ~ ( ord_less_eq_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) )
=> ( ( partial_flat_lub_nat @ B_25 @ A_40 )
= ( the_nat
@ ^ [X_3: nat] : ( member_nat @ X_3 @ ( minus_minus_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_604_flat__lub__def,axiom,
! [A_40: hoare_1167836817_state > $o,B_25: hoare_1167836817_state] :
( ( ( ord_le827224136tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) )
=> ( ( partia715677851_state @ B_25 @ A_40 )
= B_25 ) )
& ( ~ ( ord_le827224136tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) )
=> ( ( partia715677851_state @ B_25 @ A_40 )
= ( the_Ho310147232_state
@ ^ [X_3: hoare_1167836817_state] : ( member2058392318_state @ X_3 @ ( minus_2107060239tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ).
thf(fact_605_finite__nat__set__iff__bounded__le,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M: nat] :
! [X_3: nat] :
( ( member_nat @ X_3 @ N_2 )
=> ( ord_less_eq_nat @ X_3 @ M ) ) ) ).
thf(fact_606_finite__less__ub,axiom,
! [U: nat,F_1: nat > nat] :
( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F_1 @ N_1 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F_1 @ N_1 ) @ U ) ) ) ) ).
thf(fact_607_Sup__fin_Oremove,axiom,
! [X_26: nat,A_39: nat > $o] :
( ( finite_finite_nat @ A_39 )
=> ( ( member_nat @ X_26 @ A_39 )
=> ( ( ( ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ A_39 )
= X_26 ) )
& ( ( ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ A_39 )
= ( semila972727038up_nat @ X_26 @ ( big_la43341705in_nat @ ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
thf(fact_608_Sup__fin_Osingleton,axiom,
! [X_25: nat] :
( ( big_la43341705in_nat @ ( insert_nat @ X_25 @ bot_bot_nat_o ) )
= X_25 ) ).
thf(fact_609_Sup__fin_Oin__idem,axiom,
! [X_24: nat,A_38: nat > $o] :
( ( finite_finite_nat @ A_38 )
=> ( ( member_nat @ X_24 @ A_38 )
=> ( ( semila972727038up_nat @ X_24 @ ( big_la43341705in_nat @ A_38 ) )
= ( big_la43341705in_nat @ A_38 ) ) ) ) ).
thf(fact_610_Sup__fin_OF__eq,axiom,
! [A_37: nat > $o] :
( ( finite_finite_nat @ A_37 )
=> ( ( big_la43341705in_nat @ A_37 )
= ( finite_fold1_nat @ semila972727038up_nat @ A_37 ) ) ) ).
thf(fact_611_Sup__fin_Oinsert__idem,axiom,
! [X_23: nat,A_36: nat > $o] :
( ( finite_finite_nat @ A_36 )
=> ( ( A_36 != bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( insert_nat @ X_23 @ A_36 ) )
= ( semila972727038up_nat @ X_23 @ ( big_la43341705in_nat @ A_36 ) ) ) ) ) ).
thf(fact_612_Sup__fin_Oinsert,axiom,
! [X_22: nat,A_35: nat > $o] :
( ( finite_finite_nat @ A_35 )
=> ( ~ ( member_nat @ X_22 @ A_35 )
=> ( ( A_35 != bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( insert_nat @ X_22 @ A_35 ) )
= ( semila972727038up_nat @ X_22 @ ( big_la43341705in_nat @ A_35 ) ) ) ) ) ) ).
thf(fact_613_Sup__fin_Osubset__idem,axiom,
! [B_24: nat > $o,A_34: nat > $o] :
( ( finite_finite_nat @ A_34 )
=> ( ( B_24 != bot_bot_nat_o )
=> ( ( ord_less_eq_nat_o @ B_24 @ A_34 )
=> ( ( semila972727038up_nat @ ( big_la43341705in_nat @ B_24 ) @ ( big_la43341705in_nat @ A_34 ) )
= ( big_la43341705in_nat @ A_34 ) ) ) ) ) ).
thf(fact_614_Sup__fin_Ounion__idem,axiom,
! [B_23: nat > $o,A_33: nat > $o] :
( ( finite_finite_nat @ A_33 )
=> ( ( A_33 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ B_23 )
=> ( ( B_23 != bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( semila848761471_nat_o @ A_33 @ B_23 ) )
= ( semila972727038up_nat @ ( big_la43341705in_nat @ A_33 ) @ ( big_la43341705in_nat @ B_23 ) ) ) ) ) ) ) ).
thf(fact_615_Sup__fin_Oeq__fold__idem_H,axiom,
! [A_32: nat,A_31: nat > $o] :
( ( finite_finite_nat @ A_31 )
=> ( ( big_la43341705in_nat @ ( insert_nat @ A_32 @ A_31 ) )
= ( finite_fold_nat_nat @ semila972727038up_nat @ A_32 @ A_31 ) ) ) ).
thf(fact_616_Sup__fin_Oeq__fold_H,axiom,
! [X_21: nat,A_30: nat > $o] :
( ( finite_finite_nat @ A_30 )
=> ( ~ ( member_nat @ X_21 @ A_30 )
=> ( ( big_la43341705in_nat @ ( insert_nat @ X_21 @ A_30 ) )
= ( finite_fold_nat_nat @ semila972727038up_nat @ X_21 @ A_30 ) ) ) ) ).
thf(fact_617_Sup__fin_Oinsert__remove,axiom,
! [X_20: nat,A_29: nat > $o] :
( ( finite_finite_nat @ A_29 )
=> ( ( ( ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( insert_nat @ X_20 @ A_29 ) )
= X_20 ) )
& ( ( ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( insert_nat @ X_20 @ A_29 ) )
= ( semila972727038up_nat @ X_20 @ ( big_la43341705in_nat @ ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ).
thf(fact_618_Sup__fin_Ohom__commute,axiom,
! [N: nat > $o,H: nat > nat] :
( ! [X_3: nat,Y_1: nat] :
( ( H @ ( semila972727038up_nat @ X_3 @ Y_1 ) )
= ( semila972727038up_nat @ ( H @ X_3 ) @ ( H @ Y_1 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_nat_o )
=> ( ( H @ ( big_la43341705in_nat @ N ) )
= ( big_la43341705in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
thf(fact_619_Sup__fin_Oclosed,axiom,
! [A_28: nat > $o] :
( ( finite_finite_nat @ A_28 )
=> ( ( A_28 != bot_bot_nat_o )
=> ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( semila972727038up_nat @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( big_la43341705in_nat @ A_28 ) @ A_28 ) ) ) ) ).
thf(fact_620_Sup__fin_Ounion__disjoint,axiom,
! [B_22: nat > $o,A_27: nat > $o] :
( ( finite_finite_nat @ A_27 )
=> ( ( A_27 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ B_22 )
=> ( ( B_22 != bot_bot_nat_o )
=> ( ( ( semila1947288293_nat_o @ A_27 @ B_22 )
= bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( semila848761471_nat_o @ A_27 @ B_22 ) )
= ( semila972727038up_nat @ ( big_la43341705in_nat @ A_27 ) @ ( big_la43341705in_nat @ B_22 ) ) ) ) ) ) ) ) ).
thf(fact_621_IntI,axiom,
! [B_21: hoare_1775062406iple_a > $o,C_8: hoare_1775062406iple_a,A_26: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_8 @ A_26 )
=> ( ( member2122167641iple_a @ C_8 @ B_21 )
=> ( member2122167641iple_a @ C_8 @ ( semila966743401le_a_o @ A_26 @ B_21 ) ) ) ) ).
thf(fact_622_IntI,axiom,
! [B_21: nat > $o,C_8: nat,A_26: nat > $o] :
( ( member_nat @ C_8 @ A_26 )
=> ( ( member_nat @ C_8 @ B_21 )
=> ( member_nat @ C_8 @ ( semila1947288293_nat_o @ A_26 @ B_21 ) ) ) ) ).
thf(fact_623_IntE,axiom,
! [C_7: hoare_1775062406iple_a,A_25: hoare_1775062406iple_a > $o,B_20: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_7 @ ( semila966743401le_a_o @ A_25 @ B_20 ) )
=> ~ ( ( member2122167641iple_a @ C_7 @ A_25 )
=> ~ ( member2122167641iple_a @ C_7 @ B_20 ) ) ) ).
thf(fact_624_IntE,axiom,
! [C_7: nat,A_25: nat > $o,B_20: nat > $o] :
( ( member_nat @ C_7 @ ( semila1947288293_nat_o @ A_25 @ B_20 ) )
=> ~ ( ( member_nat @ C_7 @ A_25 )
=> ~ ( member_nat @ C_7 @ B_20 ) ) ) ).
thf(fact_625_finite__Int,axiom,
! [G: nat > $o,F: nat > $o] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( semila1947288293_nat_o @ F @ G ) ) ) ).
thf(fact_626_le__infE,axiom,
! [X_19: nat,A_24: nat,B_19: nat] :
( ( ord_less_eq_nat @ X_19 @ ( semila80283416nf_nat @ A_24 @ B_19 ) )
=> ~ ( ( ord_less_eq_nat @ X_19 @ A_24 )
=> ~ ( ord_less_eq_nat @ X_19 @ B_19 ) ) ) ).
thf(fact_627_inf__mono,axiom,
! [B_18: nat,D: nat,A_23: nat,C_6: nat] :
( ( ord_less_eq_nat @ A_23 @ C_6 )
=> ( ( ord_less_eq_nat @ B_18 @ D )
=> ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_23 @ B_18 ) @ ( semila80283416nf_nat @ C_6 @ D ) ) ) ) ).
thf(fact_628_inf__greatest,axiom,
! [Z_4: nat,X_18: nat,Y_12: nat] :
( ( ord_less_eq_nat @ X_18 @ Y_12 )
=> ( ( ord_less_eq_nat @ X_18 @ Z_4 )
=> ( ord_less_eq_nat @ X_18 @ ( semila80283416nf_nat @ Y_12 @ Z_4 ) ) ) ) ).
thf(fact_629_le__infI,axiom,
! [B_17: nat,X_17: nat,A_22: nat] :
( ( ord_less_eq_nat @ X_17 @ A_22 )
=> ( ( ord_less_eq_nat @ X_17 @ B_17 )
=> ( ord_less_eq_nat @ X_17 @ ( semila80283416nf_nat @ A_22 @ B_17 ) ) ) ) ).
thf(fact_630_inf__absorb2,axiom,
! [Y_11: nat,X_16: nat] :
( ( ord_less_eq_nat @ Y_11 @ X_16 )
=> ( ( semila80283416nf_nat @ X_16 @ Y_11 )
= Y_11 ) ) ).
thf(fact_631_inf__absorb1,axiom,
! [X_15: nat,Y_10: nat] :
( ( ord_less_eq_nat @ X_15 @ Y_10 )
=> ( ( semila80283416nf_nat @ X_15 @ Y_10 )
= X_15 ) ) ).
thf(fact_632_le__infI2,axiom,
! [A_21: nat,B_16: nat,X_14: nat] :
( ( ord_less_eq_nat @ B_16 @ X_14 )
=> ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_21 @ B_16 ) @ X_14 ) ) ).
thf(fact_633_le__infI1,axiom,
! [B_15: nat,A_20: nat,X_13: nat] :
( ( ord_less_eq_nat @ A_20 @ X_13 )
=> ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_20 @ B_15 ) @ X_13 ) ) ).
thf(fact_634_le__inf__iff,axiom,
! [X_12: nat,Y_9: nat,Z_3: nat] :
( ( ord_less_eq_nat @ X_12 @ ( semila80283416nf_nat @ Y_9 @ Z_3 ) )
<=> ( ( ord_less_eq_nat @ X_12 @ Y_9 )
& ( ord_less_eq_nat @ X_12 @ Z_3 ) ) ) ).
thf(fact_635_le__iff__inf,axiom,
! [X_11: nat,Y_8: nat] :
( ( ord_less_eq_nat @ X_11 @ Y_8 )
<=> ( ( semila80283416nf_nat @ X_11 @ Y_8 )
= X_11 ) ) ).
thf(fact_636_inf__le2,axiom,
! [X_10: nat,Y_7: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_10 @ Y_7 ) @ Y_7 ) ).
thf(fact_637_inf__sup__ord_I2_J,axiom,
! [X_9: nat,Y_6: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_9 @ Y_6 ) @ Y_6 ) ).
thf(fact_638_inf__le1,axiom,
! [X_8: nat,Y_5: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_8 @ Y_5 ) @ X_8 ) ).
thf(fact_639_inf__sup__ord_I1_J,axiom,
! [X_7: nat,Y_4: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_7 @ Y_4 ) @ X_7 ) ).
thf(fact_640_distrib__sup__le,axiom,
! [X_6: nat,Y_3: nat,Z_2: nat] : ( ord_less_eq_nat @ ( semila972727038up_nat @ X_6 @ ( semila80283416nf_nat @ Y_3 @ Z_2 ) ) @ ( semila80283416nf_nat @ ( semila972727038up_nat @ X_6 @ Y_3 ) @ ( semila972727038up_nat @ X_6 @ Z_2 ) ) ) ).
thf(fact_641_distrib__inf__le,axiom,
! [X_5: nat,Y_2: nat,Z_1: nat] : ( ord_less_eq_nat @ ( semila972727038up_nat @ ( semila80283416nf_nat @ X_5 @ Y_2 ) @ ( semila80283416nf_nat @ X_5 @ Z_1 ) ) @ ( semila80283416nf_nat @ X_5 @ ( semila972727038up_nat @ Y_2 @ Z_1 ) ) ) ).
thf(fact_642_Int__insert__left__if1,axiom,
! [B_14: nat > $o,A_19: nat,C_5: nat > $o] :
( ( member_nat @ A_19 @ C_5 )
=> ( ( semila1947288293_nat_o @ ( insert_nat @ A_19 @ B_14 ) @ C_5 )
= ( insert_nat @ A_19 @ ( semila1947288293_nat_o @ B_14 @ C_5 ) ) ) ) ).
thf(fact_643_Int__insert__left__if1,axiom,
! [B_14: hoare_1167836817_state > $o,A_19: hoare_1167836817_state,C_5: hoare_1167836817_state > $o] :
( ( member2058392318_state @ A_19 @ C_5 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_19 @ B_14 ) @ C_5 )
= ( insert2134838167_state @ A_19 @ ( semila179895820tate_o @ B_14 @ C_5 ) ) ) ) ).
thf(fact_644_Int__insert__left__if1,axiom,
! [B_14: hoare_1775062406iple_a > $o,A_19: hoare_1775062406iple_a,C_5: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ A_19 @ C_5 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_19 @ B_14 ) @ C_5 )
= ( insert1281456128iple_a @ A_19 @ ( semila966743401le_a_o @ B_14 @ C_5 ) ) ) ) ).
thf(fact_645_Int__insert__right__if1,axiom,
! [B_13: nat > $o,A_18: nat,A_17: nat > $o] :
( ( member_nat @ A_18 @ A_17 )
=> ( ( semila1947288293_nat_o @ A_17 @ ( insert_nat @ A_18 @ B_13 ) )
= ( insert_nat @ A_18 @ ( semila1947288293_nat_o @ A_17 @ B_13 ) ) ) ) ).
thf(fact_646_Int__insert__right__if1,axiom,
! [B_13: 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_13 ) )
= ( insert2134838167_state @ A_18 @ ( semila179895820tate_o @ A_17 @ B_13 ) ) ) ) ).
thf(fact_647_Int__insert__right__if1,axiom,
! [B_13: 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_13 ) )
= ( insert1281456128iple_a @ A_18 @ ( semila966743401le_a_o @ A_17 @ B_13 ) ) ) ) ).
thf(fact_648_Int__insert__left__if0,axiom,
! [B_12: nat > $o,A_16: nat,C_4: nat > $o] :
( ~ ( member_nat @ A_16 @ C_4 )
=> ( ( semila1947288293_nat_o @ ( insert_nat @ A_16 @ B_12 ) @ C_4 )
= ( semila1947288293_nat_o @ B_12 @ C_4 ) ) ) ).
thf(fact_649_Int__insert__left__if0,axiom,
! [B_12: hoare_1167836817_state > $o,A_16: hoare_1167836817_state,C_4: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ A_16 @ C_4 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_16 @ B_12 ) @ C_4 )
= ( semila179895820tate_o @ B_12 @ C_4 ) ) ) ).
thf(fact_650_Int__insert__left__if0,axiom,
! [B_12: hoare_1775062406iple_a > $o,A_16: hoare_1775062406iple_a,C_4: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ A_16 @ C_4 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_16 @ B_12 ) @ C_4 )
= ( semila966743401le_a_o @ B_12 @ C_4 ) ) ) ).
thf(fact_651_Int__insert__right__if0,axiom,
! [B_11: nat > $o,A_15: nat,A_14: nat > $o] :
( ~ ( member_nat @ A_15 @ A_14 )
=> ( ( semila1947288293_nat_o @ A_14 @ ( insert_nat @ A_15 @ B_11 ) )
= ( semila1947288293_nat_o @ A_14 @ B_11 ) ) ) ).
thf(fact_652_Int__insert__right__if0,axiom,
! [B_11: hoare_1167836817_state > $o,A_15: hoare_1167836817_state,A_14: hoare_1167836817_state > $o] :
( ~ ( member2058392318_state @ A_15 @ A_14 )
=> ( ( semila179895820tate_o @ A_14 @ ( insert2134838167_state @ A_15 @ B_11 ) )
= ( semila179895820tate_o @ A_14 @ B_11 ) ) ) ).
thf(fact_653_Int__insert__right__if0,axiom,
! [B_11: hoare_1775062406iple_a > $o,A_15: hoare_1775062406iple_a,A_14: hoare_1775062406iple_a > $o] :
( ~ ( member2122167641iple_a @ A_15 @ A_14 )
=> ( ( semila966743401le_a_o @ A_14 @ ( insert1281456128iple_a @ A_15 @ B_11 ) )
= ( semila966743401le_a_o @ A_14 @ B_11 ) ) ) ).
thf(fact_654_insert__inter__insert,axiom,
! [A_13: nat,A_12: nat > $o,B_10: nat > $o] :
( ( semila1947288293_nat_o @ ( insert_nat @ A_13 @ A_12 ) @ ( insert_nat @ A_13 @ B_10 ) )
= ( insert_nat @ A_13 @ ( semila1947288293_nat_o @ A_12 @ B_10 ) ) ) ).
thf(fact_655_insert__inter__insert,axiom,
! [A_13: hoare_1167836817_state,A_12: hoare_1167836817_state > $o,B_10: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ ( insert2134838167_state @ A_13 @ A_12 ) @ ( insert2134838167_state @ A_13 @ B_10 ) )
= ( insert2134838167_state @ A_13 @ ( semila179895820tate_o @ A_12 @ B_10 ) ) ) ).
thf(fact_656_insert__inter__insert,axiom,
! [A_13: hoare_1775062406iple_a,A_12: hoare_1775062406iple_a > $o,B_10: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_13 @ A_12 ) @ ( insert1281456128iple_a @ A_13 @ B_10 ) )
= ( insert1281456128iple_a @ A_13 @ ( semila966743401le_a_o @ A_12 @ B_10 ) ) ) ).
thf(fact_657_Int__insert__left,axiom,
! [B_9: nat > $o,A_11: nat,C_3: nat > $o] :
( ( ( member_nat @ A_11 @ C_3 )
=> ( ( semila1947288293_nat_o @ ( insert_nat @ A_11 @ B_9 ) @ C_3 )
= ( insert_nat @ A_11 @ ( semila1947288293_nat_o @ B_9 @ C_3 ) ) ) )
& ( ~ ( member_nat @ A_11 @ C_3 )
=> ( ( semila1947288293_nat_o @ ( insert_nat @ A_11 @ B_9 ) @ C_3 )
= ( semila1947288293_nat_o @ B_9 @ C_3 ) ) ) ) ).
thf(fact_658_Int__insert__left,axiom,
! [B_9: hoare_1167836817_state > $o,A_11: hoare_1167836817_state,C_3: hoare_1167836817_state > $o] :
( ( ( member2058392318_state @ A_11 @ C_3 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_11 @ B_9 ) @ C_3 )
= ( insert2134838167_state @ A_11 @ ( semila179895820tate_o @ B_9 @ C_3 ) ) ) )
& ( ~ ( member2058392318_state @ A_11 @ C_3 )
=> ( ( semila179895820tate_o @ ( insert2134838167_state @ A_11 @ B_9 ) @ C_3 )
= ( semila179895820tate_o @ B_9 @ C_3 ) ) ) ) ).
thf(fact_659_Int__insert__left,axiom,
! [B_9: hoare_1775062406iple_a > $o,A_11: hoare_1775062406iple_a,C_3: hoare_1775062406iple_a > $o] :
( ( ( member2122167641iple_a @ A_11 @ C_3 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_11 @ B_9 ) @ C_3 )
= ( insert1281456128iple_a @ A_11 @ ( semila966743401le_a_o @ B_9 @ C_3 ) ) ) )
& ( ~ ( member2122167641iple_a @ A_11 @ C_3 )
=> ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_11 @ B_9 ) @ C_3 )
= ( semila966743401le_a_o @ B_9 @ C_3 ) ) ) ) ).
thf(fact_660_Int__insert__right,axiom,
! [B_8: nat > $o,A_10: nat,A_9: nat > $o] :
( ( ( member_nat @ A_10 @ A_9 )
=> ( ( semila1947288293_nat_o @ A_9 @ ( insert_nat @ A_10 @ B_8 ) )
= ( insert_nat @ A_10 @ ( semila1947288293_nat_o @ A_9 @ B_8 ) ) ) )
& ( ~ ( member_nat @ A_10 @ A_9 )
=> ( ( semila1947288293_nat_o @ A_9 @ ( insert_nat @ A_10 @ B_8 ) )
= ( semila1947288293_nat_o @ A_9 @ B_8 ) ) ) ) ).
thf(fact_661_Int__insert__right,axiom,
! [B_8: hoare_1167836817_state > $o,A_10: hoare_1167836817_state,A_9: hoare_1167836817_state > $o] :
( ( ( member2058392318_state @ A_10 @ A_9 )
=> ( ( semila179895820tate_o @ A_9 @ ( insert2134838167_state @ A_10 @ B_8 ) )
= ( insert2134838167_state @ A_10 @ ( semila179895820tate_o @ A_9 @ B_8 ) ) ) )
& ( ~ ( member2058392318_state @ A_10 @ A_9 )
=> ( ( semila179895820tate_o @ A_9 @ ( insert2134838167_state @ A_10 @ B_8 ) )
= ( semila179895820tate_o @ A_9 @ B_8 ) ) ) ) ).
thf(fact_662_Int__insert__right,axiom,
! [B_8: hoare_1775062406iple_a > $o,A_10: hoare_1775062406iple_a,A_9: hoare_1775062406iple_a > $o] :
( ( ( member2122167641iple_a @ A_10 @ A_9 )
=> ( ( semila966743401le_a_o @ A_9 @ ( insert1281456128iple_a @ A_10 @ B_8 ) )
= ( insert1281456128iple_a @ A_10 @ ( semila966743401le_a_o @ A_9 @ B_8 ) ) ) )
& ( ~ ( member2122167641iple_a @ A_10 @ A_9 )
=> ( ( semila966743401le_a_o @ A_9 @ ( insert1281456128iple_a @ A_10 @ B_8 ) )
= ( semila966743401le_a_o @ A_9 @ B_8 ) ) ) ) ).
thf(fact_663_inf__Int__eq,axiom,
! [R: hoare_1775062406iple_a > $o,S_1: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
( ( semila966743401le_a_o
@ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ R )
@ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ S_1 )
@ X_3 )
<=> ( member2122167641iple_a @ X_3 @ ( semila966743401le_a_o @ R @ S_1 ) ) ) ).
thf(fact_664_inf__Int__eq,axiom,
! [R: nat > $o,S_1: nat > $o,X_3: nat] :
( ( semila1947288293_nat_o
@ ^ [Y_1: nat] : ( member_nat @ Y_1 @ R )
@ ^ [Y_1: nat] : ( member_nat @ Y_1 @ S_1 )
@ X_3 )
<=> ( member_nat @ X_3 @ ( semila1947288293_nat_o @ R @ S_1 ) ) ) ).
thf(fact_665_Collect__conj__eq,axiom,
! [P_1: hoare_1775062406iple_a > $o,Q: hoare_1775062406iple_a > $o] :
( ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( P_1 @ X_3 ) @ ( Q @ X_3 ) ) )
= ( semila966743401le_a_o @ ( collec676402587iple_a @ P_1 ) @ ( collec676402587iple_a @ Q ) ) ) ).
thf(fact_666_Collect__conj__eq,axiom,
! [P_1: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( P_1 @ X_3 ) @ ( Q @ X_3 ) ) )
= ( semila1947288293_nat_o @ ( collect_nat @ P_1 ) @ ( collect_nat @ Q ) ) ) ).
thf(fact_667_Int__Collect,axiom,
! [X_4: hoare_1775062406iple_a,A_8: hoare_1775062406iple_a > $o,P: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ X_4 @ ( semila966743401le_a_o @ A_8 @ ( collec676402587iple_a @ P ) ) )
<=> ( ( member2122167641iple_a @ X_4 @ A_8 )
& ( P @ X_4 ) ) ) ).
thf(fact_668_Int__Collect,axiom,
! [X_4: nat,A_8: nat > $o,P: nat > $o] :
( ( member_nat @ X_4 @ ( semila1947288293_nat_o @ A_8 @ ( collect_nat @ P ) ) )
<=> ( ( member_nat @ X_4 @ A_8 )
& ( P @ X_4 ) ) ) ).
thf(fact_669_Int__def,axiom,
! [A_7: hoare_1775062406iple_a > $o,B_7: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ A_7 @ B_7 )
= ( collec676402587iple_a
@ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( member2122167641iple_a @ X_3 @ A_7 ) @ ( member2122167641iple_a @ X_3 @ B_7 ) ) ) ) ).
thf(fact_670_Int__def,axiom,
! [A_7: nat > $o,B_7: nat > $o] :
( ( semila1947288293_nat_o @ A_7 @ B_7 )
= ( collect_nat
@ ^ [X_3: nat] : ( (&) @ ( member_nat @ X_3 @ A_7 ) @ ( member_nat @ X_3 @ B_7 ) ) ) ) ).
thf(fact_671_Int__iff,axiom,
! [C_2: hoare_1775062406iple_a,A_6: hoare_1775062406iple_a > $o,B_6: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_2 @ ( semila966743401le_a_o @ A_6 @ B_6 ) )
<=> ( ( member2122167641iple_a @ C_2 @ A_6 )
& ( member2122167641iple_a @ C_2 @ B_6 ) ) ) ).
thf(fact_672_Int__iff,axiom,
! [C_2: nat,A_6: nat > $o,B_6: nat > $o] :
( ( member_nat @ C_2 @ ( semila1947288293_nat_o @ A_6 @ B_6 ) )
<=> ( ( member_nat @ C_2 @ A_6 )
& ( member_nat @ C_2 @ B_6 ) ) ) ).
thf(fact_673_IntD1,axiom,
! [C_1: hoare_1775062406iple_a,A_5: hoare_1775062406iple_a > $o,B_5: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C_1 @ ( semila966743401le_a_o @ A_5 @ B_5 ) )
=> ( member2122167641iple_a @ C_1 @ A_5 ) ) ).
thf(fact_674_IntD1,axiom,
! [C_1: nat,A_5: nat > $o,B_5: nat > $o] :
( ( member_nat @ C_1 @ ( semila1947288293_nat_o @ A_5 @ B_5 ) )
=> ( member_nat @ C_1 @ A_5 ) ) ).
thf(fact_675_IntD2,axiom,
! [C: hoare_1775062406iple_a,A_4: hoare_1775062406iple_a > $o,B_4: hoare_1775062406iple_a > $o] :
( ( member2122167641iple_a @ C @ ( semila966743401le_a_o @ A_4 @ B_4 ) )
=> ( member2122167641iple_a @ C @ B_4 ) ) ).
thf(fact_676_IntD2,axiom,
! [C: nat,A_4: nat > $o,B_4: nat > $o] :
( ( member_nat @ C @ ( semila1947288293_nat_o @ A_4 @ B_4 ) )
=> ( member_nat @ C @ B_4 ) ) ).
thf(fact_677_disjoint__iff__not__equal,axiom,
! [A_3: nat > $o,B_3: nat > $o] :
( ( ( semila1947288293_nat_o @ A_3 @ B_3 )
= bot_bot_nat_o )
<=> ! [X_3: nat] :
( ( member_nat @ X_3 @ A_3 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ B_3 )
=> ( X_3 != Xa ) ) ) ) ).
thf(fact_678_disjoint__iff__not__equal,axiom,
! [A_3: hoare_1167836817_state > $o,B_3: hoare_1167836817_state > $o] :
( ( ( semila179895820tate_o @ A_3 @ B_3 )
= bot_bo70021908tate_o )
<=> ! [X_3: hoare_1167836817_state] :
( ( member2058392318_state @ X_3 @ A_3 )
=> ! [Xa: hoare_1167836817_state] :
( ( member2058392318_state @ Xa @ B_3 )
=> ( X_3 != Xa ) ) ) ) ).
thf(fact_679_disjoint__iff__not__equal,axiom,
! [A_3: hoare_1775062406iple_a > $o,B_3: hoare_1775062406iple_a > $o] :
( ( ( semila966743401le_a_o @ A_3 @ B_3 )
= bot_bo751897185le_a_o )
<=> ! [X_3: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ X_3 @ A_3 )
=> ! [Xa: hoare_1775062406iple_a] :
( ( member2122167641iple_a @ Xa @ B_3 )
=> ( X_3 != Xa ) ) ) ) ).
thf(fact_680_Int__empty__right,axiom,
! [A_2: nat > $o] :
( ( semila1947288293_nat_o @ A_2 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
thf(fact_681_Int__empty__right,axiom,
! [A_2: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ A_2 @ bot_bo70021908tate_o )
= bot_bo70021908tate_o ) ).
thf(fact_682_Int__empty__right,axiom,
! [A_2: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ A_2 @ bot_bo751897185le_a_o )
= bot_bo751897185le_a_o ) ).
thf(fact_683_Int__empty__left,axiom,
! [B_2: nat > $o] :
( ( semila1947288293_nat_o @ bot_bot_nat_o @ B_2 )
= bot_bot_nat_o ) ).
thf(fact_684_Int__empty__left,axiom,
! [B_2: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ bot_bo70021908tate_o @ B_2 )
= bot_bo70021908tate_o ) ).
thf(fact_685_Int__empty__left,axiom,
! [B_2: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ B_2 )
= bot_bo751897185le_a_o ) ).
thf(fact_686_inf__bot__left,axiom,
! [X_2: $o] :
( ( semila854092349_inf_o @ bot_bot_o @ X_2 )
<=> bot_bot_o ) ).
thf(fact_687_inf__bot__left,axiom,
! [X_2: nat > $o] :
( ( semila1947288293_nat_o @ bot_bot_nat_o @ X_2 )
= bot_bot_nat_o ) ).
thf(fact_688_inf__bot__left,axiom,
! [X_2: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ bot_bo70021908tate_o @ X_2 )
= bot_bo70021908tate_o ) ).
thf(fact_689_inf__bot__left,axiom,
! [X_2: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ X_2 )
= bot_bo751897185le_a_o ) ).
thf(fact_690_inf__bot__right,axiom,
! [X_1: $o] :
( ( semila854092349_inf_o @ X_1 @ bot_bot_o )
<=> bot_bot_o ) ).
thf(fact_691_inf__bot__right,axiom,
! [X_1: nat > $o] :
( ( semila1947288293_nat_o @ X_1 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
thf(fact_692_inf__bot__right,axiom,
! [X_1: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ X_1 @ bot_bo70021908tate_o )
= bot_bo70021908tate_o ) ).
thf(fact_693_inf__bot__right,axiom,
! [X_1: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ X_1 @ bot_bo751897185le_a_o )
= bot_bo751897185le_a_o ) ).
thf(fact_694_Diff__triv,axiom,
! [A_1: nat > $o,B_1: nat > $o] :
( ( ( semila1947288293_nat_o @ A_1 @ B_1 )
= bot_bot_nat_o )
=> ( ( minus_minus_nat_o @ A_1 @ B_1 )
= A_1 ) ) ).
thf(fact_695_Diff__triv,axiom,
! [A_1: hoare_1167836817_state > $o,B_1: hoare_1167836817_state > $o] :
( ( ( semila179895820tate_o @ A_1 @ B_1 )
= bot_bo70021908tate_o )
=> ( ( minus_2107060239tate_o @ A_1 @ B_1 )
= A_1 ) ) ).
thf(fact_696_Diff__triv,axiom,
! [A_1: hoare_1775062406iple_a > $o,B_1: hoare_1775062406iple_a > $o] :
( ( ( semila966743401le_a_o @ A_1 @ B_1 )
= bot_bo751897185le_a_o )
=> ( ( minus_1944206118le_a_o @ A_1 @ B_1 )
= A_1 ) ) ).
thf(fact_697_Diff__disjoint,axiom,
! [A: nat > $o,B: nat > $o] :
( ( semila1947288293_nat_o @ A @ ( minus_minus_nat_o @ B @ A ) )
= bot_bot_nat_o ) ).
thf(fact_698_Diff__disjoint,axiom,
! [A: hoare_1167836817_state > $o,B: hoare_1167836817_state > $o] :
( ( semila179895820tate_o @ A @ ( minus_2107060239tate_o @ B @ A ) )
= bot_bo70021908tate_o ) ).
thf(fact_699_Diff__disjoint,axiom,
! [A: hoare_1775062406iple_a > $o,B: hoare_1775062406iple_a > $o] :
( ( semila966743401le_a_o @ A @ ( minus_1944206118le_a_o @ B @ A ) )
= bot_bo751897185le_a_o ) ).
%----Helper facts (10)
thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ ( fequal_nat @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| ( fequal_nat @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
! [X: state,Y: state] :
( ~ ( fequal_state @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
! [X: state,Y: state] :
( ( X != Y )
| ( fequal_state @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
! [X: nat > $o,Y: nat > $o] :
( ~ ( fequal_nat_o @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( X != Y )
| ( fequal_nat_o @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
! [X: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
( ~ ( fequal1288209029iple_a @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
! [X: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
( ( X != Y )
| ( fequal1288209029iple_a @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
( ~ ( fequal1831255762_state @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
( ( X != Y )
| ( fequal1831255762_state @ X @ Y ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Z: x_a,S: state] : $false
@ c
@ ^ [Z: x_a,S: state] : ( (&) @ ( p @ Z @ S ) @ ( (~) @ ( b @ S ) ) ) )
@ bot_bo751897185le_a_o ) ) ).
%------------------------------------------------------------------------------