TPTP Problem File: SWW471^3.p
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%------------------------------------------------------------------------------
% File : SWW471^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 269, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_1000_thf_l269 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 1395 ( 491 unt; 185 typ; 0 def)
% Number of atoms : 4733 (1094 equ; 91 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 10204 ( 409 ~; 89 |; 188 &;8244 @)
% ( 258 <=>;1016 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 3193 (3193 >; 0 *; 0 +; 0 <<)
% Number of symbols : 183 ( 177 usr; 11 con; 0-5 aty)
% Number of variables : 3370 ( 187 ^;3113 !; 70 ?;3370 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:44:03
%------------------------------------------------------------------------------
%----Should-be-implicit typings (8)
thf(ty_ty_t__a,type,
x_a: $tType ).
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Opname,type,
pname: $tType ).
thf(ty_ty_tc__Com__Ostate,type,
state: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
hoare_2091234717iple_a: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostate_J,type,
hoare_1708887482_state: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
option_com: $tType ).
%----Explicit typings (177)
thf(sy_c_Big__Operators_Ocomm__monoid__big_000_062_Itc__Hoare____Mirabelle____nqhfsd,type,
big_co1924420859_pname: ( ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > ( ( pname > hoare_2091234717iple_a > $o ) > ( pname > $o ) > hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_I_062_Itc__Hoare____Mirabe,type,
big_la1994307886_a_o_o: ( ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_Itc__Com__Opname_M_Eo_J,type,
big_la1286884090name_o: ( ( pname > $o ) > $o ) > pname > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_Itc__Hoare____Mirabelle___,type,
big_la735727201le_a_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_Itc__Hoare____Mirabelle____001,type,
big_la1088302868tate_o: ( ( hoare_1708887482_state > $o ) > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_062_Itc__Nat__Onat_M_Eo_J,type,
big_la1658356148_nat_o: ( ( nat > $o ) > $o ) > nat > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000_Eo,type,
big_la727467310_fin_o: ( $o > $o ) > $o ).
thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000tc__Nat__Onat,type,
big_la43341705in_nat: ( nat > $o ) > nat ).
thf(sy_c_Com_Obody,type,
body_1: pname > option_com ).
thf(sy_c_Com_Ocom_OBODY,type,
body: pname > com ).
thf(sy_c_Com_Ocom_OCond,type,
cond: ( state > $o ) > com > com > com ).
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_OSemi,type,
semi: com > com > com ).
thf(sy_c_Com_Ocom_OWhile,type,
while: ( state > $o ) > com > com ).
thf(sy_c_Com_Ocom_Ocom__size,type,
com_size: com > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
finite_card_nat: ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Ot,type,
finite886417794_a_o_o: ( ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Com__Opname_M_Eo_J,type,
finite297249702name_o: ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_,type,
finite1829014797le_a_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple__002,type,
finite1329924456tate_o: ( ( hoare_1708887482_state > $o ) > $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_000_Eo,type,
finite_finite_o: ( $o > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
finite_finite_pname: ( pname > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_,type,
finite232261744iple_a: ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__C,type,
finite1625599783_state: ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_I_062_Itc__Hoare____Mirabelle____nqhfsdfvy,type,
finite2009943022_o_nat: ( ( ( hoare_2091234717iple_a > $o ) > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ) > ( nat > ( hoare_2091234717iple_a > $o ) > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > ( nat > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Com__Opname_M_Eo_J_000tc__Nat__Onat,type,
finite1427591632_o_nat: ( ( pname > $o ) > ( pname > $o ) > pname > $o ) > ( nat > pname > $o ) > ( pname > $o ) > ( nat > $o ) > pname > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otr,type,
finite903029825le_a_o: ( ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otr_003,type,
finite1290357347_pname: ( ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( pname > hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > ( pname > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otr_004,type,
finite1481787452iple_a: ( ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otr_005,type,
finite2100865449_o_nat: ( ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( nat > hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > ( nat > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otr_006,type,
finite2139561282_pname: ( ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ) > ( pname > hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > ( pname > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otr_007,type,
finite1400355848_o_nat: ( ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ) > ( nat > hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > ( nat > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Finite__Set_Ofold__image_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Nat__Onat,type,
finite141655318_o_nat: ( ( nat > $o ) > ( nat > $o ) > nat > $o ) > ( nat > nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Ot,type,
finite14499299le_a_o: ( ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,
finite1282449217_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_,type,
finite247037978iple_a: ( hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a ) > ( ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple__008,type,
finite1615457021_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > ( ( hoare_1708887482_state > $o ) > hoare_1708887482_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_000_062_Itc__Hoare____Mirabelle____nqhfsdfv,type,
finite574580006le_a_o: ( ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,
finite89670078_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Ot,type,
finite1674555159iple_a: ( hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a ) > ( ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Ot_009,type,
finite1347568576_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > ( ( hoare_1708887482_state > $o ) > hoare_1708887482_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_I_062_Itc__Hoare____Mirabelle____nqhfsd,type,
minus_1746272704_a_o_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Com__Opname_M_Eo_J,type,
minus_minus_pname_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__,type,
minus_836160335le_a_o: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv___010,type,
minus_2056855718tate_o: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_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_Ominus__class_Ominus_000_Eo,type,
minus_minus_o: $o > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_OThe_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_E,type,
the_Ho2077879471le_a_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_HOL_OThe_000tc__Com__Opname,type,
the_pname: ( pname > $o ) > pname ).
thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
the_Ho1471183438iple_a: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a ).
thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostate_,type,
the_Ho851197897_state: ( hoare_1708887482_state > $o ) > hoare_1708887482_state ).
thf(sy_c_HOL_OThe_000tc__Nat__Onat,type,
the_nat: ( nat > $o ) > nat ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_OMGT,type,
hoare_Mirabelle_MGT: com > hoare_1708887482_state ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Ohoare__derivs_000t__a,type,
hoare_1467856363rivs_a: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Ohoare__derivs_000tc__Com__Ostate,type,
hoare_90032982_state: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Ohoare__valids_000t__a,type,
hoare_1805689709lids_a: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Ohoare__valids_000tc__Com__Ostate,type,
hoare_496444244_state: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple_Otriple_000t__a,type,
hoare_657976383iple_a: ( x_a > state > $o ) > com > ( x_a > state > $o ) > hoare_2091234717iple_a ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple_Otriple_000tc__Com__Ostate,type,
hoare_858012674_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1708887482_state ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple_Otriple__size_000t__a,type,
hoare_1169027232size_a: ( x_a > nat ) > hoare_2091234717iple_a > nat ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple_Otriple__size_000tc__Com__Ostate,type,
hoare_518771297_state: ( state > nat ) > hoare_1708887482_state > nat ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple__valid_000t__a,type,
hoare_1421888935alid_a: nat > hoare_2091234717iple_a > $o ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple__valid_000tc__Com__Ostate,type,
hoare_23738522_state: nat > hoare_1708887482_state > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_I_062_Itc__Hoare____Mirabell,type,
semila1672913213_a_o_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Com__Opname_M_Eo_J,type,
semila1673364395name_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____n,type,
semila2006181266le_a_o: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____n_011,type,
semila129691299tate_o: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_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_I_062_I_062_Itc__Hoare____Mi,type,
semila484278426_o_o_o: ( ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ) > ( ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Com__Opname_M_Eo_,type,
semila181081674me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Hoare____Mirabell,type,
semila2050116131_a_o_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Hoare____Mirabell_012,type,
semila1853742644te_o_o: ( ( hoare_1708887482_state > $o ) > $o ) > ( ( hoare_1708887482_state > $o ) > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_062_Itc__Nat__Onat_M_Eo_J_,type,
semila72246288at_o_o: ( ( nat > $o ) > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_I_Eo_M_Eo_J,type,
semila2062604954up_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Com__Opname_M_Eo_J,type,
semila1780557381name_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____n,type,
semila1052848428le_a_o: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____n_013,type,
semila1122118281tate_o: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Onat__case_000_Eo,type,
nat_case_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
nat_case_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Com__Ocom,type,
size_size_com: com > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It_,type,
size_s1040486067iple_a: hoare_2091234717iple_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc,type,
size_s1186992420_state: hoare_1708887482_state > nat ).
thf(sy_c_Natural_Oevalc,type,
evalc: com > state > state > $o ).
thf(sy_c_Natural_Oevaln,type,
evaln: com > state > nat > state > $o ).
thf(sy_c_Option_Othe_000tc__Com__Ocom,type,
the_com: option_com > com ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_I_062_Itc__Hoare____Mirabelle____n,type,
bot_bo690906872_o_o_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
bot_bot_pname_o_o: ( pname > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Hoare____Mirabelle____nqhfsdf,type,
bot_bo1957696069_a_o_o: ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Hoare____Mirabelle____nqhfsdf_014,type,
bot_bo1678742418te_o_o: ( hoare_1708887482_state > $o ) > $o ).
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_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J,type,
bot_bot_pname_o: pname > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__O,type,
bot_bo1791335050le_a_o: hoare_2091234717iple_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__O_015,type,
bot_bo19817387tate_o: hoare_1708887482_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_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
collec1008234059le_a_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Set_OCollect_000tc__Com__Opname,type,
collect_pname: ( pname > $o ) > pname > $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
collec992574898iple_a: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ost,type,
collec1568722789_state: ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M,type,
image_784579955le_a_o: ( ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_016,type,
image_1908519857_pname: ( ( hoare_2091234717iple_a > $o ) > pname ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_017,type,
image_136408202iple_a: ( ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_018,type,
image_1501246093_state: ( ( hoare_2091234717iple_a > $o ) > hoare_1708887482_state ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_019,type,
image_75520503_o_nat: ( ( hoare_2091234717iple_a > $o ) > nat ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv,type,
image_742317343le_a_o: ( pname > hoare_2091234717iple_a > $o ) > ( pname > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
image_pname_pname: ( pname > pname ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otri,type,
image_231808478iple_a: ( pname > hoare_2091234717iple_a ) > ( pname > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otri_020,type,
image_1116629049_state: ( pname > hoare_1708887482_state ) > ( pname > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Nat__Onat,type,
image_pname_nat: ( pname > nat ) > ( pname > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000_062,type,
image_1642350072le_a_o: ( hoare_2091234717iple_a > hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000tc__,type,
image_924789612_pname: ( hoare_2091234717iple_a > pname ) > ( hoare_2091234717iple_a > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000tc___021,type,
image_1661191109iple_a: ( hoare_2091234717iple_a > hoare_2091234717iple_a ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000tc___022,type,
image_1884482962_state: ( hoare_2091234717iple_a > hoare_1708887482_state ) > ( hoare_2091234717iple_a > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000tc___023,type,
image_1773322034_a_nat: ( hoare_2091234717iple_a > nat ) > ( hoare_2091234717iple_a > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostat,type,
image_293283184iple_a: ( hoare_1708887482_state > hoare_2091234717iple_a ) > ( hoare_1708887482_state > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostat_024,type,
image_757158439_state: ( hoare_1708887482_state > hoare_1708887482_state ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__,type,
image_1995609573le_a_o: ( nat > hoare_2091234717iple_a > $o ) > ( nat > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname,type,
image_nat_pname: ( nat > pname ) > ( nat > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otripl,type,
image_359186840iple_a: ( nat > hoare_2091234717iple_a ) > ( nat > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otripl_025,type,
image_514827263_state: ( nat > hoare_1708887482_state ) > ( nat > $o ) > hoare_1708887482_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_I_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It,type,
insert987231145_a_o_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > ( ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Com__Opname_M_Eo_J,type,
insert_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_,type,
insert102003750le_a_o: ( hoare_2091234717iple_a > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com,type,
insert949073679tate_o: ( hoare_1708887482_state > $o ) > ( ( hoare_1708887482_state > $o ) > $o ) > ( hoare_1708887482_state > $o ) > $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_000_Eo,type,
insert_o: $o > ( $o > $o ) > $o > $o ).
thf(sy_c_Set_Oinsert_000tc__Com__Opname,type,
insert_pname: pname > ( pname > $o ) > pname > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
insert1597628439iple_a: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Osta,type,
insert528405184_state: hoare_1708887482_state > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
insert_nat: nat > ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Othe__elem_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a,type,
the_el1618277441le_a_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Othe__elem_000tc__Com__Opname,type,
the_elem_pname: ( pname > $o ) > pname ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
the_el13400124iple_a: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__O,type,
the_el864710747_state: ( hoare_1708887482_state > $o ) > hoare_1708887482_state ).
thf(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
the_elem_nat: ( nat > $o ) > nat ).
thf(sy_c_fequal_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_Eo_,type,
fequal845167073le_a_o: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_fequal_000tc__Com__Opname,type,
fequal_pname: pname > pname > $o ).
thf(sy_c_fequal_000tc__Com__Ostate,type,
fequal_state: state > state > $o ).
thf(sy_c_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
fequal1604381340iple_a: hoare_2091234717iple_a > hoare_2091234717iple_a > $o ).
thf(sy_c_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostate_J,type,
fequal224822779_state: hoare_1708887482_state > hoare_1708887482_state > $o ).
thf(sy_c_fequal_000tc__Nat__Onat,type,
fequal_nat: nat > nat > $o ).
thf(sy_c_member_000_062_I_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
member1297825410_a_o_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > ( ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Com__Opname_M_Eo_J,type,
member_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_Eo_,type,
member99268621le_a_o: ( hoare_2091234717iple_a > $o ) > ( ( hoare_2091234717iple_a > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ost,type,
member814030440tate_o: ( hoare_1708887482_state > $o ) > ( ( hoare_1708887482_state > $o ) > $o ) > $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_000_Eo,type,
member_o: $o > ( $o > $o ) > $o ).
thf(sy_c_member_000tc__Com__Opname,type,
member_pname: pname > ( pname > $o ) > $o ).
thf(sy_c_member_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
member290856304iple_a: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_member_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostate_J,type,
member451959335_state: hoare_1708887482_state > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_v_G,type,
g: hoare_2091234717iple_a > $o ).
thf(sy_v_P,type,
p: pname > x_a > state > $o ).
thf(sy_v_Procs,type,
procs: pname > $o ).
thf(sy_v_Q,type,
q: pname > x_a > state > $o ).
thf(sy_v_n,type,
n: nat ).
%----Relevant facts (1196)
thf(fact_0_triple_Oinject,axiom,
! [Fun1_4: x_a > state > $o,Com_1: com,Fun2_4: x_a > state > $o,Fun1_3: x_a > state > $o,Com: com,Fun2_3: x_a > state > $o] :
( ( ( hoare_657976383iple_a @ Fun1_4 @ Com_1 @ Fun2_4 )
= ( hoare_657976383iple_a @ Fun1_3 @ Com @ Fun2_3 ) )
<=> ( ( Fun1_4 = Fun1_3 )
& ( Com_1 = Com )
& ( Fun2_4 = Fun2_3 ) ) ) ).
thf(fact_1_triple_Oinject,axiom,
! [Fun1_4: state > state > $o,Com_1: com,Fun2_4: state > state > $o,Fun1_3: state > state > $o,Com: com,Fun2_3: state > state > $o] :
( ( ( hoare_858012674_state @ Fun1_4 @ Com_1 @ Fun2_4 )
= ( hoare_858012674_state @ Fun1_3 @ Com @ Fun2_3 ) )
<=> ( ( Fun1_4 = Fun1_3 )
& ( Com_1 = Com )
& ( Fun2_4 = Fun2_3 ) ) ) ).
thf(fact_2_hoare__valids__def,axiom,
! [G_28: hoare_1708887482_state > $o,Ts_4: hoare_1708887482_state > $o] :
( ( hoare_496444244_state @ G_28 @ Ts_4 )
<=> ! [N: nat] :
( ! [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ G_28 )
=> ( hoare_23738522_state @ N @ X ) )
=> ! [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ Ts_4 )
=> ( hoare_23738522_state @ N @ X ) ) ) ) ).
thf(fact_3_hoare__valids__def,axiom,
! [G_28: hoare_2091234717iple_a > $o,Ts_4: hoare_2091234717iple_a > $o] :
( ( hoare_1805689709lids_a @ G_28 @ Ts_4 )
<=> ! [N: nat] :
( ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ G_28 )
=> ( hoare_1421888935alid_a @ N @ X ) )
=> ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ Ts_4 )
=> ( hoare_1421888935alid_a @ N @ X ) ) ) ) ).
thf(fact_4_hoare__derivs_OBody,axiom,
! [G_27: hoare_1708887482_state > $o,P_36: pname > state > state > $o,Q_20: pname > state > state > $o,Procs_1: pname > $o] :
( ( hoare_90032982_state
@ ( semila1122118281tate_o @ G_27
@ ( image_1116629049_state
@ ^ [P_9: pname] : ( hoare_858012674_state @ ( P_36 @ P_9 ) @ ( body @ P_9 ) @ ( Q_20 @ P_9 ) )
@ Procs_1 ) )
@ ( image_1116629049_state
@ ^ [P_9: pname] : ( hoare_858012674_state @ ( P_36 @ P_9 ) @ ( the_com @ ( body_1 @ P_9 ) ) @ ( Q_20 @ P_9 ) )
@ Procs_1 ) )
=> ( hoare_90032982_state @ G_27
@ ( image_1116629049_state
@ ^ [P_9: pname] : ( hoare_858012674_state @ ( P_36 @ P_9 ) @ ( body @ P_9 ) @ ( Q_20 @ P_9 ) )
@ Procs_1 ) ) ) ).
thf(fact_5_hoare__derivs_OBody,axiom,
! [G_27: hoare_2091234717iple_a > $o,P_36: pname > x_a > state > $o,Q_20: pname > x_a > state > $o,Procs_1: pname > $o] :
( ( hoare_1467856363rivs_a
@ ( semila1052848428le_a_o @ G_27
@ ( image_231808478iple_a
@ ^ [P_9: pname] : ( hoare_657976383iple_a @ ( P_36 @ P_9 ) @ ( body @ P_9 ) @ ( Q_20 @ P_9 ) )
@ Procs_1 ) )
@ ( image_231808478iple_a
@ ^ [P_9: pname] : ( hoare_657976383iple_a @ ( P_36 @ P_9 ) @ ( the_com @ ( body_1 @ P_9 ) ) @ ( Q_20 @ P_9 ) )
@ Procs_1 ) )
=> ( hoare_1467856363rivs_a @ G_27
@ ( image_231808478iple_a
@ ^ [P_9: pname] : ( hoare_657976383iple_a @ ( P_36 @ P_9 ) @ ( body @ P_9 ) @ ( Q_20 @ P_9 ) )
@ Procs_1 ) ) ) ).
thf(fact_6_UnE,axiom,
! [C_34: nat,A_129: nat > $o,B_71: nat > $o] :
( ( member_nat @ C_34 @ ( semila848761471_nat_o @ A_129 @ B_71 ) )
=> ( ~ ( member_nat @ C_34 @ A_129 )
=> ( member_nat @ C_34 @ B_71 ) ) ) ).
thf(fact_7_UnE,axiom,
! [C_34: hoare_2091234717iple_a > $o,A_129: ( hoare_2091234717iple_a > $o ) > $o,B_71: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_34 @ ( semila2050116131_a_o_o @ A_129 @ B_71 ) )
=> ( ~ ( member99268621le_a_o @ C_34 @ A_129 )
=> ( member99268621le_a_o @ C_34 @ B_71 ) ) ) ).
thf(fact_8_UnE,axiom,
! [C_34: hoare_1708887482_state,A_129: hoare_1708887482_state > $o,B_71: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_34 @ ( semila1122118281tate_o @ A_129 @ B_71 ) )
=> ( ~ ( member451959335_state @ C_34 @ A_129 )
=> ( member451959335_state @ C_34 @ B_71 ) ) ) ).
thf(fact_9_UnE,axiom,
! [C_34: hoare_2091234717iple_a,A_129: hoare_2091234717iple_a > $o,B_71: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_34 @ ( semila1052848428le_a_o @ A_129 @ B_71 ) )
=> ( ~ ( member290856304iple_a @ C_34 @ A_129 )
=> ( member290856304iple_a @ C_34 @ B_71 ) ) ) ).
thf(fact_10_UnE,axiom,
! [C_34: pname,A_129: pname > $o,B_71: pname > $o] :
( ( member_pname @ C_34 @ ( semila1780557381name_o @ A_129 @ B_71 ) )
=> ( ~ ( member_pname @ C_34 @ A_129 )
=> ( member_pname @ C_34 @ B_71 ) ) ) ).
thf(fact_11_sup1E,axiom,
! [A_128: nat > $o,B_70: nat > $o,X_51: nat] :
( ( semila848761471_nat_o @ A_128 @ B_70 @ X_51 )
=> ( ~ ( A_128 @ X_51 )
=> ( B_70 @ X_51 ) ) ) ).
thf(fact_12_sup1E,axiom,
! [A_128: ( hoare_2091234717iple_a > $o ) > $o,B_70: ( hoare_2091234717iple_a > $o ) > $o,X_51: hoare_2091234717iple_a > $o] :
( ( semila2050116131_a_o_o @ A_128 @ B_70 @ X_51 )
=> ( ~ ( A_128 @ X_51 )
=> ( B_70 @ X_51 ) ) ) ).
thf(fact_13_sup1E,axiom,
! [A_128: hoare_1708887482_state > $o,B_70: hoare_1708887482_state > $o,X_51: hoare_1708887482_state] :
( ( semila1122118281tate_o @ A_128 @ B_70 @ X_51 )
=> ( ~ ( A_128 @ X_51 )
=> ( B_70 @ X_51 ) ) ) ).
thf(fact_14_sup1E,axiom,
! [A_128: pname > $o,B_70: pname > $o,X_51: pname] :
( ( semila1780557381name_o @ A_128 @ B_70 @ X_51 )
=> ( ~ ( A_128 @ X_51 )
=> ( B_70 @ X_51 ) ) ) ).
thf(fact_15_sup1E,axiom,
! [A_128: hoare_2091234717iple_a > $o,B_70: hoare_2091234717iple_a > $o,X_51: hoare_2091234717iple_a] :
( ( semila1052848428le_a_o @ A_128 @ B_70 @ X_51 )
=> ( ~ ( A_128 @ X_51 )
=> ( B_70 @ X_51 ) ) ) ).
thf(fact_16_sup1CI,axiom,
! [A_127: nat > $o,B_69: nat > $o,X_50: nat] :
( ( ~ ( B_69 @ X_50 )
=> ( A_127 @ X_50 ) )
=> ( semila848761471_nat_o @ A_127 @ B_69 @ X_50 ) ) ).
thf(fact_17_sup1CI,axiom,
! [A_127: ( hoare_2091234717iple_a > $o ) > $o,B_69: ( hoare_2091234717iple_a > $o ) > $o,X_50: hoare_2091234717iple_a > $o] :
( ( ~ ( B_69 @ X_50 )
=> ( A_127 @ X_50 ) )
=> ( semila2050116131_a_o_o @ A_127 @ B_69 @ X_50 ) ) ).
thf(fact_18_sup1CI,axiom,
! [A_127: hoare_1708887482_state > $o,B_69: hoare_1708887482_state > $o,X_50: hoare_1708887482_state] :
( ( ~ ( B_69 @ X_50 )
=> ( A_127 @ X_50 ) )
=> ( semila1122118281tate_o @ A_127 @ B_69 @ X_50 ) ) ).
thf(fact_19_sup1CI,axiom,
! [A_127: pname > $o,B_69: pname > $o,X_50: pname] :
( ( ~ ( B_69 @ X_50 )
=> ( A_127 @ X_50 ) )
=> ( semila1780557381name_o @ A_127 @ B_69 @ X_50 ) ) ).
thf(fact_20_sup1CI,axiom,
! [A_127: hoare_2091234717iple_a > $o,B_69: hoare_2091234717iple_a > $o,X_50: hoare_2091234717iple_a] :
( ( ~ ( B_69 @ X_50 )
=> ( A_127 @ X_50 ) )
=> ( semila1052848428le_a_o @ A_127 @ B_69 @ X_50 ) ) ).
thf(fact_21_UnCI,axiom,
! [A_126: nat > $o,C_33: nat,B_68: nat > $o] :
( ( ~ ( member_nat @ C_33 @ B_68 )
=> ( member_nat @ C_33 @ A_126 ) )
=> ( member_nat @ C_33 @ ( semila848761471_nat_o @ A_126 @ B_68 ) ) ) ).
thf(fact_22_UnCI,axiom,
! [A_126: ( hoare_2091234717iple_a > $o ) > $o,C_33: hoare_2091234717iple_a > $o,B_68: ( hoare_2091234717iple_a > $o ) > $o] :
( ( ~ ( member99268621le_a_o @ C_33 @ B_68 )
=> ( member99268621le_a_o @ C_33 @ A_126 ) )
=> ( member99268621le_a_o @ C_33 @ ( semila2050116131_a_o_o @ A_126 @ B_68 ) ) ) ).
thf(fact_23_UnCI,axiom,
! [A_126: hoare_1708887482_state > $o,C_33: hoare_1708887482_state,B_68: hoare_1708887482_state > $o] :
( ( ~ ( member451959335_state @ C_33 @ B_68 )
=> ( member451959335_state @ C_33 @ A_126 ) )
=> ( member451959335_state @ C_33 @ ( semila1122118281tate_o @ A_126 @ B_68 ) ) ) ).
thf(fact_24_UnCI,axiom,
! [A_126: hoare_2091234717iple_a > $o,C_33: hoare_2091234717iple_a,B_68: hoare_2091234717iple_a > $o] :
( ( ~ ( member290856304iple_a @ C_33 @ B_68 )
=> ( member290856304iple_a @ C_33 @ A_126 ) )
=> ( member290856304iple_a @ C_33 @ ( semila1052848428le_a_o @ A_126 @ B_68 ) ) ) ).
thf(fact_25_UnCI,axiom,
! [A_126: pname > $o,C_33: pname,B_68: pname > $o] :
( ( ~ ( member_pname @ C_33 @ B_68 )
=> ( member_pname @ C_33 @ A_126 ) )
=> ( member_pname @ C_33 @ ( semila1780557381name_o @ A_126 @ B_68 ) ) ) ).
thf(fact_26_image__eqI,axiom,
! [A_125: nat > $o,B_67: nat,F_50: nat > nat,X_49: nat] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member_nat @ X_49 @ A_125 )
=> ( member_nat @ B_67 @ ( image_nat_nat @ F_50 @ A_125 ) ) ) ) ).
thf(fact_27_image__eqI,axiom,
! [A_125: pname > $o,B_67: hoare_1708887482_state,F_50: pname > hoare_1708887482_state,X_49: pname] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member_pname @ X_49 @ A_125 )
=> ( member451959335_state @ B_67 @ ( image_1116629049_state @ F_50 @ A_125 ) ) ) ) ).
thf(fact_28_image__eqI,axiom,
! [A_125: pname > $o,B_67: nat,F_50: pname > nat,X_49: pname] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member_pname @ X_49 @ A_125 )
=> ( member_nat @ B_67 @ ( image_pname_nat @ F_50 @ A_125 ) ) ) ) ).
thf(fact_29_image__eqI,axiom,
! [A_125: pname > $o,B_67: hoare_2091234717iple_a > $o,F_50: pname > hoare_2091234717iple_a > $o,X_49: pname] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member_pname @ X_49 @ A_125 )
=> ( member99268621le_a_o @ B_67 @ ( image_742317343le_a_o @ F_50 @ A_125 ) ) ) ) ).
thf(fact_30_image__eqI,axiom,
! [A_125: nat > $o,B_67: pname,F_50: nat > pname,X_49: nat] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member_nat @ X_49 @ A_125 )
=> ( member_pname @ B_67 @ ( image_nat_pname @ F_50 @ A_125 ) ) ) ) ).
thf(fact_31_image__eqI,axiom,
! [A_125: ( hoare_2091234717iple_a > $o ) > $o,B_67: pname,F_50: ( hoare_2091234717iple_a > $o ) > pname,X_49: hoare_2091234717iple_a > $o] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member99268621le_a_o @ X_49 @ A_125 )
=> ( member_pname @ B_67 @ ( image_1908519857_pname @ F_50 @ A_125 ) ) ) ) ).
thf(fact_32_image__eqI,axiom,
! [A_125: hoare_2091234717iple_a > $o,B_67: pname,F_50: hoare_2091234717iple_a > pname,X_49: hoare_2091234717iple_a] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member290856304iple_a @ X_49 @ A_125 )
=> ( member_pname @ B_67 @ ( image_924789612_pname @ F_50 @ A_125 ) ) ) ) ).
thf(fact_33_image__eqI,axiom,
! [A_125: pname > $o,B_67: hoare_2091234717iple_a,F_50: pname > hoare_2091234717iple_a,X_49: pname] :
( ( B_67
= ( F_50 @ X_49 ) )
=> ( ( member_pname @ X_49 @ A_125 )
=> ( member290856304iple_a @ B_67 @ ( image_231808478iple_a @ F_50 @ A_125 ) ) ) ) ).
thf(fact_34_image__Un,axiom,
! [F_49: nat > nat,A_124: nat > $o,B_66: nat > $o] :
( ( image_nat_nat @ F_49 @ ( semila848761471_nat_o @ A_124 @ B_66 ) )
= ( semila848761471_nat_o @ ( image_nat_nat @ F_49 @ A_124 ) @ ( image_nat_nat @ F_49 @ B_66 ) ) ) ).
thf(fact_35_image__Un,axiom,
! [F_49: pname > hoare_1708887482_state,A_124: pname > $o,B_66: pname > $o] :
( ( image_1116629049_state @ F_49 @ ( semila1780557381name_o @ A_124 @ B_66 ) )
= ( semila1122118281tate_o @ ( image_1116629049_state @ F_49 @ A_124 ) @ ( image_1116629049_state @ F_49 @ B_66 ) ) ) ).
thf(fact_36_image__Un,axiom,
! [F_49: hoare_2091234717iple_a > nat,A_124: hoare_2091234717iple_a > $o,B_66: hoare_2091234717iple_a > $o] :
( ( image_1773322034_a_nat @ F_49 @ ( semila1052848428le_a_o @ A_124 @ B_66 ) )
= ( semila848761471_nat_o @ ( image_1773322034_a_nat @ F_49 @ A_124 ) @ ( image_1773322034_a_nat @ F_49 @ B_66 ) ) ) ).
thf(fact_37_image__Un,axiom,
! [F_49: hoare_2091234717iple_a > hoare_2091234717iple_a > $o,A_124: hoare_2091234717iple_a > $o,B_66: hoare_2091234717iple_a > $o] :
( ( image_1642350072le_a_o @ F_49 @ ( semila1052848428le_a_o @ A_124 @ B_66 ) )
= ( semila2050116131_a_o_o @ ( image_1642350072le_a_o @ F_49 @ A_124 ) @ ( image_1642350072le_a_o @ F_49 @ B_66 ) ) ) ).
thf(fact_38_image__Un,axiom,
! [F_49: hoare_2091234717iple_a > hoare_1708887482_state,A_124: hoare_2091234717iple_a > $o,B_66: hoare_2091234717iple_a > $o] :
( ( image_1884482962_state @ F_49 @ ( semila1052848428le_a_o @ A_124 @ B_66 ) )
= ( semila1122118281tate_o @ ( image_1884482962_state @ F_49 @ A_124 ) @ ( image_1884482962_state @ F_49 @ B_66 ) ) ) ).
thf(fact_39_image__Un,axiom,
! [F_49: hoare_2091234717iple_a > pname,A_124: hoare_2091234717iple_a > $o,B_66: hoare_2091234717iple_a > $o] :
( ( image_924789612_pname @ F_49 @ ( semila1052848428le_a_o @ A_124 @ B_66 ) )
= ( semila1780557381name_o @ ( image_924789612_pname @ F_49 @ A_124 ) @ ( image_924789612_pname @ F_49 @ B_66 ) ) ) ).
thf(fact_40_image__Un,axiom,
! [F_49: nat > hoare_2091234717iple_a,A_124: nat > $o,B_66: nat > $o] :
( ( image_359186840iple_a @ F_49 @ ( semila848761471_nat_o @ A_124 @ B_66 ) )
= ( semila1052848428le_a_o @ ( image_359186840iple_a @ F_49 @ A_124 ) @ ( image_359186840iple_a @ F_49 @ B_66 ) ) ) ).
thf(fact_41_image__Un,axiom,
! [F_49: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a,A_124: ( hoare_2091234717iple_a > $o ) > $o,B_66: ( hoare_2091234717iple_a > $o ) > $o] :
( ( image_136408202iple_a @ F_49 @ ( semila2050116131_a_o_o @ A_124 @ B_66 ) )
= ( semila1052848428le_a_o @ ( image_136408202iple_a @ F_49 @ A_124 ) @ ( image_136408202iple_a @ F_49 @ B_66 ) ) ) ).
thf(fact_42_image__Un,axiom,
! [F_49: hoare_1708887482_state > hoare_2091234717iple_a,A_124: hoare_1708887482_state > $o,B_66: hoare_1708887482_state > $o] :
( ( image_293283184iple_a @ F_49 @ ( semila1122118281tate_o @ A_124 @ B_66 ) )
= ( semila1052848428le_a_o @ ( image_293283184iple_a @ F_49 @ A_124 ) @ ( image_293283184iple_a @ F_49 @ B_66 ) ) ) ).
thf(fact_43_image__Un,axiom,
! [F_49: pname > hoare_2091234717iple_a,A_124: pname > $o,B_66: pname > $o] :
( ( image_231808478iple_a @ F_49 @ ( semila1780557381name_o @ A_124 @ B_66 ) )
= ( semila1052848428le_a_o @ ( image_231808478iple_a @ F_49 @ A_124 ) @ ( image_231808478iple_a @ F_49 @ B_66 ) ) ) ).
thf(fact_44_sup__fun__def,axiom,
! [F_48: nat > $o,G_26: nat > $o,X: nat] :
( ( semila848761471_nat_o @ F_48 @ G_26 @ X )
<=> ( semila10642723_sup_o @ ( F_48 @ X ) @ ( G_26 @ X ) ) ) ).
thf(fact_45_sup__fun__def,axiom,
! [F_48: ( hoare_2091234717iple_a > $o ) > $o,G_26: ( hoare_2091234717iple_a > $o ) > $o,X: hoare_2091234717iple_a > $o] :
( ( semila2050116131_a_o_o @ F_48 @ G_26 @ X )
<=> ( semila10642723_sup_o @ ( F_48 @ X ) @ ( G_26 @ X ) ) ) ).
thf(fact_46_sup__fun__def,axiom,
! [F_48: hoare_1708887482_state > $o,G_26: hoare_1708887482_state > $o,X: hoare_1708887482_state] :
( ( semila1122118281tate_o @ F_48 @ G_26 @ X )
<=> ( semila10642723_sup_o @ ( F_48 @ X ) @ ( G_26 @ X ) ) ) ).
thf(fact_47_sup__fun__def,axiom,
! [F_48: pname > $o,G_26: pname > $o,X: pname] :
( ( semila1780557381name_o @ F_48 @ G_26 @ X )
<=> ( semila10642723_sup_o @ ( F_48 @ X ) @ ( G_26 @ X ) ) ) ).
thf(fact_48_sup__fun__def,axiom,
! [F_48: hoare_2091234717iple_a > $o,G_26: hoare_2091234717iple_a > $o,X: hoare_2091234717iple_a] :
( ( semila1052848428le_a_o @ F_48 @ G_26 @ X )
<=> ( semila10642723_sup_o @ ( F_48 @ X ) @ ( G_26 @ X ) ) ) ).
thf(fact_49_sup__apply,axiom,
! [F_47: nat > $o,G_25: nat > $o,X_48: nat] :
( ( semila848761471_nat_o @ F_47 @ G_25 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_47 @ X_48 ) @ ( G_25 @ X_48 ) ) ) ).
thf(fact_50_sup__apply,axiom,
! [F_47: ( hoare_2091234717iple_a > $o ) > $o,G_25: ( hoare_2091234717iple_a > $o ) > $o,X_48: hoare_2091234717iple_a > $o] :
( ( semila2050116131_a_o_o @ F_47 @ G_25 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_47 @ X_48 ) @ ( G_25 @ X_48 ) ) ) ).
thf(fact_51_sup__apply,axiom,
! [F_47: hoare_1708887482_state > $o,G_25: hoare_1708887482_state > $o,X_48: hoare_1708887482_state] :
( ( semila1122118281tate_o @ F_47 @ G_25 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_47 @ X_48 ) @ ( G_25 @ X_48 ) ) ) ).
thf(fact_52_sup__apply,axiom,
! [F_47: pname > $o,G_25: pname > $o,X_48: pname] :
( ( semila1780557381name_o @ F_47 @ G_25 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_47 @ X_48 ) @ ( G_25 @ X_48 ) ) ) ).
thf(fact_53_sup__apply,axiom,
! [F_47: hoare_2091234717iple_a > $o,G_25: hoare_2091234717iple_a > $o,X_48: hoare_2091234717iple_a] :
( ( semila1052848428le_a_o @ F_47 @ G_25 @ X_48 )
<=> ( semila10642723_sup_o @ ( F_47 @ X_48 ) @ ( G_25 @ X_48 ) ) ) ).
thf(fact_54_cut,axiom,
! [G_24: hoare_2091234717iple_a > $o,G_23: hoare_2091234717iple_a > $o,Ts_3: hoare_2091234717iple_a > $o] :
( ( hoare_1467856363rivs_a @ G_23 @ Ts_3 )
=> ( ( hoare_1467856363rivs_a @ G_24 @ G_23 )
=> ( hoare_1467856363rivs_a @ G_24 @ Ts_3 ) ) ) ).
thf(fact_55_cut,axiom,
! [G_24: hoare_1708887482_state > $o,G_23: hoare_1708887482_state > $o,Ts_3: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ G_23 @ Ts_3 )
=> ( ( hoare_90032982_state @ G_24 @ G_23 )
=> ( hoare_90032982_state @ G_24 @ Ts_3 ) ) ) ).
thf(fact_56_sup__assoc,axiom,
! [X_47: nat > $o,Y_20: nat > $o,Z_11: nat > $o] :
( ( semila848761471_nat_o @ ( semila848761471_nat_o @ X_47 @ Y_20 ) @ Z_11 )
= ( semila848761471_nat_o @ X_47 @ ( semila848761471_nat_o @ Y_20 @ Z_11 ) ) ) ).
thf(fact_57_sup__assoc,axiom,
! [X_47: nat,Y_20: nat,Z_11: nat] :
( ( semila972727038up_nat @ ( semila972727038up_nat @ X_47 @ Y_20 ) @ Z_11 )
= ( semila972727038up_nat @ X_47 @ ( semila972727038up_nat @ Y_20 @ Z_11 ) ) ) ).
thf(fact_58_sup__assoc,axiom,
! [X_47: ( hoare_2091234717iple_a > $o ) > $o,Y_20: ( hoare_2091234717iple_a > $o ) > $o,Z_11: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ ( semila2050116131_a_o_o @ X_47 @ Y_20 ) @ Z_11 )
= ( semila2050116131_a_o_o @ X_47 @ ( semila2050116131_a_o_o @ Y_20 @ Z_11 ) ) ) ).
thf(fact_59_sup__assoc,axiom,
! [X_47: hoare_1708887482_state > $o,Y_20: hoare_1708887482_state > $o,Z_11: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ ( semila1122118281tate_o @ X_47 @ Y_20 ) @ Z_11 )
= ( semila1122118281tate_o @ X_47 @ ( semila1122118281tate_o @ Y_20 @ Z_11 ) ) ) ).
thf(fact_60_sup__assoc,axiom,
! [X_47: pname > $o,Y_20: pname > $o,Z_11: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ X_47 @ Y_20 ) @ Z_11 )
= ( semila1780557381name_o @ X_47 @ ( semila1780557381name_o @ Y_20 @ Z_11 ) ) ) ).
thf(fact_61_sup__assoc,axiom,
! [X_47: $o,Y_20: $o,Z_11: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ X_47 @ Y_20 ) @ Z_11 )
<=> ( semila10642723_sup_o @ X_47 @ ( semila10642723_sup_o @ Y_20 @ Z_11 ) ) ) ).
thf(fact_62_sup__assoc,axiom,
! [X_47: hoare_2091234717iple_a > $o,Y_20: hoare_2091234717iple_a > $o,Z_11: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ ( semila1052848428le_a_o @ X_47 @ Y_20 ) @ Z_11 )
= ( semila1052848428le_a_o @ X_47 @ ( semila1052848428le_a_o @ Y_20 @ Z_11 ) ) ) ).
thf(fact_63_inf__sup__aci_I6_J,axiom,
! [X_46: nat > $o,Y_19: nat > $o,Z_10: nat > $o] :
( ( semila848761471_nat_o @ ( semila848761471_nat_o @ X_46 @ Y_19 ) @ Z_10 )
= ( semila848761471_nat_o @ X_46 @ ( semila848761471_nat_o @ Y_19 @ Z_10 ) ) ) ).
thf(fact_64_inf__sup__aci_I6_J,axiom,
! [X_46: nat,Y_19: nat,Z_10: nat] :
( ( semila972727038up_nat @ ( semila972727038up_nat @ X_46 @ Y_19 ) @ Z_10 )
= ( semila972727038up_nat @ X_46 @ ( semila972727038up_nat @ Y_19 @ Z_10 ) ) ) ).
thf(fact_65_inf__sup__aci_I6_J,axiom,
! [X_46: ( hoare_2091234717iple_a > $o ) > $o,Y_19: ( hoare_2091234717iple_a > $o ) > $o,Z_10: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ ( semila2050116131_a_o_o @ X_46 @ Y_19 ) @ Z_10 )
= ( semila2050116131_a_o_o @ X_46 @ ( semila2050116131_a_o_o @ Y_19 @ Z_10 ) ) ) ).
thf(fact_66_inf__sup__aci_I6_J,axiom,
! [X_46: hoare_1708887482_state > $o,Y_19: hoare_1708887482_state > $o,Z_10: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ ( semila1122118281tate_o @ X_46 @ Y_19 ) @ Z_10 )
= ( semila1122118281tate_o @ X_46 @ ( semila1122118281tate_o @ Y_19 @ Z_10 ) ) ) ).
thf(fact_67_inf__sup__aci_I6_J,axiom,
! [X_46: pname > $o,Y_19: pname > $o,Z_10: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ X_46 @ Y_19 ) @ Z_10 )
= ( semila1780557381name_o @ X_46 @ ( semila1780557381name_o @ Y_19 @ Z_10 ) ) ) ).
thf(fact_68_inf__sup__aci_I6_J,axiom,
! [X_46: $o,Y_19: $o,Z_10: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ X_46 @ Y_19 ) @ Z_10 )
<=> ( semila10642723_sup_o @ X_46 @ ( semila10642723_sup_o @ Y_19 @ Z_10 ) ) ) ).
thf(fact_69_inf__sup__aci_I6_J,axiom,
! [X_46: hoare_2091234717iple_a > $o,Y_19: hoare_2091234717iple_a > $o,Z_10: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ ( semila1052848428le_a_o @ X_46 @ Y_19 ) @ Z_10 )
= ( semila1052848428le_a_o @ X_46 @ ( semila1052848428le_a_o @ Y_19 @ Z_10 ) ) ) ).
thf(fact_70_sup_Oassoc,axiom,
! [A_123: nat > $o,B_65: nat > $o,C_32: nat > $o] :
( ( semila848761471_nat_o @ ( semila848761471_nat_o @ A_123 @ B_65 ) @ C_32 )
= ( semila848761471_nat_o @ A_123 @ ( semila848761471_nat_o @ B_65 @ C_32 ) ) ) ).
thf(fact_71_sup_Oassoc,axiom,
! [A_123: nat,B_65: nat,C_32: nat] :
( ( semila972727038up_nat @ ( semila972727038up_nat @ A_123 @ B_65 ) @ C_32 )
= ( semila972727038up_nat @ A_123 @ ( semila972727038up_nat @ B_65 @ C_32 ) ) ) ).
thf(fact_72_sup_Oassoc,axiom,
! [A_123: ( hoare_2091234717iple_a > $o ) > $o,B_65: ( hoare_2091234717iple_a > $o ) > $o,C_32: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ ( semila2050116131_a_o_o @ A_123 @ B_65 ) @ C_32 )
= ( semila2050116131_a_o_o @ A_123 @ ( semila2050116131_a_o_o @ B_65 @ C_32 ) ) ) ).
thf(fact_73_sup_Oassoc,axiom,
! [A_123: hoare_1708887482_state > $o,B_65: hoare_1708887482_state > $o,C_32: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ ( semila1122118281tate_o @ A_123 @ B_65 ) @ C_32 )
= ( semila1122118281tate_o @ A_123 @ ( semila1122118281tate_o @ B_65 @ C_32 ) ) ) ).
thf(fact_74_sup_Oassoc,axiom,
! [A_123: pname > $o,B_65: pname > $o,C_32: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ A_123 @ B_65 ) @ C_32 )
= ( semila1780557381name_o @ A_123 @ ( semila1780557381name_o @ B_65 @ C_32 ) ) ) ).
thf(fact_75_sup_Oassoc,axiom,
! [A_123: $o,B_65: $o,C_32: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ A_123 @ B_65 ) @ C_32 )
<=> ( semila10642723_sup_o @ A_123 @ ( semila10642723_sup_o @ B_65 @ C_32 ) ) ) ).
thf(fact_76_sup_Oassoc,axiom,
! [A_123: hoare_2091234717iple_a > $o,B_65: hoare_2091234717iple_a > $o,C_32: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ ( semila1052848428le_a_o @ A_123 @ B_65 ) @ C_32 )
= ( semila1052848428le_a_o @ A_123 @ ( semila1052848428le_a_o @ B_65 @ C_32 ) ) ) ).
thf(fact_77_sup__left__commute,axiom,
! [X_45: nat > $o,Y_18: nat > $o,Z_9: nat > $o] :
( ( semila848761471_nat_o @ X_45 @ ( semila848761471_nat_o @ Y_18 @ Z_9 ) )
= ( semila848761471_nat_o @ Y_18 @ ( semila848761471_nat_o @ X_45 @ Z_9 ) ) ) ).
thf(fact_78_sup__left__commute,axiom,
! [X_45: nat,Y_18: nat,Z_9: nat] :
( ( semila972727038up_nat @ X_45 @ ( semila972727038up_nat @ Y_18 @ Z_9 ) )
= ( semila972727038up_nat @ Y_18 @ ( semila972727038up_nat @ X_45 @ Z_9 ) ) ) ).
thf(fact_79_sup__left__commute,axiom,
! [X_45: ( hoare_2091234717iple_a > $o ) > $o,Y_18: ( hoare_2091234717iple_a > $o ) > $o,Z_9: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_45 @ ( semila2050116131_a_o_o @ Y_18 @ Z_9 ) )
= ( semila2050116131_a_o_o @ Y_18 @ ( semila2050116131_a_o_o @ X_45 @ Z_9 ) ) ) ).
thf(fact_80_sup__left__commute,axiom,
! [X_45: hoare_1708887482_state > $o,Y_18: hoare_1708887482_state > $o,Z_9: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_45 @ ( semila1122118281tate_o @ Y_18 @ Z_9 ) )
= ( semila1122118281tate_o @ Y_18 @ ( semila1122118281tate_o @ X_45 @ Z_9 ) ) ) ).
thf(fact_81_sup__left__commute,axiom,
! [X_45: pname > $o,Y_18: pname > $o,Z_9: pname > $o] :
( ( semila1780557381name_o @ X_45 @ ( semila1780557381name_o @ Y_18 @ Z_9 ) )
= ( semila1780557381name_o @ Y_18 @ ( semila1780557381name_o @ X_45 @ Z_9 ) ) ) ).
thf(fact_82_sup__left__commute,axiom,
! [X_45: $o,Y_18: $o,Z_9: $o] :
( ( semila10642723_sup_o @ X_45 @ ( semila10642723_sup_o @ Y_18 @ Z_9 ) )
<=> ( semila10642723_sup_o @ Y_18 @ ( semila10642723_sup_o @ X_45 @ Z_9 ) ) ) ).
thf(fact_83_sup__left__commute,axiom,
! [X_45: hoare_2091234717iple_a > $o,Y_18: hoare_2091234717iple_a > $o,Z_9: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_45 @ ( semila1052848428le_a_o @ Y_18 @ Z_9 ) )
= ( semila1052848428le_a_o @ Y_18 @ ( semila1052848428le_a_o @ X_45 @ Z_9 ) ) ) ).
thf(fact_84_inf__sup__aci_I7_J,axiom,
! [X_44: nat > $o,Y_17: nat > $o,Z_8: nat > $o] :
( ( semila848761471_nat_o @ X_44 @ ( semila848761471_nat_o @ Y_17 @ Z_8 ) )
= ( semila848761471_nat_o @ Y_17 @ ( semila848761471_nat_o @ X_44 @ Z_8 ) ) ) ).
thf(fact_85_inf__sup__aci_I7_J,axiom,
! [X_44: nat,Y_17: nat,Z_8: nat] :
( ( semila972727038up_nat @ X_44 @ ( semila972727038up_nat @ Y_17 @ Z_8 ) )
= ( semila972727038up_nat @ Y_17 @ ( semila972727038up_nat @ X_44 @ Z_8 ) ) ) ).
thf(fact_86_inf__sup__aci_I7_J,axiom,
! [X_44: ( hoare_2091234717iple_a > $o ) > $o,Y_17: ( hoare_2091234717iple_a > $o ) > $o,Z_8: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_44 @ ( semila2050116131_a_o_o @ Y_17 @ Z_8 ) )
= ( semila2050116131_a_o_o @ Y_17 @ ( semila2050116131_a_o_o @ X_44 @ Z_8 ) ) ) ).
thf(fact_87_inf__sup__aci_I7_J,axiom,
! [X_44: hoare_1708887482_state > $o,Y_17: hoare_1708887482_state > $o,Z_8: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_44 @ ( semila1122118281tate_o @ Y_17 @ Z_8 ) )
= ( semila1122118281tate_o @ Y_17 @ ( semila1122118281tate_o @ X_44 @ Z_8 ) ) ) ).
thf(fact_88_inf__sup__aci_I7_J,axiom,
! [X_44: pname > $o,Y_17: pname > $o,Z_8: pname > $o] :
( ( semila1780557381name_o @ X_44 @ ( semila1780557381name_o @ Y_17 @ Z_8 ) )
= ( semila1780557381name_o @ Y_17 @ ( semila1780557381name_o @ X_44 @ Z_8 ) ) ) ).
thf(fact_89_inf__sup__aci_I7_J,axiom,
! [X_44: $o,Y_17: $o,Z_8: $o] :
( ( semila10642723_sup_o @ X_44 @ ( semila10642723_sup_o @ Y_17 @ Z_8 ) )
<=> ( semila10642723_sup_o @ Y_17 @ ( semila10642723_sup_o @ X_44 @ Z_8 ) ) ) ).
thf(fact_90_inf__sup__aci_I7_J,axiom,
! [X_44: hoare_2091234717iple_a > $o,Y_17: hoare_2091234717iple_a > $o,Z_8: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_44 @ ( semila1052848428le_a_o @ Y_17 @ Z_8 ) )
= ( semila1052848428le_a_o @ Y_17 @ ( semila1052848428le_a_o @ X_44 @ Z_8 ) ) ) ).
thf(fact_91_sup_Oleft__commute,axiom,
! [B_64: nat > $o,A_122: nat > $o,C_31: nat > $o] :
( ( semila848761471_nat_o @ B_64 @ ( semila848761471_nat_o @ A_122 @ C_31 ) )
= ( semila848761471_nat_o @ A_122 @ ( semila848761471_nat_o @ B_64 @ C_31 ) ) ) ).
thf(fact_92_sup_Oleft__commute,axiom,
! [B_64: nat,A_122: nat,C_31: nat] :
( ( semila972727038up_nat @ B_64 @ ( semila972727038up_nat @ A_122 @ C_31 ) )
= ( semila972727038up_nat @ A_122 @ ( semila972727038up_nat @ B_64 @ C_31 ) ) ) ).
thf(fact_93_sup_Oleft__commute,axiom,
! [B_64: ( hoare_2091234717iple_a > $o ) > $o,A_122: ( hoare_2091234717iple_a > $o ) > $o,C_31: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ B_64 @ ( semila2050116131_a_o_o @ A_122 @ C_31 ) )
= ( semila2050116131_a_o_o @ A_122 @ ( semila2050116131_a_o_o @ B_64 @ C_31 ) ) ) ).
thf(fact_94_sup_Oleft__commute,axiom,
! [B_64: hoare_1708887482_state > $o,A_122: hoare_1708887482_state > $o,C_31: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ B_64 @ ( semila1122118281tate_o @ A_122 @ C_31 ) )
= ( semila1122118281tate_o @ A_122 @ ( semila1122118281tate_o @ B_64 @ C_31 ) ) ) ).
thf(fact_95_sup_Oleft__commute,axiom,
! [B_64: pname > $o,A_122: pname > $o,C_31: pname > $o] :
( ( semila1780557381name_o @ B_64 @ ( semila1780557381name_o @ A_122 @ C_31 ) )
= ( semila1780557381name_o @ A_122 @ ( semila1780557381name_o @ B_64 @ C_31 ) ) ) ).
thf(fact_96_sup_Oleft__commute,axiom,
! [B_64: $o,A_122: $o,C_31: $o] :
( ( semila10642723_sup_o @ B_64 @ ( semila10642723_sup_o @ A_122 @ C_31 ) )
<=> ( semila10642723_sup_o @ A_122 @ ( semila10642723_sup_o @ B_64 @ C_31 ) ) ) ).
thf(fact_97_sup_Oleft__commute,axiom,
! [B_64: hoare_2091234717iple_a > $o,A_122: hoare_2091234717iple_a > $o,C_31: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ B_64 @ ( semila1052848428le_a_o @ A_122 @ C_31 ) )
= ( semila1052848428le_a_o @ A_122 @ ( semila1052848428le_a_o @ B_64 @ C_31 ) ) ) ).
thf(fact_98_sup__left__idem,axiom,
! [X_43: nat > $o,Y_16: nat > $o] :
( ( semila848761471_nat_o @ X_43 @ ( semila848761471_nat_o @ X_43 @ Y_16 ) )
= ( semila848761471_nat_o @ X_43 @ Y_16 ) ) ).
thf(fact_99_sup__left__idem,axiom,
! [X_43: nat,Y_16: nat] :
( ( semila972727038up_nat @ X_43 @ ( semila972727038up_nat @ X_43 @ Y_16 ) )
= ( semila972727038up_nat @ X_43 @ Y_16 ) ) ).
thf(fact_100_sup__left__idem,axiom,
! [X_43: ( hoare_2091234717iple_a > $o ) > $o,Y_16: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_43 @ ( semila2050116131_a_o_o @ X_43 @ Y_16 ) )
= ( semila2050116131_a_o_o @ X_43 @ Y_16 ) ) ).
thf(fact_101_sup__left__idem,axiom,
! [X_43: hoare_1708887482_state > $o,Y_16: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_43 @ ( semila1122118281tate_o @ X_43 @ Y_16 ) )
= ( semila1122118281tate_o @ X_43 @ Y_16 ) ) ).
thf(fact_102_sup__left__idem,axiom,
! [X_43: pname > $o,Y_16: pname > $o] :
( ( semila1780557381name_o @ X_43 @ ( semila1780557381name_o @ X_43 @ Y_16 ) )
= ( semila1780557381name_o @ X_43 @ Y_16 ) ) ).
thf(fact_103_sup__left__idem,axiom,
! [X_43: $o,Y_16: $o] :
( ( semila10642723_sup_o @ X_43 @ ( semila10642723_sup_o @ X_43 @ Y_16 ) )
<=> ( semila10642723_sup_o @ X_43 @ Y_16 ) ) ).
thf(fact_104_sup__left__idem,axiom,
! [X_43: hoare_2091234717iple_a > $o,Y_16: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_43 @ ( semila1052848428le_a_o @ X_43 @ Y_16 ) )
= ( semila1052848428le_a_o @ X_43 @ Y_16 ) ) ).
thf(fact_105_inf__sup__aci_I8_J,axiom,
! [X_42: nat > $o,Y_15: nat > $o] :
( ( semila848761471_nat_o @ X_42 @ ( semila848761471_nat_o @ X_42 @ Y_15 ) )
= ( semila848761471_nat_o @ X_42 @ Y_15 ) ) ).
thf(fact_106_inf__sup__aci_I8_J,axiom,
! [X_42: nat,Y_15: nat] :
( ( semila972727038up_nat @ X_42 @ ( semila972727038up_nat @ X_42 @ Y_15 ) )
= ( semila972727038up_nat @ X_42 @ Y_15 ) ) ).
thf(fact_107_inf__sup__aci_I8_J,axiom,
! [X_42: ( hoare_2091234717iple_a > $o ) > $o,Y_15: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_42 @ ( semila2050116131_a_o_o @ X_42 @ Y_15 ) )
= ( semila2050116131_a_o_o @ X_42 @ Y_15 ) ) ).
thf(fact_108_inf__sup__aci_I8_J,axiom,
! [X_42: hoare_1708887482_state > $o,Y_15: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_42 @ ( semila1122118281tate_o @ X_42 @ Y_15 ) )
= ( semila1122118281tate_o @ X_42 @ Y_15 ) ) ).
thf(fact_109_inf__sup__aci_I8_J,axiom,
! [X_42: pname > $o,Y_15: pname > $o] :
( ( semila1780557381name_o @ X_42 @ ( semila1780557381name_o @ X_42 @ Y_15 ) )
= ( semila1780557381name_o @ X_42 @ Y_15 ) ) ).
thf(fact_110_inf__sup__aci_I8_J,axiom,
! [X_42: $o,Y_15: $o] :
( ( semila10642723_sup_o @ X_42 @ ( semila10642723_sup_o @ X_42 @ Y_15 ) )
<=> ( semila10642723_sup_o @ X_42 @ Y_15 ) ) ).
thf(fact_111_inf__sup__aci_I8_J,axiom,
! [X_42: hoare_2091234717iple_a > $o,Y_15: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_42 @ ( semila1052848428le_a_o @ X_42 @ Y_15 ) )
= ( semila1052848428le_a_o @ X_42 @ Y_15 ) ) ).
thf(fact_112_sup_Oleft__idem,axiom,
! [A_121: nat > $o,B_63: nat > $o] :
( ( semila848761471_nat_o @ A_121 @ ( semila848761471_nat_o @ A_121 @ B_63 ) )
= ( semila848761471_nat_o @ A_121 @ B_63 ) ) ).
thf(fact_113_sup_Oleft__idem,axiom,
! [A_121: nat,B_63: nat] :
( ( semila972727038up_nat @ A_121 @ ( semila972727038up_nat @ A_121 @ B_63 ) )
= ( semila972727038up_nat @ A_121 @ B_63 ) ) ).
thf(fact_114_sup_Oleft__idem,axiom,
! [A_121: ( hoare_2091234717iple_a > $o ) > $o,B_63: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_121 @ ( semila2050116131_a_o_o @ A_121 @ B_63 ) )
= ( semila2050116131_a_o_o @ A_121 @ B_63 ) ) ).
thf(fact_115_sup_Oleft__idem,axiom,
! [A_121: hoare_1708887482_state > $o,B_63: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_121 @ ( semila1122118281tate_o @ A_121 @ B_63 ) )
= ( semila1122118281tate_o @ A_121 @ B_63 ) ) ).
thf(fact_116_sup_Oleft__idem,axiom,
! [A_121: pname > $o,B_63: pname > $o] :
( ( semila1780557381name_o @ A_121 @ ( semila1780557381name_o @ A_121 @ B_63 ) )
= ( semila1780557381name_o @ A_121 @ B_63 ) ) ).
thf(fact_117_sup_Oleft__idem,axiom,
! [A_121: $o,B_63: $o] :
( ( semila10642723_sup_o @ A_121 @ ( semila10642723_sup_o @ A_121 @ B_63 ) )
<=> ( semila10642723_sup_o @ A_121 @ B_63 ) ) ).
thf(fact_118_sup_Oleft__idem,axiom,
! [A_121: hoare_2091234717iple_a > $o,B_63: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_121 @ ( semila1052848428le_a_o @ A_121 @ B_63 ) )
= ( semila1052848428le_a_o @ A_121 @ B_63 ) ) ).
thf(fact_119_sup__commute,axiom,
! [X_41: nat > $o,Y_14: nat > $o] :
( ( semila848761471_nat_o @ X_41 @ Y_14 )
= ( semila848761471_nat_o @ Y_14 @ X_41 ) ) ).
thf(fact_120_sup__commute,axiom,
! [X_41: nat,Y_14: nat] :
( ( semila972727038up_nat @ X_41 @ Y_14 )
= ( semila972727038up_nat @ Y_14 @ X_41 ) ) ).
thf(fact_121_sup__commute,axiom,
! [X_41: ( hoare_2091234717iple_a > $o ) > $o,Y_14: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_41 @ Y_14 )
= ( semila2050116131_a_o_o @ Y_14 @ X_41 ) ) ).
thf(fact_122_sup__commute,axiom,
! [X_41: hoare_1708887482_state > $o,Y_14: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_41 @ Y_14 )
= ( semila1122118281tate_o @ Y_14 @ X_41 ) ) ).
thf(fact_123_sup__commute,axiom,
! [X_41: pname > $o,Y_14: pname > $o] :
( ( semila1780557381name_o @ X_41 @ Y_14 )
= ( semila1780557381name_o @ Y_14 @ X_41 ) ) ).
thf(fact_124_sup__commute,axiom,
! [X_41: $o,Y_14: $o] :
( ( semila10642723_sup_o @ X_41 @ Y_14 )
<=> ( semila10642723_sup_o @ Y_14 @ X_41 ) ) ).
thf(fact_125_sup__commute,axiom,
! [X_41: hoare_2091234717iple_a > $o,Y_14: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_41 @ Y_14 )
= ( semila1052848428le_a_o @ Y_14 @ X_41 ) ) ).
thf(fact_126_inf__sup__aci_I5_J,axiom,
! [X_40: nat > $o,Y_13: nat > $o] :
( ( semila848761471_nat_o @ X_40 @ Y_13 )
= ( semila848761471_nat_o @ Y_13 @ X_40 ) ) ).
thf(fact_127_inf__sup__aci_I5_J,axiom,
! [X_40: nat,Y_13: nat] :
( ( semila972727038up_nat @ X_40 @ Y_13 )
= ( semila972727038up_nat @ Y_13 @ X_40 ) ) ).
thf(fact_128_inf__sup__aci_I5_J,axiom,
! [X_40: ( hoare_2091234717iple_a > $o ) > $o,Y_13: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_40 @ Y_13 )
= ( semila2050116131_a_o_o @ Y_13 @ X_40 ) ) ).
thf(fact_129_inf__sup__aci_I5_J,axiom,
! [X_40: hoare_1708887482_state > $o,Y_13: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_40 @ Y_13 )
= ( semila1122118281tate_o @ Y_13 @ X_40 ) ) ).
thf(fact_130_inf__sup__aci_I5_J,axiom,
! [X_40: pname > $o,Y_13: pname > $o] :
( ( semila1780557381name_o @ X_40 @ Y_13 )
= ( semila1780557381name_o @ Y_13 @ X_40 ) ) ).
thf(fact_131_inf__sup__aci_I5_J,axiom,
! [X_40: $o,Y_13: $o] :
( ( semila10642723_sup_o @ X_40 @ Y_13 )
<=> ( semila10642723_sup_o @ Y_13 @ X_40 ) ) ).
thf(fact_132_inf__sup__aci_I5_J,axiom,
! [X_40: hoare_2091234717iple_a > $o,Y_13: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_40 @ Y_13 )
= ( semila1052848428le_a_o @ Y_13 @ X_40 ) ) ).
thf(fact_133_sup_Ocommute,axiom,
! [A_120: nat > $o,B_62: nat > $o] :
( ( semila848761471_nat_o @ A_120 @ B_62 )
= ( semila848761471_nat_o @ B_62 @ A_120 ) ) ).
thf(fact_134_sup_Ocommute,axiom,
! [A_120: nat,B_62: nat] :
( ( semila972727038up_nat @ A_120 @ B_62 )
= ( semila972727038up_nat @ B_62 @ A_120 ) ) ).
thf(fact_135_sup_Ocommute,axiom,
! [A_120: ( hoare_2091234717iple_a > $o ) > $o,B_62: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_120 @ B_62 )
= ( semila2050116131_a_o_o @ B_62 @ A_120 ) ) ).
thf(fact_136_sup_Ocommute,axiom,
! [A_120: hoare_1708887482_state > $o,B_62: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_120 @ B_62 )
= ( semila1122118281tate_o @ B_62 @ A_120 ) ) ).
thf(fact_137_sup_Ocommute,axiom,
! [A_120: pname > $o,B_62: pname > $o] :
( ( semila1780557381name_o @ A_120 @ B_62 )
= ( semila1780557381name_o @ B_62 @ A_120 ) ) ).
thf(fact_138_sup_Ocommute,axiom,
! [A_120: $o,B_62: $o] :
( ( semila10642723_sup_o @ A_120 @ B_62 )
<=> ( semila10642723_sup_o @ B_62 @ A_120 ) ) ).
thf(fact_139_sup_Ocommute,axiom,
! [A_120: hoare_2091234717iple_a > $o,B_62: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_120 @ B_62 )
= ( semila1052848428le_a_o @ B_62 @ A_120 ) ) ).
thf(fact_140_sup__idem,axiom,
! [X_39: nat > $o] :
( ( semila848761471_nat_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_141_sup__idem,axiom,
! [X_39: nat] :
( ( semila972727038up_nat @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_142_sup__idem,axiom,
! [X_39: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_143_sup__idem,axiom,
! [X_39: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_144_sup__idem,axiom,
! [X_39: pname > $o] :
( ( semila1780557381name_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_145_sup__idem,axiom,
! [X_39: $o] :
( ( semila10642723_sup_o @ X_39 @ X_39 )
<=> X_39 ) ).
thf(fact_146_sup__idem,axiom,
! [X_39: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_39 @ X_39 )
= X_39 ) ).
thf(fact_147_sup_Oidem,axiom,
! [A_119: nat > $o] :
( ( semila848761471_nat_o @ A_119 @ A_119 )
= A_119 ) ).
thf(fact_148_sup_Oidem,axiom,
! [A_119: nat] :
( ( semila972727038up_nat @ A_119 @ A_119 )
= A_119 ) ).
thf(fact_149_sup_Oidem,axiom,
! [A_119: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_119 @ A_119 )
= A_119 ) ).
thf(fact_150_sup_Oidem,axiom,
! [A_119: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_119 @ A_119 )
= A_119 ) ).
thf(fact_151_sup_Oidem,axiom,
! [A_119: pname > $o] :
( ( semila1780557381name_o @ A_119 @ A_119 )
= A_119 ) ).
thf(fact_152_sup_Oidem,axiom,
! [A_119: $o] :
( ( semila10642723_sup_o @ A_119 @ A_119 )
<=> A_119 ) ).
thf(fact_153_sup_Oidem,axiom,
! [A_119: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_119 @ A_119 )
= A_119 ) ).
thf(fact_154_rev__image__eqI,axiom,
! [B_61: nat,F_46: nat > nat,X_38: nat,A_118: nat > $o] :
( ( member_nat @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member_nat @ B_61 @ ( image_nat_nat @ F_46 @ A_118 ) ) ) ) ).
thf(fact_155_rev__image__eqI,axiom,
! [B_61: hoare_1708887482_state,F_46: pname > hoare_1708887482_state,X_38: pname,A_118: pname > $o] :
( ( member_pname @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member451959335_state @ B_61 @ ( image_1116629049_state @ F_46 @ A_118 ) ) ) ) ).
thf(fact_156_rev__image__eqI,axiom,
! [B_61: nat,F_46: pname > nat,X_38: pname,A_118: pname > $o] :
( ( member_pname @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member_nat @ B_61 @ ( image_pname_nat @ F_46 @ A_118 ) ) ) ) ).
thf(fact_157_rev__image__eqI,axiom,
! [B_61: hoare_2091234717iple_a > $o,F_46: pname > hoare_2091234717iple_a > $o,X_38: pname,A_118: pname > $o] :
( ( member_pname @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member99268621le_a_o @ B_61 @ ( image_742317343le_a_o @ F_46 @ A_118 ) ) ) ) ).
thf(fact_158_rev__image__eqI,axiom,
! [B_61: pname,F_46: nat > pname,X_38: nat,A_118: nat > $o] :
( ( member_nat @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member_pname @ B_61 @ ( image_nat_pname @ F_46 @ A_118 ) ) ) ) ).
thf(fact_159_rev__image__eqI,axiom,
! [B_61: pname,F_46: ( hoare_2091234717iple_a > $o ) > pname,X_38: hoare_2091234717iple_a > $o,A_118: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member_pname @ B_61 @ ( image_1908519857_pname @ F_46 @ A_118 ) ) ) ) ).
thf(fact_160_rev__image__eqI,axiom,
! [B_61: pname,F_46: hoare_2091234717iple_a > pname,X_38: hoare_2091234717iple_a,A_118: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member_pname @ B_61 @ ( image_924789612_pname @ F_46 @ A_118 ) ) ) ) ).
thf(fact_161_rev__image__eqI,axiom,
! [B_61: hoare_2091234717iple_a,F_46: pname > hoare_2091234717iple_a,X_38: pname,A_118: pname > $o] :
( ( member_pname @ X_38 @ A_118 )
=> ( ( B_61
= ( F_46 @ X_38 ) )
=> ( member290856304iple_a @ B_61 @ ( image_231808478iple_a @ F_46 @ A_118 ) ) ) ) ).
thf(fact_162_imageI,axiom,
! [F_45: nat > nat,X_37: nat,A_117: nat > $o] :
( ( member_nat @ X_37 @ A_117 )
=> ( member_nat @ ( F_45 @ X_37 ) @ ( image_nat_nat @ F_45 @ A_117 ) ) ) ).
thf(fact_163_imageI,axiom,
! [F_45: pname > hoare_1708887482_state,X_37: pname,A_117: pname > $o] :
( ( member_pname @ X_37 @ A_117 )
=> ( member451959335_state @ ( F_45 @ X_37 ) @ ( image_1116629049_state @ F_45 @ A_117 ) ) ) ).
thf(fact_164_imageI,axiom,
! [F_45: pname > nat,X_37: pname,A_117: pname > $o] :
( ( member_pname @ X_37 @ A_117 )
=> ( member_nat @ ( F_45 @ X_37 ) @ ( image_pname_nat @ F_45 @ A_117 ) ) ) ).
thf(fact_165_imageI,axiom,
! [F_45: pname > hoare_2091234717iple_a > $o,X_37: pname,A_117: pname > $o] :
( ( member_pname @ X_37 @ A_117 )
=> ( member99268621le_a_o @ ( F_45 @ X_37 ) @ ( image_742317343le_a_o @ F_45 @ A_117 ) ) ) ).
thf(fact_166_imageI,axiom,
! [F_45: nat > pname,X_37: nat,A_117: nat > $o] :
( ( member_nat @ X_37 @ A_117 )
=> ( member_pname @ ( F_45 @ X_37 ) @ ( image_nat_pname @ F_45 @ A_117 ) ) ) ).
thf(fact_167_imageI,axiom,
! [F_45: ( hoare_2091234717iple_a > $o ) > pname,X_37: hoare_2091234717iple_a > $o,A_117: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ X_37 @ A_117 )
=> ( member_pname @ ( F_45 @ X_37 ) @ ( image_1908519857_pname @ F_45 @ A_117 ) ) ) ).
thf(fact_168_imageI,axiom,
! [F_45: hoare_2091234717iple_a > pname,X_37: hoare_2091234717iple_a,A_117: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_37 @ A_117 )
=> ( member_pname @ ( F_45 @ X_37 ) @ ( image_924789612_pname @ F_45 @ A_117 ) ) ) ).
thf(fact_169_imageI,axiom,
! [F_45: pname > hoare_2091234717iple_a,X_37: pname,A_117: pname > $o] :
( ( member_pname @ X_37 @ A_117 )
=> ( member290856304iple_a @ ( F_45 @ X_37 ) @ ( image_231808478iple_a @ F_45 @ A_117 ) ) ) ).
thf(fact_170_image__iff,axiom,
! [Z_7: nat,F_44: nat > nat,A_116: nat > $o] :
( ( member_nat @ Z_7 @ ( image_nat_nat @ F_44 @ A_116 ) )
<=> ? [X: nat] :
( ( member_nat @ X @ A_116 )
& ( Z_7
= ( F_44 @ X ) ) ) ) ).
thf(fact_171_image__iff,axiom,
! [Z_7: hoare_1708887482_state,F_44: pname > hoare_1708887482_state,A_116: pname > $o] :
( ( member451959335_state @ Z_7 @ ( image_1116629049_state @ F_44 @ A_116 ) )
<=> ? [X: pname] :
( ( member_pname @ X @ A_116 )
& ( Z_7
= ( F_44 @ X ) ) ) ) ).
thf(fact_172_image__iff,axiom,
! [Z_7: hoare_2091234717iple_a,F_44: pname > hoare_2091234717iple_a,A_116: pname > $o] :
( ( member290856304iple_a @ Z_7 @ ( image_231808478iple_a @ F_44 @ A_116 ) )
<=> ? [X: pname] :
( ( member_pname @ X @ A_116 )
& ( Z_7
= ( F_44 @ X ) ) ) ) ).
thf(fact_173_UnI2,axiom,
! [A_115: nat > $o,C_30: nat,B_60: nat > $o] :
( ( member_nat @ C_30 @ B_60 )
=> ( member_nat @ C_30 @ ( semila848761471_nat_o @ A_115 @ B_60 ) ) ) ).
thf(fact_174_UnI2,axiom,
! [A_115: ( hoare_2091234717iple_a > $o ) > $o,C_30: hoare_2091234717iple_a > $o,B_60: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_30 @ B_60 )
=> ( member99268621le_a_o @ C_30 @ ( semila2050116131_a_o_o @ A_115 @ B_60 ) ) ) ).
thf(fact_175_UnI2,axiom,
! [A_115: hoare_1708887482_state > $o,C_30: hoare_1708887482_state,B_60: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_30 @ B_60 )
=> ( member451959335_state @ C_30 @ ( semila1122118281tate_o @ A_115 @ B_60 ) ) ) ).
thf(fact_176_UnI2,axiom,
! [A_115: hoare_2091234717iple_a > $o,C_30: hoare_2091234717iple_a,B_60: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_30 @ B_60 )
=> ( member290856304iple_a @ C_30 @ ( semila1052848428le_a_o @ A_115 @ B_60 ) ) ) ).
thf(fact_177_UnI2,axiom,
! [A_115: pname > $o,C_30: pname,B_60: pname > $o] :
( ( member_pname @ C_30 @ B_60 )
=> ( member_pname @ C_30 @ ( semila1780557381name_o @ A_115 @ B_60 ) ) ) ).
thf(fact_178_UnI1,axiom,
! [B_59: nat > $o,C_29: nat,A_114: nat > $o] :
( ( member_nat @ C_29 @ A_114 )
=> ( member_nat @ C_29 @ ( semila848761471_nat_o @ A_114 @ B_59 ) ) ) ).
thf(fact_179_UnI1,axiom,
! [B_59: ( hoare_2091234717iple_a > $o ) > $o,C_29: hoare_2091234717iple_a > $o,A_114: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_29 @ A_114 )
=> ( member99268621le_a_o @ C_29 @ ( semila2050116131_a_o_o @ A_114 @ B_59 ) ) ) ).
thf(fact_180_UnI1,axiom,
! [B_59: hoare_1708887482_state > $o,C_29: hoare_1708887482_state,A_114: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_29 @ A_114 )
=> ( member451959335_state @ C_29 @ ( semila1122118281tate_o @ A_114 @ B_59 ) ) ) ).
thf(fact_181_UnI1,axiom,
! [B_59: hoare_2091234717iple_a > $o,C_29: hoare_2091234717iple_a,A_114: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_29 @ A_114 )
=> ( member290856304iple_a @ C_29 @ ( semila1052848428le_a_o @ A_114 @ B_59 ) ) ) ).
thf(fact_182_UnI1,axiom,
! [B_59: pname > $o,C_29: pname,A_114: pname > $o] :
( ( member_pname @ C_29 @ A_114 )
=> ( member_pname @ C_29 @ ( semila1780557381name_o @ A_114 @ B_59 ) ) ) ).
thf(fact_183_sup1I2,axiom,
! [A_113: nat > $o,B_58: nat > $o,X_36: nat] :
( ( B_58 @ X_36 )
=> ( semila848761471_nat_o @ A_113 @ B_58 @ X_36 ) ) ).
thf(fact_184_sup1I2,axiom,
! [A_113: ( hoare_2091234717iple_a > $o ) > $o,B_58: ( hoare_2091234717iple_a > $o ) > $o,X_36: hoare_2091234717iple_a > $o] :
( ( B_58 @ X_36 )
=> ( semila2050116131_a_o_o @ A_113 @ B_58 @ X_36 ) ) ).
thf(fact_185_sup1I2,axiom,
! [A_113: hoare_1708887482_state > $o,B_58: hoare_1708887482_state > $o,X_36: hoare_1708887482_state] :
( ( B_58 @ X_36 )
=> ( semila1122118281tate_o @ A_113 @ B_58 @ X_36 ) ) ).
thf(fact_186_sup1I2,axiom,
! [A_113: pname > $o,B_58: pname > $o,X_36: pname] :
( ( B_58 @ X_36 )
=> ( semila1780557381name_o @ A_113 @ B_58 @ X_36 ) ) ).
thf(fact_187_sup1I2,axiom,
! [A_113: hoare_2091234717iple_a > $o,B_58: hoare_2091234717iple_a > $o,X_36: hoare_2091234717iple_a] :
( ( B_58 @ X_36 )
=> ( semila1052848428le_a_o @ A_113 @ B_58 @ X_36 ) ) ).
thf(fact_188_sup1I1,axiom,
! [B_57: nat > $o,A_112: nat > $o,X_35: nat] :
( ( A_112 @ X_35 )
=> ( semila848761471_nat_o @ A_112 @ B_57 @ X_35 ) ) ).
thf(fact_189_sup1I1,axiom,
! [B_57: ( hoare_2091234717iple_a > $o ) > $o,A_112: ( hoare_2091234717iple_a > $o ) > $o,X_35: hoare_2091234717iple_a > $o] :
( ( A_112 @ X_35 )
=> ( semila2050116131_a_o_o @ A_112 @ B_57 @ X_35 ) ) ).
thf(fact_190_sup1I1,axiom,
! [B_57: hoare_1708887482_state > $o,A_112: hoare_1708887482_state > $o,X_35: hoare_1708887482_state] :
( ( A_112 @ X_35 )
=> ( semila1122118281tate_o @ A_112 @ B_57 @ X_35 ) ) ).
thf(fact_191_sup1I1,axiom,
! [B_57: pname > $o,A_112: pname > $o,X_35: pname] :
( ( A_112 @ X_35 )
=> ( semila1780557381name_o @ A_112 @ B_57 @ X_35 ) ) ).
thf(fact_192_sup1I1,axiom,
! [B_57: hoare_2091234717iple_a > $o,A_112: hoare_2091234717iple_a > $o,X_35: hoare_2091234717iple_a] :
( ( A_112 @ X_35 )
=> ( semila1052848428le_a_o @ A_112 @ B_57 @ X_35 ) ) ).
thf(fact_193_ball__Un,axiom,
! [P_35: nat > $o,A_111: nat > $o,B_56: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( semila848761471_nat_o @ A_111 @ B_56 ) )
=> ( P_35 @ X ) )
<=> ( ! [X: nat] :
( ( member_nat @ X @ A_111 )
=> ( P_35 @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B_56 )
=> ( P_35 @ X ) ) ) ) ).
thf(fact_194_ball__Un,axiom,
! [P_35: ( hoare_2091234717iple_a > $o ) > $o,A_111: ( hoare_2091234717iple_a > $o ) > $o,B_56: ( hoare_2091234717iple_a > $o ) > $o] :
( ! [X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ ( semila2050116131_a_o_o @ A_111 @ B_56 ) )
=> ( P_35 @ X ) )
<=> ( ! [X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ A_111 )
=> ( P_35 @ X ) )
& ! [X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ B_56 )
=> ( P_35 @ X ) ) ) ) ).
thf(fact_195_ball__Un,axiom,
! [P_35: hoare_1708887482_state > $o,A_111: hoare_1708887482_state > $o,B_56: hoare_1708887482_state > $o] :
( ! [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ ( semila1122118281tate_o @ A_111 @ B_56 ) )
=> ( P_35 @ X ) )
<=> ( ! [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ A_111 )
=> ( P_35 @ X ) )
& ! [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ B_56 )
=> ( P_35 @ X ) ) ) ) ).
thf(fact_196_ball__Un,axiom,
! [P_35: pname > $o,A_111: pname > $o,B_56: pname > $o] :
( ! [X: pname] :
( ( member_pname @ X @ ( semila1780557381name_o @ A_111 @ B_56 ) )
=> ( P_35 @ X ) )
<=> ( ! [X: pname] :
( ( member_pname @ X @ A_111 )
=> ( P_35 @ X ) )
& ! [X: pname] :
( ( member_pname @ X @ B_56 )
=> ( P_35 @ X ) ) ) ) ).
thf(fact_197_ball__Un,axiom,
! [P_35: hoare_2091234717iple_a > $o,A_111: hoare_2091234717iple_a > $o,B_56: hoare_2091234717iple_a > $o] :
( ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ ( semila1052848428le_a_o @ A_111 @ B_56 ) )
=> ( P_35 @ X ) )
<=> ( ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ A_111 )
=> ( P_35 @ X ) )
& ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ B_56 )
=> ( P_35 @ X ) ) ) ) ).
thf(fact_198_bex__Un,axiom,
! [P_34: nat > $o,A_110: nat > $o,B_55: nat > $o] :
( ? [X: nat] :
( ( member_nat @ X @ ( semila848761471_nat_o @ A_110 @ B_55 ) )
& ( P_34 @ X ) )
<=> ( ? [X: nat] :
( ( member_nat @ X @ A_110 )
& ( P_34 @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B_55 )
& ( P_34 @ X ) ) ) ) ).
thf(fact_199_bex__Un,axiom,
! [P_34: ( hoare_2091234717iple_a > $o ) > $o,A_110: ( hoare_2091234717iple_a > $o ) > $o,B_55: ( hoare_2091234717iple_a > $o ) > $o] :
( ? [X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ ( semila2050116131_a_o_o @ A_110 @ B_55 ) )
& ( P_34 @ X ) )
<=> ( ? [X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ A_110 )
& ( P_34 @ X ) )
| ? [X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ B_55 )
& ( P_34 @ X ) ) ) ) ).
thf(fact_200_bex__Un,axiom,
! [P_34: hoare_1708887482_state > $o,A_110: hoare_1708887482_state > $o,B_55: hoare_1708887482_state > $o] :
( ? [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ ( semila1122118281tate_o @ A_110 @ B_55 ) )
& ( P_34 @ X ) )
<=> ( ? [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ A_110 )
& ( P_34 @ X ) )
| ? [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ B_55 )
& ( P_34 @ X ) ) ) ) ).
thf(fact_201_bex__Un,axiom,
! [P_34: pname > $o,A_110: pname > $o,B_55: pname > $o] :
( ? [X: pname] :
( ( member_pname @ X @ ( semila1780557381name_o @ A_110 @ B_55 ) )
& ( P_34 @ X ) )
<=> ( ? [X: pname] :
( ( member_pname @ X @ A_110 )
& ( P_34 @ X ) )
| ? [X: pname] :
( ( member_pname @ X @ B_55 )
& ( P_34 @ X ) ) ) ) ).
thf(fact_202_bex__Un,axiom,
! [P_34: hoare_2091234717iple_a > $o,A_110: hoare_2091234717iple_a > $o,B_55: hoare_2091234717iple_a > $o] :
( ? [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ ( semila1052848428le_a_o @ A_110 @ B_55 ) )
& ( P_34 @ X ) )
<=> ( ? [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ A_110 )
& ( P_34 @ X ) )
| ? [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ B_55 )
& ( P_34 @ X ) ) ) ) ).
thf(fact_203_Un__assoc,axiom,
! [A_109: nat > $o,B_54: nat > $o,C_28: nat > $o] :
( ( semila848761471_nat_o @ ( semila848761471_nat_o @ A_109 @ B_54 ) @ C_28 )
= ( semila848761471_nat_o @ A_109 @ ( semila848761471_nat_o @ B_54 @ C_28 ) ) ) ).
thf(fact_204_Un__assoc,axiom,
! [A_109: ( hoare_2091234717iple_a > $o ) > $o,B_54: ( hoare_2091234717iple_a > $o ) > $o,C_28: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ ( semila2050116131_a_o_o @ A_109 @ B_54 ) @ C_28 )
= ( semila2050116131_a_o_o @ A_109 @ ( semila2050116131_a_o_o @ B_54 @ C_28 ) ) ) ).
thf(fact_205_Un__assoc,axiom,
! [A_109: hoare_1708887482_state > $o,B_54: hoare_1708887482_state > $o,C_28: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ ( semila1122118281tate_o @ A_109 @ B_54 ) @ C_28 )
= ( semila1122118281tate_o @ A_109 @ ( semila1122118281tate_o @ B_54 @ C_28 ) ) ) ).
thf(fact_206_Un__assoc,axiom,
! [A_109: pname > $o,B_54: pname > $o,C_28: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ A_109 @ B_54 ) @ C_28 )
= ( semila1780557381name_o @ A_109 @ ( semila1780557381name_o @ B_54 @ C_28 ) ) ) ).
thf(fact_207_Un__assoc,axiom,
! [A_109: hoare_2091234717iple_a > $o,B_54: hoare_2091234717iple_a > $o,C_28: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ ( semila1052848428le_a_o @ A_109 @ B_54 ) @ C_28 )
= ( semila1052848428le_a_o @ A_109 @ ( semila1052848428le_a_o @ B_54 @ C_28 ) ) ) ).
thf(fact_208_Un__iff,axiom,
! [C_27: nat,A_108: nat > $o,B_53: nat > $o] :
( ( member_nat @ C_27 @ ( semila848761471_nat_o @ A_108 @ B_53 ) )
<=> ( ( member_nat @ C_27 @ A_108 )
| ( member_nat @ C_27 @ B_53 ) ) ) ).
thf(fact_209_Un__iff,axiom,
! [C_27: hoare_2091234717iple_a > $o,A_108: ( hoare_2091234717iple_a > $o ) > $o,B_53: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_27 @ ( semila2050116131_a_o_o @ A_108 @ B_53 ) )
<=> ( ( member99268621le_a_o @ C_27 @ A_108 )
| ( member99268621le_a_o @ C_27 @ B_53 ) ) ) ).
thf(fact_210_Un__iff,axiom,
! [C_27: hoare_1708887482_state,A_108: hoare_1708887482_state > $o,B_53: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_27 @ ( semila1122118281tate_o @ A_108 @ B_53 ) )
<=> ( ( member451959335_state @ C_27 @ A_108 )
| ( member451959335_state @ C_27 @ B_53 ) ) ) ).
thf(fact_211_Un__iff,axiom,
! [C_27: hoare_2091234717iple_a,A_108: hoare_2091234717iple_a > $o,B_53: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_27 @ ( semila1052848428le_a_o @ A_108 @ B_53 ) )
<=> ( ( member290856304iple_a @ C_27 @ A_108 )
| ( member290856304iple_a @ C_27 @ B_53 ) ) ) ).
thf(fact_212_Un__iff,axiom,
! [C_27: pname,A_108: pname > $o,B_53: pname > $o] :
( ( member_pname @ C_27 @ ( semila1780557381name_o @ A_108 @ B_53 ) )
<=> ( ( member_pname @ C_27 @ A_108 )
| ( member_pname @ C_27 @ B_53 ) ) ) ).
thf(fact_213_Un__left__commute,axiom,
! [A_107: nat > $o,B_52: nat > $o,C_26: nat > $o] :
( ( semila848761471_nat_o @ A_107 @ ( semila848761471_nat_o @ B_52 @ C_26 ) )
= ( semila848761471_nat_o @ B_52 @ ( semila848761471_nat_o @ A_107 @ C_26 ) ) ) ).
thf(fact_214_Un__left__commute,axiom,
! [A_107: ( hoare_2091234717iple_a > $o ) > $o,B_52: ( hoare_2091234717iple_a > $o ) > $o,C_26: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_107 @ ( semila2050116131_a_o_o @ B_52 @ C_26 ) )
= ( semila2050116131_a_o_o @ B_52 @ ( semila2050116131_a_o_o @ A_107 @ C_26 ) ) ) ).
thf(fact_215_Un__left__commute,axiom,
! [A_107: hoare_1708887482_state > $o,B_52: hoare_1708887482_state > $o,C_26: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_107 @ ( semila1122118281tate_o @ B_52 @ C_26 ) )
= ( semila1122118281tate_o @ B_52 @ ( semila1122118281tate_o @ A_107 @ C_26 ) ) ) ).
thf(fact_216_Un__left__commute,axiom,
! [A_107: pname > $o,B_52: pname > $o,C_26: pname > $o] :
( ( semila1780557381name_o @ A_107 @ ( semila1780557381name_o @ B_52 @ C_26 ) )
= ( semila1780557381name_o @ B_52 @ ( semila1780557381name_o @ A_107 @ C_26 ) ) ) ).
thf(fact_217_Un__left__commute,axiom,
! [A_107: hoare_2091234717iple_a > $o,B_52: hoare_2091234717iple_a > $o,C_26: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_107 @ ( semila1052848428le_a_o @ B_52 @ C_26 ) )
= ( semila1052848428le_a_o @ B_52 @ ( semila1052848428le_a_o @ A_107 @ C_26 ) ) ) ).
thf(fact_218_Un__left__absorb,axiom,
! [A_106: nat > $o,B_51: nat > $o] :
( ( semila848761471_nat_o @ A_106 @ ( semila848761471_nat_o @ A_106 @ B_51 ) )
= ( semila848761471_nat_o @ A_106 @ B_51 ) ) ).
thf(fact_219_Un__left__absorb,axiom,
! [A_106: ( hoare_2091234717iple_a > $o ) > $o,B_51: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_106 @ ( semila2050116131_a_o_o @ A_106 @ B_51 ) )
= ( semila2050116131_a_o_o @ A_106 @ B_51 ) ) ).
thf(fact_220_Un__left__absorb,axiom,
! [A_106: hoare_1708887482_state > $o,B_51: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_106 @ ( semila1122118281tate_o @ A_106 @ B_51 ) )
= ( semila1122118281tate_o @ A_106 @ B_51 ) ) ).
thf(fact_221_Un__left__absorb,axiom,
! [A_106: pname > $o,B_51: pname > $o] :
( ( semila1780557381name_o @ A_106 @ ( semila1780557381name_o @ A_106 @ B_51 ) )
= ( semila1780557381name_o @ A_106 @ B_51 ) ) ).
thf(fact_222_Un__left__absorb,axiom,
! [A_106: hoare_2091234717iple_a > $o,B_51: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_106 @ ( semila1052848428le_a_o @ A_106 @ B_51 ) )
= ( semila1052848428le_a_o @ A_106 @ B_51 ) ) ).
thf(fact_223_Un__commute,axiom,
! [A_105: nat > $o,B_50: nat > $o] :
( ( semila848761471_nat_o @ A_105 @ B_50 )
= ( semila848761471_nat_o @ B_50 @ A_105 ) ) ).
thf(fact_224_Un__commute,axiom,
! [A_105: ( hoare_2091234717iple_a > $o ) > $o,B_50: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_105 @ B_50 )
= ( semila2050116131_a_o_o @ B_50 @ A_105 ) ) ).
thf(fact_225_Un__commute,axiom,
! [A_105: hoare_1708887482_state > $o,B_50: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_105 @ B_50 )
= ( semila1122118281tate_o @ B_50 @ A_105 ) ) ).
thf(fact_226_Un__commute,axiom,
! [A_105: pname > $o,B_50: pname > $o] :
( ( semila1780557381name_o @ A_105 @ B_50 )
= ( semila1780557381name_o @ B_50 @ A_105 ) ) ).
thf(fact_227_Un__commute,axiom,
! [A_105: hoare_2091234717iple_a > $o,B_50: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_105 @ B_50 )
= ( semila1052848428le_a_o @ B_50 @ A_105 ) ) ).
thf(fact_228_Un__def,axiom,
! [A_104: ( hoare_2091234717iple_a > $o ) > $o,B_49: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_104 @ B_49 )
= ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (|) @ ( member99268621le_a_o @ X @ A_104 ) @ ( member99268621le_a_o @ X @ B_49 ) ) ) ) ).
thf(fact_229_Un__def,axiom,
! [A_104: hoare_1708887482_state > $o,B_49: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_104 @ B_49 )
= ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : ( (|) @ ( member451959335_state @ X @ A_104 ) @ ( member451959335_state @ X @ B_49 ) ) ) ) ).
thf(fact_230_Un__def,axiom,
! [A_104: hoare_2091234717iple_a > $o,B_49: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_104 @ B_49 )
= ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (|) @ ( member290856304iple_a @ X @ A_104 ) @ ( member290856304iple_a @ X @ B_49 ) ) ) ) ).
thf(fact_231_Un__def,axiom,
! [A_104: nat > $o,B_49: nat > $o] :
( ( semila848761471_nat_o @ A_104 @ B_49 )
= ( collect_nat
@ ^ [X: nat] : ( (|) @ ( member_nat @ X @ A_104 ) @ ( member_nat @ X @ B_49 ) ) ) ) ).
thf(fact_232_Un__def,axiom,
! [A_104: pname > $o,B_49: pname > $o] :
( ( semila1780557381name_o @ A_104 @ B_49 )
= ( collect_pname
@ ^ [X: pname] : ( (|) @ ( member_pname @ X @ A_104 ) @ ( member_pname @ X @ B_49 ) ) ) ) ).
thf(fact_233_Un__absorb,axiom,
! [A_103: nat > $o] :
( ( semila848761471_nat_o @ A_103 @ A_103 )
= A_103 ) ).
thf(fact_234_Un__absorb,axiom,
! [A_103: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_103 @ A_103 )
= A_103 ) ).
thf(fact_235_Un__absorb,axiom,
! [A_103: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_103 @ A_103 )
= A_103 ) ).
thf(fact_236_Un__absorb,axiom,
! [A_103: pname > $o] :
( ( semila1780557381name_o @ A_103 @ A_103 )
= A_103 ) ).
thf(fact_237_Un__absorb,axiom,
! [A_103: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_103 @ A_103 )
= A_103 ) ).
thf(fact_238_image__ident,axiom,
! [Y_12: nat > $o] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y_12 )
= Y_12 ) ).
thf(fact_239_image__image,axiom,
! [F_43: hoare_2091234717iple_a > hoare_1708887482_state,G_22: pname > hoare_2091234717iple_a,A_102: pname > $o] :
( ( image_1884482962_state @ F_43 @ ( image_231808478iple_a @ G_22 @ A_102 ) )
= ( image_1116629049_state
@ ^ [X: pname] : ( F_43 @ ( G_22 @ X ) )
@ A_102 ) ) ).
thf(fact_240_image__image,axiom,
! [F_43: hoare_1708887482_state > hoare_2091234717iple_a,G_22: pname > hoare_1708887482_state,A_102: pname > $o] :
( ( image_293283184iple_a @ F_43 @ ( image_1116629049_state @ G_22 @ A_102 ) )
= ( image_231808478iple_a
@ ^ [X: pname] : ( F_43 @ ( G_22 @ X ) )
@ A_102 ) ) ).
thf(fact_241_sup__Un__eq,axiom,
! [R_2: nat > $o,S_5: nat > $o,X: nat] :
( ( semila848761471_nat_o
@ ^ [Y_7: nat] : ( member_nat @ Y_7 @ R_2 )
@ ^ [Y_7: nat] : ( member_nat @ Y_7 @ S_5 )
@ X )
<=> ( member_nat @ X @ ( semila848761471_nat_o @ R_2 @ S_5 ) ) ) ).
thf(fact_242_sup__Un__eq,axiom,
! [R_2: ( hoare_2091234717iple_a > $o ) > $o,S_5: ( hoare_2091234717iple_a > $o ) > $o,X: hoare_2091234717iple_a > $o] :
( ( semila2050116131_a_o_o
@ ^ [Y_7: hoare_2091234717iple_a > $o] : ( member99268621le_a_o @ Y_7 @ R_2 )
@ ^ [Y_7: hoare_2091234717iple_a > $o] : ( member99268621le_a_o @ Y_7 @ S_5 )
@ X )
<=> ( member99268621le_a_o @ X @ ( semila2050116131_a_o_o @ R_2 @ S_5 ) ) ) ).
thf(fact_243_sup__Un__eq,axiom,
! [R_2: hoare_1708887482_state > $o,S_5: hoare_1708887482_state > $o,X: hoare_1708887482_state] :
( ( semila1122118281tate_o
@ ^ [Y_7: hoare_1708887482_state] : ( member451959335_state @ Y_7 @ R_2 )
@ ^ [Y_7: hoare_1708887482_state] : ( member451959335_state @ Y_7 @ S_5 )
@ X )
<=> ( member451959335_state @ X @ ( semila1122118281tate_o @ R_2 @ S_5 ) ) ) ).
thf(fact_244_sup__Un__eq,axiom,
! [R_2: hoare_2091234717iple_a > $o,S_5: hoare_2091234717iple_a > $o,X: hoare_2091234717iple_a] :
( ( semila1052848428le_a_o
@ ^ [Y_7: hoare_2091234717iple_a] : ( member290856304iple_a @ Y_7 @ R_2 )
@ ^ [Y_7: hoare_2091234717iple_a] : ( member290856304iple_a @ Y_7 @ S_5 )
@ X )
<=> ( member290856304iple_a @ X @ ( semila1052848428le_a_o @ R_2 @ S_5 ) ) ) ).
thf(fact_245_sup__Un__eq,axiom,
! [R_2: pname > $o,S_5: pname > $o,X: pname] :
( ( semila1780557381name_o
@ ^ [Y_7: pname] : ( member_pname @ Y_7 @ R_2 )
@ ^ [Y_7: pname] : ( member_pname @ Y_7 @ S_5 )
@ X )
<=> ( member_pname @ X @ ( semila1780557381name_o @ R_2 @ S_5 ) ) ) ).
thf(fact_246_Collect__disj__eq,axiom,
! [P_33: pname > $o,Q_19: pname > $o] :
( ( collect_pname
@ ^ [X: pname] : ( (|) @ ( P_33 @ X ) @ ( Q_19 @ X ) ) )
= ( semila1780557381name_o @ ( collect_pname @ P_33 ) @ ( collect_pname @ Q_19 ) ) ) ).
thf(fact_247_Collect__disj__eq,axiom,
! [P_33: ( hoare_2091234717iple_a > $o ) > $o,Q_19: ( hoare_2091234717iple_a > $o ) > $o] :
( ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (|) @ ( P_33 @ X ) @ ( Q_19 @ X ) ) )
= ( semila2050116131_a_o_o @ ( collec1008234059le_a_o @ P_33 ) @ ( collec1008234059le_a_o @ Q_19 ) ) ) ).
thf(fact_248_Collect__disj__eq,axiom,
! [P_33: hoare_1708887482_state > $o,Q_19: hoare_1708887482_state > $o] :
( ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : ( (|) @ ( P_33 @ X ) @ ( Q_19 @ X ) ) )
= ( semila1122118281tate_o @ ( collec1568722789_state @ P_33 ) @ ( collec1568722789_state @ Q_19 ) ) ) ).
thf(fact_249_Collect__disj__eq,axiom,
! [P_33: hoare_2091234717iple_a > $o,Q_19: hoare_2091234717iple_a > $o] :
( ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (|) @ ( P_33 @ X ) @ ( Q_19 @ X ) ) )
= ( semila1052848428le_a_o @ ( collec992574898iple_a @ P_33 ) @ ( collec992574898iple_a @ Q_19 ) ) ) ).
thf(fact_250_Collect__disj__eq,axiom,
! [P_33: nat > $o,Q_19: nat > $o] :
( ( collect_nat
@ ^ [X: nat] : ( (|) @ ( P_33 @ X ) @ ( Q_19 @ X ) ) )
= ( semila848761471_nat_o @ ( collect_nat @ P_33 ) @ ( collect_nat @ Q_19 ) ) ) ).
thf(fact_251_imageE,axiom,
! [B_48: nat,F_42: nat > nat,A_101: nat > $o] :
( ( member_nat @ B_48 @ ( image_nat_nat @ F_42 @ A_101 ) )
=> ~ ! [X: nat] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member_nat @ X @ A_101 ) ) ) ).
thf(fact_252_imageE,axiom,
! [B_48: pname,F_42: nat > pname,A_101: nat > $o] :
( ( member_pname @ B_48 @ ( image_nat_pname @ F_42 @ A_101 ) )
=> ~ ! [X: nat] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member_nat @ X @ A_101 ) ) ) ).
thf(fact_253_imageE,axiom,
! [B_48: pname,F_42: ( hoare_2091234717iple_a > $o ) > pname,A_101: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member_pname @ B_48 @ ( image_1908519857_pname @ F_42 @ A_101 ) )
=> ~ ! [X: hoare_2091234717iple_a > $o] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member99268621le_a_o @ X @ A_101 ) ) ) ).
thf(fact_254_imageE,axiom,
! [B_48: pname,F_42: hoare_2091234717iple_a > pname,A_101: hoare_2091234717iple_a > $o] :
( ( member_pname @ B_48 @ ( image_924789612_pname @ F_42 @ A_101 ) )
=> ~ ! [X: hoare_2091234717iple_a] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member290856304iple_a @ X @ A_101 ) ) ) ).
thf(fact_255_imageE,axiom,
! [B_48: hoare_1708887482_state,F_42: pname > hoare_1708887482_state,A_101: pname > $o] :
( ( member451959335_state @ B_48 @ ( image_1116629049_state @ F_42 @ A_101 ) )
=> ~ ! [X: pname] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member_pname @ X @ A_101 ) ) ) ).
thf(fact_256_imageE,axiom,
! [B_48: nat,F_42: pname > nat,A_101: pname > $o] :
( ( member_nat @ B_48 @ ( image_pname_nat @ F_42 @ A_101 ) )
=> ~ ! [X: pname] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member_pname @ X @ A_101 ) ) ) ).
thf(fact_257_imageE,axiom,
! [B_48: hoare_2091234717iple_a > $o,F_42: pname > hoare_2091234717iple_a > $o,A_101: pname > $o] :
( ( member99268621le_a_o @ B_48 @ ( image_742317343le_a_o @ F_42 @ A_101 ) )
=> ~ ! [X: pname] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member_pname @ X @ A_101 ) ) ) ).
thf(fact_258_imageE,axiom,
! [B_48: hoare_2091234717iple_a,F_42: pname > hoare_2091234717iple_a,A_101: pname > $o] :
( ( member290856304iple_a @ B_48 @ ( image_231808478iple_a @ F_42 @ A_101 ) )
=> ~ ! [X: pname] :
( ( B_48
= ( F_42 @ X ) )
=> ~ ( member_pname @ X @ A_101 ) ) ) ).
thf(fact_259_Body__triple__valid__Suc,axiom,
! [N_8: nat,P_32: state > state > $o,Pn_6: pname,Q_18: state > state > $o] :
( ( hoare_23738522_state @ N_8 @ ( hoare_858012674_state @ P_32 @ ( the_com @ ( body_1 @ Pn_6 ) ) @ Q_18 ) )
<=> ( hoare_23738522_state @ ( suc @ N_8 ) @ ( hoare_858012674_state @ P_32 @ ( body @ Pn_6 ) @ Q_18 ) ) ) ).
thf(fact_260_Body__triple__valid__Suc,axiom,
! [N_8: nat,P_32: x_a > state > $o,Pn_6: pname,Q_18: x_a > state > $o] :
( ( hoare_1421888935alid_a @ N_8 @ ( hoare_657976383iple_a @ P_32 @ ( the_com @ ( body_1 @ Pn_6 ) ) @ Q_18 ) )
<=> ( hoare_1421888935alid_a @ ( suc @ N_8 ) @ ( hoare_657976383iple_a @ P_32 @ ( body @ Pn_6 ) @ Q_18 ) ) ) ).
thf(fact_261_triple_Oexhaust,axiom,
! [Y_11: hoare_2091234717iple_a] :
~ ! [Fun1_2: x_a > state > $o,Com_4: com,Fun2_2: x_a > state > $o] :
( Y_11
!= ( hoare_657976383iple_a @ Fun1_2 @ Com_4 @ Fun2_2 ) ) ).
thf(fact_262_triple_Oexhaust,axiom,
! [Y_11: hoare_1708887482_state] :
~ ! [Fun1_2: state > state > $o,Com_4: com,Fun2_2: state > state > $o] :
( Y_11
!= ( hoare_858012674_state @ Fun1_2 @ Com_4 @ Fun2_2 ) ) ).
thf(fact_263_Body1,axiom,
! [Pn_5: pname,G_21: hoare_1708887482_state > $o,P_31: pname > state > state > $o,Q_17: pname > state > state > $o,Procs: pname > $o] :
( ( hoare_90032982_state
@ ( semila1122118281tate_o @ G_21
@ ( image_1116629049_state
@ ^ [P_9: pname] : ( hoare_858012674_state @ ( P_31 @ P_9 ) @ ( body @ P_9 ) @ ( Q_17 @ P_9 ) )
@ Procs ) )
@ ( image_1116629049_state
@ ^ [P_9: pname] : ( hoare_858012674_state @ ( P_31 @ P_9 ) @ ( the_com @ ( body_1 @ P_9 ) ) @ ( Q_17 @ P_9 ) )
@ Procs ) )
=> ( ( member_pname @ Pn_5 @ Procs )
=> ( hoare_90032982_state @ G_21 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_31 @ Pn_5 ) @ ( body @ Pn_5 ) @ ( Q_17 @ Pn_5 ) ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_264_Body1,axiom,
! [Pn_5: pname,G_21: hoare_2091234717iple_a > $o,P_31: pname > x_a > state > $o,Q_17: pname > x_a > state > $o,Procs: pname > $o] :
( ( hoare_1467856363rivs_a
@ ( semila1052848428le_a_o @ G_21
@ ( image_231808478iple_a
@ ^ [P_9: pname] : ( hoare_657976383iple_a @ ( P_31 @ P_9 ) @ ( body @ P_9 ) @ ( Q_17 @ P_9 ) )
@ Procs ) )
@ ( image_231808478iple_a
@ ^ [P_9: pname] : ( hoare_657976383iple_a @ ( P_31 @ P_9 ) @ ( the_com @ ( body_1 @ P_9 ) ) @ ( Q_17 @ P_9 ) )
@ Procs ) )
=> ( ( member_pname @ Pn_5 @ Procs )
=> ( hoare_1467856363rivs_a @ G_21 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_31 @ Pn_5 ) @ ( body @ Pn_5 ) @ ( Q_17 @ Pn_5 ) ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_265_image__cong,axiom,
! [F_41: nat > nat,G_20: nat > nat,M_3: nat > $o,N_7: nat > $o] :
( ( M_3 = N_7 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N_7 )
=> ( ( F_41 @ X )
= ( G_20 @ X ) ) )
=> ( ( image_nat_nat @ F_41 @ M_3 )
= ( image_nat_nat @ G_20 @ N_7 ) ) ) ) ).
thf(fact_266_image__cong,axiom,
! [F_41: pname > hoare_1708887482_state,G_20: pname > hoare_1708887482_state,M_3: pname > $o,N_7: pname > $o] :
( ( M_3 = N_7 )
=> ( ! [X: pname] :
( ( member_pname @ X @ N_7 )
=> ( ( F_41 @ X )
= ( G_20 @ X ) ) )
=> ( ( image_1116629049_state @ F_41 @ M_3 )
= ( image_1116629049_state @ G_20 @ N_7 ) ) ) ) ).
thf(fact_267_image__cong,axiom,
! [F_41: pname > hoare_2091234717iple_a,G_20: pname > hoare_2091234717iple_a,M_3: pname > $o,N_7: pname > $o] :
( ( M_3 = N_7 )
=> ( ! [X: pname] :
( ( member_pname @ X @ N_7 )
=> ( ( F_41 @ X )
= ( G_20 @ X ) ) )
=> ( ( image_231808478iple_a @ F_41 @ M_3 )
= ( image_231808478iple_a @ G_20 @ N_7 ) ) ) ) ).
thf(fact_268_Body__triple__valid__0,axiom,
! [P_30: state > state > $o,Pn_4: pname,Q_16: state > state > $o] : ( hoare_23738522_state @ zero_zero_nat @ ( hoare_858012674_state @ P_30 @ ( body @ Pn_4 ) @ Q_16 ) ) ).
thf(fact_269_Body__triple__valid__0,axiom,
! [P_30: x_a > state > $o,Pn_4: pname,Q_16: x_a > state > $o] : ( hoare_1421888935alid_a @ zero_zero_nat @ ( hoare_657976383iple_a @ P_30 @ ( body @ Pn_4 ) @ Q_16 ) ) ).
thf(fact_270_com_Osimps_I6_J,axiom,
! [Pname_1: pname,Pname: pname] :
( ( ( body @ Pname_1 )
= ( body @ Pname ) )
<=> ( Pname_1 = Pname ) ) ).
thf(fact_271_evalc_OBody,axiom,
! [Pn_1: pname,S0: state,S1: state] :
( ( evalc @ ( the_com @ ( body_1 @ Pn_1 ) ) @ S0 @ S1 )
=> ( evalc @ ( body @ Pn_1 ) @ S0 @ S1 ) ) ).
thf(fact_272_evalc__elim__cases_I6_J,axiom,
! [P: pname,S: state,S1: state] :
( ( evalc @ ( body @ P ) @ S @ S1 )
=> ( evalc @ ( the_com @ ( body_1 @ P ) ) @ S @ S1 ) ) ).
thf(fact_273_Sup__fin_Oidem,axiom,
! [X_34: nat > $o] :
( ( semila848761471_nat_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_274_Sup__fin_Oidem,axiom,
! [X_34: nat] :
( ( semila972727038up_nat @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_275_Sup__fin_Oidem,axiom,
! [X_34: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_276_Sup__fin_Oidem,axiom,
! [X_34: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_277_Sup__fin_Oidem,axiom,
! [X_34: pname > $o] :
( ( semila1780557381name_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_278_Sup__fin_Oidem,axiom,
! [X_34: $o] :
( ( semila10642723_sup_o @ X_34 @ X_34 )
<=> X_34 ) ).
thf(fact_279_Sup__fin_Oidem,axiom,
! [X_34: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_34 @ X_34 )
= X_34 ) ).
thf(fact_280_triples__valid__Suc,axiom,
! [N_6: nat,Ts_2: hoare_1708887482_state > $o] :
( ! [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ Ts_2 )
=> ( hoare_23738522_state @ ( suc @ N_6 ) @ X ) )
=> ! [X: hoare_1708887482_state] :
( ( member451959335_state @ X @ Ts_2 )
=> ( hoare_23738522_state @ N_6 @ X ) ) ) ).
thf(fact_281_triples__valid__Suc,axiom,
! [N_6: nat,Ts_2: hoare_2091234717iple_a > $o] :
( ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ Ts_2 )
=> ( hoare_1421888935alid_a @ ( suc @ N_6 ) @ X ) )
=> ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ Ts_2 )
=> ( hoare_1421888935alid_a @ N_6 @ X ) ) ) ).
thf(fact_282_emptyE,axiom,
! [A_100: nat] :
~ ( member_nat @ A_100 @ bot_bot_nat_o ) ).
thf(fact_283_emptyE,axiom,
! [A_100: hoare_2091234717iple_a > $o] :
~ ( member99268621le_a_o @ A_100 @ bot_bo1957696069_a_o_o ) ).
thf(fact_284_emptyE,axiom,
! [A_100: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ A_100 @ bot_bo1791335050le_a_o ) ).
thf(fact_285_emptyE,axiom,
! [A_100: hoare_1708887482_state] :
~ ( member451959335_state @ A_100 @ bot_bo19817387tate_o ) ).
thf(fact_286_emptyE,axiom,
! [A_100: pname] :
~ ( member_pname @ A_100 @ bot_bot_pname_o ) ).
thf(fact_287_insertE,axiom,
! [A_99: nat,B_47: nat,A_98: nat > $o] :
( ( member_nat @ A_99 @ ( insert_nat @ B_47 @ A_98 ) )
=> ( ( A_99 != B_47 )
=> ( member_nat @ A_99 @ A_98 ) ) ) ).
thf(fact_288_insertE,axiom,
! [A_99: hoare_2091234717iple_a > $o,B_47: hoare_2091234717iple_a > $o,A_98: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ A_99 @ ( insert102003750le_a_o @ B_47 @ A_98 ) )
=> ( ( A_99 != B_47 )
=> ( member99268621le_a_o @ A_99 @ A_98 ) ) ) ).
thf(fact_289_insertE,axiom,
! [A_99: hoare_2091234717iple_a,B_47: hoare_2091234717iple_a,A_98: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_99 @ ( insert1597628439iple_a @ B_47 @ A_98 ) )
=> ( ( A_99 != B_47 )
=> ( member290856304iple_a @ A_99 @ A_98 ) ) ) ).
thf(fact_290_insertE,axiom,
! [A_99: hoare_1708887482_state,B_47: hoare_1708887482_state,A_98: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_99 @ ( insert528405184_state @ B_47 @ A_98 ) )
=> ( ( A_99 != B_47 )
=> ( member451959335_state @ A_99 @ A_98 ) ) ) ).
thf(fact_291_insertE,axiom,
! [A_99: pname,B_47: pname,A_98: pname > $o] :
( ( member_pname @ A_99 @ ( insert_pname @ B_47 @ A_98 ) )
=> ( ( A_99 != B_47 )
=> ( member_pname @ A_99 @ A_98 ) ) ) ).
thf(fact_292_insertCI,axiom,
! [B_46: nat,A_97: nat,B_45: nat > $o] :
( ( ~ ( member_nat @ A_97 @ B_45 )
=> ( A_97 = B_46 ) )
=> ( member_nat @ A_97 @ ( insert_nat @ B_46 @ B_45 ) ) ) ).
thf(fact_293_insertCI,axiom,
! [B_46: hoare_2091234717iple_a > $o,A_97: hoare_2091234717iple_a > $o,B_45: ( hoare_2091234717iple_a > $o ) > $o] :
( ( ~ ( member99268621le_a_o @ A_97 @ B_45 )
=> ( A_97 = B_46 ) )
=> ( member99268621le_a_o @ A_97 @ ( insert102003750le_a_o @ B_46 @ B_45 ) ) ) ).
thf(fact_294_insertCI,axiom,
! [B_46: hoare_2091234717iple_a,A_97: hoare_2091234717iple_a,B_45: hoare_2091234717iple_a > $o] :
( ( ~ ( member290856304iple_a @ A_97 @ B_45 )
=> ( A_97 = B_46 ) )
=> ( member290856304iple_a @ A_97 @ ( insert1597628439iple_a @ B_46 @ B_45 ) ) ) ).
thf(fact_295_insertCI,axiom,
! [B_46: hoare_1708887482_state,A_97: hoare_1708887482_state,B_45: hoare_1708887482_state > $o] :
( ( ~ ( member451959335_state @ A_97 @ B_45 )
=> ( A_97 = B_46 ) )
=> ( member451959335_state @ A_97 @ ( insert528405184_state @ B_46 @ B_45 ) ) ) ).
thf(fact_296_insertCI,axiom,
! [B_46: pname,A_97: pname,B_45: pname > $o] :
( ( ~ ( member_pname @ A_97 @ B_45 )
=> ( A_97 = B_46 ) )
=> ( member_pname @ A_97 @ ( insert_pname @ B_46 @ B_45 ) ) ) ).
thf(fact_297_empty__not__insert,axiom,
! [A_96: nat,A_95: nat > $o] :
( bot_bot_nat_o
!= ( insert_nat @ A_96 @ A_95 ) ) ).
thf(fact_298_empty__not__insert,axiom,
! [A_96: hoare_2091234717iple_a > $o,A_95: ( hoare_2091234717iple_a > $o ) > $o] :
( bot_bo1957696069_a_o_o
!= ( insert102003750le_a_o @ A_96 @ A_95 ) ) ).
thf(fact_299_empty__not__insert,axiom,
! [A_96: pname,A_95: pname > $o] :
( bot_bot_pname_o
!= ( insert_pname @ A_96 @ A_95 ) ) ).
thf(fact_300_empty__not__insert,axiom,
! [A_96: hoare_2091234717iple_a,A_95: hoare_2091234717iple_a > $o] :
( bot_bo1791335050le_a_o
!= ( insert1597628439iple_a @ A_96 @ A_95 ) ) ).
thf(fact_301_empty__not__insert,axiom,
! [A_96: hoare_1708887482_state,A_95: hoare_1708887482_state > $o] :
( bot_bo19817387tate_o
!= ( insert528405184_state @ A_96 @ A_95 ) ) ).
thf(fact_302_insert__not__empty,axiom,
! [A_94: nat,A_93: nat > $o] :
( ( insert_nat @ A_94 @ A_93 )
!= bot_bot_nat_o ) ).
thf(fact_303_insert__not__empty,axiom,
! [A_94: hoare_2091234717iple_a > $o,A_93: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ A_94 @ A_93 )
!= bot_bo1957696069_a_o_o ) ).
thf(fact_304_insert__not__empty,axiom,
! [A_94: pname,A_93: pname > $o] :
( ( insert_pname @ A_94 @ A_93 )
!= bot_bot_pname_o ) ).
thf(fact_305_insert__not__empty,axiom,
! [A_94: hoare_2091234717iple_a,A_93: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_94 @ A_93 )
!= bot_bo1791335050le_a_o ) ).
thf(fact_306_insert__not__empty,axiom,
! [A_94: hoare_1708887482_state,A_93: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_94 @ A_93 )
!= bot_bo19817387tate_o ) ).
thf(fact_307_bot__empty__eq,axiom,
! [X: nat] :
( ( bot_bot_nat_o @ X )
<=> ( member_nat @ X @ bot_bot_nat_o ) ) ).
thf(fact_308_bot__empty__eq,axiom,
! [X: hoare_2091234717iple_a > $o] :
( ( bot_bo1957696069_a_o_o @ X )
<=> ( member99268621le_a_o @ X @ bot_bo1957696069_a_o_o ) ) ).
thf(fact_309_bot__empty__eq,axiom,
! [X: hoare_2091234717iple_a] :
( ( bot_bo1791335050le_a_o @ X )
<=> ( member290856304iple_a @ X @ bot_bo1791335050le_a_o ) ) ).
thf(fact_310_bot__empty__eq,axiom,
! [X: hoare_1708887482_state] :
( ( bot_bo19817387tate_o @ X )
<=> ( member451959335_state @ X @ bot_bo19817387tate_o ) ) ).
thf(fact_311_bot__empty__eq,axiom,
! [X: pname] :
( ( bot_bot_pname_o @ X )
<=> ( member_pname @ X @ bot_bot_pname_o ) ) ).
thf(fact_312_empty__def,axiom,
( bot_bot_pname_o
= ( collect_pname
@ ^ [X: pname] : $false ) ) ).
thf(fact_313_empty__def,axiom,
( bot_bo1791335050le_a_o
= ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : $false ) ) ).
thf(fact_314_empty__def,axiom,
( bot_bo1957696069_a_o_o
= ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : $false ) ) ).
thf(fact_315_empty__def,axiom,
( bot_bo19817387tate_o
= ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : $false ) ) ).
thf(fact_316_empty__def,axiom,
( bot_bot_nat_o
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
thf(fact_317_insertI1,axiom,
! [A_92: nat,B_44: nat > $o] : ( member_nat @ A_92 @ ( insert_nat @ A_92 @ B_44 ) ) ).
thf(fact_318_insertI1,axiom,
! [A_92: hoare_2091234717iple_a > $o,B_44: ( hoare_2091234717iple_a > $o ) > $o] : ( member99268621le_a_o @ A_92 @ ( insert102003750le_a_o @ A_92 @ B_44 ) ) ).
thf(fact_319_insertI1,axiom,
! [A_92: hoare_2091234717iple_a,B_44: hoare_2091234717iple_a > $o] : ( member290856304iple_a @ A_92 @ ( insert1597628439iple_a @ A_92 @ B_44 ) ) ).
thf(fact_320_insertI1,axiom,
! [A_92: hoare_1708887482_state,B_44: hoare_1708887482_state > $o] : ( member451959335_state @ A_92 @ ( insert528405184_state @ A_92 @ B_44 ) ) ).
thf(fact_321_insertI1,axiom,
! [A_92: pname,B_44: pname > $o] : ( member_pname @ A_92 @ ( insert_pname @ A_92 @ B_44 ) ) ).
thf(fact_322_all__not__in__conv,axiom,
! [A_91: nat > $o] :
( ! [X: nat] :
~ ( member_nat @ X @ A_91 )
<=> ( A_91 = bot_bot_nat_o ) ) ).
thf(fact_323_all__not__in__conv,axiom,
! [A_91: ( hoare_2091234717iple_a > $o ) > $o] :
( ! [X: hoare_2091234717iple_a > $o] :
~ ( member99268621le_a_o @ X @ A_91 )
<=> ( A_91 = bot_bo1957696069_a_o_o ) ) ).
thf(fact_324_all__not__in__conv,axiom,
! [A_91: hoare_2091234717iple_a > $o] :
( ! [X: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ X @ A_91 )
<=> ( A_91 = bot_bo1791335050le_a_o ) ) ).
thf(fact_325_all__not__in__conv,axiom,
! [A_91: hoare_1708887482_state > $o] :
( ! [X: hoare_1708887482_state] :
~ ( member451959335_state @ X @ A_91 )
<=> ( A_91 = bot_bo19817387tate_o ) ) ).
thf(fact_326_all__not__in__conv,axiom,
! [A_91: pname > $o] :
( ! [X: pname] :
~ ( member_pname @ X @ A_91 )
<=> ( A_91 = bot_bot_pname_o ) ) ).
thf(fact_327_singleton__conv2,axiom,
! [A_90: hoare_2091234717iple_a > $o] :
( ( collec1008234059le_a_o @ ( fequal845167073le_a_o @ A_90 ) )
= ( insert102003750le_a_o @ A_90 @ bot_bo1957696069_a_o_o ) ) ).
thf(fact_328_singleton__conv2,axiom,
! [A_90: pname] :
( ( collect_pname @ ( fequal_pname @ A_90 ) )
= ( insert_pname @ A_90 @ bot_bot_pname_o ) ) ).
thf(fact_329_singleton__conv2,axiom,
! [A_90: hoare_2091234717iple_a] :
( ( collec992574898iple_a @ ( fequal1604381340iple_a @ A_90 ) )
= ( insert1597628439iple_a @ A_90 @ bot_bo1791335050le_a_o ) ) ).
thf(fact_330_singleton__conv2,axiom,
! [A_90: hoare_1708887482_state] :
( ( collec1568722789_state @ ( fequal224822779_state @ A_90 ) )
= ( insert528405184_state @ A_90 @ bot_bo19817387tate_o ) ) ).
thf(fact_331_singleton__conv2,axiom,
! [A_90: nat] :
( ( collect_nat @ ( fequal_nat @ A_90 ) )
= ( insert_nat @ A_90 @ bot_bot_nat_o ) ) ).
thf(fact_332_ex__in__conv,axiom,
! [A_89: nat > $o] :
( ? [X: nat] : ( member_nat @ X @ A_89 )
<=> ( A_89 != bot_bot_nat_o ) ) ).
thf(fact_333_ex__in__conv,axiom,
! [A_89: ( hoare_2091234717iple_a > $o ) > $o] :
( ? [X: hoare_2091234717iple_a > $o] : ( member99268621le_a_o @ X @ A_89 )
<=> ( A_89 != bot_bo1957696069_a_o_o ) ) ).
thf(fact_334_ex__in__conv,axiom,
! [A_89: hoare_2091234717iple_a > $o] :
( ? [X: hoare_2091234717iple_a] : ( member290856304iple_a @ X @ A_89 )
<=> ( A_89 != bot_bo1791335050le_a_o ) ) ).
thf(fact_335_ex__in__conv,axiom,
! [A_89: hoare_1708887482_state > $o] :
( ? [X: hoare_1708887482_state] : ( member451959335_state @ X @ A_89 )
<=> ( A_89 != bot_bo19817387tate_o ) ) ).
thf(fact_336_ex__in__conv,axiom,
! [A_89: pname > $o] :
( ? [X: pname] : ( member_pname @ X @ A_89 )
<=> ( A_89 != bot_bot_pname_o ) ) ).
thf(fact_337_singleton__conv,axiom,
! [A_88: hoare_2091234717iple_a > $o] :
( ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : X = A_88 )
= ( insert102003750le_a_o @ A_88 @ bot_bo1957696069_a_o_o ) ) ).
thf(fact_338_singleton__conv,axiom,
! [A_88: pname] :
( ( collect_pname
@ ^ [X: pname] : X = A_88 )
= ( insert_pname @ A_88 @ bot_bot_pname_o ) ) ).
thf(fact_339_singleton__conv,axiom,
! [A_88: hoare_2091234717iple_a] :
( ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : X = A_88 )
= ( insert1597628439iple_a @ A_88 @ bot_bo1791335050le_a_o ) ) ).
thf(fact_340_singleton__conv,axiom,
! [A_88: hoare_1708887482_state] :
( ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : X = A_88 )
= ( insert528405184_state @ A_88 @ bot_bo19817387tate_o ) ) ).
thf(fact_341_singleton__conv,axiom,
! [A_88: nat] :
( ( collect_nat
@ ^ [X: nat] : X = A_88 )
= ( insert_nat @ A_88 @ bot_bot_nat_o ) ) ).
thf(fact_342_Collect__conv__if2,axiom,
! [P_29: ( hoare_2091234717iple_a > $o ) > $o,A_87: hoare_2091234717iple_a > $o] :
( ( ( P_29 @ A_87 )
=> ( ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= ( insert102003750le_a_o @ A_87 @ bot_bo1957696069_a_o_o ) ) )
& ( ~ ( P_29 @ A_87 )
=> ( ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= bot_bo1957696069_a_o_o ) ) ) ).
thf(fact_343_Collect__conv__if2,axiom,
! [P_29: pname > $o,A_87: pname] :
( ( ( P_29 @ A_87 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= ( insert_pname @ A_87 @ bot_bot_pname_o ) ) )
& ( ~ ( P_29 @ A_87 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_344_Collect__conv__if2,axiom,
! [P_29: hoare_2091234717iple_a > $o,A_87: hoare_2091234717iple_a] :
( ( ( P_29 @ A_87 )
=> ( ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= ( insert1597628439iple_a @ A_87 @ bot_bo1791335050le_a_o ) ) )
& ( ~ ( P_29 @ A_87 )
=> ( ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= bot_bo1791335050le_a_o ) ) ) ).
thf(fact_345_Collect__conv__if2,axiom,
! [P_29: hoare_1708887482_state > $o,A_87: hoare_1708887482_state] :
( ( ( P_29 @ A_87 )
=> ( ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= ( insert528405184_state @ A_87 @ bot_bo19817387tate_o ) ) )
& ( ~ ( P_29 @ A_87 )
=> ( ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= bot_bo19817387tate_o ) ) ) ).
thf(fact_346_Collect__conv__if2,axiom,
! [P_29: nat > $o,A_87: nat] :
( ( ( P_29 @ A_87 )
=> ( ( collect_nat
@ ^ [X: nat] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= ( insert_nat @ A_87 @ bot_bot_nat_o ) ) )
& ( ~ ( P_29 @ A_87 )
=> ( ( collect_nat
@ ^ [X: nat] : ( (&) @ ( A_87 = X ) @ ( P_29 @ X ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_347_Collect__conv__if,axiom,
! [P_28: ( hoare_2091234717iple_a > $o ) > $o,A_86: hoare_2091234717iple_a > $o] :
( ( ( P_28 @ A_86 )
=> ( ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= ( insert102003750le_a_o @ A_86 @ bot_bo1957696069_a_o_o ) ) )
& ( ~ ( P_28 @ A_86 )
=> ( ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= bot_bo1957696069_a_o_o ) ) ) ).
thf(fact_348_Collect__conv__if,axiom,
! [P_28: pname > $o,A_86: pname] :
( ( ( P_28 @ A_86 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= ( insert_pname @ A_86 @ bot_bot_pname_o ) ) )
& ( ~ ( P_28 @ A_86 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_349_Collect__conv__if,axiom,
! [P_28: hoare_2091234717iple_a > $o,A_86: hoare_2091234717iple_a] :
( ( ( P_28 @ A_86 )
=> ( ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= ( insert1597628439iple_a @ A_86 @ bot_bo1791335050le_a_o ) ) )
& ( ~ ( P_28 @ A_86 )
=> ( ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= bot_bo1791335050le_a_o ) ) ) ).
thf(fact_350_Collect__conv__if,axiom,
! [P_28: hoare_1708887482_state > $o,A_86: hoare_1708887482_state] :
( ( ( P_28 @ A_86 )
=> ( ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= ( insert528405184_state @ A_86 @ bot_bo19817387tate_o ) ) )
& ( ~ ( P_28 @ A_86 )
=> ( ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= bot_bo19817387tate_o ) ) ) ).
thf(fact_351_Collect__conv__if,axiom,
! [P_28: nat > $o,A_86: nat] :
( ( ( P_28 @ A_86 )
=> ( ( collect_nat
@ ^ [X: nat] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= ( insert_nat @ A_86 @ bot_bot_nat_o ) ) )
& ( ~ ( P_28 @ A_86 )
=> ( ( collect_nat
@ ^ [X: nat] : ( (&) @ ( X = A_86 ) @ ( P_28 @ X ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_352_mem__def,axiom,
! [X_33: nat,A_85: nat > $o] :
( ( member_nat @ X_33 @ A_85 )
<=> ( A_85 @ X_33 ) ) ).
thf(fact_353_mem__def,axiom,
! [X_33: hoare_2091234717iple_a > $o,A_85: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ X_33 @ A_85 )
<=> ( A_85 @ X_33 ) ) ).
thf(fact_354_mem__def,axiom,
! [X_33: hoare_2091234717iple_a,A_85: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_33 @ A_85 )
<=> ( A_85 @ X_33 ) ) ).
thf(fact_355_mem__def,axiom,
! [X_33: pname,A_85: pname > $o] :
( ( member_pname @ X_33 @ A_85 )
<=> ( A_85 @ X_33 ) ) ).
thf(fact_356_Collect__def,axiom,
! [P_27: pname > $o] :
( ( collect_pname @ P_27 )
= P_27 ) ).
thf(fact_357_Collect__def,axiom,
! [P_27: hoare_2091234717iple_a > $o] :
( ( collec992574898iple_a @ P_27 )
= P_27 ) ).
thf(fact_358_Collect__def,axiom,
! [P_27: nat > $o] :
( ( collect_nat @ P_27 )
= P_27 ) ).
thf(fact_359_empty__Collect__eq,axiom,
! [P_26: pname > $o] :
( ( bot_bot_pname_o
= ( collect_pname @ P_26 ) )
<=> ! [X: pname] :
~ ( P_26 @ X ) ) ).
thf(fact_360_empty__Collect__eq,axiom,
! [P_26: hoare_2091234717iple_a > $o] :
( ( bot_bo1791335050le_a_o
= ( collec992574898iple_a @ P_26 ) )
<=> ! [X: hoare_2091234717iple_a] :
~ ( P_26 @ X ) ) ).
thf(fact_361_empty__Collect__eq,axiom,
! [P_26: ( hoare_2091234717iple_a > $o ) > $o] :
( ( bot_bo1957696069_a_o_o
= ( collec1008234059le_a_o @ P_26 ) )
<=> ! [X: hoare_2091234717iple_a > $o] :
~ ( P_26 @ X ) ) ).
thf(fact_362_empty__Collect__eq,axiom,
! [P_26: hoare_1708887482_state > $o] :
( ( bot_bo19817387tate_o
= ( collec1568722789_state @ P_26 ) )
<=> ! [X: hoare_1708887482_state] :
~ ( P_26 @ X ) ) ).
thf(fact_363_empty__Collect__eq,axiom,
! [P_26: nat > $o] :
( ( bot_bot_nat_o
= ( collect_nat @ P_26 ) )
<=> ! [X: nat] :
~ ( P_26 @ X ) ) ).
thf(fact_364_empty__iff,axiom,
! [C_25: nat] :
~ ( member_nat @ C_25 @ bot_bot_nat_o ) ).
thf(fact_365_empty__iff,axiom,
! [C_25: hoare_2091234717iple_a > $o] :
~ ( member99268621le_a_o @ C_25 @ bot_bo1957696069_a_o_o ) ).
thf(fact_366_empty__iff,axiom,
! [C_25: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ C_25 @ bot_bo1791335050le_a_o ) ).
thf(fact_367_empty__iff,axiom,
! [C_25: hoare_1708887482_state] :
~ ( member451959335_state @ C_25 @ bot_bo19817387tate_o ) ).
thf(fact_368_empty__iff,axiom,
! [C_25: pname] :
~ ( member_pname @ C_25 @ bot_bot_pname_o ) ).
thf(fact_369_insert__compr,axiom,
! [A_84: hoare_2091234717iple_a > $o,B_43: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ A_84 @ B_43 )
= ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (|) @ ( X = A_84 ) @ ( member99268621le_a_o @ X @ B_43 ) ) ) ) ).
thf(fact_370_insert__compr,axiom,
! [A_84: hoare_2091234717iple_a,B_43: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_84 @ B_43 )
= ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (|) @ ( X = A_84 ) @ ( member290856304iple_a @ X @ B_43 ) ) ) ) ).
thf(fact_371_insert__compr,axiom,
! [A_84: hoare_1708887482_state,B_43: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_84 @ B_43 )
= ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : ( (|) @ ( X = A_84 ) @ ( member451959335_state @ X @ B_43 ) ) ) ) ).
thf(fact_372_insert__compr,axiom,
! [A_84: nat,B_43: nat > $o] :
( ( insert_nat @ A_84 @ B_43 )
= ( collect_nat
@ ^ [X: nat] : ( (|) @ ( X = A_84 ) @ ( member_nat @ X @ B_43 ) ) ) ) ).
thf(fact_373_insert__compr,axiom,
! [A_84: pname,B_43: pname > $o] :
( ( insert_pname @ A_84 @ B_43 )
= ( collect_pname
@ ^ [X: pname] : ( (|) @ ( X = A_84 ) @ ( member_pname @ X @ B_43 ) ) ) ) ).
thf(fact_374_insert__Collect,axiom,
! [A_83: hoare_2091234717iple_a > $o,P_25: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ A_83 @ ( collec1008234059le_a_o @ P_25 ) )
= ( collec1008234059le_a_o
@ ^ [U_2: hoare_2091234717iple_a > $o] : ( (=>) @ ( (~) @ ( U_2 = A_83 ) ) @ ( P_25 @ U_2 ) ) ) ) ).
thf(fact_375_insert__Collect,axiom,
! [A_83: pname,P_25: pname > $o] :
( ( insert_pname @ A_83 @ ( collect_pname @ P_25 ) )
= ( collect_pname
@ ^ [U_2: pname] : ( (=>) @ ( (~) @ ( U_2 = A_83 ) ) @ ( P_25 @ U_2 ) ) ) ) ).
thf(fact_376_insert__Collect,axiom,
! [A_83: hoare_2091234717iple_a,P_25: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_83 @ ( collec992574898iple_a @ P_25 ) )
= ( collec992574898iple_a
@ ^ [U_2: hoare_2091234717iple_a] : ( (=>) @ ( (~) @ ( U_2 = A_83 ) ) @ ( P_25 @ U_2 ) ) ) ) ).
thf(fact_377_insert__Collect,axiom,
! [A_83: hoare_1708887482_state,P_25: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_83 @ ( collec1568722789_state @ P_25 ) )
= ( collec1568722789_state
@ ^ [U_2: hoare_1708887482_state] : ( (=>) @ ( (~) @ ( U_2 = A_83 ) ) @ ( P_25 @ U_2 ) ) ) ) ).
thf(fact_378_insert__Collect,axiom,
! [A_83: nat,P_25: nat > $o] :
( ( insert_nat @ A_83 @ ( collect_nat @ P_25 ) )
= ( collect_nat
@ ^ [U_2: nat] : ( (=>) @ ( (~) @ ( U_2 = A_83 ) ) @ ( P_25 @ U_2 ) ) ) ) ).
thf(fact_379_singleton__iff,axiom,
! [B_42: nat,A_82: nat] :
( ( member_nat @ B_42 @ ( insert_nat @ A_82 @ bot_bot_nat_o ) )
<=> ( B_42 = A_82 ) ) ).
thf(fact_380_singleton__iff,axiom,
! [B_42: hoare_2091234717iple_a > $o,A_82: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ B_42 @ ( insert102003750le_a_o @ A_82 @ bot_bo1957696069_a_o_o ) )
<=> ( B_42 = A_82 ) ) ).
thf(fact_381_singleton__iff,axiom,
! [B_42: hoare_2091234717iple_a,A_82: hoare_2091234717iple_a] :
( ( member290856304iple_a @ B_42 @ ( insert1597628439iple_a @ A_82 @ bot_bo1791335050le_a_o ) )
<=> ( B_42 = A_82 ) ) ).
thf(fact_382_singleton__iff,axiom,
! [B_42: hoare_1708887482_state,A_82: hoare_1708887482_state] :
( ( member451959335_state @ B_42 @ ( insert528405184_state @ A_82 @ bot_bo19817387tate_o ) )
<=> ( B_42 = A_82 ) ) ).
thf(fact_383_singleton__iff,axiom,
! [B_42: pname,A_82: pname] :
( ( member_pname @ B_42 @ ( insert_pname @ A_82 @ bot_bot_pname_o ) )
<=> ( B_42 = A_82 ) ) ).
thf(fact_384_insert__absorb2,axiom,
! [X_32: nat,A_81: nat > $o] :
( ( insert_nat @ X_32 @ ( insert_nat @ X_32 @ A_81 ) )
= ( insert_nat @ X_32 @ A_81 ) ) ).
thf(fact_385_insert__absorb2,axiom,
! [X_32: hoare_2091234717iple_a > $o,A_81: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ X_32 @ ( insert102003750le_a_o @ X_32 @ A_81 ) )
= ( insert102003750le_a_o @ X_32 @ A_81 ) ) ).
thf(fact_386_insert__absorb2,axiom,
! [X_32: pname,A_81: pname > $o] :
( ( insert_pname @ X_32 @ ( insert_pname @ X_32 @ A_81 ) )
= ( insert_pname @ X_32 @ A_81 ) ) ).
thf(fact_387_insert__absorb2,axiom,
! [X_32: hoare_2091234717iple_a,A_81: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ X_32 @ ( insert1597628439iple_a @ X_32 @ A_81 ) )
= ( insert1597628439iple_a @ X_32 @ A_81 ) ) ).
thf(fact_388_insert__absorb2,axiom,
! [X_32: hoare_1708887482_state,A_81: hoare_1708887482_state > $o] :
( ( insert528405184_state @ X_32 @ ( insert528405184_state @ X_32 @ A_81 ) )
= ( insert528405184_state @ X_32 @ A_81 ) ) ).
thf(fact_389_insert__commute,axiom,
! [X_31: nat,Y_10: nat,A_80: nat > $o] :
( ( insert_nat @ X_31 @ ( insert_nat @ Y_10 @ A_80 ) )
= ( insert_nat @ Y_10 @ ( insert_nat @ X_31 @ A_80 ) ) ) ).
thf(fact_390_insert__commute,axiom,
! [X_31: hoare_2091234717iple_a > $o,Y_10: hoare_2091234717iple_a > $o,A_80: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ X_31 @ ( insert102003750le_a_o @ Y_10 @ A_80 ) )
= ( insert102003750le_a_o @ Y_10 @ ( insert102003750le_a_o @ X_31 @ A_80 ) ) ) ).
thf(fact_391_insert__commute,axiom,
! [X_31: pname,Y_10: pname,A_80: pname > $o] :
( ( insert_pname @ X_31 @ ( insert_pname @ Y_10 @ A_80 ) )
= ( insert_pname @ Y_10 @ ( insert_pname @ X_31 @ A_80 ) ) ) ).
thf(fact_392_insert__commute,axiom,
! [X_31: hoare_2091234717iple_a,Y_10: hoare_2091234717iple_a,A_80: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ X_31 @ ( insert1597628439iple_a @ Y_10 @ A_80 ) )
= ( insert1597628439iple_a @ Y_10 @ ( insert1597628439iple_a @ X_31 @ A_80 ) ) ) ).
thf(fact_393_insert__commute,axiom,
! [X_31: hoare_1708887482_state,Y_10: hoare_1708887482_state,A_80: hoare_1708887482_state > $o] :
( ( insert528405184_state @ X_31 @ ( insert528405184_state @ Y_10 @ A_80 ) )
= ( insert528405184_state @ Y_10 @ ( insert528405184_state @ X_31 @ A_80 ) ) ) ).
thf(fact_394_insert__iff,axiom,
! [A_79: nat,B_41: nat,A_78: nat > $o] :
( ( member_nat @ A_79 @ ( insert_nat @ B_41 @ A_78 ) )
<=> ( ( A_79 = B_41 )
| ( member_nat @ A_79 @ A_78 ) ) ) ).
thf(fact_395_insert__iff,axiom,
! [A_79: hoare_2091234717iple_a > $o,B_41: hoare_2091234717iple_a > $o,A_78: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ A_79 @ ( insert102003750le_a_o @ B_41 @ A_78 ) )
<=> ( ( A_79 = B_41 )
| ( member99268621le_a_o @ A_79 @ A_78 ) ) ) ).
thf(fact_396_insert__iff,axiom,
! [A_79: hoare_2091234717iple_a,B_41: hoare_2091234717iple_a,A_78: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_79 @ ( insert1597628439iple_a @ B_41 @ A_78 ) )
<=> ( ( A_79 = B_41 )
| ( member290856304iple_a @ A_79 @ A_78 ) ) ) ).
thf(fact_397_insert__iff,axiom,
! [A_79: hoare_1708887482_state,B_41: hoare_1708887482_state,A_78: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_79 @ ( insert528405184_state @ B_41 @ A_78 ) )
<=> ( ( A_79 = B_41 )
| ( member451959335_state @ A_79 @ A_78 ) ) ) ).
thf(fact_398_insert__iff,axiom,
! [A_79: pname,B_41: pname,A_78: pname > $o] :
( ( member_pname @ A_79 @ ( insert_pname @ B_41 @ A_78 ) )
<=> ( ( A_79 = B_41 )
| ( member_pname @ A_79 @ A_78 ) ) ) ).
thf(fact_399_Collect__empty__eq,axiom,
! [P_24: pname > $o] :
( ( ( collect_pname @ P_24 )
= bot_bot_pname_o )
<=> ! [X: pname] :
~ ( P_24 @ X ) ) ).
thf(fact_400_Collect__empty__eq,axiom,
! [P_24: hoare_2091234717iple_a > $o] :
( ( ( collec992574898iple_a @ P_24 )
= bot_bo1791335050le_a_o )
<=> ! [X: hoare_2091234717iple_a] :
~ ( P_24 @ X ) ) ).
thf(fact_401_Collect__empty__eq,axiom,
! [P_24: ( hoare_2091234717iple_a > $o ) > $o] :
( ( ( collec1008234059le_a_o @ P_24 )
= bot_bo1957696069_a_o_o )
<=> ! [X: hoare_2091234717iple_a > $o] :
~ ( P_24 @ X ) ) ).
thf(fact_402_Collect__empty__eq,axiom,
! [P_24: hoare_1708887482_state > $o] :
( ( ( collec1568722789_state @ P_24 )
= bot_bo19817387tate_o )
<=> ! [X: hoare_1708887482_state] :
~ ( P_24 @ X ) ) ).
thf(fact_403_Collect__empty__eq,axiom,
! [P_24: nat > $o] :
( ( ( collect_nat @ P_24 )
= bot_bot_nat_o )
<=> ! [X: nat] :
~ ( P_24 @ X ) ) ).
thf(fact_404_doubleton__eq__iff,axiom,
! [A_77: nat,B_40: nat,C_24: nat,D_1: nat] :
( ( ( insert_nat @ A_77 @ ( insert_nat @ B_40 @ bot_bot_nat_o ) )
= ( insert_nat @ C_24 @ ( insert_nat @ D_1 @ bot_bot_nat_o ) ) )
<=> ( ( ( A_77 = C_24 )
& ( B_40 = D_1 ) )
| ( ( A_77 = D_1 )
& ( B_40 = C_24 ) ) ) ) ).
thf(fact_405_doubleton__eq__iff,axiom,
! [A_77: hoare_2091234717iple_a > $o,B_40: hoare_2091234717iple_a > $o,C_24: hoare_2091234717iple_a > $o,D_1: hoare_2091234717iple_a > $o] :
( ( ( insert102003750le_a_o @ A_77 @ ( insert102003750le_a_o @ B_40 @ bot_bo1957696069_a_o_o ) )
= ( insert102003750le_a_o @ C_24 @ ( insert102003750le_a_o @ D_1 @ bot_bo1957696069_a_o_o ) ) )
<=> ( ( ( A_77 = C_24 )
& ( B_40 = D_1 ) )
| ( ( A_77 = D_1 )
& ( B_40 = C_24 ) ) ) ) ).
thf(fact_406_doubleton__eq__iff,axiom,
! [A_77: pname,B_40: pname,C_24: pname,D_1: pname] :
( ( ( insert_pname @ A_77 @ ( insert_pname @ B_40 @ bot_bot_pname_o ) )
= ( insert_pname @ C_24 @ ( insert_pname @ D_1 @ bot_bot_pname_o ) ) )
<=> ( ( ( A_77 = C_24 )
& ( B_40 = D_1 ) )
| ( ( A_77 = D_1 )
& ( B_40 = C_24 ) ) ) ) ).
thf(fact_407_doubleton__eq__iff,axiom,
! [A_77: hoare_2091234717iple_a,B_40: hoare_2091234717iple_a,C_24: hoare_2091234717iple_a,D_1: hoare_2091234717iple_a] :
( ( ( insert1597628439iple_a @ A_77 @ ( insert1597628439iple_a @ B_40 @ bot_bo1791335050le_a_o ) )
= ( insert1597628439iple_a @ C_24 @ ( insert1597628439iple_a @ D_1 @ bot_bo1791335050le_a_o ) ) )
<=> ( ( ( A_77 = C_24 )
& ( B_40 = D_1 ) )
| ( ( A_77 = D_1 )
& ( B_40 = C_24 ) ) ) ) ).
thf(fact_408_doubleton__eq__iff,axiom,
! [A_77: hoare_1708887482_state,B_40: hoare_1708887482_state,C_24: hoare_1708887482_state,D_1: hoare_1708887482_state] :
( ( ( insert528405184_state @ A_77 @ ( insert528405184_state @ B_40 @ bot_bo19817387tate_o ) )
= ( insert528405184_state @ C_24 @ ( insert528405184_state @ D_1 @ bot_bo19817387tate_o ) ) )
<=> ( ( ( A_77 = C_24 )
& ( B_40 = D_1 ) )
| ( ( A_77 = D_1 )
& ( B_40 = C_24 ) ) ) ) ).
thf(fact_409_insert__code,axiom,
! [Y_9: nat,A_76: nat > $o,X_30: nat] :
( ( insert_nat @ Y_9 @ A_76 @ X_30 )
<=> ( ( Y_9 = X_30 )
| ( A_76 @ X_30 ) ) ) ).
thf(fact_410_insert__code,axiom,
! [Y_9: hoare_2091234717iple_a > $o,A_76: ( hoare_2091234717iple_a > $o ) > $o,X_30: hoare_2091234717iple_a > $o] :
( ( insert102003750le_a_o @ Y_9 @ A_76 @ X_30 )
<=> ( ( Y_9 = X_30 )
| ( A_76 @ X_30 ) ) ) ).
thf(fact_411_insert__code,axiom,
! [Y_9: pname,A_76: pname > $o,X_30: pname] :
( ( insert_pname @ Y_9 @ A_76 @ X_30 )
<=> ( ( Y_9 = X_30 )
| ( A_76 @ X_30 ) ) ) ).
thf(fact_412_insert__code,axiom,
! [Y_9: hoare_2091234717iple_a,A_76: hoare_2091234717iple_a > $o,X_30: hoare_2091234717iple_a] :
( ( insert1597628439iple_a @ Y_9 @ A_76 @ X_30 )
<=> ( ( Y_9 = X_30 )
| ( A_76 @ X_30 ) ) ) ).
thf(fact_413_insert__code,axiom,
! [Y_9: hoare_1708887482_state,A_76: hoare_1708887482_state > $o,X_30: hoare_1708887482_state] :
( ( insert528405184_state @ Y_9 @ A_76 @ X_30 )
<=> ( ( Y_9 = X_30 )
| ( A_76 @ X_30 ) ) ) ).
thf(fact_414_insert__ident,axiom,
! [B_39: nat > $o,X_29: nat,A_75: nat > $o] :
( ~ ( member_nat @ X_29 @ A_75 )
=> ( ~ ( member_nat @ X_29 @ B_39 )
=> ( ( ( insert_nat @ X_29 @ A_75 )
= ( insert_nat @ X_29 @ B_39 ) )
<=> ( A_75 = B_39 ) ) ) ) ).
thf(fact_415_insert__ident,axiom,
! [B_39: ( hoare_2091234717iple_a > $o ) > $o,X_29: hoare_2091234717iple_a > $o,A_75: ( hoare_2091234717iple_a > $o ) > $o] :
( ~ ( member99268621le_a_o @ X_29 @ A_75 )
=> ( ~ ( member99268621le_a_o @ X_29 @ B_39 )
=> ( ( ( insert102003750le_a_o @ X_29 @ A_75 )
= ( insert102003750le_a_o @ X_29 @ B_39 ) )
<=> ( A_75 = B_39 ) ) ) ) ).
thf(fact_416_insert__ident,axiom,
! [B_39: hoare_2091234717iple_a > $o,X_29: hoare_2091234717iple_a,A_75: hoare_2091234717iple_a > $o] :
( ~ ( member290856304iple_a @ X_29 @ A_75 )
=> ( ~ ( member290856304iple_a @ X_29 @ B_39 )
=> ( ( ( insert1597628439iple_a @ X_29 @ A_75 )
= ( insert1597628439iple_a @ X_29 @ B_39 ) )
<=> ( A_75 = B_39 ) ) ) ) ).
thf(fact_417_insert__ident,axiom,
! [B_39: hoare_1708887482_state > $o,X_29: hoare_1708887482_state,A_75: hoare_1708887482_state > $o] :
( ~ ( member451959335_state @ X_29 @ A_75 )
=> ( ~ ( member451959335_state @ X_29 @ B_39 )
=> ( ( ( insert528405184_state @ X_29 @ A_75 )
= ( insert528405184_state @ X_29 @ B_39 ) )
<=> ( A_75 = B_39 ) ) ) ) ).
thf(fact_418_insert__ident,axiom,
! [B_39: pname > $o,X_29: pname,A_75: pname > $o] :
( ~ ( member_pname @ X_29 @ A_75 )
=> ( ~ ( member_pname @ X_29 @ B_39 )
=> ( ( ( insert_pname @ X_29 @ A_75 )
= ( insert_pname @ X_29 @ B_39 ) )
<=> ( A_75 = B_39 ) ) ) ) ).
thf(fact_419_equals0D,axiom,
! [A_74: nat,A_73: nat > $o] :
( ( A_73 = bot_bot_nat_o )
=> ~ ( member_nat @ A_74 @ A_73 ) ) ).
thf(fact_420_equals0D,axiom,
! [A_74: hoare_2091234717iple_a > $o,A_73: ( hoare_2091234717iple_a > $o ) > $o] :
( ( A_73 = bot_bo1957696069_a_o_o )
=> ~ ( member99268621le_a_o @ A_74 @ A_73 ) ) ).
thf(fact_421_equals0D,axiom,
! [A_74: hoare_2091234717iple_a,A_73: hoare_2091234717iple_a > $o] :
( ( A_73 = bot_bo1791335050le_a_o )
=> ~ ( member290856304iple_a @ A_74 @ A_73 ) ) ).
thf(fact_422_equals0D,axiom,
! [A_74: hoare_1708887482_state,A_73: hoare_1708887482_state > $o] :
( ( A_73 = bot_bo19817387tate_o )
=> ~ ( member451959335_state @ A_74 @ A_73 ) ) ).
thf(fact_423_equals0D,axiom,
! [A_74: pname,A_73: pname > $o] :
( ( A_73 = bot_bot_pname_o )
=> ~ ( member_pname @ A_74 @ A_73 ) ) ).
thf(fact_424_insertI2,axiom,
! [B_38: nat,A_72: nat,B_37: nat > $o] :
( ( member_nat @ A_72 @ B_37 )
=> ( member_nat @ A_72 @ ( insert_nat @ B_38 @ B_37 ) ) ) ).
thf(fact_425_insertI2,axiom,
! [B_38: hoare_2091234717iple_a > $o,A_72: hoare_2091234717iple_a > $o,B_37: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ A_72 @ B_37 )
=> ( member99268621le_a_o @ A_72 @ ( insert102003750le_a_o @ B_38 @ B_37 ) ) ) ).
thf(fact_426_insertI2,axiom,
! [B_38: hoare_2091234717iple_a,A_72: hoare_2091234717iple_a,B_37: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_72 @ B_37 )
=> ( member290856304iple_a @ A_72 @ ( insert1597628439iple_a @ B_38 @ B_37 ) ) ) ).
thf(fact_427_insertI2,axiom,
! [B_38: hoare_1708887482_state,A_72: hoare_1708887482_state,B_37: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_72 @ B_37 )
=> ( member451959335_state @ A_72 @ ( insert528405184_state @ B_38 @ B_37 ) ) ) ).
thf(fact_428_insertI2,axiom,
! [B_38: pname,A_72: pname,B_37: pname > $o] :
( ( member_pname @ A_72 @ B_37 )
=> ( member_pname @ A_72 @ ( insert_pname @ B_38 @ B_37 ) ) ) ).
thf(fact_429_insert__absorb,axiom,
! [A_71: nat,A_70: nat > $o] :
( ( member_nat @ A_71 @ A_70 )
=> ( ( insert_nat @ A_71 @ A_70 )
= A_70 ) ) ).
thf(fact_430_insert__absorb,axiom,
! [A_71: hoare_2091234717iple_a > $o,A_70: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ A_71 @ A_70 )
=> ( ( insert102003750le_a_o @ A_71 @ A_70 )
= A_70 ) ) ).
thf(fact_431_insert__absorb,axiom,
! [A_71: hoare_2091234717iple_a,A_70: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_71 @ A_70 )
=> ( ( insert1597628439iple_a @ A_71 @ A_70 )
= A_70 ) ) ).
thf(fact_432_insert__absorb,axiom,
! [A_71: hoare_1708887482_state,A_70: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_71 @ A_70 )
=> ( ( insert528405184_state @ A_71 @ A_70 )
= A_70 ) ) ).
thf(fact_433_insert__absorb,axiom,
! [A_71: pname,A_70: pname > $o] :
( ( member_pname @ A_71 @ A_70 )
=> ( ( insert_pname @ A_71 @ A_70 )
= A_70 ) ) ).
thf(fact_434_singletonE,axiom,
! [B_36: nat,A_69: nat] :
( ( member_nat @ B_36 @ ( insert_nat @ A_69 @ bot_bot_nat_o ) )
=> ( B_36 = A_69 ) ) ).
thf(fact_435_singletonE,axiom,
! [B_36: hoare_2091234717iple_a > $o,A_69: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ B_36 @ ( insert102003750le_a_o @ A_69 @ bot_bo1957696069_a_o_o ) )
=> ( B_36 = A_69 ) ) ).
thf(fact_436_singletonE,axiom,
! [B_36: hoare_2091234717iple_a,A_69: hoare_2091234717iple_a] :
( ( member290856304iple_a @ B_36 @ ( insert1597628439iple_a @ A_69 @ bot_bo1791335050le_a_o ) )
=> ( B_36 = A_69 ) ) ).
thf(fact_437_singletonE,axiom,
! [B_36: hoare_1708887482_state,A_69: hoare_1708887482_state] :
( ( member451959335_state @ B_36 @ ( insert528405184_state @ A_69 @ bot_bo19817387tate_o ) )
=> ( B_36 = A_69 ) ) ).
thf(fact_438_singletonE,axiom,
! [B_36: pname,A_69: pname] :
( ( member_pname @ B_36 @ ( insert_pname @ A_69 @ bot_bot_pname_o ) )
=> ( B_36 = A_69 ) ) ).
thf(fact_439_singleton__inject,axiom,
! [A_68: nat,B_35: nat] :
( ( ( insert_nat @ A_68 @ bot_bot_nat_o )
= ( insert_nat @ B_35 @ bot_bot_nat_o ) )
=> ( A_68 = B_35 ) ) ).
thf(fact_440_singleton__inject,axiom,
! [A_68: hoare_2091234717iple_a > $o,B_35: hoare_2091234717iple_a > $o] :
( ( ( insert102003750le_a_o @ A_68 @ bot_bo1957696069_a_o_o )
= ( insert102003750le_a_o @ B_35 @ bot_bo1957696069_a_o_o ) )
=> ( A_68 = B_35 ) ) ).
thf(fact_441_singleton__inject,axiom,
! [A_68: pname,B_35: pname] :
( ( ( insert_pname @ A_68 @ bot_bot_pname_o )
= ( insert_pname @ B_35 @ bot_bot_pname_o ) )
=> ( A_68 = B_35 ) ) ).
thf(fact_442_singleton__inject,axiom,
! [A_68: hoare_2091234717iple_a,B_35: hoare_2091234717iple_a] :
( ( ( insert1597628439iple_a @ A_68 @ bot_bo1791335050le_a_o )
= ( insert1597628439iple_a @ B_35 @ bot_bo1791335050le_a_o ) )
=> ( A_68 = B_35 ) ) ).
thf(fact_443_singleton__inject,axiom,
! [A_68: hoare_1708887482_state,B_35: hoare_1708887482_state] :
( ( ( insert528405184_state @ A_68 @ bot_bo19817387tate_o )
= ( insert528405184_state @ B_35 @ bot_bo19817387tate_o ) )
=> ( A_68 = B_35 ) ) ).
thf(fact_444_com__det,axiom,
! [U: state,C: com,S: state,T: state] :
( ( evalc @ C @ S @ T )
=> ( ( evalc @ C @ S @ U )
=> ( U = T ) ) ) ).
thf(fact_445_insert__is__Un,axiom,
! [A_67: nat,A_66: nat > $o] :
( ( insert_nat @ A_67 @ A_66 )
= ( semila848761471_nat_o @ ( insert_nat @ A_67 @ bot_bot_nat_o ) @ A_66 ) ) ).
thf(fact_446_insert__is__Un,axiom,
! [A_67: hoare_2091234717iple_a > $o,A_66: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ A_67 @ A_66 )
= ( semila2050116131_a_o_o @ ( insert102003750le_a_o @ A_67 @ bot_bo1957696069_a_o_o ) @ A_66 ) ) ).
thf(fact_447_insert__is__Un,axiom,
! [A_67: pname,A_66: pname > $o] :
( ( insert_pname @ A_67 @ A_66 )
= ( semila1780557381name_o @ ( insert_pname @ A_67 @ bot_bot_pname_o ) @ A_66 ) ) ).
thf(fact_448_insert__is__Un,axiom,
! [A_67: hoare_1708887482_state,A_66: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_67 @ A_66 )
= ( semila1122118281tate_o @ ( insert528405184_state @ A_67 @ bot_bo19817387tate_o ) @ A_66 ) ) ).
thf(fact_449_insert__is__Un,axiom,
! [A_67: hoare_2091234717iple_a,A_66: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_67 @ A_66 )
= ( semila1052848428le_a_o @ ( insert1597628439iple_a @ A_67 @ bot_bo1791335050le_a_o ) @ A_66 ) ) ).
thf(fact_450_insert__compr__raw,axiom,
! [X: hoare_2091234717iple_a > $o,Xa: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ X @ Xa )
= ( collec1008234059le_a_o
@ ^ [Y_7: hoare_2091234717iple_a > $o] : ( (|) @ ( Y_7 = X ) @ ( member99268621le_a_o @ Y_7 @ Xa ) ) ) ) ).
thf(fact_451_insert__compr__raw,axiom,
! [X: hoare_2091234717iple_a,Xa: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ X @ Xa )
= ( collec992574898iple_a
@ ^ [Y_7: hoare_2091234717iple_a] : ( (|) @ ( Y_7 = X ) @ ( member290856304iple_a @ Y_7 @ Xa ) ) ) ) ).
thf(fact_452_insert__compr__raw,axiom,
! [X: hoare_1708887482_state,Xa: hoare_1708887482_state > $o] :
( ( insert528405184_state @ X @ Xa )
= ( collec1568722789_state
@ ^ [Y_7: hoare_1708887482_state] : ( (|) @ ( Y_7 = X ) @ ( member451959335_state @ Y_7 @ Xa ) ) ) ) ).
thf(fact_453_insert__compr__raw,axiom,
! [X: nat,Xa: nat > $o] :
( ( insert_nat @ X @ Xa )
= ( collect_nat
@ ^ [Y_7: nat] : ( (|) @ ( Y_7 = X ) @ ( member_nat @ Y_7 @ Xa ) ) ) ) ).
thf(fact_454_insert__compr__raw,axiom,
! [X: pname,Xa: pname > $o] :
( ( insert_pname @ X @ Xa )
= ( collect_pname
@ ^ [Y_7: pname] : ( (|) @ ( Y_7 = X ) @ ( member_pname @ Y_7 @ Xa ) ) ) ) ).
thf(fact_455_derivs__insertD,axiom,
! [G_19: hoare_2091234717iple_a > $o,T_3: hoare_2091234717iple_a,Ts_1: hoare_2091234717iple_a > $o] :
( ( hoare_1467856363rivs_a @ G_19 @ ( insert1597628439iple_a @ T_3 @ Ts_1 ) )
=> ( ( hoare_1467856363rivs_a @ G_19 @ ( insert1597628439iple_a @ T_3 @ bot_bo1791335050le_a_o ) )
& ( hoare_1467856363rivs_a @ G_19 @ Ts_1 ) ) ) ).
thf(fact_456_derivs__insertD,axiom,
! [G_19: hoare_1708887482_state > $o,T_3: hoare_1708887482_state,Ts_1: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ G_19 @ ( insert528405184_state @ T_3 @ Ts_1 ) )
=> ( ( hoare_90032982_state @ G_19 @ ( insert528405184_state @ T_3 @ bot_bo19817387tate_o ) )
& ( hoare_90032982_state @ G_19 @ Ts_1 ) ) ) ).
thf(fact_457_hoare__derivs_Oinsert,axiom,
! [Ts: hoare_2091234717iple_a > $o,G_18: hoare_2091234717iple_a > $o,T_2: hoare_2091234717iple_a] :
( ( hoare_1467856363rivs_a @ G_18 @ ( insert1597628439iple_a @ T_2 @ bot_bo1791335050le_a_o ) )
=> ( ( hoare_1467856363rivs_a @ G_18 @ Ts )
=> ( hoare_1467856363rivs_a @ G_18 @ ( insert1597628439iple_a @ T_2 @ Ts ) ) ) ) ).
thf(fact_458_hoare__derivs_Oinsert,axiom,
! [Ts: hoare_1708887482_state > $o,G_18: hoare_1708887482_state > $o,T_2: hoare_1708887482_state] :
( ( hoare_90032982_state @ G_18 @ ( insert528405184_state @ T_2 @ bot_bo19817387tate_o ) )
=> ( ( hoare_90032982_state @ G_18 @ Ts )
=> ( hoare_90032982_state @ G_18 @ ( insert528405184_state @ T_2 @ Ts ) ) ) ) ).
thf(fact_459_image__constant__conv,axiom,
! [C_23: nat,A_65: nat > $o] :
( ( ( A_65 = bot_bot_nat_o )
=> ( ( image_nat_nat
@ ^ [X: nat] : C_23
@ A_65 )
= bot_bot_nat_o ) )
& ( ( A_65 != bot_bot_nat_o )
=> ( ( image_nat_nat
@ ^ [X: nat] : C_23
@ A_65 )
= ( insert_nat @ C_23 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_460_image__constant__conv,axiom,
! [C_23: hoare_1708887482_state,A_65: pname > $o] :
( ( ( A_65 = bot_bot_pname_o )
=> ( ( image_1116629049_state
@ ^ [X: pname] : C_23
@ A_65 )
= bot_bo19817387tate_o ) )
& ( ( A_65 != bot_bot_pname_o )
=> ( ( image_1116629049_state
@ ^ [X: pname] : C_23
@ A_65 )
= ( insert528405184_state @ C_23 @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_461_image__constant__conv,axiom,
! [C_23: hoare_2091234717iple_a,A_65: pname > $o] :
( ( ( A_65 = bot_bot_pname_o )
=> ( ( image_231808478iple_a
@ ^ [X: pname] : C_23
@ A_65 )
= bot_bo1791335050le_a_o ) )
& ( ( A_65 != bot_bot_pname_o )
=> ( ( image_231808478iple_a
@ ^ [X: pname] : C_23
@ A_65 )
= ( insert1597628439iple_a @ C_23 @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_462_image__constant,axiom,
! [C_22: nat,X_28: nat,A_64: nat > $o] :
( ( member_nat @ X_28 @ A_64 )
=> ( ( image_nat_nat
@ ^ [X: nat] : C_22
@ A_64 )
= ( insert_nat @ C_22 @ bot_bot_nat_o ) ) ) ).
thf(fact_463_image__constant,axiom,
! [C_22: hoare_1708887482_state,X_28: pname,A_64: pname > $o] :
( ( member_pname @ X_28 @ A_64 )
=> ( ( image_1116629049_state
@ ^ [X: pname] : C_22
@ A_64 )
= ( insert528405184_state @ C_22 @ bot_bo19817387tate_o ) ) ) ).
thf(fact_464_image__constant,axiom,
! [C_22: nat,X_28: pname,A_64: pname > $o] :
( ( member_pname @ X_28 @ A_64 )
=> ( ( image_pname_nat
@ ^ [X: pname] : C_22
@ A_64 )
= ( insert_nat @ C_22 @ bot_bot_nat_o ) ) ) ).
thf(fact_465_image__constant,axiom,
! [C_22: hoare_2091234717iple_a > $o,X_28: pname,A_64: pname > $o] :
( ( member_pname @ X_28 @ A_64 )
=> ( ( image_742317343le_a_o
@ ^ [X: pname] : C_22
@ A_64 )
= ( insert102003750le_a_o @ C_22 @ bot_bo1957696069_a_o_o ) ) ) ).
thf(fact_466_image__constant,axiom,
! [C_22: pname,X_28: pname,A_64: pname > $o] :
( ( member_pname @ X_28 @ A_64 )
=> ( ( image_pname_pname
@ ^ [X: pname] : C_22
@ A_64 )
= ( insert_pname @ C_22 @ bot_bot_pname_o ) ) ) ).
thf(fact_467_image__constant,axiom,
! [C_22: hoare_2091234717iple_a,X_28: pname,A_64: pname > $o] :
( ( member_pname @ X_28 @ A_64 )
=> ( ( image_231808478iple_a
@ ^ [X: pname] : C_22
@ A_64 )
= ( insert1597628439iple_a @ C_22 @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_468_image__insert,axiom,
! [F_40: nat > nat,A_63: nat,B_34: nat > $o] :
( ( image_nat_nat @ F_40 @ ( insert_nat @ A_63 @ B_34 ) )
= ( insert_nat @ ( F_40 @ A_63 ) @ ( image_nat_nat @ F_40 @ B_34 ) ) ) ).
thf(fact_469_image__insert,axiom,
! [F_40: pname > hoare_1708887482_state,A_63: pname,B_34: pname > $o] :
( ( image_1116629049_state @ F_40 @ ( insert_pname @ A_63 @ B_34 ) )
= ( insert528405184_state @ ( F_40 @ A_63 ) @ ( image_1116629049_state @ F_40 @ B_34 ) ) ) ).
thf(fact_470_image__insert,axiom,
! [F_40: pname > hoare_2091234717iple_a,A_63: pname,B_34: pname > $o] :
( ( image_231808478iple_a @ F_40 @ ( insert_pname @ A_63 @ B_34 ) )
= ( insert1597628439iple_a @ ( F_40 @ A_63 ) @ ( image_231808478iple_a @ F_40 @ B_34 ) ) ) ).
thf(fact_471_insert__image,axiom,
! [F_39: nat > nat,X_27: nat,A_62: nat > $o] :
( ( member_nat @ X_27 @ A_62 )
=> ( ( insert_nat @ ( F_39 @ X_27 ) @ ( image_nat_nat @ F_39 @ A_62 ) )
= ( image_nat_nat @ F_39 @ A_62 ) ) ) ).
thf(fact_472_insert__image,axiom,
! [F_39: pname > hoare_1708887482_state,X_27: pname,A_62: pname > $o] :
( ( member_pname @ X_27 @ A_62 )
=> ( ( insert528405184_state @ ( F_39 @ X_27 ) @ ( image_1116629049_state @ F_39 @ A_62 ) )
= ( image_1116629049_state @ F_39 @ A_62 ) ) ) ).
thf(fact_473_insert__image,axiom,
! [F_39: pname > nat,X_27: pname,A_62: pname > $o] :
( ( member_pname @ X_27 @ A_62 )
=> ( ( insert_nat @ ( F_39 @ X_27 ) @ ( image_pname_nat @ F_39 @ A_62 ) )
= ( image_pname_nat @ F_39 @ A_62 ) ) ) ).
thf(fact_474_insert__image,axiom,
! [F_39: pname > hoare_2091234717iple_a > $o,X_27: pname,A_62: pname > $o] :
( ( member_pname @ X_27 @ A_62 )
=> ( ( insert102003750le_a_o @ ( F_39 @ X_27 ) @ ( image_742317343le_a_o @ F_39 @ A_62 ) )
= ( image_742317343le_a_o @ F_39 @ A_62 ) ) ) ).
thf(fact_475_insert__image,axiom,
! [F_39: pname > pname,X_27: pname,A_62: pname > $o] :
( ( member_pname @ X_27 @ A_62 )
=> ( ( insert_pname @ ( F_39 @ X_27 ) @ ( image_pname_pname @ F_39 @ A_62 ) )
= ( image_pname_pname @ F_39 @ A_62 ) ) ) ).
thf(fact_476_insert__image,axiom,
! [F_39: pname > hoare_2091234717iple_a,X_27: pname,A_62: pname > $o] :
( ( member_pname @ X_27 @ A_62 )
=> ( ( insert1597628439iple_a @ ( F_39 @ X_27 ) @ ( image_231808478iple_a @ F_39 @ A_62 ) )
= ( image_231808478iple_a @ F_39 @ A_62 ) ) ) ).
thf(fact_477_Un__insert__right,axiom,
! [A_61: nat > $o,A_60: nat,B_33: nat > $o] :
( ( semila848761471_nat_o @ A_61 @ ( insert_nat @ A_60 @ B_33 ) )
= ( insert_nat @ A_60 @ ( semila848761471_nat_o @ A_61 @ B_33 ) ) ) ).
thf(fact_478_Un__insert__right,axiom,
! [A_61: ( hoare_2091234717iple_a > $o ) > $o,A_60: hoare_2091234717iple_a > $o,B_33: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_61 @ ( insert102003750le_a_o @ A_60 @ B_33 ) )
= ( insert102003750le_a_o @ A_60 @ ( semila2050116131_a_o_o @ A_61 @ B_33 ) ) ) ).
thf(fact_479_Un__insert__right,axiom,
! [A_61: pname > $o,A_60: pname,B_33: pname > $o] :
( ( semila1780557381name_o @ A_61 @ ( insert_pname @ A_60 @ B_33 ) )
= ( insert_pname @ A_60 @ ( semila1780557381name_o @ A_61 @ B_33 ) ) ) ).
thf(fact_480_Un__insert__right,axiom,
! [A_61: hoare_1708887482_state > $o,A_60: hoare_1708887482_state,B_33: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_61 @ ( insert528405184_state @ A_60 @ B_33 ) )
= ( insert528405184_state @ A_60 @ ( semila1122118281tate_o @ A_61 @ B_33 ) ) ) ).
thf(fact_481_Un__insert__right,axiom,
! [A_61: hoare_2091234717iple_a > $o,A_60: hoare_2091234717iple_a,B_33: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_61 @ ( insert1597628439iple_a @ A_60 @ B_33 ) )
= ( insert1597628439iple_a @ A_60 @ ( semila1052848428le_a_o @ A_61 @ B_33 ) ) ) ).
thf(fact_482_Un__insert__left,axiom,
! [A_59: nat,B_32: nat > $o,C_21: nat > $o] :
( ( semila848761471_nat_o @ ( insert_nat @ A_59 @ B_32 ) @ C_21 )
= ( insert_nat @ A_59 @ ( semila848761471_nat_o @ B_32 @ C_21 ) ) ) ).
thf(fact_483_Un__insert__left,axiom,
! [A_59: hoare_2091234717iple_a > $o,B_32: ( hoare_2091234717iple_a > $o ) > $o,C_21: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ ( insert102003750le_a_o @ A_59 @ B_32 ) @ C_21 )
= ( insert102003750le_a_o @ A_59 @ ( semila2050116131_a_o_o @ B_32 @ C_21 ) ) ) ).
thf(fact_484_Un__insert__left,axiom,
! [A_59: pname,B_32: pname > $o,C_21: pname > $o] :
( ( semila1780557381name_o @ ( insert_pname @ A_59 @ B_32 ) @ C_21 )
= ( insert_pname @ A_59 @ ( semila1780557381name_o @ B_32 @ C_21 ) ) ) ).
thf(fact_485_Un__insert__left,axiom,
! [A_59: hoare_1708887482_state,B_32: hoare_1708887482_state > $o,C_21: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ ( insert528405184_state @ A_59 @ B_32 ) @ C_21 )
= ( insert528405184_state @ A_59 @ ( semila1122118281tate_o @ B_32 @ C_21 ) ) ) ).
thf(fact_486_Un__insert__left,axiom,
! [A_59: hoare_2091234717iple_a,B_32: hoare_2091234717iple_a > $o,C_21: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ ( insert1597628439iple_a @ A_59 @ B_32 ) @ C_21 )
= ( insert1597628439iple_a @ A_59 @ ( semila1052848428le_a_o @ B_32 @ C_21 ) ) ) ).
thf(fact_487_empty__is__image,axiom,
! [F_38: nat > nat,A_58: nat > $o] :
( ( bot_bot_nat_o
= ( image_nat_nat @ F_38 @ A_58 ) )
<=> ( A_58 = bot_bot_nat_o ) ) ).
thf(fact_488_empty__is__image,axiom,
! [F_38: pname > hoare_1708887482_state,A_58: pname > $o] :
( ( bot_bo19817387tate_o
= ( image_1116629049_state @ F_38 @ A_58 ) )
<=> ( A_58 = bot_bot_pname_o ) ) ).
thf(fact_489_empty__is__image,axiom,
! [F_38: pname > hoare_2091234717iple_a,A_58: pname > $o] :
( ( bot_bo1791335050le_a_o
= ( image_231808478iple_a @ F_38 @ A_58 ) )
<=> ( A_58 = bot_bot_pname_o ) ) ).
thf(fact_490_image__empty,axiom,
! [F_37: nat > nat] :
( ( image_nat_nat @ F_37 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
thf(fact_491_image__empty,axiom,
! [F_37: pname > hoare_1708887482_state] :
( ( image_1116629049_state @ F_37 @ bot_bot_pname_o )
= bot_bo19817387tate_o ) ).
thf(fact_492_image__empty,axiom,
! [F_37: pname > hoare_2091234717iple_a] :
( ( image_231808478iple_a @ F_37 @ bot_bot_pname_o )
= bot_bo1791335050le_a_o ) ).
thf(fact_493_image__is__empty,axiom,
! [F_36: nat > nat,A_57: nat > $o] :
( ( ( image_nat_nat @ F_36 @ A_57 )
= bot_bot_nat_o )
<=> ( A_57 = bot_bot_nat_o ) ) ).
thf(fact_494_image__is__empty,axiom,
! [F_36: pname > hoare_1708887482_state,A_57: pname > $o] :
( ( ( image_1116629049_state @ F_36 @ A_57 )
= bot_bo19817387tate_o )
<=> ( A_57 = bot_bot_pname_o ) ) ).
thf(fact_495_image__is__empty,axiom,
! [F_36: pname > hoare_2091234717iple_a,A_57: pname > $o] :
( ( ( image_231808478iple_a @ F_36 @ A_57 )
= bot_bo1791335050le_a_o )
<=> ( A_57 = bot_bot_pname_o ) ) ).
thf(fact_496_ball__empty,axiom,
! [P_23: nat > $o,X: nat] :
( ( member_nat @ X @ bot_bot_nat_o )
=> ( P_23 @ X ) ) ).
thf(fact_497_ball__empty,axiom,
! [P_23: hoare_2091234717iple_a > $o,X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ bot_bo1791335050le_a_o )
=> ( P_23 @ X ) ) ).
thf(fact_498_ball__empty,axiom,
! [P_23: ( hoare_2091234717iple_a > $o ) > $o,X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ bot_bo1957696069_a_o_o )
=> ( P_23 @ X ) ) ).
thf(fact_499_ball__empty,axiom,
! [P_23: pname > $o,X: pname] :
( ( member_pname @ X @ bot_bot_pname_o )
=> ( P_23 @ X ) ) ).
thf(fact_500_ball__empty,axiom,
! [P_23: hoare_1708887482_state > $o,X: hoare_1708887482_state] :
( ( member451959335_state @ X @ bot_bo19817387tate_o )
=> ( P_23 @ X ) ) ).
thf(fact_501_Un__empty__left,axiom,
! [B_31: nat > $o] :
( ( semila848761471_nat_o @ bot_bot_nat_o @ B_31 )
= B_31 ) ).
thf(fact_502_Un__empty__left,axiom,
! [B_31: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ bot_bo1957696069_a_o_o @ B_31 )
= B_31 ) ).
thf(fact_503_Un__empty__left,axiom,
! [B_31: pname > $o] :
( ( semila1780557381name_o @ bot_bot_pname_o @ B_31 )
= B_31 ) ).
thf(fact_504_Un__empty__left,axiom,
! [B_31: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ bot_bo19817387tate_o @ B_31 )
= B_31 ) ).
thf(fact_505_Un__empty__left,axiom,
! [B_31: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ bot_bo1791335050le_a_o @ B_31 )
= B_31 ) ).
thf(fact_506_Un__empty__right,axiom,
! [A_56: nat > $o] :
( ( semila848761471_nat_o @ A_56 @ bot_bot_nat_o )
= A_56 ) ).
thf(fact_507_Un__empty__right,axiom,
! [A_56: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ A_56 @ bot_bo1957696069_a_o_o )
= A_56 ) ).
thf(fact_508_Un__empty__right,axiom,
! [A_56: pname > $o] :
( ( semila1780557381name_o @ A_56 @ bot_bot_pname_o )
= A_56 ) ).
thf(fact_509_Un__empty__right,axiom,
! [A_56: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ A_56 @ bot_bo19817387tate_o )
= A_56 ) ).
thf(fact_510_Un__empty__right,axiom,
! [A_56: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ A_56 @ bot_bo1791335050le_a_o )
= A_56 ) ).
thf(fact_511_Un__empty,axiom,
! [A_55: nat > $o,B_30: nat > $o] :
( ( ( semila848761471_nat_o @ A_55 @ B_30 )
= bot_bot_nat_o )
<=> ( ( A_55 = bot_bot_nat_o )
& ( B_30 = bot_bot_nat_o ) ) ) ).
thf(fact_512_Un__empty,axiom,
! [A_55: ( hoare_2091234717iple_a > $o ) > $o,B_30: ( hoare_2091234717iple_a > $o ) > $o] :
( ( ( semila2050116131_a_o_o @ A_55 @ B_30 )
= bot_bo1957696069_a_o_o )
<=> ( ( A_55 = bot_bo1957696069_a_o_o )
& ( B_30 = bot_bo1957696069_a_o_o ) ) ) ).
thf(fact_513_Un__empty,axiom,
! [A_55: pname > $o,B_30: pname > $o] :
( ( ( semila1780557381name_o @ A_55 @ B_30 )
= bot_bot_pname_o )
<=> ( ( A_55 = bot_bot_pname_o )
& ( B_30 = bot_bot_pname_o ) ) ) ).
thf(fact_514_Un__empty,axiom,
! [A_55: hoare_1708887482_state > $o,B_30: hoare_1708887482_state > $o] :
( ( ( semila1122118281tate_o @ A_55 @ B_30 )
= bot_bo19817387tate_o )
<=> ( ( A_55 = bot_bo19817387tate_o )
& ( B_30 = bot_bo19817387tate_o ) ) ) ).
thf(fact_515_Un__empty,axiom,
! [A_55: hoare_2091234717iple_a > $o,B_30: hoare_2091234717iple_a > $o] :
( ( ( semila1052848428le_a_o @ A_55 @ B_30 )
= bot_bo1791335050le_a_o )
<=> ( ( A_55 = bot_bo1791335050le_a_o )
& ( B_30 = bot_bo1791335050le_a_o ) ) ) ).
thf(fact_516_constant,axiom,
! [G_17: hoare_2091234717iple_a > $o,P_22: x_a > state > $o,C_20: com,Q_15: x_a > state > $o,C_19: $o] :
( ( C_19
=> ( hoare_1467856363rivs_a @ G_17 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_22 @ C_20 @ Q_15 ) @ bot_bo1791335050le_a_o ) ) )
=> ( hoare_1467856363rivs_a @ G_17
@ ( insert1597628439iple_a
@ ( hoare_657976383iple_a
@ ^ [Z_5: x_a,S_2: state] : ( (&) @ ( P_22 @ Z_5 @ S_2 ) @ C_19 )
@ C_20
@ Q_15 )
@ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_517_constant,axiom,
! [G_17: hoare_1708887482_state > $o,P_22: state > state > $o,C_20: com,Q_15: state > state > $o,C_19: $o] :
( ( C_19
=> ( hoare_90032982_state @ G_17 @ ( insert528405184_state @ ( hoare_858012674_state @ P_22 @ C_20 @ Q_15 ) @ bot_bo19817387tate_o ) ) )
=> ( hoare_90032982_state @ G_17
@ ( insert528405184_state
@ ( hoare_858012674_state
@ ^ [Z_5: state,S_2: state] : ( (&) @ ( P_22 @ Z_5 @ S_2 ) @ C_19 )
@ C_20
@ Q_15 )
@ bot_bo19817387tate_o ) ) ) ).
thf(fact_518_empty,axiom,
! [G_16: hoare_2091234717iple_a > $o] : ( hoare_1467856363rivs_a @ G_16 @ bot_bo1791335050le_a_o ) ).
thf(fact_519_empty,axiom,
! [G_16: hoare_1708887482_state > $o] : ( hoare_90032982_state @ G_16 @ bot_bo19817387tate_o ) ).
thf(fact_520_sup__bot__left,axiom,
! [X_26: nat > $o] :
( ( semila848761471_nat_o @ bot_bot_nat_o @ X_26 )
= X_26 ) ).
thf(fact_521_sup__bot__left,axiom,
! [X_26: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ bot_bo1957696069_a_o_o @ X_26 )
= X_26 ) ).
thf(fact_522_sup__bot__left,axiom,
! [X_26: pname > $o] :
( ( semila1780557381name_o @ bot_bot_pname_o @ X_26 )
= X_26 ) ).
thf(fact_523_sup__bot__left,axiom,
! [X_26: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ bot_bo19817387tate_o @ X_26 )
= X_26 ) ).
thf(fact_524_sup__bot__left,axiom,
! [X_26: $o] :
( ( semila10642723_sup_o @ bot_bot_o @ X_26 )
<=> X_26 ) ).
thf(fact_525_sup__bot__left,axiom,
! [X_26: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ bot_bo1791335050le_a_o @ X_26 )
= X_26 ) ).
thf(fact_526_sup__bot__right,axiom,
! [X_25: nat > $o] :
( ( semila848761471_nat_o @ X_25 @ bot_bot_nat_o )
= X_25 ) ).
thf(fact_527_sup__bot__right,axiom,
! [X_25: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ X_25 @ bot_bo1957696069_a_o_o )
= X_25 ) ).
thf(fact_528_sup__bot__right,axiom,
! [X_25: pname > $o] :
( ( semila1780557381name_o @ X_25 @ bot_bot_pname_o )
= X_25 ) ).
thf(fact_529_sup__bot__right,axiom,
! [X_25: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ X_25 @ bot_bo19817387tate_o )
= X_25 ) ).
thf(fact_530_sup__bot__right,axiom,
! [X_25: $o] :
( ( semila10642723_sup_o @ X_25 @ bot_bot_o )
<=> X_25 ) ).
thf(fact_531_sup__bot__right,axiom,
! [X_25: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ X_25 @ bot_bo1791335050le_a_o )
= X_25 ) ).
thf(fact_532_sup__eq__bot__iff,axiom,
! [X_24: nat > $o,Y_8: nat > $o] :
( ( ( semila848761471_nat_o @ X_24 @ Y_8 )
= bot_bot_nat_o )
<=> ( ( X_24 = bot_bot_nat_o )
& ( Y_8 = bot_bot_nat_o ) ) ) ).
thf(fact_533_sup__eq__bot__iff,axiom,
! [X_24: ( hoare_2091234717iple_a > $o ) > $o,Y_8: ( hoare_2091234717iple_a > $o ) > $o] :
( ( ( semila2050116131_a_o_o @ X_24 @ Y_8 )
= bot_bo1957696069_a_o_o )
<=> ( ( X_24 = bot_bo1957696069_a_o_o )
& ( Y_8 = bot_bo1957696069_a_o_o ) ) ) ).
thf(fact_534_sup__eq__bot__iff,axiom,
! [X_24: pname > $o,Y_8: pname > $o] :
( ( ( semila1780557381name_o @ X_24 @ Y_8 )
= bot_bot_pname_o )
<=> ( ( X_24 = bot_bot_pname_o )
& ( Y_8 = bot_bot_pname_o ) ) ) ).
thf(fact_535_sup__eq__bot__iff,axiom,
! [X_24: hoare_1708887482_state > $o,Y_8: hoare_1708887482_state > $o] :
( ( ( semila1122118281tate_o @ X_24 @ Y_8 )
= bot_bo19817387tate_o )
<=> ( ( X_24 = bot_bo19817387tate_o )
& ( Y_8 = bot_bo19817387tate_o ) ) ) ).
thf(fact_536_sup__eq__bot__iff,axiom,
! [X_24: $o,Y_8: $o] :
( ( ( semila10642723_sup_o @ X_24 @ Y_8 )
<=> bot_bot_o )
<=> ( ( X_24
<=> bot_bot_o )
& ( Y_8
<=> bot_bot_o ) ) ) ).
thf(fact_537_sup__eq__bot__iff,axiom,
! [X_24: hoare_2091234717iple_a > $o,Y_8: hoare_2091234717iple_a > $o] :
( ( ( semila1052848428le_a_o @ X_24 @ Y_8 )
= bot_bo1791335050le_a_o )
<=> ( ( X_24 = bot_bo1791335050le_a_o )
& ( Y_8 = bot_bo1791335050le_a_o ) ) ) ).
thf(fact_538_triple__valid__Suc,axiom,
! [N_5: nat,T_1: hoare_1708887482_state] :
( ( hoare_23738522_state @ ( suc @ N_5 ) @ T_1 )
=> ( hoare_23738522_state @ N_5 @ T_1 ) ) ).
thf(fact_539_triple__valid__Suc,axiom,
! [N_5: nat,T_1: hoare_2091234717iple_a] :
( ( hoare_1421888935alid_a @ ( suc @ N_5 ) @ T_1 )
=> ( hoare_1421888935alid_a @ N_5 @ T_1 ) ) ).
thf(fact_540_insert__def,axiom,
! [A_54: hoare_2091234717iple_a > $o,B_29: ( hoare_2091234717iple_a > $o ) > $o] :
( ( insert102003750le_a_o @ A_54 @ B_29 )
= ( semila2050116131_a_o_o
@ ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : X = A_54 )
@ B_29 ) ) ).
thf(fact_541_insert__def,axiom,
! [A_54: pname,B_29: pname > $o] :
( ( insert_pname @ A_54 @ B_29 )
= ( semila1780557381name_o
@ ( collect_pname
@ ^ [X: pname] : X = A_54 )
@ B_29 ) ) ).
thf(fact_542_insert__def,axiom,
! [A_54: hoare_1708887482_state,B_29: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_54 @ B_29 )
= ( semila1122118281tate_o
@ ( collec1568722789_state
@ ^ [X: hoare_1708887482_state] : X = A_54 )
@ B_29 ) ) ).
thf(fact_543_insert__def,axiom,
! [A_54: hoare_2091234717iple_a,B_29: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_54 @ B_29 )
= ( semila1052848428le_a_o
@ ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : X = A_54 )
@ B_29 ) ) ).
thf(fact_544_insert__def,axiom,
! [A_54: nat,B_29: nat > $o] :
( ( insert_nat @ A_54 @ B_29 )
= ( semila848761471_nat_o
@ ( collect_nat
@ ^ [X: nat] : X = A_54 )
@ B_29 ) ) ).
thf(fact_545_weak__Body,axiom,
! [G_15: hoare_2091234717iple_a > $o,P_21: x_a > state > $o,Pn_3: pname,Q_14: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_15 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_21 @ ( the_com @ ( body_1 @ Pn_3 ) ) @ Q_14 ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_15 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_21 @ ( body @ Pn_3 ) @ Q_14 ) @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_546_weak__Body,axiom,
! [G_15: hoare_1708887482_state > $o,P_21: state > state > $o,Pn_3: pname,Q_14: state > state > $o] :
( ( hoare_90032982_state @ G_15 @ ( insert528405184_state @ ( hoare_858012674_state @ P_21 @ ( the_com @ ( body_1 @ Pn_3 ) ) @ Q_14 ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_15 @ ( insert528405184_state @ ( hoare_858012674_state @ P_21 @ ( body @ Pn_3 ) @ Q_14 ) @ bot_bo19817387tate_o ) ) ) ).
thf(fact_547_BodyN,axiom,
! [P_20: x_a > state > $o,Pn_2: pname,Q_13: x_a > state > $o,G_14: hoare_2091234717iple_a > $o] :
( ( hoare_1467856363rivs_a @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_20 @ ( body @ Pn_2 ) @ Q_13 ) @ G_14 ) @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_20 @ ( the_com @ ( body_1 @ Pn_2 ) ) @ Q_13 ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_14 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_20 @ ( body @ Pn_2 ) @ Q_13 ) @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_548_BodyN,axiom,
! [P_20: state > state > $o,Pn_2: pname,Q_13: state > state > $o,G_14: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ ( insert528405184_state @ ( hoare_858012674_state @ P_20 @ ( body @ Pn_2 ) @ Q_13 ) @ G_14 ) @ ( insert528405184_state @ ( hoare_858012674_state @ P_20 @ ( the_com @ ( body_1 @ Pn_2 ) ) @ Q_13 ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_14 @ ( insert528405184_state @ ( hoare_858012674_state @ P_20 @ ( body @ Pn_2 ) @ Q_13 ) @ bot_bo19817387tate_o ) ) ) ).
thf(fact_549_escape,axiom,
! [G_13: hoare_2091234717iple_a > $o,C_18: com,Q_12: x_a > state > $o,P_19: x_a > state > $o] :
( ! [Z_5: x_a,S_2: state] :
( ( P_19 @ Z_5 @ S_2 )
=> ( hoare_1467856363rivs_a @ G_13
@ ( insert1597628439iple_a
@ ( hoare_657976383iple_a
@ ^ [Za: x_a,S_3: state] : S_3 = S_2
@ C_18
@ ^ [Z_6: x_a] : ( Q_12 @ Z_5 ) )
@ bot_bo1791335050le_a_o ) ) )
=> ( hoare_1467856363rivs_a @ G_13 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_19 @ C_18 @ Q_12 ) @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_550_escape,axiom,
! [G_13: hoare_1708887482_state > $o,C_18: com,Q_12: state > state > $o,P_19: state > state > $o] :
( ! [Z_5: state,S_2: state] :
( ( P_19 @ Z_5 @ S_2 )
=> ( hoare_90032982_state @ G_13
@ ( insert528405184_state
@ ( hoare_858012674_state
@ ^ [Za: state,S_3: state] : S_3 = S_2
@ C_18
@ ^ [Z_6: state] : ( Q_12 @ Z_5 ) )
@ bot_bo19817387tate_o ) ) )
=> ( hoare_90032982_state @ G_13 @ ( insert528405184_state @ ( hoare_858012674_state @ P_19 @ C_18 @ Q_12 ) @ bot_bo19817387tate_o ) ) ) ).
thf(fact_551_conseq1,axiom,
! [P_18: x_a > state > $o,G_12: hoare_2091234717iple_a > $o,P_17: x_a > state > $o,C_17: com,Q_11: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_12 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_17 @ C_17 @ Q_11 ) @ bot_bo1791335050le_a_o ) )
=> ( ! [Z_5: x_a,S_2: state] :
( ( P_18 @ Z_5 @ S_2 )
=> ( P_17 @ Z_5 @ S_2 ) )
=> ( hoare_1467856363rivs_a @ G_12 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_18 @ C_17 @ Q_11 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_552_conseq1,axiom,
! [P_18: state > state > $o,G_12: hoare_1708887482_state > $o,P_17: state > state > $o,C_17: com,Q_11: state > state > $o] :
( ( hoare_90032982_state @ G_12 @ ( insert528405184_state @ ( hoare_858012674_state @ P_17 @ C_17 @ Q_11 ) @ bot_bo19817387tate_o ) )
=> ( ! [Z_5: state,S_2: state] :
( ( P_18 @ Z_5 @ S_2 )
=> ( P_17 @ Z_5 @ S_2 ) )
=> ( hoare_90032982_state @ G_12 @ ( insert528405184_state @ ( hoare_858012674_state @ P_18 @ C_17 @ Q_11 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_553_conseq2,axiom,
! [Q_10: x_a > state > $o,G_11: hoare_2091234717iple_a > $o,P_16: x_a > state > $o,C_16: com,Q_9: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_11 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_16 @ C_16 @ Q_9 ) @ bot_bo1791335050le_a_o ) )
=> ( ! [Z_5: x_a,S_2: state] :
( ( Q_9 @ Z_5 @ S_2 )
=> ( Q_10 @ Z_5 @ S_2 ) )
=> ( hoare_1467856363rivs_a @ G_11 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_16 @ C_16 @ Q_10 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_554_conseq2,axiom,
! [Q_10: state > state > $o,G_11: hoare_1708887482_state > $o,P_16: state > state > $o,C_16: com,Q_9: state > state > $o] :
( ( hoare_90032982_state @ G_11 @ ( insert528405184_state @ ( hoare_858012674_state @ P_16 @ C_16 @ Q_9 ) @ bot_bo19817387tate_o ) )
=> ( ! [Z_5: state,S_2: state] :
( ( Q_9 @ Z_5 @ S_2 )
=> ( Q_10 @ Z_5 @ S_2 ) )
=> ( hoare_90032982_state @ G_11 @ ( insert528405184_state @ ( hoare_858012674_state @ P_16 @ C_16 @ Q_10 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_555_triple_Osize_I1_J,axiom,
! [Fa: x_a > nat,Fun1_1: x_a > state > $o,Com_3: com,Fun2_1: x_a > state > $o] :
( ( hoare_1169027232size_a @ Fa @ ( hoare_657976383iple_a @ Fun1_1 @ Com_3 @ Fun2_1 ) )
= zero_zero_nat ) ).
thf(fact_556_triple_Osize_I1_J,axiom,
! [Fa: state > nat,Fun1_1: state > state > $o,Com_3: com,Fun2_1: state > state > $o] :
( ( hoare_518771297_state @ Fa @ ( hoare_858012674_state @ Fun1_1 @ Com_3 @ Fun2_1 ) )
= zero_zero_nat ) ).
thf(fact_557_MGT__def,axiom,
! [C: com] :
( ( hoare_Mirabelle_MGT @ C )
= ( hoare_858012674_state @ fequal_state @ C @ ( evalc @ C ) ) ) ).
thf(fact_558_triple_Osize_I2_J,axiom,
! [Fun1: x_a > state > $o,Com_2: com,Fun2: x_a > state > $o] :
( ( size_s1040486067iple_a @ ( hoare_657976383iple_a @ Fun1 @ Com_2 @ Fun2 ) )
= zero_zero_nat ) ).
thf(fact_559_triple_Osize_I2_J,axiom,
! [Fun1: state > state > $o,Com_2: com,Fun2: state > state > $o] :
( ( size_s1186992420_state @ ( hoare_858012674_state @ Fun1 @ Com_2 @ Fun2 ) )
= zero_zero_nat ) ).
thf(fact_560_conseq12,axiom,
! [Q_8: x_a > state > $o,P_15: x_a > state > $o,G_10: hoare_2091234717iple_a > $o,P_14: x_a > state > $o,C_15: com,Q_7: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_10 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_14 @ C_15 @ Q_7 ) @ bot_bo1791335050le_a_o ) )
=> ( ! [Z_5: x_a,S_2: state] :
( ( P_15 @ Z_5 @ S_2 )
=> ! [S_3: state] :
( ! [Z_6: x_a] :
( ( P_14 @ Z_6 @ S_2 )
=> ( Q_7 @ Z_6 @ S_3 ) )
=> ( Q_8 @ Z_5 @ S_3 ) ) )
=> ( hoare_1467856363rivs_a @ G_10 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_15 @ C_15 @ Q_8 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_561_conseq12,axiom,
! [Q_8: state > state > $o,P_15: state > state > $o,G_10: hoare_1708887482_state > $o,P_14: state > state > $o,C_15: com,Q_7: state > state > $o] :
( ( hoare_90032982_state @ G_10 @ ( insert528405184_state @ ( hoare_858012674_state @ P_14 @ C_15 @ Q_7 ) @ bot_bo19817387tate_o ) )
=> ( ! [Z_5: state,S_2: state] :
( ( P_15 @ Z_5 @ S_2 )
=> ! [S_3: state] :
( ! [Z_6: state] :
( ( P_14 @ Z_6 @ S_2 )
=> ( Q_7 @ Z_6 @ S_3 ) )
=> ( Q_8 @ Z_5 @ S_3 ) ) )
=> ( hoare_90032982_state @ G_10 @ ( insert528405184_state @ ( hoare_858012674_state @ P_15 @ C_15 @ Q_8 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_562_the__elem__eq,axiom,
! [X_23: nat] :
( ( the_elem_nat @ ( insert_nat @ X_23 @ bot_bot_nat_o ) )
= X_23 ) ).
thf(fact_563_the__elem__eq,axiom,
! [X_23: hoare_2091234717iple_a > $o] :
( ( the_el1618277441le_a_o @ ( insert102003750le_a_o @ X_23 @ bot_bo1957696069_a_o_o ) )
= X_23 ) ).
thf(fact_564_the__elem__eq,axiom,
! [X_23: pname] :
( ( the_elem_pname @ ( insert_pname @ X_23 @ bot_bot_pname_o ) )
= X_23 ) ).
thf(fact_565_the__elem__eq,axiom,
! [X_23: hoare_2091234717iple_a] :
( ( the_el13400124iple_a @ ( insert1597628439iple_a @ X_23 @ bot_bo1791335050le_a_o ) )
= X_23 ) ).
thf(fact_566_the__elem__eq,axiom,
! [X_23: hoare_1708887482_state] :
( ( the_el864710747_state @ ( insert528405184_state @ X_23 @ bot_bo19817387tate_o ) )
= X_23 ) ).
thf(fact_567_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_568_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
thf(fact_569_n__not__Suc__n,axiom,
! [N_1: nat] :
( N_1
!= ( suc @ N_1 ) ) ).
thf(fact_570_Suc__n__not__n,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
!= N_1 ) ).
thf(fact_571_nat_Oinject,axiom,
! [Nat_3: nat,Nat_2: nat] :
( ( ( suc @ Nat_3 )
= ( suc @ Nat_2 ) )
<=> ( Nat_3 = Nat_2 ) ) ).
thf(fact_572_Suc__inject,axiom,
! [X_1: nat,Y: nat] :
( ( ( suc @ X_1 )
= ( suc @ Y ) )
=> ( X_1 = Y ) ) ).
thf(fact_573_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_574_nat_Osimps_I2_J,axiom,
! [Nat_2: nat] :
( zero_zero_nat
!= ( suc @ Nat_2 ) ) ).
thf(fact_575_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_576_nat_Osimps_I3_J,axiom,
! [Nat_1: nat] :
( ( suc @ Nat_1 )
!= zero_zero_nat ) ).
thf(fact_577_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_578_not0__implies__Suc,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
=> ? [M_1: nat] :
( N_1
= ( suc @ M_1 ) ) ) ).
thf(fact_579_nat__induct,axiom,
! [N_1: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N_1 ) ) ) ).
thf(fact_580_zero__induct,axiom,
! [P: nat > $o,K_1: nat] :
( ( P @ K_1 )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_581_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
thf(fact_582_bot__fun__def,axiom,
! [X: nat] :
( ( bot_bot_nat_o @ X )
<=> bot_bot_o ) ).
thf(fact_583_bot__fun__def,axiom,
! [X: hoare_2091234717iple_a] :
( ( bot_bo1791335050le_a_o @ X )
<=> bot_bot_o ) ).
thf(fact_584_bot__fun__def,axiom,
! [X: hoare_2091234717iple_a > $o] :
( ( bot_bo1957696069_a_o_o @ X )
<=> bot_bot_o ) ).
thf(fact_585_bot__fun__def,axiom,
! [X: pname] :
( ( bot_bot_pname_o @ X )
<=> bot_bot_o ) ).
thf(fact_586_bot__fun__def,axiom,
! [X: hoare_1708887482_state] :
( ( bot_bo19817387tate_o @ X )
<=> bot_bot_o ) ).
thf(fact_587_bot__apply,axiom,
! [X_22: nat] :
( ( bot_bot_nat_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_588_bot__apply,axiom,
! [X_22: hoare_2091234717iple_a] :
( ( bot_bo1791335050le_a_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_589_bot__apply,axiom,
! [X_22: hoare_2091234717iple_a > $o] :
( ( bot_bo1957696069_a_o_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_590_bot__apply,axiom,
! [X_22: pname] :
( ( bot_bot_pname_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_591_bot__apply,axiom,
! [X_22: hoare_1708887482_state] :
( ( bot_bo19817387tate_o @ X_22 )
<=> bot_bot_o ) ).
thf(fact_592_evaln_OBody,axiom,
! [Pn_1: pname,S0: state,N_1: nat,S1: state] :
( ( evaln @ ( the_com @ ( body_1 @ Pn_1 ) ) @ S0 @ N_1 @ S1 )
=> ( evaln @ ( body @ Pn_1 ) @ S0 @ ( suc @ N_1 ) @ S1 ) ) ).
thf(fact_593_hoare__derivs_OSkip,axiom,
! [G_9: hoare_2091234717iple_a > $o,P_13: x_a > state > $o] : ( hoare_1467856363rivs_a @ G_9 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_13 @ skip @ P_13 ) @ bot_bo1791335050le_a_o ) ) ).
thf(fact_594_hoare__derivs_OSkip,axiom,
! [G_9: hoare_1708887482_state > $o,P_13: state > state > $o] : ( hoare_90032982_state @ G_9 @ ( insert528405184_state @ ( hoare_858012674_state @ P_13 @ skip @ P_13 ) @ bot_bo19817387tate_o ) ) ).
thf(fact_595_LoopF,axiom,
! [G_8: hoare_2091234717iple_a > $o,P_12: x_a > state > $o,B_28: state > $o,C_14: com] :
( hoare_1467856363rivs_a @ G_8
@ ( insert1597628439iple_a
@ ( hoare_657976383iple_a
@ ^ [Z_5: x_a,S_2: state] : ( (&) @ ( P_12 @ Z_5 @ S_2 ) @ ( (~) @ ( B_28 @ S_2 ) ) )
@ ( while @ B_28 @ C_14 )
@ P_12 )
@ bot_bo1791335050le_a_o ) ) ).
thf(fact_596_LoopF,axiom,
! [G_8: hoare_1708887482_state > $o,P_12: state > state > $o,B_28: state > $o,C_14: com] :
( hoare_90032982_state @ G_8
@ ( insert528405184_state
@ ( hoare_858012674_state
@ ^ [Z_5: state,S_2: state] : ( (&) @ ( P_12 @ Z_5 @ S_2 ) @ ( (~) @ ( B_28 @ S_2 ) ) )
@ ( while @ B_28 @ C_14 )
@ P_12 )
@ bot_bo19817387tate_o ) ) ).
thf(fact_597_evaln_OWhileFalse,axiom,
! [C: com,N_1: nat,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( evaln @ ( while @ B @ C ) @ S @ N_1 @ S ) ) ).
thf(fact_598_evaln_OWhileTrue,axiom,
! [S2: state,C: com,N_1: nat,S1: state,B: state > $o,S0: state] :
( ( B @ S0 )
=> ( ( evaln @ C @ S0 @ N_1 @ S1 )
=> ( ( evaln @ ( while @ B @ C ) @ S1 @ N_1 @ S2 )
=> ( evaln @ ( while @ B @ C ) @ S0 @ N_1 @ S2 ) ) ) ) ).
thf(fact_599_evalc_OWhileTrue,axiom,
! [S2: state,C: com,S1: state,B: state > $o,S0: state] :
( ( B @ S0 )
=> ( ( evalc @ C @ S0 @ S1 )
=> ( ( evalc @ ( while @ B @ C ) @ S1 @ S2 )
=> ( evalc @ ( while @ B @ C ) @ S0 @ S2 ) ) ) ) ).
thf(fact_600_evalc_OWhileFalse,axiom,
! [C: com,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( evalc @ ( while @ B @ C ) @ S @ S ) ) ).
thf(fact_601_evaln_OSkip,axiom,
! [S: state,N_1: nat] : ( evaln @ skip @ S @ N_1 @ S ) ).
thf(fact_602_evaln__elim__cases_I1_J,axiom,
! [S: state,N_1: nat,T: state] :
( ( evaln @ skip @ S @ N_1 @ T )
=> ( T = S ) ) ).
thf(fact_603_evalc__elim__cases_I1_J,axiom,
! [S: state,T: state] :
( ( evalc @ skip @ S @ T )
=> ( T = S ) ) ).
thf(fact_604_evalc_OSkip,axiom,
! [S: state] : ( evalc @ skip @ S @ S ) ).
thf(fact_605_com_Osimps_I16_J,axiom,
! [Fun: state > $o,Com: com] :
( skip
!= ( while @ Fun @ Com ) ) ).
thf(fact_606_com_Osimps_I17_J,axiom,
! [Fun: state > $o,Com: com] :
( ( while @ Fun @ Com )
!= skip ) ).
thf(fact_607_com_Osimps_I5_J,axiom,
! [Fun_1: state > $o,Com_1: com,Fun: state > $o,Com: com] :
( ( ( while @ Fun_1 @ Com_1 )
= ( while @ Fun @ Com ) )
<=> ( ( Fun_1 = Fun )
& ( Com_1 = Com ) ) ) ).
thf(fact_608_evaln__Suc,axiom,
! [C: com,S: state,N_1: nat,S_4: state] :
( ( evaln @ C @ S @ N_1 @ S_4 )
=> ( evaln @ C @ S @ ( suc @ N_1 ) @ S_4 ) ) ).
thf(fact_609_evaln__evalc,axiom,
! [C: com,S: state,N_1: nat,T: state] :
( ( evaln @ C @ S @ N_1 @ T )
=> ( evalc @ C @ S @ T ) ) ).
thf(fact_610_eval__eq,axiom,
! [C: com,S: state,T: state] :
( ( evalc @ C @ S @ T )
<=> ? [N: nat] : ( evaln @ C @ S @ N @ T ) ) ).
thf(fact_611_com_Osimps_I59_J,axiom,
! [Pname: pname,Fun_1: state > $o,Com_1: com] :
( ( body @ Pname )
!= ( while @ Fun_1 @ Com_1 ) ) ).
thf(fact_612_com_Osimps_I58_J,axiom,
! [Fun_1: state > $o,Com_1: com,Pname: pname] :
( ( while @ Fun_1 @ Com_1 )
!= ( body @ Pname ) ) ).
thf(fact_613_com_Osimps_I18_J,axiom,
! [Pname: pname] :
( skip
!= ( body @ Pname ) ) ).
thf(fact_614_com_Osimps_I19_J,axiom,
! [Pname: pname] :
( ( body @ Pname )
!= skip ) ).
thf(fact_615_triple__valid__def2,axiom,
! [N_4: nat,P_11: state > state > $o,C_13: com,Q_6: state > state > $o] :
( ( hoare_23738522_state @ N_4 @ ( hoare_858012674_state @ P_11 @ C_13 @ Q_6 ) )
<=> ! [Z_5: state,S_2: state] :
( ( P_11 @ Z_5 @ S_2 )
=> ! [S_3: state] :
( ( evaln @ C_13 @ S_2 @ N_4 @ S_3 )
=> ( Q_6 @ Z_5 @ S_3 ) ) ) ) ).
thf(fact_616_triple__valid__def2,axiom,
! [N_4: nat,P_11: x_a > state > $o,C_13: com,Q_6: x_a > state > $o] :
( ( hoare_1421888935alid_a @ N_4 @ ( hoare_657976383iple_a @ P_11 @ C_13 @ Q_6 ) )
<=> ! [Z_5: x_a,S_2: state] :
( ( P_11 @ Z_5 @ S_2 )
=> ! [S_3: state] :
( ( evaln @ C_13 @ S_2 @ N_4 @ S_3 )
=> ( Q_6 @ Z_5 @ S_3 ) ) ) ) ).
thf(fact_617_evaln__elim__cases_I6_J,axiom,
! [P: pname,S: state,N_1: nat,S1: state] :
( ( evaln @ ( body @ P ) @ S @ N_1 @ S1 )
=> ~ ! [N: nat] :
( ( N_1
= ( suc @ N ) )
=> ~ ( evaln @ ( the_com @ ( body_1 @ P ) ) @ S @ N @ S1 ) ) ) ).
thf(fact_618_evalc__WHILE__case,axiom,
! [B: state > $o,C: com,S: state,T: state] :
( ( evalc @ ( while @ B @ C ) @ S @ T )
=> ( ( ( T = S )
=> ( B @ S ) )
=> ~ ( ( B @ S )
=> ! [S1_1: state] :
( ( evalc @ C @ S @ S1_1 )
=> ~ ( evalc @ ( while @ B @ C ) @ S1_1 @ T ) ) ) ) ) ).
thf(fact_619_evaln__WHILE__case,axiom,
! [B: state > $o,C: com,S: state,N_1: nat,T: state] :
( ( evaln @ ( while @ B @ C ) @ S @ N_1 @ T )
=> ( ( ( T = S )
=> ( B @ S ) )
=> ~ ( ( B @ S )
=> ! [S1_1: state] :
( ( evaln @ C @ S @ N_1 @ S1_1 )
=> ~ ( evaln @ ( while @ B @ C ) @ S1_1 @ N_1 @ T ) ) ) ) ) ).
thf(fact_620_evalc__evaln,axiom,
! [C: com,S: state,T: state] :
( ( evalc @ C @ S @ T )
=> ? [N: nat] : ( evaln @ C @ S @ N @ T ) ) ).
thf(fact_621_Comp,axiom,
! [D: com,R_1: x_a > state > $o,G_7: hoare_2091234717iple_a > $o,P_10: x_a > state > $o,C_12: com,Q_5: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_7 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_10 @ C_12 @ Q_5 ) @ bot_bo1791335050le_a_o ) )
=> ( ( hoare_1467856363rivs_a @ G_7 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ Q_5 @ D @ R_1 ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_7 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_10 @ ( semi @ C_12 @ D ) @ R_1 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_622_Comp,axiom,
! [D: com,R_1: state > state > $o,G_7: hoare_1708887482_state > $o,P_10: state > state > $o,C_12: com,Q_5: state > state > $o] :
( ( hoare_90032982_state @ G_7 @ ( insert528405184_state @ ( hoare_858012674_state @ P_10 @ C_12 @ Q_5 ) @ bot_bo19817387tate_o ) )
=> ( ( hoare_90032982_state @ G_7 @ ( insert528405184_state @ ( hoare_858012674_state @ Q_5 @ D @ R_1 ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_7 @ ( insert528405184_state @ ( hoare_858012674_state @ P_10 @ ( semi @ C_12 @ D ) @ R_1 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_623_the__elem__def,axiom,
! [X_21: nat > $o] :
( ( the_elem_nat @ X_21 )
= ( the_nat
@ ^ [X: nat] :
( X_21
= ( insert_nat @ X @ bot_bot_nat_o ) ) ) ) ).
thf(fact_624_the__elem__def,axiom,
! [X_21: ( hoare_2091234717iple_a > $o ) > $o] :
( ( the_el1618277441le_a_o @ X_21 )
= ( the_Ho2077879471le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] :
( X_21
= ( insert102003750le_a_o @ X @ bot_bo1957696069_a_o_o ) ) ) ) ).
thf(fact_625_the__elem__def,axiom,
! [X_21: pname > $o] :
( ( the_elem_pname @ X_21 )
= ( the_pname
@ ^ [X: pname] :
( X_21
= ( insert_pname @ X @ bot_bot_pname_o ) ) ) ) ).
thf(fact_626_the__elem__def,axiom,
! [X_21: hoare_2091234717iple_a > $o] :
( ( the_el13400124iple_a @ X_21 )
= ( the_Ho1471183438iple_a
@ ^ [X: hoare_2091234717iple_a] :
( X_21
= ( insert1597628439iple_a @ X @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_627_the__elem__def,axiom,
! [X_21: hoare_1708887482_state > $o] :
( ( the_el864710747_state @ X_21 )
= ( the_Ho851197897_state
@ ^ [X: hoare_1708887482_state] :
( X_21
= ( insert528405184_state @ X @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_628_finite__pointwise,axiom,
! [P_8: hoare_2091234717iple_a > x_a > state > $o,Q_4: hoare_2091234717iple_a > x_a > state > $o,G_6: hoare_2091234717iple_a > $o,P_7: hoare_2091234717iple_a > x_a > state > $o,C0_1: hoare_2091234717iple_a > com,Q_3: hoare_2091234717iple_a > x_a > state > $o,U_1: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ U_1 )
=> ( ! [P_9: hoare_2091234717iple_a] :
( ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo1791335050le_a_o ) ) )
=> ( ( hoare_1467856363rivs_a @ G_6
@ ( image_1661191109iple_a
@ ^ [P_9: hoare_2091234717iple_a] : ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_1467856363rivs_a @ G_6
@ ( image_1661191109iple_a
@ ^ [P_9: hoare_2091234717iple_a] : ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_629_finite__pointwise,axiom,
! [P_8: ( hoare_2091234717iple_a > $o ) > x_a > state > $o,Q_4: ( hoare_2091234717iple_a > $o ) > x_a > state > $o,G_6: hoare_2091234717iple_a > $o,P_7: ( hoare_2091234717iple_a > $o ) > x_a > state > $o,C0_1: ( hoare_2091234717iple_a > $o ) > com,Q_3: ( hoare_2091234717iple_a > $o ) > x_a > state > $o,U_1: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ U_1 )
=> ( ! [P_9: hoare_2091234717iple_a > $o] :
( ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo1791335050le_a_o ) ) )
=> ( ( hoare_1467856363rivs_a @ G_6
@ ( image_136408202iple_a
@ ^ [P_9: hoare_2091234717iple_a > $o] : ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_1467856363rivs_a @ G_6
@ ( image_136408202iple_a
@ ^ [P_9: hoare_2091234717iple_a > $o] : ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_630_finite__pointwise,axiom,
! [P_8: pname > state > state > $o,Q_4: pname > state > state > $o,G_6: hoare_1708887482_state > $o,P_7: pname > state > state > $o,C0_1: pname > com,Q_3: pname > state > state > $o,U_1: pname > $o] :
( ( finite_finite_pname @ U_1 )
=> ( ! [P_9: pname] :
( ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo19817387tate_o ) ) )
=> ( ( hoare_90032982_state @ G_6
@ ( image_1116629049_state
@ ^ [P_9: pname] : ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_90032982_state @ G_6
@ ( image_1116629049_state
@ ^ [P_9: pname] : ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_631_finite__pointwise,axiom,
! [P_8: hoare_2091234717iple_a > state > state > $o,Q_4: hoare_2091234717iple_a > state > state > $o,G_6: hoare_1708887482_state > $o,P_7: hoare_2091234717iple_a > state > state > $o,C0_1: hoare_2091234717iple_a > com,Q_3: hoare_2091234717iple_a > state > state > $o,U_1: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ U_1 )
=> ( ! [P_9: hoare_2091234717iple_a] :
( ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo19817387tate_o ) ) )
=> ( ( hoare_90032982_state @ G_6
@ ( image_1884482962_state
@ ^ [P_9: hoare_2091234717iple_a] : ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_90032982_state @ G_6
@ ( image_1884482962_state
@ ^ [P_9: hoare_2091234717iple_a] : ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_632_finite__pointwise,axiom,
! [P_8: ( hoare_2091234717iple_a > $o ) > state > state > $o,Q_4: ( hoare_2091234717iple_a > $o ) > state > state > $o,G_6: hoare_1708887482_state > $o,P_7: ( hoare_2091234717iple_a > $o ) > state > state > $o,C0_1: ( hoare_2091234717iple_a > $o ) > com,Q_3: ( hoare_2091234717iple_a > $o ) > state > state > $o,U_1: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ U_1 )
=> ( ! [P_9: hoare_2091234717iple_a > $o] :
( ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo19817387tate_o ) ) )
=> ( ( hoare_90032982_state @ G_6
@ ( image_1501246093_state
@ ^ [P_9: hoare_2091234717iple_a > $o] : ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_90032982_state @ G_6
@ ( image_1501246093_state
@ ^ [P_9: hoare_2091234717iple_a > $o] : ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_633_finite__pointwise,axiom,
! [P_8: nat > x_a > state > $o,Q_4: nat > x_a > state > $o,G_6: hoare_2091234717iple_a > $o,P_7: nat > x_a > state > $o,C0_1: nat > com,Q_3: nat > x_a > state > $o,U_1: nat > $o] :
( ( finite_finite_nat @ U_1 )
=> ( ! [P_9: nat] :
( ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo1791335050le_a_o ) ) )
=> ( ( hoare_1467856363rivs_a @ G_6
@ ( image_359186840iple_a
@ ^ [P_9: nat] : ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_1467856363rivs_a @ G_6
@ ( image_359186840iple_a
@ ^ [P_9: nat] : ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_634_finite__pointwise,axiom,
! [P_8: nat > state > state > $o,Q_4: nat > state > state > $o,G_6: hoare_1708887482_state > $o,P_7: nat > state > state > $o,C0_1: nat > com,Q_3: nat > state > state > $o,U_1: nat > $o] :
( ( finite_finite_nat @ U_1 )
=> ( ! [P_9: nat] :
( ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_6 @ ( insert528405184_state @ ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo19817387tate_o ) ) )
=> ( ( hoare_90032982_state @ G_6
@ ( image_514827263_state
@ ^ [P_9: nat] : ( hoare_858012674_state @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_90032982_state @ G_6
@ ( image_514827263_state
@ ^ [P_9: nat] : ( hoare_858012674_state @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_635_finite__pointwise,axiom,
! [P_8: pname > x_a > state > $o,Q_4: pname > x_a > state > $o,G_6: hoare_2091234717iple_a > $o,P_7: pname > x_a > state > $o,C0_1: pname > com,Q_3: pname > x_a > state > $o,U_1: pname > $o] :
( ( finite_finite_pname @ U_1 )
=> ( ! [P_9: pname] :
( ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_6 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) ) @ bot_bo1791335050le_a_o ) ) )
=> ( ( hoare_1467856363rivs_a @ G_6
@ ( image_231808478iple_a
@ ^ [P_9: pname] : ( hoare_657976383iple_a @ ( P_7 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_3 @ P_9 ) )
@ U_1 ) )
=> ( hoare_1467856363rivs_a @ G_6
@ ( image_231808478iple_a
@ ^ [P_9: pname] : ( hoare_657976383iple_a @ ( P_8 @ P_9 ) @ ( C0_1 @ P_9 ) @ ( Q_4 @ P_9 ) )
@ U_1 ) ) ) ) ) ).
thf(fact_636_evaln__max2,axiom,
! [C2: com,S2: state,N2: nat,T2: state,C1: com,S1: state,N1: nat,T1: state] :
( ( evaln @ C1 @ S1 @ N1 @ T1 )
=> ( ( evaln @ C2 @ S2 @ N2 @ T2 )
=> ? [N: nat] :
( ( evaln @ C1 @ S1 @ N @ T1 )
& ( evaln @ C2 @ S2 @ N @ T2 ) ) ) ) ).
thf(fact_637_mk__disjoint__insert,axiom,
! [A_53: nat,A_52: nat > $o] :
( ( member_nat @ A_53 @ A_52 )
=> ? [B_26: nat > $o] :
( ( A_52
= ( insert_nat @ A_53 @ B_26 ) )
& ~ ( member_nat @ A_53 @ B_26 ) ) ) ).
thf(fact_638_mk__disjoint__insert,axiom,
! [A_53: hoare_2091234717iple_a > $o,A_52: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ A_53 @ A_52 )
=> ? [B_26: ( hoare_2091234717iple_a > $o ) > $o] :
( ( A_52
= ( insert102003750le_a_o @ A_53 @ B_26 ) )
& ~ ( member99268621le_a_o @ A_53 @ B_26 ) ) ) ).
thf(fact_639_mk__disjoint__insert,axiom,
! [A_53: hoare_2091234717iple_a,A_52: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_53 @ A_52 )
=> ? [B_26: hoare_2091234717iple_a > $o] :
( ( A_52
= ( insert1597628439iple_a @ A_53 @ B_26 ) )
& ~ ( member290856304iple_a @ A_53 @ B_26 ) ) ) ).
thf(fact_640_mk__disjoint__insert,axiom,
! [A_53: hoare_1708887482_state,A_52: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_53 @ A_52 )
=> ? [B_26: hoare_1708887482_state > $o] :
( ( A_52
= ( insert528405184_state @ A_53 @ B_26 ) )
& ~ ( member451959335_state @ A_53 @ B_26 ) ) ) ).
thf(fact_641_mk__disjoint__insert,axiom,
! [A_53: pname,A_52: pname > $o] :
( ( member_pname @ A_53 @ A_52 )
=> ? [B_26: pname > $o] :
( ( A_52
= ( insert_pname @ A_53 @ B_26 ) )
& ~ ( member_pname @ A_53 @ B_26 ) ) ) ).
thf(fact_642_evaln_OSemi,axiom,
! [C1: com,S2: state,C0: com,S0: state,N_1: nat,S1: state] :
( ( evaln @ C0 @ S0 @ N_1 @ S1 )
=> ( ( evaln @ C1 @ S1 @ N_1 @ S2 )
=> ( evaln @ ( semi @ C0 @ C1 ) @ S0 @ N_1 @ S2 ) ) ) ).
thf(fact_643_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_644_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_645_com_Osimps_I48_J,axiom,
! [Com1_1: com,Com2_1: com,Pname: pname] :
( ( semi @ Com1_1 @ Com2_1 )
!= ( body @ Pname ) ) ).
thf(fact_646_com_Osimps_I49_J,axiom,
! [Pname: pname,Com1_1: com,Com2_1: com] :
( ( body @ Pname )
!= ( semi @ Com1_1 @ Com2_1 ) ) ).
thf(fact_647_com_Osimps_I47_J,axiom,
! [Fun: state > $o,Com: com,Com1_1: com,Com2_1: com] :
( ( while @ Fun @ Com )
!= ( semi @ Com1_1 @ Com2_1 ) ) ).
thf(fact_648_com_Osimps_I46_J,axiom,
! [Com1_1: com,Com2_1: com,Fun: state > $o,Com: com] :
( ( semi @ Com1_1 @ Com2_1 )
!= ( while @ Fun @ Com ) ) ).
thf(fact_649_com_Osimps_I13_J,axiom,
! [Com1: com,Com2: com] :
( ( semi @ Com1 @ Com2 )
!= skip ) ).
thf(fact_650_com_Osimps_I12_J,axiom,
! [Com1: com,Com2: com] :
( skip
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_651_evalc__elim__cases_I4_J,axiom,
! [C1: com,C2: com,S: state,T: state] :
( ( evalc @ ( semi @ C1 @ C2 ) @ S @ T )
=> ~ ! [S1_1: state] :
( ( evalc @ C1 @ S @ S1_1 )
=> ~ ( evalc @ C2 @ S1_1 @ T ) ) ) ).
thf(fact_652_evaln__elim__cases_I4_J,axiom,
! [C1: com,C2: com,S: state,N_1: nat,T: state] :
( ( evaln @ ( semi @ C1 @ C2 ) @ S @ N_1 @ T )
=> ~ ! [S1_1: state] :
( ( evaln @ C1 @ S @ N_1 @ S1_1 )
=> ~ ( evaln @ C2 @ S1_1 @ N_1 @ T ) ) ) ).
thf(fact_653_finite__imageI,axiom,
! [H_1: pname > hoare_1708887482_state,F_35: pname > $o] :
( ( finite_finite_pname @ F_35 )
=> ( finite1625599783_state @ ( image_1116629049_state @ H_1 @ F_35 ) ) ) ).
thf(fact_654_finite__imageI,axiom,
! [H_1: nat > hoare_2091234717iple_a,F_35: nat > $o] :
( ( finite_finite_nat @ F_35 )
=> ( finite232261744iple_a @ ( image_359186840iple_a @ H_1 @ F_35 ) ) ) ).
thf(fact_655_finite__imageI,axiom,
! [H_1: nat > hoare_2091234717iple_a > $o,F_35: nat > $o] :
( ( finite_finite_nat @ F_35 )
=> ( finite1829014797le_a_o @ ( image_1995609573le_a_o @ H_1 @ F_35 ) ) ) ).
thf(fact_656_finite__imageI,axiom,
! [H_1: nat > pname,F_35: nat > $o] :
( ( finite_finite_nat @ F_35 )
=> ( finite_finite_pname @ ( image_nat_pname @ H_1 @ F_35 ) ) ) ).
thf(fact_657_finite__imageI,axiom,
! [H_1: nat > nat,F_35: nat > $o] :
( ( finite_finite_nat @ F_35 )
=> ( finite_finite_nat @ ( image_nat_nat @ H_1 @ F_35 ) ) ) ).
thf(fact_658_finite__imageI,axiom,
! [H_1: hoare_2091234717iple_a > nat,F_35: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_35 )
=> ( finite_finite_nat @ ( image_1773322034_a_nat @ H_1 @ F_35 ) ) ) ).
thf(fact_659_finite__imageI,axiom,
! [H_1: ( hoare_2091234717iple_a > $o ) > nat,F_35: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ F_35 )
=> ( finite_finite_nat @ ( image_75520503_o_nat @ H_1 @ F_35 ) ) ) ).
thf(fact_660_finite__imageI,axiom,
! [H_1: pname > nat,F_35: pname > $o] :
( ( finite_finite_pname @ F_35 )
=> ( finite_finite_nat @ ( image_pname_nat @ H_1 @ F_35 ) ) ) ).
thf(fact_661_finite__imageI,axiom,
! [H_1: pname > hoare_2091234717iple_a,F_35: pname > $o] :
( ( finite_finite_pname @ F_35 )
=> ( finite232261744iple_a @ ( image_231808478iple_a @ H_1 @ F_35 ) ) ) ).
thf(fact_662_finite_OinsertI,axiom,
! [A_51: hoare_2091234717iple_a > $o,A_50: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_50 )
=> ( finite1829014797le_a_o @ ( insert102003750le_a_o @ A_51 @ A_50 ) ) ) ).
thf(fact_663_finite_OinsertI,axiom,
! [A_51: pname,A_50: pname > $o] :
( ( finite_finite_pname @ A_50 )
=> ( finite_finite_pname @ ( insert_pname @ A_51 @ A_50 ) ) ) ).
thf(fact_664_finite_OinsertI,axiom,
! [A_51: hoare_2091234717iple_a,A_50: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_50 )
=> ( finite232261744iple_a @ ( insert1597628439iple_a @ A_51 @ A_50 ) ) ) ).
thf(fact_665_finite_OinsertI,axiom,
! [A_51: hoare_1708887482_state,A_50: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_50 )
=> ( finite1625599783_state @ ( insert528405184_state @ A_51 @ A_50 ) ) ) ).
thf(fact_666_finite_OinsertI,axiom,
! [A_51: nat,A_50: nat > $o] :
( ( finite_finite_nat @ A_50 )
=> ( finite_finite_nat @ ( insert_nat @ A_51 @ A_50 ) ) ) ).
thf(fact_667_finite_OemptyI,axiom,
finite232261744iple_a @ bot_bo1791335050le_a_o ).
thf(fact_668_finite_OemptyI,axiom,
finite1829014797le_a_o @ bot_bo1957696069_a_o_o ).
thf(fact_669_finite_OemptyI,axiom,
finite_finite_pname @ bot_bot_pname_o ).
thf(fact_670_finite_OemptyI,axiom,
finite1625599783_state @ bot_bo19817387tate_o ).
thf(fact_671_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_nat_o ).
thf(fact_672_finite__Collect__conjI,axiom,
! [Q_2: pname > $o,P_6: pname > $o] :
( ( ( finite_finite_pname @ ( collect_pname @ P_6 ) )
| ( finite_finite_pname @ ( collect_pname @ Q_2 ) ) )
=> ( finite_finite_pname
@ ( collect_pname
@ ^ [X: pname] : ( (&) @ ( P_6 @ X ) @ ( Q_2 @ X ) ) ) ) ) ).
thf(fact_673_finite__Collect__conjI,axiom,
! [Q_2: hoare_2091234717iple_a > $o,P_6: hoare_2091234717iple_a > $o] :
( ( ( finite232261744iple_a @ ( collec992574898iple_a @ P_6 ) )
| ( finite232261744iple_a @ ( collec992574898iple_a @ Q_2 ) ) )
=> ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( P_6 @ X ) @ ( Q_2 @ X ) ) ) ) ) ).
thf(fact_674_finite__Collect__conjI,axiom,
! [Q_2: ( hoare_2091234717iple_a > $o ) > $o,P_6: ( hoare_2091234717iple_a > $o ) > $o] :
( ( ( finite1829014797le_a_o @ ( collec1008234059le_a_o @ P_6 ) )
| ( finite1829014797le_a_o @ ( collec1008234059le_a_o @ Q_2 ) ) )
=> ( finite1829014797le_a_o
@ ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (&) @ ( P_6 @ X ) @ ( Q_2 @ X ) ) ) ) ) ).
thf(fact_675_finite__Collect__conjI,axiom,
! [Q_2: nat > $o,P_6: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P_6 ) )
| ( finite_finite_nat @ ( collect_nat @ Q_2 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( (&) @ ( P_6 @ X ) @ ( Q_2 @ X ) ) ) ) ) ).
thf(fact_676_finite__Un,axiom,
! [F_34: ( hoare_2091234717iple_a > $o ) > $o,G_5: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ ( semila2050116131_a_o_o @ F_34 @ G_5 ) )
<=> ( ( finite1829014797le_a_o @ F_34 )
& ( finite1829014797le_a_o @ G_5 ) ) ) ).
thf(fact_677_finite__Un,axiom,
! [F_34: pname > $o,G_5: pname > $o] :
( ( finite_finite_pname @ ( semila1780557381name_o @ F_34 @ G_5 ) )
<=> ( ( finite_finite_pname @ F_34 )
& ( finite_finite_pname @ G_5 ) ) ) ).
thf(fact_678_finite__Un,axiom,
! [F_34: hoare_1708887482_state > $o,G_5: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ ( semila1122118281tate_o @ F_34 @ G_5 ) )
<=> ( ( finite1625599783_state @ F_34 )
& ( finite1625599783_state @ G_5 ) ) ) ).
thf(fact_679_finite__Un,axiom,
! [F_34: hoare_2091234717iple_a > $o,G_5: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ ( semila1052848428le_a_o @ F_34 @ G_5 ) )
<=> ( ( finite232261744iple_a @ F_34 )
& ( finite232261744iple_a @ G_5 ) ) ) ).
thf(fact_680_finite__Un,axiom,
! [F_34: nat > $o,G_5: nat > $o] :
( ( finite_finite_nat @ ( semila848761471_nat_o @ F_34 @ G_5 ) )
<=> ( ( finite_finite_nat @ F_34 )
& ( finite_finite_nat @ G_5 ) ) ) ).
thf(fact_681_finite__UnI,axiom,
! [G_4: ( hoare_2091234717iple_a > $o ) > $o,F_33: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ F_33 )
=> ( ( finite1829014797le_a_o @ G_4 )
=> ( finite1829014797le_a_o @ ( semila2050116131_a_o_o @ F_33 @ G_4 ) ) ) ) ).
thf(fact_682_finite__UnI,axiom,
! [G_4: pname > $o,F_33: pname > $o] :
( ( finite_finite_pname @ F_33 )
=> ( ( finite_finite_pname @ G_4 )
=> ( finite_finite_pname @ ( semila1780557381name_o @ F_33 @ G_4 ) ) ) ) ).
thf(fact_683_finite__UnI,axiom,
! [G_4: hoare_1708887482_state > $o,F_33: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_33 )
=> ( ( finite1625599783_state @ G_4 )
=> ( finite1625599783_state @ ( semila1122118281tate_o @ F_33 @ G_4 ) ) ) ) ).
thf(fact_684_finite__UnI,axiom,
! [G_4: hoare_2091234717iple_a > $o,F_33: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_33 )
=> ( ( finite232261744iple_a @ G_4 )
=> ( finite232261744iple_a @ ( semila1052848428le_a_o @ F_33 @ G_4 ) ) ) ) ).
thf(fact_685_finite__UnI,axiom,
! [G_4: nat > $o,F_33: nat > $o] :
( ( finite_finite_nat @ F_33 )
=> ( ( finite_finite_nat @ G_4 )
=> ( finite_finite_nat @ ( semila848761471_nat_o @ F_33 @ G_4 ) ) ) ) ).
thf(fact_686_finite__Collect__disjI,axiom,
! [P_5: pname > $o,Q_1: pname > $o] :
( ( finite_finite_pname
@ ( collect_pname
@ ^ [X: pname] : ( (|) @ ( P_5 @ X ) @ ( Q_1 @ X ) ) ) )
<=> ( ( finite_finite_pname @ ( collect_pname @ P_5 ) )
& ( finite_finite_pname @ ( collect_pname @ Q_1 ) ) ) ) ).
thf(fact_687_finite__Collect__disjI,axiom,
! [P_5: hoare_2091234717iple_a > $o,Q_1: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (|) @ ( P_5 @ X ) @ ( Q_1 @ X ) ) ) )
<=> ( ( finite232261744iple_a @ ( collec992574898iple_a @ P_5 ) )
& ( finite232261744iple_a @ ( collec992574898iple_a @ Q_1 ) ) ) ) ).
thf(fact_688_finite__Collect__disjI,axiom,
! [P_5: ( hoare_2091234717iple_a > $o ) > $o,Q_1: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o
@ ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (|) @ ( P_5 @ X ) @ ( Q_1 @ X ) ) ) )
<=> ( ( finite1829014797le_a_o @ ( collec1008234059le_a_o @ P_5 ) )
& ( finite1829014797le_a_o @ ( collec1008234059le_a_o @ Q_1 ) ) ) ) ).
thf(fact_689_finite__Collect__disjI,axiom,
! [P_5: nat > $o,Q_1: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( (|) @ ( P_5 @ X ) @ ( Q_1 @ X ) ) ) )
<=> ( ( finite_finite_nat @ ( collect_nat @ P_5 ) )
& ( finite_finite_nat @ ( collect_nat @ Q_1 ) ) ) ) ).
thf(fact_690_finite__insert,axiom,
! [A_49: hoare_2091234717iple_a > $o,A_48: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ ( insert102003750le_a_o @ A_49 @ A_48 ) )
<=> ( finite1829014797le_a_o @ A_48 ) ) ).
thf(fact_691_finite__insert,axiom,
! [A_49: pname,A_48: pname > $o] :
( ( finite_finite_pname @ ( insert_pname @ A_49 @ A_48 ) )
<=> ( finite_finite_pname @ A_48 ) ) ).
thf(fact_692_finite__insert,axiom,
! [A_49: hoare_2091234717iple_a,A_48: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ ( insert1597628439iple_a @ A_49 @ A_48 ) )
<=> ( finite232261744iple_a @ A_48 ) ) ).
thf(fact_693_finite__insert,axiom,
! [A_49: hoare_1708887482_state,A_48: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ ( insert528405184_state @ A_49 @ A_48 ) )
<=> ( finite1625599783_state @ A_48 ) ) ).
thf(fact_694_finite__insert,axiom,
! [A_49: nat,A_48: nat > $o] :
( ( finite_finite_nat @ ( insert_nat @ A_49 @ A_48 ) )
<=> ( finite_finite_nat @ A_48 ) ) ).
thf(fact_695_finite__induct,axiom,
! [P_4: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o,F_32: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ F_32 )
=> ( ( P_4 @ bot_bo1957696069_a_o_o )
=> ( ! [X: hoare_2091234717iple_a > $o,F_25: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ F_25 )
=> ( ~ ( member99268621le_a_o @ X @ F_25 )
=> ( ( P_4 @ F_25 )
=> ( P_4 @ ( insert102003750le_a_o @ X @ F_25 ) ) ) ) )
=> ( P_4 @ F_32 ) ) ) ) ).
thf(fact_696_finite__induct,axiom,
! [P_4: ( hoare_2091234717iple_a > $o ) > $o,F_32: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_32 )
=> ( ( P_4 @ bot_bo1791335050le_a_o )
=> ( ! [X: hoare_2091234717iple_a,F_25: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_25 )
=> ( ~ ( member290856304iple_a @ X @ F_25 )
=> ( ( P_4 @ F_25 )
=> ( P_4 @ ( insert1597628439iple_a @ X @ F_25 ) ) ) ) )
=> ( P_4 @ F_32 ) ) ) ) ).
thf(fact_697_finite__induct,axiom,
! [P_4: ( hoare_1708887482_state > $o ) > $o,F_32: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_32 )
=> ( ( P_4 @ bot_bo19817387tate_o )
=> ( ! [X: hoare_1708887482_state,F_25: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_25 )
=> ( ~ ( member451959335_state @ X @ F_25 )
=> ( ( P_4 @ F_25 )
=> ( P_4 @ ( insert528405184_state @ X @ F_25 ) ) ) ) )
=> ( P_4 @ F_32 ) ) ) ) ).
thf(fact_698_finite__induct,axiom,
! [P_4: ( nat > $o ) > $o,F_32: nat > $o] :
( ( finite_finite_nat @ F_32 )
=> ( ( P_4 @ bot_bot_nat_o )
=> ( ! [X: nat,F_25: nat > $o] :
( ( finite_finite_nat @ F_25 )
=> ( ~ ( member_nat @ X @ F_25 )
=> ( ( P_4 @ F_25 )
=> ( P_4 @ ( insert_nat @ X @ F_25 ) ) ) ) )
=> ( P_4 @ F_32 ) ) ) ) ).
thf(fact_699_finite__induct,axiom,
! [P_4: ( pname > $o ) > $o,F_32: pname > $o] :
( ( finite_finite_pname @ F_32 )
=> ( ( P_4 @ bot_bot_pname_o )
=> ( ! [X: pname,F_25: pname > $o] :
( ( finite_finite_pname @ F_25 )
=> ( ~ ( member_pname @ X @ F_25 )
=> ( ( P_4 @ F_25 )
=> ( P_4 @ ( insert_pname @ X @ F_25 ) ) ) ) )
=> ( P_4 @ F_32 ) ) ) ) ).
thf(fact_700_finite_Osimps,axiom,
! [A_46: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_46 )
<=> ( ( A_46 = bot_bo1957696069_a_o_o )
| ? [A_47: ( hoare_2091234717iple_a > $o ) > $o,A_45: hoare_2091234717iple_a > $o] :
( ( A_46
= ( insert102003750le_a_o @ A_45 @ A_47 ) )
& ( finite1829014797le_a_o @ A_47 ) ) ) ) ).
thf(fact_701_finite_Osimps,axiom,
! [A_46: pname > $o] :
( ( finite_finite_pname @ A_46 )
<=> ( ( A_46 = bot_bot_pname_o )
| ? [A_47: pname > $o,A_45: pname] :
( ( A_46
= ( insert_pname @ A_45 @ A_47 ) )
& ( finite_finite_pname @ A_47 ) ) ) ) ).
thf(fact_702_finite_Osimps,axiom,
! [A_46: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_46 )
<=> ( ( A_46 = bot_bo1791335050le_a_o )
| ? [A_47: hoare_2091234717iple_a > $o,A_45: hoare_2091234717iple_a] :
( ( A_46
= ( insert1597628439iple_a @ A_45 @ A_47 ) )
& ( finite232261744iple_a @ A_47 ) ) ) ) ).
thf(fact_703_finite_Osimps,axiom,
! [A_46: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_46 )
<=> ( ( A_46 = bot_bo19817387tate_o )
| ? [A_47: hoare_1708887482_state > $o,A_45: hoare_1708887482_state] :
( ( A_46
= ( insert528405184_state @ A_45 @ A_47 ) )
& ( finite1625599783_state @ A_47 ) ) ) ) ).
thf(fact_704_finite_Osimps,axiom,
! [A_46: nat > $o] :
( ( finite_finite_nat @ A_46 )
<=> ( ( A_46 = bot_bot_nat_o )
| ? [A_47: nat > $o,A_45: nat] :
( ( A_46
= ( insert_nat @ A_45 @ A_47 ) )
& ( finite_finite_nat @ A_47 ) ) ) ) ).
thf(fact_705_pigeonhole__infinite,axiom,
! [F_31: ( hoare_2091234717iple_a > $o ) > nat,A_44: ( hoare_2091234717iple_a > $o ) > $o] :
( ~ ( finite1829014797le_a_o @ A_44 )
=> ( ( finite_finite_nat @ ( image_75520503_o_nat @ F_31 @ A_44 ) )
=> ? [X: hoare_2091234717iple_a > $o] :
( ( member99268621le_a_o @ X @ A_44 )
& ~ ( finite1829014797le_a_o
@ ( collec1008234059le_a_o
@ ^ [A_45: hoare_2091234717iple_a > $o] :
( (&) @ ( member99268621le_a_o @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_706_pigeonhole__infinite,axiom,
! [F_31: hoare_2091234717iple_a > nat,A_44: hoare_2091234717iple_a > $o] :
( ~ ( finite232261744iple_a @ A_44 )
=> ( ( finite_finite_nat @ ( image_1773322034_a_nat @ F_31 @ A_44 ) )
=> ? [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ A_44 )
& ~ ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [A_45: hoare_2091234717iple_a] :
( (&) @ ( member290856304iple_a @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_707_pigeonhole__infinite,axiom,
! [F_31: pname > nat,A_44: pname > $o] :
( ~ ( finite_finite_pname @ A_44 )
=> ( ( finite_finite_nat @ ( image_pname_nat @ F_31 @ A_44 ) )
=> ? [X: pname] :
( ( member_pname @ X @ A_44 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_45: pname] :
( (&) @ ( member_pname @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_708_pigeonhole__infinite,axiom,
! [F_31: nat > nat,A_44: nat > $o] :
( ~ ( finite_finite_nat @ A_44 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F_31 @ A_44 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A_44 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_45: nat] :
( (&) @ ( member_nat @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_709_pigeonhole__infinite,axiom,
! [F_31: nat > hoare_2091234717iple_a,A_44: nat > $o] :
( ~ ( finite_finite_nat @ A_44 )
=> ( ( finite232261744iple_a @ ( image_359186840iple_a @ F_31 @ A_44 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A_44 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_45: nat] :
( (&) @ ( member_nat @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_710_pigeonhole__infinite,axiom,
! [F_31: nat > hoare_2091234717iple_a > $o,A_44: nat > $o] :
( ~ ( finite_finite_nat @ A_44 )
=> ( ( finite1829014797le_a_o @ ( image_1995609573le_a_o @ F_31 @ A_44 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A_44 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_45: nat] :
( (&) @ ( member_nat @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_711_pigeonhole__infinite,axiom,
! [F_31: nat > pname,A_44: nat > $o] :
( ~ ( finite_finite_nat @ A_44 )
=> ( ( finite_finite_pname @ ( image_nat_pname @ F_31 @ A_44 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A_44 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_45: nat] :
( (&) @ ( member_nat @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_712_pigeonhole__infinite,axiom,
! [F_31: pname > hoare_1708887482_state,A_44: pname > $o] :
( ~ ( finite_finite_pname @ A_44 )
=> ( ( finite1625599783_state @ ( image_1116629049_state @ F_31 @ A_44 ) )
=> ? [X: pname] :
( ( member_pname @ X @ A_44 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_45: pname] :
( (&) @ ( member_pname @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_713_pigeonhole__infinite,axiom,
! [F_31: pname > hoare_2091234717iple_a > $o,A_44: pname > $o] :
( ~ ( finite_finite_pname @ A_44 )
=> ( ( finite1829014797le_a_o @ ( image_742317343le_a_o @ F_31 @ A_44 ) )
=> ? [X: pname] :
( ( member_pname @ X @ A_44 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_45: pname] :
( (&) @ ( member_pname @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_714_pigeonhole__infinite,axiom,
! [F_31: pname > pname,A_44: pname > $o] :
( ~ ( finite_finite_pname @ A_44 )
=> ( ( finite_finite_pname @ ( image_pname_pname @ F_31 @ A_44 ) )
=> ? [X: pname] :
( ( member_pname @ X @ A_44 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_45: pname] :
( (&) @ ( member_pname @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_715_pigeonhole__infinite,axiom,
! [F_31: pname > hoare_2091234717iple_a,A_44: pname > $o] :
( ~ ( finite_finite_pname @ A_44 )
=> ( ( finite232261744iple_a @ ( image_231808478iple_a @ F_31 @ A_44 ) )
=> ? [X: pname] :
( ( member_pname @ X @ A_44 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_45: pname] :
( (&) @ ( member_pname @ A_45 @ A_44 )
@ ( ( F_31 @ A_45 )
= ( F_31 @ X ) ) ) ) ) ) ) ) ).
thf(fact_716_nonempty__iff,axiom,
! [A_43: nat > $o] :
( ( A_43 != bot_bot_nat_o )
<=> ? [X: nat,B_26: nat > $o] :
( ( A_43
= ( insert_nat @ X @ B_26 ) )
& ~ ( member_nat @ X @ B_26 ) ) ) ).
thf(fact_717_nonempty__iff,axiom,
! [A_43: ( hoare_2091234717iple_a > $o ) > $o] :
( ( A_43 != bot_bo1957696069_a_o_o )
<=> ? [X: hoare_2091234717iple_a > $o,B_26: ( hoare_2091234717iple_a > $o ) > $o] :
( ( A_43
= ( insert102003750le_a_o @ X @ B_26 ) )
& ~ ( member99268621le_a_o @ X @ B_26 ) ) ) ).
thf(fact_718_nonempty__iff,axiom,
! [A_43: hoare_2091234717iple_a > $o] :
( ( A_43 != bot_bo1791335050le_a_o )
<=> ? [X: hoare_2091234717iple_a,B_26: hoare_2091234717iple_a > $o] :
( ( A_43
= ( insert1597628439iple_a @ X @ B_26 ) )
& ~ ( member290856304iple_a @ X @ B_26 ) ) ) ).
thf(fact_719_nonempty__iff,axiom,
! [A_43: hoare_1708887482_state > $o] :
( ( A_43 != bot_bo19817387tate_o )
<=> ? [X: hoare_1708887482_state,B_26: hoare_1708887482_state > $o] :
( ( A_43
= ( insert528405184_state @ X @ B_26 ) )
& ~ ( member451959335_state @ X @ B_26 ) ) ) ).
thf(fact_720_nonempty__iff,axiom,
! [A_43: pname > $o] :
( ( A_43 != bot_bot_pname_o )
<=> ? [X: pname,B_26: pname > $o] :
( ( A_43
= ( insert_pname @ X @ B_26 ) )
& ~ ( member_pname @ X @ B_26 ) ) ) ).
thf(fact_721_folding__one__idem_Ounion__idem,axiom,
! [B_27: ( hoare_2091234717iple_a > $o ) > $o,A_42: ( hoare_2091234717iple_a > $o ) > $o,F_30: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_29: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite574580006le_a_o @ F_30 @ F_29 )
=> ( ( finite1829014797le_a_o @ A_42 )
=> ( ( A_42 != bot_bo1957696069_a_o_o )
=> ( ( finite1829014797le_a_o @ B_27 )
=> ( ( B_27 != bot_bo1957696069_a_o_o )
=> ( ( F_29 @ ( semila2050116131_a_o_o @ A_42 @ B_27 ) )
= ( F_30 @ ( F_29 @ A_42 ) @ ( F_29 @ B_27 ) ) ) ) ) ) ) ) ).
thf(fact_722_folding__one__idem_Ounion__idem,axiom,
! [B_27: pname > $o,A_42: pname > $o,F_30: pname > pname > pname,F_29: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_30 @ F_29 )
=> ( ( finite_finite_pname @ A_42 )
=> ( ( A_42 != bot_bot_pname_o )
=> ( ( finite_finite_pname @ B_27 )
=> ( ( B_27 != bot_bot_pname_o )
=> ( ( F_29 @ ( semila1780557381name_o @ A_42 @ B_27 ) )
= ( F_30 @ ( F_29 @ A_42 ) @ ( F_29 @ B_27 ) ) ) ) ) ) ) ) ).
thf(fact_723_folding__one__idem_Ounion__idem,axiom,
! [B_27: hoare_1708887482_state > $o,A_42: hoare_1708887482_state > $o,F_30: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_29: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_30 @ F_29 )
=> ( ( finite1625599783_state @ A_42 )
=> ( ( A_42 != bot_bo19817387tate_o )
=> ( ( finite1625599783_state @ B_27 )
=> ( ( B_27 != bot_bo19817387tate_o )
=> ( ( F_29 @ ( semila1122118281tate_o @ A_42 @ B_27 ) )
= ( F_30 @ ( F_29 @ A_42 ) @ ( F_29 @ B_27 ) ) ) ) ) ) ) ) ).
thf(fact_724_folding__one__idem_Ounion__idem,axiom,
! [B_27: hoare_2091234717iple_a > $o,A_42: hoare_2091234717iple_a > $o,F_30: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_29: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_30 @ F_29 )
=> ( ( finite232261744iple_a @ A_42 )
=> ( ( A_42 != bot_bo1791335050le_a_o )
=> ( ( finite232261744iple_a @ B_27 )
=> ( ( B_27 != bot_bo1791335050le_a_o )
=> ( ( F_29 @ ( semila1052848428le_a_o @ A_42 @ B_27 ) )
= ( F_30 @ ( F_29 @ A_42 ) @ ( F_29 @ B_27 ) ) ) ) ) ) ) ) ).
thf(fact_725_folding__one__idem_Ounion__idem,axiom,
! [B_27: nat > $o,A_42: nat > $o,F_30: nat > nat > nat,F_29: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_30 @ F_29 )
=> ( ( finite_finite_nat @ A_42 )
=> ( ( A_42 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ B_27 )
=> ( ( B_27 != bot_bot_nat_o )
=> ( ( F_29 @ ( semila848761471_nat_o @ A_42 @ B_27 ) )
= ( F_30 @ ( F_29 @ A_42 ) @ ( F_29 @ B_27 ) ) ) ) ) ) ) ) ).
thf(fact_726_folding__one__idem_Oinsert__idem,axiom,
! [X_20: hoare_2091234717iple_a > $o,A_41: ( hoare_2091234717iple_a > $o ) > $o,F_28: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_27: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite574580006le_a_o @ F_28 @ F_27 )
=> ( ( finite1829014797le_a_o @ A_41 )
=> ( ( A_41 != bot_bo1957696069_a_o_o )
=> ( ( F_27 @ ( insert102003750le_a_o @ X_20 @ A_41 ) )
= ( F_28 @ X_20 @ ( F_27 @ A_41 ) ) ) ) ) ) ).
thf(fact_727_folding__one__idem_Oinsert__idem,axiom,
! [X_20: pname,A_41: pname > $o,F_28: pname > pname > pname,F_27: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_28 @ F_27 )
=> ( ( finite_finite_pname @ A_41 )
=> ( ( A_41 != bot_bot_pname_o )
=> ( ( F_27 @ ( insert_pname @ X_20 @ A_41 ) )
= ( F_28 @ X_20 @ ( F_27 @ A_41 ) ) ) ) ) ) ).
thf(fact_728_folding__one__idem_Oinsert__idem,axiom,
! [X_20: hoare_2091234717iple_a,A_41: hoare_2091234717iple_a > $o,F_28: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_27: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_28 @ F_27 )
=> ( ( finite232261744iple_a @ A_41 )
=> ( ( A_41 != bot_bo1791335050le_a_o )
=> ( ( F_27 @ ( insert1597628439iple_a @ X_20 @ A_41 ) )
= ( F_28 @ X_20 @ ( F_27 @ A_41 ) ) ) ) ) ) ).
thf(fact_729_folding__one__idem_Oinsert__idem,axiom,
! [X_20: hoare_1708887482_state,A_41: hoare_1708887482_state > $o,F_28: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_27: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_28 @ F_27 )
=> ( ( finite1625599783_state @ A_41 )
=> ( ( A_41 != bot_bo19817387tate_o )
=> ( ( F_27 @ ( insert528405184_state @ X_20 @ A_41 ) )
= ( F_28 @ X_20 @ ( F_27 @ A_41 ) ) ) ) ) ) ).
thf(fact_730_folding__one__idem_Oinsert__idem,axiom,
! [X_20: nat,A_41: nat > $o,F_28: nat > nat > nat,F_27: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_28 @ F_27 )
=> ( ( finite_finite_nat @ A_41 )
=> ( ( A_41 != bot_bot_nat_o )
=> ( ( F_27 @ ( insert_nat @ X_20 @ A_41 ) )
= ( F_28 @ X_20 @ ( F_27 @ A_41 ) ) ) ) ) ) ).
thf(fact_731_image__eq__fold__image,axiom,
! [F_26: pname > hoare_1708887482_state,A_40: pname > $o] :
( ( finite_finite_pname @ A_40 )
=> ( ( image_1116629049_state @ F_26 @ A_40 )
= ( finite2139561282_pname @ semila1122118281tate_o
@ ^ [X: pname] : ( insert528405184_state @ ( F_26 @ X ) @ bot_bo19817387tate_o )
@ bot_bo19817387tate_o
@ A_40 ) ) ) ).
thf(fact_732_image__eq__fold__image,axiom,
! [F_26: hoare_2091234717iple_a > hoare_2091234717iple_a,A_40: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_40 )
=> ( ( image_1661191109iple_a @ F_26 @ A_40 )
= ( finite1481787452iple_a @ semila1052848428le_a_o
@ ^ [X: hoare_2091234717iple_a] : ( insert1597628439iple_a @ ( F_26 @ X ) @ bot_bo1791335050le_a_o )
@ bot_bo1791335050le_a_o
@ A_40 ) ) ) ).
thf(fact_733_image__eq__fold__image,axiom,
! [F_26: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a,A_40: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_40 )
=> ( ( image_136408202iple_a @ F_26 @ A_40 )
= ( finite903029825le_a_o @ semila1052848428le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( insert1597628439iple_a @ ( F_26 @ X ) @ bot_bo1791335050le_a_o )
@ bot_bo1791335050le_a_o
@ A_40 ) ) ) ).
thf(fact_734_image__eq__fold__image,axiom,
! [F_26: nat > nat,A_40: nat > $o] :
( ( finite_finite_nat @ A_40 )
=> ( ( image_nat_nat @ F_26 @ A_40 )
= ( finite141655318_o_nat @ semila848761471_nat_o
@ ^ [X: nat] : ( insert_nat @ ( F_26 @ X ) @ bot_bot_nat_o )
@ bot_bot_nat_o
@ A_40 ) ) ) ).
thf(fact_735_image__eq__fold__image,axiom,
! [F_26: nat > hoare_2091234717iple_a > $o,A_40: nat > $o] :
( ( finite_finite_nat @ A_40 )
=> ( ( image_1995609573le_a_o @ F_26 @ A_40 )
= ( finite2009943022_o_nat @ semila2050116131_a_o_o
@ ^ [X: nat] : ( insert102003750le_a_o @ ( F_26 @ X ) @ bot_bo1957696069_a_o_o )
@ bot_bo1957696069_a_o_o
@ A_40 ) ) ) ).
thf(fact_736_image__eq__fold__image,axiom,
! [F_26: nat > pname,A_40: nat > $o] :
( ( finite_finite_nat @ A_40 )
=> ( ( image_nat_pname @ F_26 @ A_40 )
= ( finite1427591632_o_nat @ semila1780557381name_o
@ ^ [X: nat] : ( insert_pname @ ( F_26 @ X ) @ bot_bot_pname_o )
@ bot_bot_pname_o
@ A_40 ) ) ) ).
thf(fact_737_image__eq__fold__image,axiom,
! [F_26: nat > hoare_2091234717iple_a,A_40: nat > $o] :
( ( finite_finite_nat @ A_40 )
=> ( ( image_359186840iple_a @ F_26 @ A_40 )
= ( finite2100865449_o_nat @ semila1052848428le_a_o
@ ^ [X: nat] : ( insert1597628439iple_a @ ( F_26 @ X ) @ bot_bo1791335050le_a_o )
@ bot_bo1791335050le_a_o
@ A_40 ) ) ) ).
thf(fact_738_image__eq__fold__image,axiom,
! [F_26: nat > hoare_1708887482_state,A_40: nat > $o] :
( ( finite_finite_nat @ A_40 )
=> ( ( image_514827263_state @ F_26 @ A_40 )
= ( finite1400355848_o_nat @ semila1122118281tate_o
@ ^ [X: nat] : ( insert528405184_state @ ( F_26 @ X ) @ bot_bo19817387tate_o )
@ bot_bo19817387tate_o
@ A_40 ) ) ) ).
thf(fact_739_image__eq__fold__image,axiom,
! [F_26: pname > hoare_2091234717iple_a,A_40: pname > $o] :
( ( finite_finite_pname @ A_40 )
=> ( ( image_231808478iple_a @ F_26 @ A_40 )
= ( finite1290357347_pname @ semila1052848428le_a_o
@ ^ [X: pname] : ( insert1597628439iple_a @ ( F_26 @ X ) @ bot_bo1791335050le_a_o )
@ bot_bo1791335050le_a_o
@ A_40 ) ) ) ).
thf(fact_740_finite__ne__induct,axiom,
! [P_3: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o,F_24: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ F_24 )
=> ( ( F_24 != bot_bo1957696069_a_o_o )
=> ( ! [X: hoare_2091234717iple_a > $o] : ( P_3 @ ( insert102003750le_a_o @ X @ bot_bo1957696069_a_o_o ) )
=> ( ! [X: hoare_2091234717iple_a > $o,F_25: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ F_25 )
=> ( ( F_25 != bot_bo1957696069_a_o_o )
=> ( ~ ( member99268621le_a_o @ X @ F_25 )
=> ( ( P_3 @ F_25 )
=> ( P_3 @ ( insert102003750le_a_o @ X @ F_25 ) ) ) ) ) )
=> ( P_3 @ F_24 ) ) ) ) ) ).
thf(fact_741_finite__ne__induct,axiom,
! [P_3: ( hoare_2091234717iple_a > $o ) > $o,F_24: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_24 )
=> ( ( F_24 != bot_bo1791335050le_a_o )
=> ( ! [X: hoare_2091234717iple_a] : ( P_3 @ ( insert1597628439iple_a @ X @ bot_bo1791335050le_a_o ) )
=> ( ! [X: hoare_2091234717iple_a,F_25: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_25 )
=> ( ( F_25 != bot_bo1791335050le_a_o )
=> ( ~ ( member290856304iple_a @ X @ F_25 )
=> ( ( P_3 @ F_25 )
=> ( P_3 @ ( insert1597628439iple_a @ X @ F_25 ) ) ) ) ) )
=> ( P_3 @ F_24 ) ) ) ) ) ).
thf(fact_742_finite__ne__induct,axiom,
! [P_3: ( hoare_1708887482_state > $o ) > $o,F_24: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_24 )
=> ( ( F_24 != bot_bo19817387tate_o )
=> ( ! [X: hoare_1708887482_state] : ( P_3 @ ( insert528405184_state @ X @ bot_bo19817387tate_o ) )
=> ( ! [X: hoare_1708887482_state,F_25: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_25 )
=> ( ( F_25 != bot_bo19817387tate_o )
=> ( ~ ( member451959335_state @ X @ F_25 )
=> ( ( P_3 @ F_25 )
=> ( P_3 @ ( insert528405184_state @ X @ F_25 ) ) ) ) ) )
=> ( P_3 @ F_24 ) ) ) ) ) ).
thf(fact_743_finite__ne__induct,axiom,
! [P_3: ( nat > $o ) > $o,F_24: nat > $o] :
( ( finite_finite_nat @ F_24 )
=> ( ( F_24 != bot_bot_nat_o )
=> ( ! [X: nat] : ( P_3 @ ( insert_nat @ X @ bot_bot_nat_o ) )
=> ( ! [X: nat,F_25: nat > $o] :
( ( finite_finite_nat @ F_25 )
=> ( ( F_25 != bot_bot_nat_o )
=> ( ~ ( member_nat @ X @ F_25 )
=> ( ( P_3 @ F_25 )
=> ( P_3 @ ( insert_nat @ X @ F_25 ) ) ) ) ) )
=> ( P_3 @ F_24 ) ) ) ) ) ).
thf(fact_744_finite__ne__induct,axiom,
! [P_3: ( pname > $o ) > $o,F_24: pname > $o] :
( ( finite_finite_pname @ F_24 )
=> ( ( F_24 != bot_bot_pname_o )
=> ( ! [X: pname] : ( P_3 @ ( insert_pname @ X @ bot_bot_pname_o ) )
=> ( ! [X: pname,F_25: pname > $o] :
( ( finite_finite_pname @ F_25 )
=> ( ( F_25 != bot_bot_pname_o )
=> ( ~ ( member_pname @ X @ F_25 )
=> ( ( P_3 @ F_25 )
=> ( P_3 @ ( insert_pname @ X @ F_25 ) ) ) ) ) )
=> ( P_3 @ F_24 ) ) ) ) ) ).
thf(fact_745_folding__one__idem_Oidem,axiom,
! [X_19: nat,F_23: nat > nat > nat,F_22: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_23 @ F_22 )
=> ( ( F_23 @ X_19 @ X_19 )
= X_19 ) ) ).
thf(fact_746_folding__one__idem_Oidem,axiom,
! [X_19: pname,F_23: pname > pname > pname,F_22: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_23 @ F_22 )
=> ( ( F_23 @ X_19 @ X_19 )
= X_19 ) ) ).
thf(fact_747_folding__one__idem_Oidem,axiom,
! [X_19: hoare_2091234717iple_a,F_23: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_22: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_23 @ F_22 )
=> ( ( F_23 @ X_19 @ X_19 )
= X_19 ) ) ).
thf(fact_748_fold__image__empty,axiom,
! [F_21: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,G_3: pname > hoare_2091234717iple_a > $o,Z_4: hoare_2091234717iple_a > $o] :
( ( finite1290357347_pname @ F_21 @ G_3 @ Z_4 @ bot_bot_pname_o )
= Z_4 ) ).
thf(fact_749_folding__one__idem_Oin__idem,axiom,
! [X_18: hoare_2091234717iple_a > $o,A_39: ( hoare_2091234717iple_a > $o ) > $o,F_20: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_19: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite574580006le_a_o @ F_20 @ F_19 )
=> ( ( finite1829014797le_a_o @ A_39 )
=> ( ( member99268621le_a_o @ X_18 @ A_39 )
=> ( ( F_20 @ X_18 @ ( F_19 @ A_39 ) )
= ( F_19 @ A_39 ) ) ) ) ) ).
thf(fact_750_folding__one__idem_Oin__idem,axiom,
! [X_18: hoare_2091234717iple_a,A_39: hoare_2091234717iple_a > $o,F_20: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_19: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_20 @ F_19 )
=> ( ( finite232261744iple_a @ A_39 )
=> ( ( member290856304iple_a @ X_18 @ A_39 )
=> ( ( F_20 @ X_18 @ ( F_19 @ A_39 ) )
= ( F_19 @ A_39 ) ) ) ) ) ).
thf(fact_751_folding__one__idem_Oin__idem,axiom,
! [X_18: nat,A_39: nat > $o,F_20: nat > nat > nat,F_19: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_20 @ F_19 )
=> ( ( finite_finite_nat @ A_39 )
=> ( ( member_nat @ X_18 @ A_39 )
=> ( ( F_20 @ X_18 @ ( F_19 @ A_39 ) )
= ( F_19 @ A_39 ) ) ) ) ) ).
thf(fact_752_folding__one__idem_Oin__idem,axiom,
! [X_18: pname,A_39: pname > $o,F_20: pname > pname > pname,F_19: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_20 @ F_19 )
=> ( ( finite_finite_pname @ A_39 )
=> ( ( member_pname @ X_18 @ A_39 )
=> ( ( F_20 @ X_18 @ ( F_19 @ A_39 ) )
= ( F_19 @ A_39 ) ) ) ) ) ).
thf(fact_753_folding__one__idem_Ohom__commute,axiom,
! [N_3: hoare_2091234717iple_a > $o,H: hoare_2091234717iple_a > hoare_2091234717iple_a,F_18: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_17: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_18 @ F_17 )
=> ( ! [X: hoare_2091234717iple_a,Y_7: hoare_2091234717iple_a] :
( ( H @ ( F_18 @ X @ Y_7 ) )
= ( F_18 @ ( H @ X ) @ ( H @ Y_7 ) ) )
=> ( ( finite232261744iple_a @ N_3 )
=> ( ( N_3 != bot_bo1791335050le_a_o )
=> ( ( H @ ( F_17 @ N_3 ) )
= ( F_17 @ ( image_1661191109iple_a @ H @ N_3 ) ) ) ) ) ) ) ).
thf(fact_754_folding__one__idem_Ohom__commute,axiom,
! [N_3: ( hoare_2091234717iple_a > $o ) > $o,H: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_18: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_17: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite574580006le_a_o @ F_18 @ F_17 )
=> ( ! [X: hoare_2091234717iple_a > $o,Y_7: hoare_2091234717iple_a > $o] :
( ( H @ ( F_18 @ X @ Y_7 ) )
= ( F_18 @ ( H @ X ) @ ( H @ Y_7 ) ) )
=> ( ( finite1829014797le_a_o @ N_3 )
=> ( ( N_3 != bot_bo1957696069_a_o_o )
=> ( ( H @ ( F_17 @ N_3 ) )
= ( F_17 @ ( image_784579955le_a_o @ H @ N_3 ) ) ) ) ) ) ) ).
thf(fact_755_folding__one__idem_Ohom__commute,axiom,
! [N_3: pname > $o,H: pname > pname,F_18: pname > pname > pname,F_17: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_18 @ F_17 )
=> ( ! [X: pname,Y_7: pname] :
( ( H @ ( F_18 @ X @ Y_7 ) )
= ( F_18 @ ( H @ X ) @ ( H @ Y_7 ) ) )
=> ( ( finite_finite_pname @ N_3 )
=> ( ( N_3 != bot_bot_pname_o )
=> ( ( H @ ( F_17 @ N_3 ) )
= ( F_17 @ ( image_pname_pname @ H @ N_3 ) ) ) ) ) ) ) ).
thf(fact_756_folding__one__idem_Ohom__commute,axiom,
! [N_3: hoare_1708887482_state > $o,H: hoare_1708887482_state > hoare_1708887482_state,F_18: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_17: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_18 @ F_17 )
=> ( ! [X: hoare_1708887482_state,Y_7: hoare_1708887482_state] :
( ( H @ ( F_18 @ X @ Y_7 ) )
= ( F_18 @ ( H @ X ) @ ( H @ Y_7 ) ) )
=> ( ( finite1625599783_state @ N_3 )
=> ( ( N_3 != bot_bo19817387tate_o )
=> ( ( H @ ( F_17 @ N_3 ) )
= ( F_17 @ ( image_757158439_state @ H @ N_3 ) ) ) ) ) ) ) ).
thf(fact_757_folding__one__idem_Ohom__commute,axiom,
! [N_3: nat > $o,H: nat > nat,F_18: nat > nat > nat,F_17: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_18 @ F_17 )
=> ( ! [X: nat,Y_7: nat] :
( ( H @ ( F_18 @ X @ Y_7 ) )
= ( F_18 @ ( H @ X ) @ ( H @ Y_7 ) ) )
=> ( ( finite_finite_nat @ N_3 )
=> ( ( N_3 != bot_bot_nat_o )
=> ( ( H @ ( F_17 @ N_3 ) )
= ( F_17 @ ( image_nat_nat @ H @ N_3 ) ) ) ) ) ) ) ).
thf(fact_758_comm__monoid__big_OF__eq,axiom,
! [G_2: pname > hoare_2091234717iple_a > $o,A_38: pname > $o,F_16: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,Z_3: hoare_2091234717iple_a > $o,F_15: ( pname > hoare_2091234717iple_a > $o ) > ( pname > $o ) > hoare_2091234717iple_a > $o] :
( ( big_co1924420859_pname @ F_16 @ Z_3 @ F_15 )
=> ( ( ( finite_finite_pname @ A_38 )
=> ( ( F_15 @ G_2 @ A_38 )
= ( finite1290357347_pname @ F_16 @ G_2 @ Z_3 @ A_38 ) ) )
& ( ~ ( finite_finite_pname @ A_38 )
=> ( ( F_15 @ G_2 @ A_38 )
= Z_3 ) ) ) ) ).
thf(fact_759_folding__one_Oinsert,axiom,
! [X_17: hoare_2091234717iple_a > $o,A_37: ( hoare_2091234717iple_a > $o ) > $o,F_14: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_13: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite14499299le_a_o @ F_14 @ F_13 )
=> ( ( finite1829014797le_a_o @ A_37 )
=> ( ~ ( member99268621le_a_o @ X_17 @ A_37 )
=> ( ( A_37 != bot_bo1957696069_a_o_o )
=> ( ( F_13 @ ( insert102003750le_a_o @ X_17 @ A_37 ) )
= ( F_14 @ X_17 @ ( F_13 @ A_37 ) ) ) ) ) ) ) ).
thf(fact_760_folding__one_Oinsert,axiom,
! [X_17: hoare_2091234717iple_a,A_37: hoare_2091234717iple_a > $o,F_14: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_13: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_14 @ F_13 )
=> ( ( finite232261744iple_a @ A_37 )
=> ( ~ ( member290856304iple_a @ X_17 @ A_37 )
=> ( ( A_37 != bot_bo1791335050le_a_o )
=> ( ( F_13 @ ( insert1597628439iple_a @ X_17 @ A_37 ) )
= ( F_14 @ X_17 @ ( F_13 @ A_37 ) ) ) ) ) ) ) ).
thf(fact_761_folding__one_Oinsert,axiom,
! [X_17: hoare_1708887482_state,A_37: hoare_1708887482_state > $o,F_14: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_13: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_14 @ F_13 )
=> ( ( finite1625599783_state @ A_37 )
=> ( ~ ( member451959335_state @ X_17 @ A_37 )
=> ( ( A_37 != bot_bo19817387tate_o )
=> ( ( F_13 @ ( insert528405184_state @ X_17 @ A_37 ) )
= ( F_14 @ X_17 @ ( F_13 @ A_37 ) ) ) ) ) ) ) ).
thf(fact_762_folding__one_Oinsert,axiom,
! [X_17: nat,A_37: nat > $o,F_14: nat > nat > nat,F_13: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_14 @ F_13 )
=> ( ( finite_finite_nat @ A_37 )
=> ( ~ ( member_nat @ X_17 @ A_37 )
=> ( ( A_37 != bot_bot_nat_o )
=> ( ( F_13 @ ( insert_nat @ X_17 @ A_37 ) )
= ( F_14 @ X_17 @ ( F_13 @ A_37 ) ) ) ) ) ) ) ).
thf(fact_763_folding__one_Oinsert,axiom,
! [X_17: pname,A_37: pname > $o,F_14: pname > pname > pname,F_13: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_14 @ F_13 )
=> ( ( finite_finite_pname @ A_37 )
=> ( ~ ( member_pname @ X_17 @ A_37 )
=> ( ( A_37 != bot_bot_pname_o )
=> ( ( F_13 @ ( insert_pname @ X_17 @ A_37 ) )
= ( F_14 @ X_17 @ ( F_13 @ A_37 ) ) ) ) ) ) ) ).
thf(fact_764_folding__one_Osingleton,axiom,
! [X_16: nat,F_12: nat > nat > nat,F_11: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert_nat @ X_16 @ bot_bot_nat_o ) )
= X_16 ) ) ).
thf(fact_765_folding__one_Osingleton,axiom,
! [X_16: hoare_2091234717iple_a > $o,F_12: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_11: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite14499299le_a_o @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert102003750le_a_o @ X_16 @ bot_bo1957696069_a_o_o ) )
= X_16 ) ) ).
thf(fact_766_folding__one_Osingleton,axiom,
! [X_16: pname,F_12: pname > pname > pname,F_11: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert_pname @ X_16 @ bot_bot_pname_o ) )
= X_16 ) ) ).
thf(fact_767_folding__one_Osingleton,axiom,
! [X_16: hoare_2091234717iple_a,F_12: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_11: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert1597628439iple_a @ X_16 @ bot_bo1791335050le_a_o ) )
= X_16 ) ) ).
thf(fact_768_folding__one_Osingleton,axiom,
! [X_16: hoare_1708887482_state,F_12: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_11: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert528405184_state @ X_16 @ bot_bo19817387tate_o ) )
= X_16 ) ) ).
thf(fact_769_folding__one_Oclosed,axiom,
! [A_36: ( hoare_2091234717iple_a > $o ) > $o,F_10: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_9: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite14499299le_a_o @ F_10 @ F_9 )
=> ( ( finite1829014797le_a_o @ A_36 )
=> ( ( A_36 != bot_bo1957696069_a_o_o )
=> ( ! [X: hoare_2091234717iple_a > $o,Y_7: hoare_2091234717iple_a > $o] : ( member99268621le_a_o @ ( F_10 @ X @ Y_7 ) @ ( insert102003750le_a_o @ X @ ( insert102003750le_a_o @ Y_7 @ bot_bo1957696069_a_o_o ) ) )
=> ( member99268621le_a_o @ ( F_9 @ A_36 ) @ A_36 ) ) ) ) ) ).
thf(fact_770_folding__one_Oclosed,axiom,
! [A_36: hoare_2091234717iple_a > $o,F_10: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_9: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_10 @ F_9 )
=> ( ( finite232261744iple_a @ A_36 )
=> ( ( A_36 != bot_bo1791335050le_a_o )
=> ( ! [X: hoare_2091234717iple_a,Y_7: hoare_2091234717iple_a] : ( member290856304iple_a @ ( F_10 @ X @ Y_7 ) @ ( insert1597628439iple_a @ X @ ( insert1597628439iple_a @ Y_7 @ bot_bo1791335050le_a_o ) ) )
=> ( member290856304iple_a @ ( F_9 @ A_36 ) @ A_36 ) ) ) ) ) ).
thf(fact_771_folding__one_Oclosed,axiom,
! [A_36: hoare_1708887482_state > $o,F_10: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_9: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_10 @ F_9 )
=> ( ( finite1625599783_state @ A_36 )
=> ( ( A_36 != bot_bo19817387tate_o )
=> ( ! [X: hoare_1708887482_state,Y_7: hoare_1708887482_state] : ( member451959335_state @ ( F_10 @ X @ Y_7 ) @ ( insert528405184_state @ X @ ( insert528405184_state @ Y_7 @ bot_bo19817387tate_o ) ) )
=> ( member451959335_state @ ( F_9 @ A_36 ) @ A_36 ) ) ) ) ) ).
thf(fact_772_folding__one_Oclosed,axiom,
! [A_36: nat > $o,F_10: nat > nat > nat,F_9: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_10 @ F_9 )
=> ( ( finite_finite_nat @ A_36 )
=> ( ( A_36 != bot_bot_nat_o )
=> ( ! [X: nat,Y_7: nat] : ( member_nat @ ( F_10 @ X @ Y_7 ) @ ( insert_nat @ X @ ( insert_nat @ Y_7 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( F_9 @ A_36 ) @ A_36 ) ) ) ) ) ).
thf(fact_773_folding__one_Oclosed,axiom,
! [A_36: pname > $o,F_10: pname > pname > pname,F_9: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_10 @ F_9 )
=> ( ( finite_finite_pname @ A_36 )
=> ( ( A_36 != bot_bot_pname_o )
=> ( ! [X: pname,Y_7: pname] : ( member_pname @ ( F_10 @ X @ Y_7 ) @ ( insert_pname @ X @ ( insert_pname @ Y_7 @ bot_bot_pname_o ) ) )
=> ( member_pname @ ( F_9 @ A_36 ) @ A_36 ) ) ) ) ) ).
thf(fact_774_Set_Oset__insert,axiom,
! [X_15: nat,A_35: nat > $o] :
( ( member_nat @ X_15 @ A_35 )
=> ~ ! [B_26: nat > $o] :
( ( A_35
= ( insert_nat @ X_15 @ B_26 ) )
=> ( member_nat @ X_15 @ B_26 ) ) ) ).
thf(fact_775_Set_Oset__insert,axiom,
! [X_15: hoare_2091234717iple_a > $o,A_35: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ X_15 @ A_35 )
=> ~ ! [B_26: ( hoare_2091234717iple_a > $o ) > $o] :
( ( A_35
= ( insert102003750le_a_o @ X_15 @ B_26 ) )
=> ( member99268621le_a_o @ X_15 @ B_26 ) ) ) ).
thf(fact_776_Set_Oset__insert,axiom,
! [X_15: hoare_2091234717iple_a,A_35: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_15 @ A_35 )
=> ~ ! [B_26: hoare_2091234717iple_a > $o] :
( ( A_35
= ( insert1597628439iple_a @ X_15 @ B_26 ) )
=> ( member290856304iple_a @ X_15 @ B_26 ) ) ) ).
thf(fact_777_Set_Oset__insert,axiom,
! [X_15: hoare_1708887482_state,A_35: hoare_1708887482_state > $o] :
( ( member451959335_state @ X_15 @ A_35 )
=> ~ ! [B_26: hoare_1708887482_state > $o] :
( ( A_35
= ( insert528405184_state @ X_15 @ B_26 ) )
=> ( member451959335_state @ X_15 @ B_26 ) ) ) ).
thf(fact_778_Set_Oset__insert,axiom,
! [X_15: pname,A_35: pname > $o] :
( ( member_pname @ X_15 @ A_35 )
=> ~ ! [B_26: pname > $o] :
( ( A_35
= ( insert_pname @ X_15 @ B_26 ) )
=> ( member_pname @ X_15 @ B_26 ) ) ) ).
thf(fact_779_equals0I,axiom,
! [A_34: nat > $o] :
( ! [Y_7: nat] :
~ ( member_nat @ Y_7 @ A_34 )
=> ( A_34 = bot_bot_nat_o ) ) ).
thf(fact_780_equals0I,axiom,
! [A_34: ( hoare_2091234717iple_a > $o ) > $o] :
( ! [Y_7: hoare_2091234717iple_a > $o] :
~ ( member99268621le_a_o @ Y_7 @ A_34 )
=> ( A_34 = bot_bo1957696069_a_o_o ) ) ).
thf(fact_781_equals0I,axiom,
! [A_34: hoare_2091234717iple_a > $o] :
( ! [Y_7: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ Y_7 @ A_34 )
=> ( A_34 = bot_bo1791335050le_a_o ) ) ).
thf(fact_782_equals0I,axiom,
! [A_34: hoare_1708887482_state > $o] :
( ! [Y_7: hoare_1708887482_state] :
~ ( member451959335_state @ Y_7 @ A_34 )
=> ( A_34 = bot_bo19817387tate_o ) ) ).
thf(fact_783_equals0I,axiom,
! [A_34: pname > $o] :
( ! [Y_7: pname] :
~ ( member_pname @ Y_7 @ A_34 )
=> ( A_34 = bot_bot_pname_o ) ) ).
thf(fact_784_Sup__fin_Ounion__idem,axiom,
! [B_25: ( nat > $o ) > $o,A_33: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_33 )
=> ( ( A_33 != bot_bot_nat_o_o )
=> ( ( finite_finite_nat_o @ B_25 )
=> ( ( B_25 != bot_bot_nat_o_o )
=> ( ( big_la1658356148_nat_o @ ( semila72246288at_o_o @ A_33 @ B_25 ) )
= ( semila848761471_nat_o @ ( big_la1658356148_nat_o @ A_33 ) @ ( big_la1658356148_nat_o @ B_25 ) ) ) ) ) ) ) ).
thf(fact_785_Sup__fin_Ounion__idem,axiom,
! [B_25: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o,A_33: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o] :
( ( finite886417794_a_o_o @ A_33 )
=> ( ( A_33 != bot_bo690906872_o_o_o )
=> ( ( finite886417794_a_o_o @ B_25 )
=> ( ( B_25 != bot_bo690906872_o_o_o )
=> ( ( big_la1994307886_a_o_o @ ( semila484278426_o_o_o @ A_33 @ B_25 ) )
= ( semila2050116131_a_o_o @ ( big_la1994307886_a_o_o @ A_33 ) @ ( big_la1994307886_a_o_o @ B_25 ) ) ) ) ) ) ) ).
thf(fact_786_Sup__fin_Ounion__idem,axiom,
! [B_25: ( hoare_1708887482_state > $o ) > $o,A_33: ( hoare_1708887482_state > $o ) > $o] :
( ( finite1329924456tate_o @ A_33 )
=> ( ( A_33 != bot_bo1678742418te_o_o )
=> ( ( finite1329924456tate_o @ B_25 )
=> ( ( B_25 != bot_bo1678742418te_o_o )
=> ( ( big_la1088302868tate_o @ ( semila1853742644te_o_o @ A_33 @ B_25 ) )
= ( semila1122118281tate_o @ ( big_la1088302868tate_o @ A_33 ) @ ( big_la1088302868tate_o @ B_25 ) ) ) ) ) ) ) ).
thf(fact_787_Sup__fin_Ounion__idem,axiom,
! [B_25: ( pname > $o ) > $o,A_33: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_33 )
=> ( ( A_33 != bot_bot_pname_o_o )
=> ( ( finite297249702name_o @ B_25 )
=> ( ( B_25 != bot_bot_pname_o_o )
=> ( ( big_la1286884090name_o @ ( semila181081674me_o_o @ A_33 @ B_25 ) )
= ( semila1780557381name_o @ ( big_la1286884090name_o @ A_33 ) @ ( big_la1286884090name_o @ B_25 ) ) ) ) ) ) ) ).
thf(fact_788_Sup__fin_Ounion__idem,axiom,
! [B_25: $o > $o,A_33: $o > $o] :
( ( finite_finite_o @ A_33 )
=> ( ( A_33 != bot_bot_o_o )
=> ( ( finite_finite_o @ B_25 )
=> ( ( B_25 != bot_bot_o_o )
=> ( ( big_la727467310_fin_o @ ( semila2062604954up_o_o @ A_33 @ B_25 ) )
<=> ( semila10642723_sup_o @ ( big_la727467310_fin_o @ A_33 ) @ ( big_la727467310_fin_o @ B_25 ) ) ) ) ) ) ) ).
thf(fact_789_Sup__fin_Ounion__idem,axiom,
! [B_25: ( hoare_2091234717iple_a > $o ) > $o,A_33: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_33 )
=> ( ( A_33 != bot_bo1957696069_a_o_o )
=> ( ( finite1829014797le_a_o @ B_25 )
=> ( ( B_25 != bot_bo1957696069_a_o_o )
=> ( ( big_la735727201le_a_o @ ( semila2050116131_a_o_o @ A_33 @ B_25 ) )
= ( semila1052848428le_a_o @ ( big_la735727201le_a_o @ A_33 ) @ ( big_la735727201le_a_o @ B_25 ) ) ) ) ) ) ) ).
thf(fact_790_Sup__fin_Ounion__idem,axiom,
! [B_25: nat > $o,A_33: nat > $o] :
( ( finite_finite_nat @ A_33 )
=> ( ( A_33 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ B_25 )
=> ( ( B_25 != bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( semila848761471_nat_o @ A_33 @ B_25 ) )
= ( semila972727038up_nat @ ( big_la43341705in_nat @ A_33 ) @ ( big_la43341705in_nat @ B_25 ) ) ) ) ) ) ) ).
thf(fact_791_Sup__fin_Oinsert,axiom,
! [X_14: nat > $o,A_32: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_32 )
=> ( ~ ( member_nat_o @ X_14 @ A_32 )
=> ( ( A_32 != bot_bot_nat_o_o )
=> ( ( big_la1658356148_nat_o @ ( insert_nat_o @ X_14 @ A_32 ) )
= ( semila848761471_nat_o @ X_14 @ ( big_la1658356148_nat_o @ A_32 ) ) ) ) ) ) ).
thf(fact_792_Sup__fin_Oinsert,axiom,
! [X_14: ( hoare_2091234717iple_a > $o ) > $o,A_32: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o] :
( ( finite886417794_a_o_o @ A_32 )
=> ( ~ ( member1297825410_a_o_o @ X_14 @ A_32 )
=> ( ( A_32 != bot_bo690906872_o_o_o )
=> ( ( big_la1994307886_a_o_o @ ( insert987231145_a_o_o @ X_14 @ A_32 ) )
= ( semila2050116131_a_o_o @ X_14 @ ( big_la1994307886_a_o_o @ A_32 ) ) ) ) ) ) ).
thf(fact_793_Sup__fin_Oinsert,axiom,
! [X_14: hoare_1708887482_state > $o,A_32: ( hoare_1708887482_state > $o ) > $o] :
( ( finite1329924456tate_o @ A_32 )
=> ( ~ ( member814030440tate_o @ X_14 @ A_32 )
=> ( ( A_32 != bot_bo1678742418te_o_o )
=> ( ( big_la1088302868tate_o @ ( insert949073679tate_o @ X_14 @ A_32 ) )
= ( semila1122118281tate_o @ X_14 @ ( big_la1088302868tate_o @ A_32 ) ) ) ) ) ) ).
thf(fact_794_Sup__fin_Oinsert,axiom,
! [X_14: pname > $o,A_32: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_32 )
=> ( ~ ( member_pname_o @ X_14 @ A_32 )
=> ( ( A_32 != bot_bot_pname_o_o )
=> ( ( big_la1286884090name_o @ ( insert_pname_o @ X_14 @ A_32 ) )
= ( semila1780557381name_o @ X_14 @ ( big_la1286884090name_o @ A_32 ) ) ) ) ) ) ).
thf(fact_795_Sup__fin_Oinsert,axiom,
! [X_14: $o,A_32: $o > $o] :
( ( finite_finite_o @ A_32 )
=> ( ~ ( member_o @ X_14 @ A_32 )
=> ( ( A_32 != bot_bot_o_o )
=> ( ( big_la727467310_fin_o @ ( insert_o @ X_14 @ A_32 ) )
<=> ( semila10642723_sup_o @ X_14 @ ( big_la727467310_fin_o @ A_32 ) ) ) ) ) ) ).
thf(fact_796_Sup__fin_Oinsert,axiom,
! [X_14: hoare_2091234717iple_a > $o,A_32: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_32 )
=> ( ~ ( member99268621le_a_o @ X_14 @ A_32 )
=> ( ( A_32 != bot_bo1957696069_a_o_o )
=> ( ( big_la735727201le_a_o @ ( insert102003750le_a_o @ X_14 @ A_32 ) )
= ( semila1052848428le_a_o @ X_14 @ ( big_la735727201le_a_o @ A_32 ) ) ) ) ) ) ).
thf(fact_797_Sup__fin_Oinsert,axiom,
! [X_14: nat,A_32: nat > $o] :
( ( finite_finite_nat @ A_32 )
=> ( ~ ( member_nat @ X_14 @ A_32 )
=> ( ( A_32 != bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( insert_nat @ X_14 @ A_32 ) )
= ( semila972727038up_nat @ X_14 @ ( big_la43341705in_nat @ A_32 ) ) ) ) ) ) ).
thf(fact_798_Sup__fin_Oinsert__idem,axiom,
! [X_13: nat > $o,A_31: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_31 )
=> ( ( A_31 != bot_bot_nat_o_o )
=> ( ( big_la1658356148_nat_o @ ( insert_nat_o @ X_13 @ A_31 ) )
= ( semila848761471_nat_o @ X_13 @ ( big_la1658356148_nat_o @ A_31 ) ) ) ) ) ).
thf(fact_799_Sup__fin_Oinsert__idem,axiom,
! [X_13: ( hoare_2091234717iple_a > $o ) > $o,A_31: ( ( hoare_2091234717iple_a > $o ) > $o ) > $o] :
( ( finite886417794_a_o_o @ A_31 )
=> ( ( A_31 != bot_bo690906872_o_o_o )
=> ( ( big_la1994307886_a_o_o @ ( insert987231145_a_o_o @ X_13 @ A_31 ) )
= ( semila2050116131_a_o_o @ X_13 @ ( big_la1994307886_a_o_o @ A_31 ) ) ) ) ) ).
thf(fact_800_Sup__fin_Oinsert__idem,axiom,
! [X_13: hoare_1708887482_state > $o,A_31: ( hoare_1708887482_state > $o ) > $o] :
( ( finite1329924456tate_o @ A_31 )
=> ( ( A_31 != bot_bo1678742418te_o_o )
=> ( ( big_la1088302868tate_o @ ( insert949073679tate_o @ X_13 @ A_31 ) )
= ( semila1122118281tate_o @ X_13 @ ( big_la1088302868tate_o @ A_31 ) ) ) ) ) ).
thf(fact_801_Sup__fin_Oinsert__idem,axiom,
! [X_13: pname > $o,A_31: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_31 )
=> ( ( A_31 != bot_bot_pname_o_o )
=> ( ( big_la1286884090name_o @ ( insert_pname_o @ X_13 @ A_31 ) )
= ( semila1780557381name_o @ X_13 @ ( big_la1286884090name_o @ A_31 ) ) ) ) ) ).
thf(fact_802_Sup__fin_Oinsert__idem,axiom,
! [X_13: $o,A_31: $o > $o] :
( ( finite_finite_o @ A_31 )
=> ( ( A_31 != bot_bot_o_o )
=> ( ( big_la727467310_fin_o @ ( insert_o @ X_13 @ A_31 ) )
<=> ( semila10642723_sup_o @ X_13 @ ( big_la727467310_fin_o @ A_31 ) ) ) ) ) ).
thf(fact_803_Sup__fin_Oinsert__idem,axiom,
! [X_13: hoare_2091234717iple_a > $o,A_31: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_31 )
=> ( ( A_31 != bot_bo1957696069_a_o_o )
=> ( ( big_la735727201le_a_o @ ( insert102003750le_a_o @ X_13 @ A_31 ) )
= ( semila1052848428le_a_o @ X_13 @ ( big_la735727201le_a_o @ A_31 ) ) ) ) ) ).
thf(fact_804_Sup__fin_Oinsert__idem,axiom,
! [X_13: nat,A_31: nat > $o] :
( ( finite_finite_nat @ A_31 )
=> ( ( A_31 != bot_bot_nat_o )
=> ( ( big_la43341705in_nat @ ( insert_nat @ X_13 @ A_31 ) )
= ( semila972727038up_nat @ X_13 @ ( big_la43341705in_nat @ A_31 ) ) ) ) ) ).
thf(fact_805_folding__one_Ounion__disjoint,axiom,
! [B_24: ( hoare_2091234717iple_a > $o ) > $o,A_30: ( hoare_2091234717iple_a > $o ) > $o,F_8: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_7: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite14499299le_a_o @ F_8 @ F_7 )
=> ( ( finite1829014797le_a_o @ A_30 )
=> ( ( A_30 != bot_bo1957696069_a_o_o )
=> ( ( finite1829014797le_a_o @ B_24 )
=> ( ( B_24 != bot_bo1957696069_a_o_o )
=> ( ( ( semila1672913213_a_o_o @ A_30 @ B_24 )
= bot_bo1957696069_a_o_o )
=> ( ( F_7 @ ( semila2050116131_a_o_o @ A_30 @ B_24 ) )
= ( F_8 @ ( F_7 @ A_30 ) @ ( F_7 @ B_24 ) ) ) ) ) ) ) ) ) ).
thf(fact_806_folding__one_Ounion__disjoint,axiom,
! [B_24: pname > $o,A_30: pname > $o,F_8: pname > pname > pname,F_7: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_8 @ F_7 )
=> ( ( finite_finite_pname @ A_30 )
=> ( ( A_30 != bot_bot_pname_o )
=> ( ( finite_finite_pname @ B_24 )
=> ( ( B_24 != bot_bot_pname_o )
=> ( ( ( semila1673364395name_o @ A_30 @ B_24 )
= bot_bot_pname_o )
=> ( ( F_7 @ ( semila1780557381name_o @ A_30 @ B_24 ) )
= ( F_8 @ ( F_7 @ A_30 ) @ ( F_7 @ B_24 ) ) ) ) ) ) ) ) ) ).
thf(fact_807_folding__one_Ounion__disjoint,axiom,
! [B_24: hoare_1708887482_state > $o,A_30: hoare_1708887482_state > $o,F_8: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_7: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_8 @ F_7 )
=> ( ( finite1625599783_state @ A_30 )
=> ( ( A_30 != bot_bo19817387tate_o )
=> ( ( finite1625599783_state @ B_24 )
=> ( ( B_24 != bot_bo19817387tate_o )
=> ( ( ( semila129691299tate_o @ A_30 @ B_24 )
= bot_bo19817387tate_o )
=> ( ( F_7 @ ( semila1122118281tate_o @ A_30 @ B_24 ) )
= ( F_8 @ ( F_7 @ A_30 ) @ ( F_7 @ B_24 ) ) ) ) ) ) ) ) ) ).
thf(fact_808_folding__one_Ounion__disjoint,axiom,
! [B_24: hoare_2091234717iple_a > $o,A_30: hoare_2091234717iple_a > $o,F_8: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_7: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_8 @ F_7 )
=> ( ( finite232261744iple_a @ A_30 )
=> ( ( A_30 != bot_bo1791335050le_a_o )
=> ( ( finite232261744iple_a @ B_24 )
=> ( ( B_24 != bot_bo1791335050le_a_o )
=> ( ( ( semila2006181266le_a_o @ A_30 @ B_24 )
= bot_bo1791335050le_a_o )
=> ( ( F_7 @ ( semila1052848428le_a_o @ A_30 @ B_24 ) )
= ( F_8 @ ( F_7 @ A_30 ) @ ( F_7 @ B_24 ) ) ) ) ) ) ) ) ) ).
thf(fact_809_folding__one_Ounion__disjoint,axiom,
! [B_24: nat > $o,A_30: nat > $o,F_8: nat > nat > nat,F_7: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_8 @ F_7 )
=> ( ( finite_finite_nat @ A_30 )
=> ( ( A_30 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ B_24 )
=> ( ( B_24 != bot_bot_nat_o )
=> ( ( ( semila1947288293_nat_o @ A_30 @ B_24 )
= bot_bot_nat_o )
=> ( ( F_7 @ ( semila848761471_nat_o @ A_30 @ B_24 ) )
= ( F_8 @ ( F_7 @ A_30 ) @ ( F_7 @ B_24 ) ) ) ) ) ) ) ) ) ).
thf(fact_810_folding__one_Ounion__inter,axiom,
! [B_23: ( hoare_2091234717iple_a > $o ) > $o,A_29: ( hoare_2091234717iple_a > $o ) > $o,F_6: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_5: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite14499299le_a_o @ F_6 @ F_5 )
=> ( ( finite1829014797le_a_o @ A_29 )
=> ( ( finite1829014797le_a_o @ B_23 )
=> ( ( ( semila1672913213_a_o_o @ A_29 @ B_23 )
!= bot_bo1957696069_a_o_o )
=> ( ( F_6 @ ( F_5 @ ( semila2050116131_a_o_o @ A_29 @ B_23 ) ) @ ( F_5 @ ( semila1672913213_a_o_o @ A_29 @ B_23 ) ) )
= ( F_6 @ ( F_5 @ A_29 ) @ ( F_5 @ B_23 ) ) ) ) ) ) ) ).
thf(fact_811_folding__one_Ounion__inter,axiom,
! [B_23: pname > $o,A_29: pname > $o,F_6: pname > pname > pname,F_5: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_6 @ F_5 )
=> ( ( finite_finite_pname @ A_29 )
=> ( ( finite_finite_pname @ B_23 )
=> ( ( ( semila1673364395name_o @ A_29 @ B_23 )
!= bot_bot_pname_o )
=> ( ( F_6 @ ( F_5 @ ( semila1780557381name_o @ A_29 @ B_23 ) ) @ ( F_5 @ ( semila1673364395name_o @ A_29 @ B_23 ) ) )
= ( F_6 @ ( F_5 @ A_29 ) @ ( F_5 @ B_23 ) ) ) ) ) ) ) ).
thf(fact_812_folding__one_Ounion__inter,axiom,
! [B_23: hoare_1708887482_state > $o,A_29: hoare_1708887482_state > $o,F_6: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_5: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_6 @ F_5 )
=> ( ( finite1625599783_state @ A_29 )
=> ( ( finite1625599783_state @ B_23 )
=> ( ( ( semila129691299tate_o @ A_29 @ B_23 )
!= bot_bo19817387tate_o )
=> ( ( F_6 @ ( F_5 @ ( semila1122118281tate_o @ A_29 @ B_23 ) ) @ ( F_5 @ ( semila129691299tate_o @ A_29 @ B_23 ) ) )
= ( F_6 @ ( F_5 @ A_29 ) @ ( F_5 @ B_23 ) ) ) ) ) ) ) ).
thf(fact_813_folding__one_Ounion__inter,axiom,
! [B_23: hoare_2091234717iple_a > $o,A_29: hoare_2091234717iple_a > $o,F_6: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_5: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_6 @ F_5 )
=> ( ( finite232261744iple_a @ A_29 )
=> ( ( finite232261744iple_a @ B_23 )
=> ( ( ( semila2006181266le_a_o @ A_29 @ B_23 )
!= bot_bo1791335050le_a_o )
=> ( ( F_6 @ ( F_5 @ ( semila1052848428le_a_o @ A_29 @ B_23 ) ) @ ( F_5 @ ( semila2006181266le_a_o @ A_29 @ B_23 ) ) )
= ( F_6 @ ( F_5 @ A_29 ) @ ( F_5 @ B_23 ) ) ) ) ) ) ) ).
thf(fact_814_folding__one_Ounion__inter,axiom,
! [B_23: nat > $o,A_29: nat > $o,F_6: nat > nat > nat,F_5: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_6 @ F_5 )
=> ( ( finite_finite_nat @ A_29 )
=> ( ( finite_finite_nat @ B_23 )
=> ( ( ( semila1947288293_nat_o @ A_29 @ B_23 )
!= bot_bot_nat_o )
=> ( ( F_6 @ ( F_5 @ ( semila848761471_nat_o @ A_29 @ B_23 ) ) @ ( F_5 @ ( semila1947288293_nat_o @ A_29 @ B_23 ) ) )
= ( F_6 @ ( F_5 @ A_29 ) @ ( F_5 @ B_23 ) ) ) ) ) ) ) ).
thf(fact_815_folding__one_Oinsert__remove,axiom,
! [X_12: hoare_2091234717iple_a > $o,A_28: ( hoare_2091234717iple_a > $o ) > $o,F_4: ( hoare_2091234717iple_a > $o ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,F_3: ( ( hoare_2091234717iple_a > $o ) > $o ) > hoare_2091234717iple_a > $o] :
( ( finite14499299le_a_o @ F_4 @ F_3 )
=> ( ( finite1829014797le_a_o @ A_28 )
=> ( ( ( ( minus_1746272704_a_o_o @ A_28 @ ( insert102003750le_a_o @ X_12 @ bot_bo1957696069_a_o_o ) )
= bot_bo1957696069_a_o_o )
=> ( ( F_3 @ ( insert102003750le_a_o @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_1746272704_a_o_o @ A_28 @ ( insert102003750le_a_o @ X_12 @ bot_bo1957696069_a_o_o ) )
!= bot_bo1957696069_a_o_o )
=> ( ( F_3 @ ( insert102003750le_a_o @ X_12 @ A_28 ) )
= ( F_4 @ X_12 @ ( F_3 @ ( minus_1746272704_a_o_o @ A_28 @ ( insert102003750le_a_o @ X_12 @ bot_bo1957696069_a_o_o ) ) ) ) ) ) ) ) ) ).
thf(fact_816_folding__one_Oinsert__remove,axiom,
! [X_12: pname,A_28: pname > $o,F_4: pname > pname > pname,F_3: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_4 @ F_3 )
=> ( ( finite_finite_pname @ A_28 )
=> ( ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
= bot_bot_pname_o )
=> ( ( F_3 @ ( insert_pname @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
!= bot_bot_pname_o )
=> ( ( F_3 @ ( insert_pname @ X_12 @ A_28 ) )
= ( F_4 @ X_12 @ ( F_3 @ ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ).
thf(fact_817_folding__one_Oinsert__remove,axiom,
! [X_12: hoare_2091234717iple_a,A_28: hoare_2091234717iple_a > $o,F_4: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_3: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_4 @ F_3 )
=> ( ( finite232261744iple_a @ A_28 )
=> ( ( ( ( minus_836160335le_a_o @ A_28 @ ( insert1597628439iple_a @ X_12 @ bot_bo1791335050le_a_o ) )
= bot_bo1791335050le_a_o )
=> ( ( F_3 @ ( insert1597628439iple_a @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_836160335le_a_o @ A_28 @ ( insert1597628439iple_a @ X_12 @ bot_bo1791335050le_a_o ) )
!= bot_bo1791335050le_a_o )
=> ( ( F_3 @ ( insert1597628439iple_a @ X_12 @ A_28 ) )
= ( F_4 @ X_12 @ ( F_3 @ ( minus_836160335le_a_o @ A_28 @ ( insert1597628439iple_a @ X_12 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ) ) ).
thf(fact_818_folding__one_Oinsert__remove,axiom,
! [X_12: hoare_1708887482_state,A_28: hoare_1708887482_state > $o,F_4: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_3: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_4 @ F_3 )
=> ( ( finite1625599783_state @ A_28 )
=> ( ( ( ( minus_2056855718tate_o @ A_28 @ ( insert528405184_state @ X_12 @ bot_bo19817387tate_o ) )
= bot_bo19817387tate_o )
=> ( ( F_3 @ ( insert528405184_state @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_2056855718tate_o @ A_28 @ ( insert528405184_state @ X_12 @ bot_bo19817387tate_o ) )
!= bot_bo19817387tate_o )
=> ( ( F_3 @ ( insert528405184_state @ X_12 @ A_28 ) )
= ( F_4 @ X_12 @ ( F_3 @ ( minus_2056855718tate_o @ A_28 @ ( insert528405184_state @ X_12 @ bot_bo19817387tate_o ) ) ) ) ) ) ) ) ) ).
thf(fact_819_folding__one_Oinsert__remove,axiom,
! [X_12: nat,A_28: nat > $o,F_4: nat > nat > nat,F_3: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_4 @ F_3 )
=> ( ( finite_finite_nat @ A_28 )
=> ( ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_3 @ ( insert_nat @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_3 @ ( insert_nat @ X_12 @ A_28 ) )
= ( F_4 @ X_12 @ ( F_3 @ ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
thf(fact_820_inf1I,axiom,
! [B_22: nat > $o,A_27: nat > $o,X_11: nat] :
( ( A_27 @ X_11 )
=> ( ( B_22 @ X_11 )
=> ( semila1947288293_nat_o @ A_27 @ B_22 @ X_11 ) ) ) ).
thf(fact_821_inf1I,axiom,
! [B_22: hoare_2091234717iple_a > $o,A_27: hoare_2091234717iple_a > $o,X_11: hoare_2091234717iple_a] :
( ( A_27 @ X_11 )
=> ( ( B_22 @ X_11 )
=> ( semila2006181266le_a_o @ A_27 @ B_22 @ X_11 ) ) ) ).
thf(fact_822_inf1I,axiom,
! [B_22: pname > $o,A_27: pname > $o,X_11: pname] :
( ( A_27 @ X_11 )
=> ( ( B_22 @ X_11 )
=> ( semila1673364395name_o @ A_27 @ B_22 @ X_11 ) ) ) ).
thf(fact_823_IntI,axiom,
! [B_21: nat > $o,C_11: nat,A_26: nat > $o] :
( ( member_nat @ C_11 @ A_26 )
=> ( ( member_nat @ C_11 @ B_21 )
=> ( member_nat @ C_11 @ ( semila1947288293_nat_o @ A_26 @ B_21 ) ) ) ) ).
thf(fact_824_IntI,axiom,
! [B_21: ( hoare_2091234717iple_a > $o ) > $o,C_11: hoare_2091234717iple_a > $o,A_26: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_11 @ A_26 )
=> ( ( member99268621le_a_o @ C_11 @ B_21 )
=> ( member99268621le_a_o @ C_11 @ ( semila1672913213_a_o_o @ A_26 @ B_21 ) ) ) ) ).
thf(fact_825_IntI,axiom,
! [B_21: hoare_2091234717iple_a > $o,C_11: hoare_2091234717iple_a,A_26: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_11 @ A_26 )
=> ( ( member290856304iple_a @ C_11 @ B_21 )
=> ( member290856304iple_a @ C_11 @ ( semila2006181266le_a_o @ A_26 @ B_21 ) ) ) ) ).
thf(fact_826_IntI,axiom,
! [B_21: pname > $o,C_11: pname,A_26: pname > $o] :
( ( member_pname @ C_11 @ A_26 )
=> ( ( member_pname @ C_11 @ B_21 )
=> ( member_pname @ C_11 @ ( semila1673364395name_o @ A_26 @ B_21 ) ) ) ) ).
thf(fact_827_IntE,axiom,
! [C_10: nat,A_25: nat > $o,B_20: nat > $o] :
( ( member_nat @ C_10 @ ( semila1947288293_nat_o @ A_25 @ B_20 ) )
=> ~ ( ( member_nat @ C_10 @ A_25 )
=> ~ ( member_nat @ C_10 @ B_20 ) ) ) ).
thf(fact_828_IntE,axiom,
! [C_10: hoare_2091234717iple_a > $o,A_25: ( hoare_2091234717iple_a > $o ) > $o,B_20: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_10 @ ( semila1672913213_a_o_o @ A_25 @ B_20 ) )
=> ~ ( ( member99268621le_a_o @ C_10 @ A_25 )
=> ~ ( member99268621le_a_o @ C_10 @ B_20 ) ) ) ).
thf(fact_829_IntE,axiom,
! [C_10: hoare_2091234717iple_a,A_25: hoare_2091234717iple_a > $o,B_20: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_10 @ ( semila2006181266le_a_o @ A_25 @ B_20 ) )
=> ~ ( ( member290856304iple_a @ C_10 @ A_25 )
=> ~ ( member290856304iple_a @ C_10 @ B_20 ) ) ) ).
thf(fact_830_IntE,axiom,
! [C_10: pname,A_25: pname > $o,B_20: pname > $o] :
( ( member_pname @ C_10 @ ( semila1673364395name_o @ A_25 @ B_20 ) )
=> ~ ( ( member_pname @ C_10 @ A_25 )
=> ~ ( member_pname @ C_10 @ B_20 ) ) ) ).
thf(fact_831_inf1E,axiom,
! [A_24: nat > $o,B_19: nat > $o,X_10: nat] :
( ( semila1947288293_nat_o @ A_24 @ B_19 @ X_10 )
=> ~ ( ( A_24 @ X_10 )
=> ~ ( B_19 @ X_10 ) ) ) ).
thf(fact_832_inf1E,axiom,
! [A_24: hoare_2091234717iple_a > $o,B_19: hoare_2091234717iple_a > $o,X_10: hoare_2091234717iple_a] :
( ( semila2006181266le_a_o @ A_24 @ B_19 @ X_10 )
=> ~ ( ( A_24 @ X_10 )
=> ~ ( B_19 @ X_10 ) ) ) ).
thf(fact_833_inf1E,axiom,
! [A_24: pname > $o,B_19: pname > $o,X_10: pname] :
( ( semila1673364395name_o @ A_24 @ B_19 @ X_10 )
=> ~ ( ( A_24 @ X_10 )
=> ~ ( B_19 @ X_10 ) ) ) ).
thf(fact_834_DiffI,axiom,
! [B_18: nat > $o,C_9: nat,A_23: nat > $o] :
( ( member_nat @ C_9 @ A_23 )
=> ( ~ ( member_nat @ C_9 @ B_18 )
=> ( member_nat @ C_9 @ ( minus_minus_nat_o @ A_23 @ B_18 ) ) ) ) ).
thf(fact_835_DiffI,axiom,
! [B_18: ( hoare_2091234717iple_a > $o ) > $o,C_9: hoare_2091234717iple_a > $o,A_23: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_9 @ A_23 )
=> ( ~ ( member99268621le_a_o @ C_9 @ B_18 )
=> ( member99268621le_a_o @ C_9 @ ( minus_1746272704_a_o_o @ A_23 @ B_18 ) ) ) ) ).
thf(fact_836_DiffI,axiom,
! [B_18: hoare_2091234717iple_a > $o,C_9: hoare_2091234717iple_a,A_23: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_9 @ A_23 )
=> ( ~ ( member290856304iple_a @ C_9 @ B_18 )
=> ( member290856304iple_a @ C_9 @ ( minus_836160335le_a_o @ A_23 @ B_18 ) ) ) ) ).
thf(fact_837_DiffI,axiom,
! [B_18: pname > $o,C_9: pname,A_23: pname > $o] :
( ( member_pname @ C_9 @ A_23 )
=> ( ~ ( member_pname @ C_9 @ B_18 )
=> ( member_pname @ C_9 @ ( minus_minus_pname_o @ A_23 @ B_18 ) ) ) ) ).
thf(fact_838_DiffE,axiom,
! [C_8: nat,A_22: nat > $o,B_17: nat > $o] :
( ( member_nat @ C_8 @ ( minus_minus_nat_o @ A_22 @ B_17 ) )
=> ~ ( ( member_nat @ C_8 @ A_22 )
=> ( member_nat @ C_8 @ B_17 ) ) ) ).
thf(fact_839_DiffE,axiom,
! [C_8: hoare_2091234717iple_a > $o,A_22: ( hoare_2091234717iple_a > $o ) > $o,B_17: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_8 @ ( minus_1746272704_a_o_o @ A_22 @ B_17 ) )
=> ~ ( ( member99268621le_a_o @ C_8 @ A_22 )
=> ( member99268621le_a_o @ C_8 @ B_17 ) ) ) ).
thf(fact_840_DiffE,axiom,
! [C_8: hoare_2091234717iple_a,A_22: hoare_2091234717iple_a > $o,B_17: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_8 @ ( minus_836160335le_a_o @ A_22 @ B_17 ) )
=> ~ ( ( member290856304iple_a @ C_8 @ A_22 )
=> ( member290856304iple_a @ C_8 @ B_17 ) ) ) ).
thf(fact_841_DiffE,axiom,
! [C_8: pname,A_22: pname > $o,B_17: pname > $o] :
( ( member_pname @ C_8 @ ( minus_minus_pname_o @ A_22 @ B_17 ) )
=> ~ ( ( member_pname @ C_8 @ A_22 )
=> ( member_pname @ C_8 @ B_17 ) ) ) ).
thf(fact_842_finite__Int,axiom,
! [G_1: hoare_2091234717iple_a > $o,F_2: hoare_2091234717iple_a > $o] :
( ( ( finite232261744iple_a @ F_2 )
| ( finite232261744iple_a @ G_1 ) )
=> ( finite232261744iple_a @ ( semila2006181266le_a_o @ F_2 @ G_1 ) ) ) ).
thf(fact_843_finite__Int,axiom,
! [G_1: ( hoare_2091234717iple_a > $o ) > $o,F_2: ( hoare_2091234717iple_a > $o ) > $o] :
( ( ( finite1829014797le_a_o @ F_2 )
| ( finite1829014797le_a_o @ G_1 ) )
=> ( finite1829014797le_a_o @ ( semila1672913213_a_o_o @ F_2 @ G_1 ) ) ) ).
thf(fact_844_finite__Int,axiom,
! [G_1: pname > $o,F_2: pname > $o] :
( ( ( finite_finite_pname @ F_2 )
| ( finite_finite_pname @ G_1 ) )
=> ( finite_finite_pname @ ( semila1673364395name_o @ F_2 @ G_1 ) ) ) ).
thf(fact_845_finite__Int,axiom,
! [G_1: nat > $o,F_2: nat > $o] :
( ( ( finite_finite_nat @ F_2 )
| ( finite_finite_nat @ G_1 ) )
=> ( finite_finite_nat @ ( semila1947288293_nat_o @ F_2 @ G_1 ) ) ) ).
thf(fact_846_finite__Diff,axiom,
! [B_16: hoare_2091234717iple_a > $o,A_21: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_21 )
=> ( finite232261744iple_a @ ( minus_836160335le_a_o @ A_21 @ B_16 ) ) ) ).
thf(fact_847_finite__Diff,axiom,
! [B_16: ( hoare_2091234717iple_a > $o ) > $o,A_21: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_21 )
=> ( finite1829014797le_a_o @ ( minus_1746272704_a_o_o @ A_21 @ B_16 ) ) ) ).
thf(fact_848_finite__Diff,axiom,
! [B_16: pname > $o,A_21: pname > $o] :
( ( finite_finite_pname @ A_21 )
=> ( finite_finite_pname @ ( minus_minus_pname_o @ A_21 @ B_16 ) ) ) ).
thf(fact_849_finite__Diff,axiom,
! [B_16: nat > $o,A_21: nat > $o] :
( ( finite_finite_nat @ A_21 )
=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_21 @ B_16 ) ) ) ).
thf(fact_850_inf__Sup__absorb,axiom,
! [A_20: hoare_2091234717iple_a > $o,A_19: ( hoare_2091234717iple_a > $o ) > $o] :
( ( finite1829014797le_a_o @ A_19 )
=> ( ( member99268621le_a_o @ A_20 @ A_19 )
=> ( ( semila2006181266le_a_o @ A_20 @ ( big_la735727201le_a_o @ A_19 ) )
= A_20 ) ) ) ).
thf(fact_851_inf__Sup__absorb,axiom,
! [A_20: nat > $o,A_19: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_19 )
=> ( ( member_nat_o @ A_20 @ A_19 )
=> ( ( semila1947288293_nat_o @ A_20 @ ( big_la1658356148_nat_o @ A_19 ) )
= A_20 ) ) ) ).
thf(fact_852_inf__Sup__absorb,axiom,
! [A_20: pname > $o,A_19: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_19 )
=> ( ( member_pname_o @ A_20 @ A_19 )
=> ( ( semila1673364395name_o @ A_20 @ ( big_la1286884090name_o @ A_19 ) )
= A_20 ) ) ) ).
thf(fact_853_inf__Sup__absorb,axiom,
! [A_20: nat,A_19: nat > $o] :
( ( finite_finite_nat @ A_19 )
=> ( ( member_nat @ A_20 @ A_19 )
=> ( ( semila80283416nf_nat @ A_20 @ ( big_la43341705in_nat @ A_19 ) )
= A_20 ) ) ) ).
thf(fact_854_Diff__Int,axiom,
! [A_18: nat > $o,B_15: nat > $o,C_7: nat > $o] :
( ( minus_minus_nat_o @ A_18 @ ( semila1947288293_nat_o @ B_15 @ C_7 ) )
= ( semila848761471_nat_o @ ( minus_minus_nat_o @ A_18 @ B_15 ) @ ( minus_minus_nat_o @ A_18 @ C_7 ) ) ) ).
thf(fact_855_Diff__Int,axiom,
! [A_18: pname > $o,B_15: pname > $o,C_7: pname > $o] :
( ( minus_minus_pname_o @ A_18 @ ( semila1673364395name_o @ B_15 @ C_7 ) )
= ( semila1780557381name_o @ ( minus_minus_pname_o @ A_18 @ B_15 ) @ ( minus_minus_pname_o @ A_18 @ C_7 ) ) ) ).
thf(fact_856_Diff__Int,axiom,
! [A_18: ( hoare_2091234717iple_a > $o ) > $o,B_15: ( hoare_2091234717iple_a > $o ) > $o,C_7: ( hoare_2091234717iple_a > $o ) > $o] :
( ( minus_1746272704_a_o_o @ A_18 @ ( semila1672913213_a_o_o @ B_15 @ C_7 ) )
= ( semila2050116131_a_o_o @ ( minus_1746272704_a_o_o @ A_18 @ B_15 ) @ ( minus_1746272704_a_o_o @ A_18 @ C_7 ) ) ) ).
thf(fact_857_Diff__Int,axiom,
! [A_18: hoare_1708887482_state > $o,B_15: hoare_1708887482_state > $o,C_7: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ A_18 @ ( semila129691299tate_o @ B_15 @ C_7 ) )
= ( semila1122118281tate_o @ ( minus_2056855718tate_o @ A_18 @ B_15 ) @ ( minus_2056855718tate_o @ A_18 @ C_7 ) ) ) ).
thf(fact_858_Diff__Int,axiom,
! [A_18: hoare_2091234717iple_a > $o,B_15: hoare_2091234717iple_a > $o,C_7: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_18 @ ( semila2006181266le_a_o @ B_15 @ C_7 ) )
= ( semila1052848428le_a_o @ ( minus_836160335le_a_o @ A_18 @ B_15 ) @ ( minus_836160335le_a_o @ A_18 @ C_7 ) ) ) ).
thf(fact_859_Diff__Un,axiom,
! [A_17: nat > $o,B_14: nat > $o,C_6: nat > $o] :
( ( minus_minus_nat_o @ A_17 @ ( semila848761471_nat_o @ B_14 @ C_6 ) )
= ( semila1947288293_nat_o @ ( minus_minus_nat_o @ A_17 @ B_14 ) @ ( minus_minus_nat_o @ A_17 @ C_6 ) ) ) ).
thf(fact_860_Diff__Un,axiom,
! [A_17: pname > $o,B_14: pname > $o,C_6: pname > $o] :
( ( minus_minus_pname_o @ A_17 @ ( semila1780557381name_o @ B_14 @ C_6 ) )
= ( semila1673364395name_o @ ( minus_minus_pname_o @ A_17 @ B_14 ) @ ( minus_minus_pname_o @ A_17 @ C_6 ) ) ) ).
thf(fact_861_Diff__Un,axiom,
! [A_17: ( hoare_2091234717iple_a > $o ) > $o,B_14: ( hoare_2091234717iple_a > $o ) > $o,C_6: ( hoare_2091234717iple_a > $o ) > $o] :
( ( minus_1746272704_a_o_o @ A_17 @ ( semila2050116131_a_o_o @ B_14 @ C_6 ) )
= ( semila1672913213_a_o_o @ ( minus_1746272704_a_o_o @ A_17 @ B_14 ) @ ( minus_1746272704_a_o_o @ A_17 @ C_6 ) ) ) ).
thf(fact_862_Diff__Un,axiom,
! [A_17: hoare_1708887482_state > $o,B_14: hoare_1708887482_state > $o,C_6: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ A_17 @ ( semila1122118281tate_o @ B_14 @ C_6 ) )
= ( semila129691299tate_o @ ( minus_2056855718tate_o @ A_17 @ B_14 ) @ ( minus_2056855718tate_o @ A_17 @ C_6 ) ) ) ).
thf(fact_863_Diff__Un,axiom,
! [A_17: hoare_2091234717iple_a > $o,B_14: hoare_2091234717iple_a > $o,C_6: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_17 @ ( semila1052848428le_a_o @ B_14 @ C_6 ) )
= ( semila2006181266le_a_o @ ( minus_836160335le_a_o @ A_17 @ B_14 ) @ ( minus_836160335le_a_o @ A_17 @ C_6 ) ) ) ).
thf(fact_864_Un__Diff__Int,axiom,
! [A_16: nat > $o,B_13: nat > $o] :
( ( semila848761471_nat_o @ ( minus_minus_nat_o @ A_16 @ B_13 ) @ ( semila1947288293_nat_o @ A_16 @ B_13 ) )
= A_16 ) ).
thf(fact_865_Un__Diff__Int,axiom,
! [A_16: pname > $o,B_13: pname > $o] :
( ( semila1780557381name_o @ ( minus_minus_pname_o @ A_16 @ B_13 ) @ ( semila1673364395name_o @ A_16 @ B_13 ) )
= A_16 ) ).
thf(fact_866_Un__Diff__Int,axiom,
! [A_16: ( hoare_2091234717iple_a > $o ) > $o,B_13: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila2050116131_a_o_o @ ( minus_1746272704_a_o_o @ A_16 @ B_13 ) @ ( semila1672913213_a_o_o @ A_16 @ B_13 ) )
= A_16 ) ).
thf(fact_867_Un__Diff__Int,axiom,
! [A_16: hoare_1708887482_state > $o,B_13: hoare_1708887482_state > $o] :
( ( semila1122118281tate_o @ ( minus_2056855718tate_o @ A_16 @ B_13 ) @ ( semila129691299tate_o @ A_16 @ B_13 ) )
= A_16 ) ).
thf(fact_868_Un__Diff__Int,axiom,
! [A_16: hoare_2091234717iple_a > $o,B_13: hoare_2091234717iple_a > $o] :
( ( semila1052848428le_a_o @ ( minus_836160335le_a_o @ A_16 @ B_13 ) @ ( semila2006181266le_a_o @ A_16 @ B_13 ) )
= A_16 ) ).
thf(fact_869_Collect__conj__eq,axiom,
! [P_2: pname > $o,Q: pname > $o] :
( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( P_2 @ X ) @ ( Q @ X ) ) )
= ( semila1673364395name_o @ ( collect_pname @ P_2 ) @ ( collect_pname @ Q ) ) ) ).
thf(fact_870_Collect__conj__eq,axiom,
! [P_2: hoare_2091234717iple_a > $o,Q: hoare_2091234717iple_a > $o] :
( ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( P_2 @ X ) @ ( Q @ X ) ) )
= ( semila2006181266le_a_o @ ( collec992574898iple_a @ P_2 ) @ ( collec992574898iple_a @ Q ) ) ) ).
thf(fact_871_Collect__conj__eq,axiom,
! [P_2: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] : ( (&) @ ( P_2 @ X ) @ ( Q @ X ) ) )
= ( semila1947288293_nat_o @ ( collect_nat @ P_2 ) @ ( collect_nat @ Q ) ) ) ).
thf(fact_872_Int__Collect,axiom,
! [X_9: hoare_2091234717iple_a > $o,A_15: ( hoare_2091234717iple_a > $o ) > $o,P_1: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ X_9 @ ( semila1672913213_a_o_o @ A_15 @ ( collec1008234059le_a_o @ P_1 ) ) )
<=> ( ( member99268621le_a_o @ X_9 @ A_15 )
& ( P_1 @ X_9 ) ) ) ).
thf(fact_873_Int__Collect,axiom,
! [X_9: nat,A_15: nat > $o,P_1: nat > $o] :
( ( member_nat @ X_9 @ ( semila1947288293_nat_o @ A_15 @ ( collect_nat @ P_1 ) ) )
<=> ( ( member_nat @ X_9 @ A_15 )
& ( P_1 @ X_9 ) ) ) ).
thf(fact_874_Int__Collect,axiom,
! [X_9: hoare_2091234717iple_a,A_15: hoare_2091234717iple_a > $o,P_1: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_9 @ ( semila2006181266le_a_o @ A_15 @ ( collec992574898iple_a @ P_1 ) ) )
<=> ( ( member290856304iple_a @ X_9 @ A_15 )
& ( P_1 @ X_9 ) ) ) ).
thf(fact_875_Int__Collect,axiom,
! [X_9: pname,A_15: pname > $o,P_1: pname > $o] :
( ( member_pname @ X_9 @ ( semila1673364395name_o @ A_15 @ ( collect_pname @ P_1 ) ) )
<=> ( ( member_pname @ X_9 @ A_15 )
& ( P_1 @ X_9 ) ) ) ).
thf(fact_876_inf__Int__eq,axiom,
! [R: ( hoare_2091234717iple_a > $o ) > $o,S_1: ( hoare_2091234717iple_a > $o ) > $o,X: hoare_2091234717iple_a > $o] :
( ( semila1672913213_a_o_o
@ ^ [Y_7: hoare_2091234717iple_a > $o] : ( member99268621le_a_o @ Y_7 @ R )
@ ^ [Y_7: hoare_2091234717iple_a > $o] : ( member99268621le_a_o @ Y_7 @ S_1 )
@ X )
<=> ( member99268621le_a_o @ X @ ( semila1672913213_a_o_o @ R @ S_1 ) ) ) ).
thf(fact_877_inf__Int__eq,axiom,
! [R: nat > $o,S_1: nat > $o,X: nat] :
( ( semila1947288293_nat_o
@ ^ [Y_7: nat] : ( member_nat @ Y_7 @ R )
@ ^ [Y_7: nat] : ( member_nat @ Y_7 @ S_1 )
@ X )
<=> ( member_nat @ X @ ( semila1947288293_nat_o @ R @ S_1 ) ) ) ).
thf(fact_878_inf__Int__eq,axiom,
! [R: hoare_2091234717iple_a > $o,S_1: hoare_2091234717iple_a > $o,X: hoare_2091234717iple_a] :
( ( semila2006181266le_a_o
@ ^ [Y_7: hoare_2091234717iple_a] : ( member290856304iple_a @ Y_7 @ R )
@ ^ [Y_7: hoare_2091234717iple_a] : ( member290856304iple_a @ Y_7 @ S_1 )
@ X )
<=> ( member290856304iple_a @ X @ ( semila2006181266le_a_o @ R @ S_1 ) ) ) ).
thf(fact_879_inf__Int__eq,axiom,
! [R: pname > $o,S_1: pname > $o,X: pname] :
( ( semila1673364395name_o
@ ^ [Y_7: pname] : ( member_pname @ Y_7 @ R )
@ ^ [Y_7: pname] : ( member_pname @ Y_7 @ S_1 )
@ X )
<=> ( member_pname @ X @ ( semila1673364395name_o @ R @ S_1 ) ) ) ).
thf(fact_880_Int__absorb,axiom,
! [A_14: nat > $o] :
( ( semila1947288293_nat_o @ A_14 @ A_14 )
= A_14 ) ).
thf(fact_881_Int__absorb,axiom,
! [A_14: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_14 @ A_14 )
= A_14 ) ).
thf(fact_882_Int__absorb,axiom,
! [A_14: pname > $o] :
( ( semila1673364395name_o @ A_14 @ A_14 )
= A_14 ) ).
thf(fact_883_inf_Oidem,axiom,
! [A_13: nat > $o] :
( ( semila1947288293_nat_o @ A_13 @ A_13 )
= A_13 ) ).
thf(fact_884_inf_Oidem,axiom,
! [A_13: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_13 @ A_13 )
= A_13 ) ).
thf(fact_885_inf_Oidem,axiom,
! [A_13: nat] :
( ( semila80283416nf_nat @ A_13 @ A_13 )
= A_13 ) ).
thf(fact_886_inf_Oidem,axiom,
! [A_13: pname > $o] :
( ( semila1673364395name_o @ A_13 @ A_13 )
= A_13 ) ).
thf(fact_887_inf__idem,axiom,
! [X_8: nat > $o] :
( ( semila1947288293_nat_o @ X_8 @ X_8 )
= X_8 ) ).
thf(fact_888_inf__idem,axiom,
! [X_8: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ X_8 @ X_8 )
= X_8 ) ).
thf(fact_889_inf__idem,axiom,
! [X_8: nat] :
( ( semila80283416nf_nat @ X_8 @ X_8 )
= X_8 ) ).
thf(fact_890_inf__idem,axiom,
! [X_8: pname > $o] :
( ( semila1673364395name_o @ X_8 @ X_8 )
= X_8 ) ).
thf(fact_891_fun__diff__def,axiom,
! [A_12: hoare_2091234717iple_a > $o,B_12: hoare_2091234717iple_a > $o,X: hoare_2091234717iple_a] :
( ( minus_836160335le_a_o @ A_12 @ B_12 @ X )
<=> ( minus_minus_o @ ( A_12 @ X ) @ ( B_12 @ X ) ) ) ).
thf(fact_892_fun__diff__def,axiom,
! [A_12: pname > $o,B_12: pname > $o,X: pname] :
( ( minus_minus_pname_o @ A_12 @ B_12 @ X )
<=> ( minus_minus_o @ ( A_12 @ X ) @ ( B_12 @ X ) ) ) ).
thf(fact_893_fun__diff__def,axiom,
! [A_12: nat > $o,B_12: nat > $o,X: nat] :
( ( minus_minus_nat_o @ A_12 @ B_12 @ X )
<=> ( minus_minus_o @ ( A_12 @ X ) @ ( B_12 @ X ) ) ) ).
thf(fact_894_inf__fun__def,axiom,
! [F_1: nat > $o,G: nat > $o,X: nat] :
( ( semila1947288293_nat_o @ F_1 @ G @ X )
<=> ( semila854092349_inf_o @ ( F_1 @ X ) @ ( G @ X ) ) ) ).
thf(fact_895_inf__fun__def,axiom,
! [F_1: hoare_2091234717iple_a > $o,G: hoare_2091234717iple_a > $o,X: hoare_2091234717iple_a] :
( ( semila2006181266le_a_o @ F_1 @ G @ X )
<=> ( semila854092349_inf_o @ ( F_1 @ X ) @ ( G @ X ) ) ) ).
thf(fact_896_inf__fun__def,axiom,
! [F_1: pname > $o,G: pname > $o,X: pname] :
( ( semila1673364395name_o @ F_1 @ G @ X )
<=> ( semila854092349_inf_o @ ( F_1 @ X ) @ ( G @ X ) ) ) ).
thf(fact_897_set__diff__eq,axiom,
! [A_11: ( hoare_2091234717iple_a > $o ) > $o,B_11: ( hoare_2091234717iple_a > $o ) > $o] :
( ( minus_1746272704_a_o_o @ A_11 @ B_11 )
= ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (&) @ ( member99268621le_a_o @ X @ A_11 ) @ ( (~) @ ( member99268621le_a_o @ X @ B_11 ) ) ) ) ) ).
thf(fact_898_set__diff__eq,axiom,
! [A_11: nat > $o,B_11: nat > $o] :
( ( minus_minus_nat_o @ A_11 @ B_11 )
= ( collect_nat
@ ^ [X: nat] : ( (&) @ ( member_nat @ X @ A_11 ) @ ( (~) @ ( member_nat @ X @ B_11 ) ) ) ) ) ).
thf(fact_899_set__diff__eq,axiom,
! [A_11: hoare_2091234717iple_a > $o,B_11: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_11 @ B_11 )
= ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( member290856304iple_a @ X @ A_11 ) @ ( (~) @ ( member290856304iple_a @ X @ B_11 ) ) ) ) ) ).
thf(fact_900_set__diff__eq,axiom,
! [A_11: pname > $o,B_11: pname > $o] :
( ( minus_minus_pname_o @ A_11 @ B_11 )
= ( collect_pname
@ ^ [X: pname] : ( (&) @ ( member_pname @ X @ A_11 ) @ ( (~) @ ( member_pname @ X @ B_11 ) ) ) ) ) ).
thf(fact_901_Int__def,axiom,
! [A_10: ( hoare_2091234717iple_a > $o ) > $o,B_10: ( hoare_2091234717iple_a > $o ) > $o] :
( ( semila1672913213_a_o_o @ A_10 @ B_10 )
= ( collec1008234059le_a_o
@ ^ [X: hoare_2091234717iple_a > $o] : ( (&) @ ( member99268621le_a_o @ X @ A_10 ) @ ( member99268621le_a_o @ X @ B_10 ) ) ) ) ).
thf(fact_902_Int__def,axiom,
! [A_10: nat > $o,B_10: nat > $o] :
( ( semila1947288293_nat_o @ A_10 @ B_10 )
= ( collect_nat
@ ^ [X: nat] : ( (&) @ ( member_nat @ X @ A_10 ) @ ( member_nat @ X @ B_10 ) ) ) ) ).
thf(fact_903_Int__def,axiom,
! [A_10: hoare_2091234717iple_a > $o,B_10: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_10 @ B_10 )
= ( collec992574898iple_a
@ ^ [X: hoare_2091234717iple_a] : ( (&) @ ( member290856304iple_a @ X @ A_10 ) @ ( member290856304iple_a @ X @ B_10 ) ) ) ) ).
thf(fact_904_Int__def,axiom,
! [A_10: pname > $o,B_10: pname > $o] :
( ( semila1673364395name_o @ A_10 @ B_10 )
= ( collect_pname
@ ^ [X: pname] : ( (&) @ ( member_pname @ X @ A_10 ) @ ( member_pname @ X @ B_10 ) ) ) ) ).
thf(fact_905_Int__commute,axiom,
! [A_9: nat > $o,B_9: nat > $o] :
( ( semila1947288293_nat_o @ A_9 @ B_9 )
= ( semila1947288293_nat_o @ B_9 @ A_9 ) ) ).
thf(fact_906_Int__commute,axiom,
! [A_9: hoare_2091234717iple_a > $o,B_9: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_9 @ B_9 )
= ( semila2006181266le_a_o @ B_9 @ A_9 ) ) ).
thf(fact_907_Int__commute,axiom,
! [A_9: pname > $o,B_9: pname > $o] :
( ( semila1673364395name_o @ A_9 @ B_9 )
= ( semila1673364395name_o @ B_9 @ A_9 ) ) ).
thf(fact_908_inf_Ocommute,axiom,
! [A_8: nat > $o,B_8: nat > $o] :
( ( semila1947288293_nat_o @ A_8 @ B_8 )
= ( semila1947288293_nat_o @ B_8 @ A_8 ) ) ).
thf(fact_909_inf_Ocommute,axiom,
! [A_8: hoare_2091234717iple_a > $o,B_8: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_8 @ B_8 )
= ( semila2006181266le_a_o @ B_8 @ A_8 ) ) ).
thf(fact_910_inf_Ocommute,axiom,
! [A_8: nat,B_8: nat] :
( ( semila80283416nf_nat @ A_8 @ B_8 )
= ( semila80283416nf_nat @ B_8 @ A_8 ) ) ).
thf(fact_911_inf_Ocommute,axiom,
! [A_8: pname > $o,B_8: pname > $o] :
( ( semila1673364395name_o @ A_8 @ B_8 )
= ( semila1673364395name_o @ B_8 @ A_8 ) ) ).
thf(fact_912_inf__sup__aci_I1_J,axiom,
! [X_7: nat > $o,Y_6: nat > $o] :
( ( semila1947288293_nat_o @ X_7 @ Y_6 )
= ( semila1947288293_nat_o @ Y_6 @ X_7 ) ) ).
thf(fact_913_inf__sup__aci_I1_J,axiom,
! [X_7: hoare_2091234717iple_a > $o,Y_6: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ X_7 @ Y_6 )
= ( semila2006181266le_a_o @ Y_6 @ X_7 ) ) ).
thf(fact_914_inf__sup__aci_I1_J,axiom,
! [X_7: nat,Y_6: nat] :
( ( semila80283416nf_nat @ X_7 @ Y_6 )
= ( semila80283416nf_nat @ Y_6 @ X_7 ) ) ).
thf(fact_915_inf__sup__aci_I1_J,axiom,
! [X_7: pname > $o,Y_6: pname > $o] :
( ( semila1673364395name_o @ X_7 @ Y_6 )
= ( semila1673364395name_o @ Y_6 @ X_7 ) ) ).
thf(fact_916_inf__commute,axiom,
! [X_6: nat > $o,Y_5: nat > $o] :
( ( semila1947288293_nat_o @ X_6 @ Y_5 )
= ( semila1947288293_nat_o @ Y_5 @ X_6 ) ) ).
thf(fact_917_inf__commute,axiom,
! [X_6: hoare_2091234717iple_a > $o,Y_5: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ X_6 @ Y_5 )
= ( semila2006181266le_a_o @ Y_5 @ X_6 ) ) ).
thf(fact_918_inf__commute,axiom,
! [X_6: nat,Y_5: nat] :
( ( semila80283416nf_nat @ X_6 @ Y_5 )
= ( semila80283416nf_nat @ Y_5 @ X_6 ) ) ).
thf(fact_919_inf__commute,axiom,
! [X_6: pname > $o,Y_5: pname > $o] :
( ( semila1673364395name_o @ X_6 @ Y_5 )
= ( semila1673364395name_o @ Y_5 @ X_6 ) ) ).
thf(fact_920_Int__left__absorb,axiom,
! [A_7: nat > $o,B_7: nat > $o] :
( ( semila1947288293_nat_o @ A_7 @ ( semila1947288293_nat_o @ A_7 @ B_7 ) )
= ( semila1947288293_nat_o @ A_7 @ B_7 ) ) ).
thf(fact_921_Int__left__absorb,axiom,
! [A_7: hoare_2091234717iple_a > $o,B_7: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_7 @ ( semila2006181266le_a_o @ A_7 @ B_7 ) )
= ( semila2006181266le_a_o @ A_7 @ B_7 ) ) ).
thf(fact_922_Int__left__absorb,axiom,
! [A_7: pname > $o,B_7: pname > $o] :
( ( semila1673364395name_o @ A_7 @ ( semila1673364395name_o @ A_7 @ B_7 ) )
= ( semila1673364395name_o @ A_7 @ B_7 ) ) ).
thf(fact_923_inf_Oleft__idem,axiom,
! [A_6: nat > $o,B_6: nat > $o] :
( ( semila1947288293_nat_o @ A_6 @ ( semila1947288293_nat_o @ A_6 @ B_6 ) )
= ( semila1947288293_nat_o @ A_6 @ B_6 ) ) ).
thf(fact_924_inf_Oleft__idem,axiom,
! [A_6: hoare_2091234717iple_a > $o,B_6: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_6 @ ( semila2006181266le_a_o @ A_6 @ B_6 ) )
= ( semila2006181266le_a_o @ A_6 @ B_6 ) ) ).
thf(fact_925_inf_Oleft__idem,axiom,
! [A_6: nat,B_6: nat] :
( ( semila80283416nf_nat @ A_6 @ ( semila80283416nf_nat @ A_6 @ B_6 ) )
= ( semila80283416nf_nat @ A_6 @ B_6 ) ) ).
thf(fact_926_inf_Oleft__idem,axiom,
! [A_6: pname > $o,B_6: pname > $o] :
( ( semila1673364395name_o @ A_6 @ ( semila1673364395name_o @ A_6 @ B_6 ) )
= ( semila1673364395name_o @ A_6 @ B_6 ) ) ).
thf(fact_927_inf__sup__aci_I4_J,axiom,
! [X_5: nat > $o,Y_4: nat > $o] :
( ( semila1947288293_nat_o @ X_5 @ ( semila1947288293_nat_o @ X_5 @ Y_4 ) )
= ( semila1947288293_nat_o @ X_5 @ Y_4 ) ) ).
thf(fact_928_inf__sup__aci_I4_J,axiom,
! [X_5: hoare_2091234717iple_a > $o,Y_4: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ X_5 @ ( semila2006181266le_a_o @ X_5 @ Y_4 ) )
= ( semila2006181266le_a_o @ X_5 @ Y_4 ) ) ).
thf(fact_929_inf__sup__aci_I4_J,axiom,
! [X_5: nat,Y_4: nat] :
( ( semila80283416nf_nat @ X_5 @ ( semila80283416nf_nat @ X_5 @ Y_4 ) )
= ( semila80283416nf_nat @ X_5 @ Y_4 ) ) ).
thf(fact_930_inf__sup__aci_I4_J,axiom,
! [X_5: pname > $o,Y_4: pname > $o] :
( ( semila1673364395name_o @ X_5 @ ( semila1673364395name_o @ X_5 @ Y_4 ) )
= ( semila1673364395name_o @ X_5 @ Y_4 ) ) ).
thf(fact_931_inf__left__idem,axiom,
! [X_4: nat > $o,Y_3: nat > $o] :
( ( semila1947288293_nat_o @ X_4 @ ( semila1947288293_nat_o @ X_4 @ Y_3 ) )
= ( semila1947288293_nat_o @ X_4 @ Y_3 ) ) ).
thf(fact_932_inf__left__idem,axiom,
! [X_4: hoare_2091234717iple_a > $o,Y_3: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ X_4 @ ( semila2006181266le_a_o @ X_4 @ Y_3 ) )
= ( semila2006181266le_a_o @ X_4 @ Y_3 ) ) ).
thf(fact_933_inf__left__idem,axiom,
! [X_4: nat,Y_3: nat] :
( ( semila80283416nf_nat @ X_4 @ ( semila80283416nf_nat @ X_4 @ Y_3 ) )
= ( semila80283416nf_nat @ X_4 @ Y_3 ) ) ).
thf(fact_934_inf__left__idem,axiom,
! [X_4: pname > $o,Y_3: pname > $o] :
( ( semila1673364395name_o @ X_4 @ ( semila1673364395name_o @ X_4 @ Y_3 ) )
= ( semila1673364395name_o @ X_4 @ Y_3 ) ) ).
thf(fact_935_Int__left__commute,axiom,
! [A_5: nat > $o,B_5: nat > $o,C_5: nat > $o] :
( ( semila1947288293_nat_o @ A_5 @ ( semila1947288293_nat_o @ B_5 @ C_5 ) )
= ( semila1947288293_nat_o @ B_5 @ ( semila1947288293_nat_o @ A_5 @ C_5 ) ) ) ).
thf(fact_936_Int__left__commute,axiom,
! [A_5: hoare_2091234717iple_a > $o,B_5: hoare_2091234717iple_a > $o,C_5: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ A_5 @ ( semila2006181266le_a_o @ B_5 @ C_5 ) )
= ( semila2006181266le_a_o @ B_5 @ ( semila2006181266le_a_o @ A_5 @ C_5 ) ) ) ).
thf(fact_937_Int__left__commute,axiom,
! [A_5: pname > $o,B_5: pname > $o,C_5: pname > $o] :
( ( semila1673364395name_o @ A_5 @ ( semila1673364395name_o @ B_5 @ C_5 ) )
= ( semila1673364395name_o @ B_5 @ ( semila1673364395name_o @ A_5 @ C_5 ) ) ) ).
thf(fact_938_inf_Oleft__commute,axiom,
! [B_4: nat > $o,A_4: nat > $o,C_4: nat > $o] :
( ( semila1947288293_nat_o @ B_4 @ ( semila1947288293_nat_o @ A_4 @ C_4 ) )
= ( semila1947288293_nat_o @ A_4 @ ( semila1947288293_nat_o @ B_4 @ C_4 ) ) ) ).
thf(fact_939_inf_Oleft__commute,axiom,
! [B_4: hoare_2091234717iple_a > $o,A_4: hoare_2091234717iple_a > $o,C_4: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ B_4 @ ( semila2006181266le_a_o @ A_4 @ C_4 ) )
= ( semila2006181266le_a_o @ A_4 @ ( semila2006181266le_a_o @ B_4 @ C_4 ) ) ) ).
thf(fact_940_inf_Oleft__commute,axiom,
! [B_4: nat,A_4: nat,C_4: nat] :
( ( semila80283416nf_nat @ B_4 @ ( semila80283416nf_nat @ A_4 @ C_4 ) )
= ( semila80283416nf_nat @ A_4 @ ( semila80283416nf_nat @ B_4 @ C_4 ) ) ) ).
thf(fact_941_inf_Oleft__commute,axiom,
! [B_4: pname > $o,A_4: pname > $o,C_4: pname > $o] :
( ( semila1673364395name_o @ B_4 @ ( semila1673364395name_o @ A_4 @ C_4 ) )
= ( semila1673364395name_o @ A_4 @ ( semila1673364395name_o @ B_4 @ C_4 ) ) ) ).
thf(fact_942_inf__sup__aci_I3_J,axiom,
! [X_3: nat > $o,Y_2: nat > $o,Z_2: nat > $o] :
( ( semila1947288293_nat_o @ X_3 @ ( semila1947288293_nat_o @ Y_2 @ Z_2 ) )
= ( semila1947288293_nat_o @ Y_2 @ ( semila1947288293_nat_o @ X_3 @ Z_2 ) ) ) ).
thf(fact_943_inf__sup__aci_I3_J,axiom,
! [X_3: hoare_2091234717iple_a > $o,Y_2: hoare_2091234717iple_a > $o,Z_2: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ X_3 @ ( semila2006181266le_a_o @ Y_2 @ Z_2 ) )
= ( semila2006181266le_a_o @ Y_2 @ ( semila2006181266le_a_o @ X_3 @ Z_2 ) ) ) ).
thf(fact_944_inf__sup__aci_I3_J,axiom,
! [X_3: nat,Y_2: nat,Z_2: nat] :
( ( semila80283416nf_nat @ X_3 @ ( semila80283416nf_nat @ Y_2 @ Z_2 ) )
= ( semila80283416nf_nat @ Y_2 @ ( semila80283416nf_nat @ X_3 @ Z_2 ) ) ) ).
thf(fact_945_inf__sup__aci_I3_J,axiom,
! [X_3: pname > $o,Y_2: pname > $o,Z_2: pname > $o] :
( ( semila1673364395name_o @ X_3 @ ( semila1673364395name_o @ Y_2 @ Z_2 ) )
= ( semila1673364395name_o @ Y_2 @ ( semila1673364395name_o @ X_3 @ Z_2 ) ) ) ).
thf(fact_946_inf__left__commute,axiom,
! [X_2: nat > $o,Y_1: nat > $o,Z_1: nat > $o] :
( ( semila1947288293_nat_o @ X_2 @ ( semila1947288293_nat_o @ Y_1 @ Z_1 ) )
= ( semila1947288293_nat_o @ Y_1 @ ( semila1947288293_nat_o @ X_2 @ Z_1 ) ) ) ).
thf(fact_947_inf__left__commute,axiom,
! [X_2: hoare_2091234717iple_a > $o,Y_1: hoare_2091234717iple_a > $o,Z_1: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ X_2 @ ( semila2006181266le_a_o @ Y_1 @ Z_1 ) )
= ( semila2006181266le_a_o @ Y_1 @ ( semila2006181266le_a_o @ X_2 @ Z_1 ) ) ) ).
thf(fact_948_inf__left__commute,axiom,
! [X_2: nat,Y_1: nat,Z_1: nat] :
( ( semila80283416nf_nat @ X_2 @ ( semila80283416nf_nat @ Y_1 @ Z_1 ) )
= ( semila80283416nf_nat @ Y_1 @ ( semila80283416nf_nat @ X_2 @ Z_1 ) ) ) ).
thf(fact_949_inf__left__commute,axiom,
! [X_2: pname > $o,Y_1: pname > $o,Z_1: pname > $o] :
( ( semila1673364395name_o @ X_2 @ ( semila1673364395name_o @ Y_1 @ Z_1 ) )
= ( semila1673364395name_o @ Y_1 @ ( semila1673364395name_o @ X_2 @ Z_1 ) ) ) ).
thf(fact_950_Diff__iff,axiom,
! [C_3: hoare_2091234717iple_a > $o,A_3: ( hoare_2091234717iple_a > $o ) > $o,B_3: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_3 @ ( minus_1746272704_a_o_o @ A_3 @ B_3 ) )
<=> ( ( member99268621le_a_o @ C_3 @ A_3 )
& ~ ( member99268621le_a_o @ C_3 @ B_3 ) ) ) ).
thf(fact_951_Diff__iff,axiom,
! [C_3: nat,A_3: nat > $o,B_3: nat > $o] :
( ( member_nat @ C_3 @ ( minus_minus_nat_o @ A_3 @ B_3 ) )
<=> ( ( member_nat @ C_3 @ A_3 )
& ~ ( member_nat @ C_3 @ B_3 ) ) ) ).
thf(fact_952_Diff__iff,axiom,
! [C_3: hoare_2091234717iple_a,A_3: hoare_2091234717iple_a > $o,B_3: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_3 @ ( minus_836160335le_a_o @ A_3 @ B_3 ) )
<=> ( ( member290856304iple_a @ C_3 @ A_3 )
& ~ ( member290856304iple_a @ C_3 @ B_3 ) ) ) ).
thf(fact_953_Diff__iff,axiom,
! [C_3: pname,A_3: pname > $o,B_3: pname > $o] :
( ( member_pname @ C_3 @ ( minus_minus_pname_o @ A_3 @ B_3 ) )
<=> ( ( member_pname @ C_3 @ A_3 )
& ~ ( member_pname @ C_3 @ B_3 ) ) ) ).
thf(fact_954_Int__iff,axiom,
! [C_2: hoare_2091234717iple_a > $o,A_2: ( hoare_2091234717iple_a > $o ) > $o,B_2: ( hoare_2091234717iple_a > $o ) > $o] :
( ( member99268621le_a_o @ C_2 @ ( semila1672913213_a_o_o @ A_2 @ B_2 ) )
<=> ( ( member99268621le_a_o @ C_2 @ A_2 )
& ( member99268621le_a_o @ C_2 @ B_2 ) ) ) ).
thf(fact_955_Int__iff,axiom,
! [C_2: nat,A_2: nat > $o,B_2: nat > $o] :
( ( member_nat @ C_2 @ ( semila1947288293_nat_o @ A_2 @ B_2 ) )
<=> ( ( member_nat @ C_2 @ A_2 )
& ( member_nat @ C_2 @ B_2 ) ) ) ).
thf(fact_956_Int__iff,axiom,
! [C_2: hoare_2091234717iple_a,A_2: hoare_2091234717iple_a > $o,B_2: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_2 @ ( semila2006181266le_a_o @ A_2 @ B_2 ) )
<=> ( ( member290856304iple_a @ C_2 @ A_2 )
& ( member290856304iple_a @ C_2 @ B_2 ) ) ) ).
thf(fact_957_Int__iff,axiom,
! [C_2: pname,A_2: pname > $o,B_2: pname > $o] :
( ( member_pname @ C_2 @ ( semila1673364395name_o @ A_2 @ B_2 ) )
<=> ( ( member_pname @ C_2 @ A_2 )
& ( member_pname @ C_2 @ B_2 ) ) ) ).
thf(fact_958_Diff__Int__distrib,axiom,
! [C_1: hoare_2091234717iple_a > $o,A_1: hoare_2091234717iple_a > $o,B_1: hoare_2091234717iple_a > $o] :
( ( semila2006181266le_a_o @ C_1 @ ( minus_836160335le_a_o @ A_1 @ B_1 ) )
= ( minus_836160335le_a_o @ ( semila2006181266le_a_o @ C_1 @ A_1 ) @ ( semila2006181266le_a_o @ C_1 @ B_1 ) ) ) ).
thf(fact_959_Diff__Int__distrib,axiom,
! [C_1: pname > $o,A_1: pname > $o,B_1: pname > $o] :
( ( semila1673364395name_o @ C_1 @ ( minus_minus_pname_o @ A_1 @ B_1 ) )
= ( minus_minus_pname_o @ ( semila1673364395name_o @ C_1 @ A_1 ) @ ( semila1673364395name_o @ C_1 @ B_1 ) ) ) ).
thf(fact_960_Diff__Int__distrib,axiom,
! [C_1: nat > $o,A_1: nat > $o,B_1: nat > $o] :
( ( semila1947288293_nat_o @ C_1 @ ( minus_minus_nat_o @ A_1 @ B_1 ) )
= ( minus_minus_nat_o @ ( semila1947288293_nat_o @ C_1 @ A_1 ) @ ( semila1947288293_nat_o @ C_1 @ B_1 ) ) ) ).
thf(fact_961_diff__0__eq__0,axiom,
! [N_1: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N_1 )
= zero_zero_nat ) ).
thf(fact_962_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_963_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_964_diffs0__imp__equal,axiom,
! [M: nat,N_1: nat] :
( ( ( minus_minus_nat @ M @ N_1 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N_1 @ M )
= zero_zero_nat )
=> ( M = N_1 ) ) ) ).
thf(fact_965_diff__Suc__Suc,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_966_Suc__diff__diff,axiom,
! [M: nat,N_1: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N_1 ) @ ( suc @ K_1 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N_1 ) @ K_1 ) ) ).
thf(fact_967_diff__commute,axiom,
! [I_1: nat,J_1: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ K_1 )
= ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ K_1 ) @ J_1 ) ) ).
thf(fact_968_zero__induct__lemma,axiom,
! [I_1: nat,P: nat > $o,K_1: nat] :
( ( P @ K_1 )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ ( minus_minus_nat @ K_1 @ I_1 ) ) ) ) ).
thf(fact_969_diff__Suc,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ M @ ( suc @ N_1 ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [K: nat] : K
@ ( minus_minus_nat @ M @ N_1 ) ) ) ).
thf(fact_970_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_971_diff__Suc__1,axiom,
! [N_1: nat] :
( ( minus_minus_nat @ ( suc @ N_1 ) @ one_one_nat )
= N_1 ) ).
thf(fact_972_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ M @ ( suc @ N_1 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ).
thf(fact_973_plus__nat_Oadd__0,axiom,
! [N_1: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N_1 )
= N_1 ) ).
thf(fact_974_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_975_add__is__0,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N_1 = zero_zero_nat ) ) ) ).
thf(fact_976_add__eq__self__zero,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= M )
=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_977_add__Suc__right,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ M @ ( suc @ N_1 ) )
= ( suc @ ( plus_plus_nat @ M @ N_1 ) ) ) ).
thf(fact_978_add__Suc,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_1 )
= ( suc @ ( plus_plus_nat @ M @ N_1 ) ) ) ).
thf(fact_979_add__Suc__shift,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_1 )
= ( plus_plus_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_980_nat__add__commute,axiom,
! [M: nat,N_1: nat] :
( ( plus_plus_nat @ M @ N_1 )
= ( plus_plus_nat @ N_1 @ M ) ) ).
thf(fact_981_nat__add__left__commute,axiom,
! [X_1: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X_1 @ ( plus_plus_nat @ Y @ Z ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X_1 @ Z ) ) ) ).
thf(fact_982_nat__add__assoc,axiom,
! [M: nat,N_1: nat,K_1: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N_1 ) @ K_1 )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_983_nat__add__left__cancel,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ( plus_plus_nat @ K_1 @ M )
= ( plus_plus_nat @ K_1 @ N_1 ) )
<=> ( M = N_1 ) ) ).
thf(fact_984_nat__add__right__cancel,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ K_1 )
= ( plus_plus_nat @ N_1 @ K_1 ) )
<=> ( M = N_1 ) ) ).
thf(fact_985_diff__add__inverse2,axiom,
! [M: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N_1 ) @ N_1 )
= M ) ).
thf(fact_986_diff__add__inverse,axiom,
! [N_1: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N_1 @ M ) @ N_1 )
= M ) ).
thf(fact_987_diff__diff__left,axiom,
! [I_1: nat,J_1: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ K_1 )
= ( minus_minus_nat @ I_1 @ ( plus_plus_nat @ J_1 @ K_1 ) ) ) ).
thf(fact_988_diff__cancel,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_989_diff__cancel2,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K_1 ) @ ( plus_plus_nat @ N_1 @ K_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ).
thf(fact_990_add__is__1,axiom,
! [M: nat,N_1: nat] :
( ( ( plus_plus_nat @ M @ N_1 )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_991_one__is__add,axiom,
! [M: nat,N_1: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N_1 ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_992_diff__add__0,axiom,
! [N_1: nat,M: nat] :
( ( minus_minus_nat @ N_1 @ ( plus_plus_nat @ N_1 @ M ) )
= zero_zero_nat ) ).
thf(fact_993_Suc__eq__plus1,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
= ( plus_plus_nat @ N_1 @ one_one_nat ) ) ).
thf(fact_994_Suc__eq__plus1__left,axiom,
! [N_1: nat] :
( ( suc @ N_1 )
= ( plus_plus_nat @ one_one_nat @ N_1 ) ) ).
thf(fact_995_add__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_1 )
= N_1 ) )
& ( ( M != zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_1 )
= ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ) ) ) ).
thf(fact_996_com_Osize_I4_J,axiom,
! [Com1_1: com,Com2_1: com] :
( ( com_size @ ( semi @ Com1_1 @ Com2_1 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( com_size @ Com1_1 ) @ ( com_size @ Com2_1 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_997_com_Osize_I6_J,axiom,
! [Fun_1: state > $o,Com_1: com] :
( ( com_size @ ( while @ Fun_1 @ Com_1 ) )
= ( plus_plus_nat @ ( com_size @ Com_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_998_com_Osize_I7_J,axiom,
! [Pname_1: pname] :
( ( com_size @ ( body @ Pname_1 ) )
= zero_zero_nat ) ).
thf(fact_999_com_Osize_I1_J,axiom,
( ( com_size @ skip )
= zero_zero_nat ) ).
thf(fact_1000_com_Osize_I12_J,axiom,
! [Com1_1: com,Com2_1: com] :
( ( size_size_com @ ( semi @ Com1_1 @ Com2_1 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ Com1_1 ) @ ( size_size_com @ Com2_1 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1001_com_Osize_I14_J,axiom,
! [Fun_1: state > $o,Com_1: com] :
( ( size_size_com @ ( while @ Fun_1 @ Com_1 ) )
= ( plus_plus_nat @ ( size_size_com @ Com_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1002_add__mult__distrib2,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( times_times_nat @ K_1 @ ( plus_plus_nat @ M @ N_1 ) )
= ( plus_plus_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N_1 ) ) ) ).
thf(fact_1003_add__mult__distrib,axiom,
! [M: nat,N_1: nat,K_1: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N_1 ) @ K_1 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_1004_nat__mult__eq__1__iff,axiom,
! [M: nat,N_1: nat] :
( ( ( times_times_nat @ M @ N_1 )
= one_one_nat )
<=> ( ( M = one_one_nat )
& ( N_1 = one_one_nat ) ) ) ).
thf(fact_1005_nat__mult__1__right,axiom,
! [N_1: nat] :
( ( times_times_nat @ N_1 @ one_one_nat )
= N_1 ) ).
thf(fact_1006_nat__1__eq__mult__iff,axiom,
! [M: nat,N_1: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N_1 ) )
<=> ( ( M = one_one_nat )
& ( N_1 = one_one_nat ) ) ) ).
thf(fact_1007_nat__mult__1,axiom,
! [N_1: nat] :
( ( times_times_nat @ one_one_nat @ N_1 )
= N_1 ) ).
thf(fact_1008_mult__0,axiom,
! [N_1: nat] :
( ( times_times_nat @ zero_zero_nat @ N_1 )
= zero_zero_nat ) ).
thf(fact_1009_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_1010_mult__is__0,axiom,
! [M: nat,N_1: nat] :
( ( ( times_times_nat @ M @ N_1 )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N_1 = zero_zero_nat ) ) ) ).
thf(fact_1011_mult__cancel1,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N_1 ) )
<=> ( ( M = N_1 )
| ( K_1 = zero_zero_nat ) ) ) ).
thf(fact_1012_mult__cancel2,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( ( times_times_nat @ M @ K_1 )
= ( times_times_nat @ N_1 @ K_1 ) )
<=> ( ( M = N_1 )
| ( K_1 = zero_zero_nat ) ) ) ).
thf(fact_1013_Suc__mult__cancel1,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ( times_times_nat @ ( suc @ K_1 ) @ M )
= ( times_times_nat @ ( suc @ K_1 ) @ N_1 ) )
<=> ( M = N_1 ) ) ).
thf(fact_1014_diff__mult__distrib,axiom,
! [M: nat,N_1: nat,K_1: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N_1 ) @ K_1 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_1015_diff__mult__distrib2,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( times_times_nat @ K_1 @ ( minus_minus_nat @ M @ N_1 ) )
= ( minus_minus_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N_1 ) ) ) ).
thf(fact_1016_mult__eq__1__iff,axiom,
! [M: nat,N_1: nat] :
( ( ( times_times_nat @ M @ N_1 )
= ( suc @ zero_zero_nat ) )
<=> ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_1
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_1017_mult__Suc,axiom,
! [M: nat,N_1: nat] :
( ( times_times_nat @ ( suc @ M ) @ N_1 )
= ( plus_plus_nat @ N_1 @ ( times_times_nat @ M @ N_1 ) ) ) ).
thf(fact_1018_mult__Suc__right,axiom,
! [M: nat,N_1: nat] :
( ( times_times_nat @ M @ ( suc @ N_1 ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N_1 ) ) ) ).
thf(fact_1019_mult__eq__self__implies__10,axiom,
! [M: nat,N_1: nat] :
( ( M
= ( times_times_nat @ M @ N_1 ) )
=> ( ( N_1 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
thf(fact_1020_com_Osize_I15_J,axiom,
! [Pname_1: pname] :
( ( size_size_com @ ( body @ Pname_1 ) )
= zero_zero_nat ) ).
thf(fact_1021_com_Osize_I9_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
thf(fact_1022_mult__eq__if,axiom,
! [N_1: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( times_times_nat @ M @ N_1 )
= zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( times_times_nat @ M @ N_1 )
= ( plus_plus_nat @ N_1 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_1 ) ) ) ) ) ).
thf(fact_1023_nat__mult__commute,axiom,
! [M: nat,N_1: nat] :
( ( times_times_nat @ M @ N_1 )
= ( times_times_nat @ N_1 @ M ) ) ).
thf(fact_1024_nat__mult__assoc,axiom,
! [M: nat,N_1: nat,K_1: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N_1 ) @ K_1 )
= ( times_times_nat @ M @ ( times_times_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_1025_left__add__mult__distrib,axiom,
! [I_1: nat,U: nat,J_1: nat,K_1: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ K_1 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ U ) @ K_1 ) ) ).
thf(fact_1026_nat__mult__eq__cancel__disj,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N_1 ) )
<=> ( ( K_1 = zero_zero_nat )
| ( M = N_1 ) ) ) ).
thf(fact_1027_com_Osize_I13_J,axiom,
! [Fun_1: state > $o,Com1_1: com,Com2_1: com] :
( ( size_size_com @ ( cond @ Fun_1 @ Com1_1 @ Com2_1 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ Com1_1 ) @ ( size_size_com @ Com2_1 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1028_com_Osize_I5_J,axiom,
! [Fun_1: state > $o,Com1_1: com,Com2_1: com] :
( ( com_size @ ( cond @ Fun_1 @ Com1_1 @ Com2_1 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( com_size @ Com1_1 ) @ ( com_size @ Com2_1 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1029_finite__Collect__le__nat,axiom,
! [K_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K_1 ) ) ) ).
thf(fact_1030_le0,axiom,
! [N_1: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N_1 ) ).
thf(fact_1031_evaln__elim__cases_I5_J,axiom,
! [B: state > $o,C1: com,C2: com,S: state,N_1: nat,T: state] :
( ( evaln @ ( cond @ B @ C1 @ C2 ) @ S @ N_1 @ T )
=> ( ( ( B @ S )
=> ~ ( evaln @ C1 @ S @ N_1 @ T ) )
=> ~ ( ~ ( B @ S )
=> ~ ( evaln @ C2 @ S @ N_1 @ T ) ) ) ) ).
thf(fact_1032_evaln_OIfTrue,axiom,
! [C1: com,C0: com,N_1: nat,S1: state,B: state > $o,S: state] :
( ( B @ S )
=> ( ( evaln @ C0 @ S @ N_1 @ S1 )
=> ( evaln @ ( cond @ B @ C0 @ C1 ) @ S @ N_1 @ S1 ) ) ) ).
thf(fact_1033_evaln_OIfFalse,axiom,
! [C0: com,C1: com,N_1: nat,S1: state,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( ( evaln @ C1 @ S @ N_1 @ S1 )
=> ( evaln @ ( cond @ B @ C0 @ C1 ) @ S @ N_1 @ S1 ) ) ) ).
thf(fact_1034_evalc_OIfFalse,axiom,
! [C0: com,C1: com,S1: state,B: state > $o,S: state] :
( ~ ( B @ S )
=> ( ( evalc @ C1 @ S @ S1 )
=> ( evalc @ ( cond @ B @ C0 @ C1 ) @ S @ S1 ) ) ) ).
thf(fact_1035_evalc_OIfTrue,axiom,
! [C1: com,C0: com,S1: state,B: state > $o,S: state] :
( ( B @ S )
=> ( ( evalc @ C0 @ S @ S1 )
=> ( evalc @ ( cond @ B @ C0 @ C1 ) @ S @ S1 ) ) ) ).
thf(fact_1036_evalc__elim__cases_I5_J,axiom,
! [B: state > $o,C1: com,C2: com,S: state,T: state] :
( ( evalc @ ( cond @ B @ C1 @ C2 ) @ S @ T )
=> ( ( ( B @ S )
=> ~ ( evalc @ C1 @ S @ T ) )
=> ~ ( ~ ( B @ S )
=> ~ ( evalc @ C2 @ S @ T ) ) ) ) ).
thf(fact_1037_com_Osimps_I55_J,axiom,
! [Pname: pname,Fun_1: state > $o,Com1_1: com,Com2_1: com] :
( ( body @ Pname )
!= ( cond @ Fun_1 @ Com1_1 @ Com2_1 ) ) ).
thf(fact_1038_com_Osimps_I54_J,axiom,
! [Fun_1: state > $o,Com1_1: com,Com2_1: com,Pname: pname] :
( ( cond @ Fun_1 @ Com1_1 @ Com2_1 )
!= ( body @ Pname ) ) ).
thf(fact_1039_com_Osimps_I4_J,axiom,
! [Fun_1: state > $o,Com1_1: com,Com2_1: com,Fun: state > $o,Com1: com,Com2: com] :
( ( ( cond @ Fun_1 @ Com1_1 @ Com2_1 )
= ( cond @ Fun @ Com1 @ Com2 ) )
<=> ( ( Fun_1 = Fun )
& ( Com1_1 = Com1 )
& ( Com2_1 = Com2 ) ) ) ).
thf(fact_1040_le__antisym,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( ord_less_eq_nat @ N_1 @ M )
=> ( M = N_1 ) ) ) ).
thf(fact_1041_le__trans,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ J_1 @ K_1 )
=> ( ord_less_eq_nat @ I_1 @ K_1 ) ) ) ).
thf(fact_1042_eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( M = N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1043_nat__le__linear,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
| ( ord_less_eq_nat @ N_1 @ M ) ) ).
thf(fact_1044_le__refl,axiom,
! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ N_1 ) ).
thf(fact_1045_Suc__leD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1046_le__SucE,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N_1 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N_1 )
=> ( M
= ( suc @ N_1 ) ) ) ) ).
thf(fact_1047_le__SucI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_1048_Suc__le__mono,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N_1 ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N_1 @ M ) ) ).
thf(fact_1049_le__Suc__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N_1 ) )
<=> ( ( ord_less_eq_nat @ M @ N_1 )
| ( M
= ( suc @ N_1 ) ) ) ) ).
thf(fact_1050_not__less__eq__eq,axiom,
! [M: nat,N_1: nat] :
( ~ ( ord_less_eq_nat @ M @ N_1 )
<=> ( ord_less_eq_nat @ ( suc @ N_1 ) @ M ) ) ).
thf(fact_1051_Suc__n__not__le__n,axiom,
! [N_1: nat] :
~ ( ord_less_eq_nat @ ( suc @ N_1 ) @ N_1 ) ).
thf(fact_1052_le__0__eq,axiom,
! [N_1: nat] :
( ( ord_less_eq_nat @ N_1 @ zero_zero_nat )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_1053_less__eq__nat_Osimps_I1_J,axiom,
! [N_1: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N_1 ) ).
thf(fact_1054_evaln__nonstrict,axiom,
! [M: nat,C: com,S: state,N_1: nat,T: state] :
( ( evaln @ C @ S @ N_1 @ T )
=> ( ( ord_less_eq_nat @ N_1 @ M )
=> ( evaln @ C @ S @ M @ T ) ) ) ).
thf(fact_1055_diff__le__self,axiom,
! [M: nat,N_1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N_1 ) @ M ) ).
thf(fact_1056_diff__le__mono2,axiom,
! [L: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N_1 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_1057_diff__le__mono,axiom,
! [L: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N_1 @ L ) ) ) ).
thf(fact_1058_diff__diff__cancel,axiom,
! [I_1: nat,N_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N_1 )
=> ( ( minus_minus_nat @ N_1 @ ( minus_minus_nat @ N_1 @ I_1 ) )
= I_1 ) ) ).
thf(fact_1059_eq__diff__iff,axiom,
! [N_1: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ( ( minus_minus_nat @ M @ K_1 )
= ( minus_minus_nat @ N_1 @ K_1 ) )
<=> ( M = N_1 ) ) ) ) ).
thf(fact_1060_Nat_Odiff__diff__eq,axiom,
! [N_1: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N_1 @ K_1 ) )
= ( minus_minus_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1061_le__diff__iff,axiom,
! [N_1: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N_1 @ K_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1062_add__leE,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N_1 )
=> ~ ( ( ord_less_eq_nat @ M @ N_1 )
=> ~ ( ord_less_eq_nat @ K_1 @ N_1 ) ) ) ).
thf(fact_1063_add__leD1,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1064_add__leD2,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N_1 )
=> ( ord_less_eq_nat @ K_1 @ N_1 ) ) ).
thf(fact_1065_add__le__mono,axiom,
! [K_1: nat,L: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ K_1 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K_1 ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_1066_add__le__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K_1 ) @ ( plus_plus_nat @ J_1 @ K_1 ) ) ) ).
thf(fact_1067_trans__le__add2,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ M @ J_1 ) ) ) ).
thf(fact_1068_trans__le__add1,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ J_1 @ M ) ) ) ).
thf(fact_1069_nat__add__left__cancel__le,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1070_le__iff__add,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
<=> ? [K: nat] :
( N_1
= ( plus_plus_nat @ M @ K ) ) ) ).
thf(fact_1071_le__add1,axiom,
! [N_1: nat,M: nat] : ( ord_less_eq_nat @ N_1 @ ( plus_plus_nat @ N_1 @ M ) ) ).
thf(fact_1072_le__add2,axiom,
! [N_1: nat,M: nat] : ( ord_less_eq_nat @ N_1 @ ( plus_plus_nat @ M @ N_1 ) ) ).
thf(fact_1073_card__Collect__le__nat,axiom,
! [N_1: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N_1 ) ) )
= ( suc @ N_1 ) ) ).
thf(fact_1074_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N_1 )
<=> ( nat_case_o @ $false @ ( ord_less_eq_nat @ M ) @ N_1 ) ) ).
thf(fact_1075_mult__le__mono,axiom,
! [K_1: nat,L: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ K_1 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K_1 ) @ ( times_times_nat @ J_1 @ L ) ) ) ) ).
thf(fact_1076_mult__le__mono2,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K_1 @ I_1 ) @ ( times_times_nat @ K_1 @ J_1 ) ) ) ).
thf(fact_1077_mult__le__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K_1 ) @ ( times_times_nat @ J_1 @ K_1 ) ) ) ).
thf(fact_1078_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_1079_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_1080_com_Osimps_I52_J,axiom,
! [Fun_1: state > $o,Com1_1: com,Com2_1: com,Fun: state > $o,Com: com] :
( ( cond @ Fun_1 @ Com1_1 @ Com2_1 )
!= ( while @ Fun @ Com ) ) ).
thf(fact_1081_com_Osimps_I53_J,axiom,
! [Fun: state > $o,Com: com,Fun_1: state > $o,Com1_1: com,Com2_1: com] :
( ( while @ Fun @ Com )
!= ( cond @ Fun_1 @ Com1_1 @ Com2_1 ) ) ).
thf(fact_1082_com_Osimps_I45_J,axiom,
! [Fun: state > $o,Com1: com,Com2: com,Com1_1: com,Com2_1: com] :
( ( cond @ Fun @ Com1 @ Com2 )
!= ( semi @ Com1_1 @ Com2_1 ) ) ).
thf(fact_1083_com_Osimps_I44_J,axiom,
! [Com1_1: com,Com2_1: com,Fun: state > $o,Com1: com,Com2: com] :
( ( semi @ Com1_1 @ Com2_1 )
!= ( cond @ Fun @ Com1 @ Com2 ) ) ).
thf(fact_1084_com_Osimps_I15_J,axiom,
! [Fun: state > $o,Com1: com,Com2: com] :
( ( cond @ Fun @ Com1 @ Com2 )
!= skip ) ).
thf(fact_1085_com_Osimps_I14_J,axiom,
! [Fun: state > $o,Com1: com,Com2: com] :
( skip
!= ( cond @ Fun @ Com1 @ Com2 ) ) ).
thf(fact_1086_diff__is__0__eq_H,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( minus_minus_nat @ M @ N_1 )
= zero_zero_nat ) ) ).
thf(fact_1087_diff__is__0__eq,axiom,
! [M: nat,N_1: nat] :
( ( ( minus_minus_nat @ M @ N_1 )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1088_Suc__diff__le,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N_1 )
= ( suc @ ( minus_minus_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1089_Suc__mult__le__cancel1,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K_1 ) @ M ) @ ( times_times_nat @ ( suc @ K_1 ) @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1090_diff__diff__right,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( minus_minus_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ K_1 ) @ J_1 ) ) ) ).
thf(fact_1091_le__diff__conv,axiom,
! [J_1: nat,K_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J_1 @ K_1 ) @ I_1 )
<=> ( ord_less_eq_nat @ J_1 @ ( plus_plus_nat @ I_1 @ K_1 ) ) ) ).
thf(fact_1092_le__add__diff,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N_1 @ M ) @ K_1 ) ) ) ).
thf(fact_1093_le__add__diff__inverse,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( plus_plus_nat @ N_1 @ ( minus_minus_nat @ M @ N_1 ) )
= M ) ) ).
thf(fact_1094_add__diff__assoc,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ K_1 ) ) ) ).
thf(fact_1095_le__diff__conv2,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( ord_less_eq_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K_1 ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K_1 ) @ J_1 ) ) ) ).
thf(fact_1096_le__add__diff__inverse2,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_eq_nat @ N_1 @ M )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N_1 ) @ N_1 )
= M ) ) ).
thf(fact_1097_le__imp__diff__is__add,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ( minus_minus_nat @ J_1 @ I_1 )
= K_1 )
<=> ( J_1
= ( plus_plus_nat @ K_1 @ I_1 ) ) ) ) ).
thf(fact_1098_diff__add__assoc,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ K_1 )
= ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K_1 ) ) ) ) ).
thf(fact_1099_add__diff__assoc2,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J_1 @ K_1 ) @ I_1 )
= ( minus_minus_nat @ ( plus_plus_nat @ J_1 @ I_1 ) @ K_1 ) ) ) ).
thf(fact_1100_diff__add__assoc2,axiom,
! [I_1: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J_1 @ I_1 ) @ K_1 )
= ( plus_plus_nat @ ( minus_minus_nat @ J_1 @ K_1 ) @ I_1 ) ) ) ).
thf(fact_1101_one__le__mult__iff,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N_1 ) )
<=> ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N_1 ) ) ) ).
thf(fact_1102_diff__Suc__diff__eq2,axiom,
! [M: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J_1 @ K_1 ) ) @ M )
= ( minus_minus_nat @ ( suc @ J_1 ) @ ( plus_plus_nat @ K_1 @ M ) ) ) ) ).
thf(fact_1103_diff__Suc__diff__eq1,axiom,
! [M: nat,K_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ K_1 @ J_1 )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J_1 @ K_1 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K_1 ) @ ( suc @ J_1 ) ) ) ) ).
thf(fact_1104_nat__le__add__iff1,axiom,
! [U: nat,M: nat,N_1: nat,J_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ J_1 @ I_1 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N_1 ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ U ) @ M ) @ N_1 ) ) ) ).
thf(fact_1105_nat__diff__add__eq1,axiom,
! [U: nat,M: nat,N_1: nat,J_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ J_1 @ I_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ U ) @ M ) @ N_1 ) ) ) ).
thf(fact_1106_nat__eq__add__iff1,axiom,
! [U: nat,M: nat,N_1: nat,J_1: nat,I_1: nat] :
( ( ord_less_eq_nat @ J_1 @ I_1 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N_1 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J_1 ) @ U ) @ M )
= N_1 ) ) ) ).
thf(fact_1107_nat__le__add__iff2,axiom,
! [U: nat,M: nat,N_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J_1 @ I_1 ) @ U ) @ N_1 ) ) ) ) ).
thf(fact_1108_nat__diff__add__eq2,axiom,
! [U: nat,M: nat,N_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N_1 ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J_1 @ I_1 ) @ U ) @ N_1 ) ) ) ) ).
thf(fact_1109_nat__eq__add__iff2,axiom,
! [U: nat,M: nat,N_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_1 @ J_1 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J_1 @ U ) @ N_1 ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J_1 @ I_1 ) @ U ) @ N_1 ) ) ) ) ).
thf(fact_1110_Suc__le__D,axiom,
! [N_1: nat,M_2: nat] :
( ( ord_less_eq_nat @ ( suc @ N_1 ) @ M_2 )
=> ? [M_1: nat] :
( M_2
= ( suc @ M_1 ) ) ) ).
thf(fact_1111_Suc__le__D__lemma,axiom,
! [P: nat > $o,N_1: nat,M_2: nat] :
( ( ord_less_eq_nat @ ( suc @ N_1 ) @ M_2 )
=> ( ! [M_1: nat] :
( ( ord_less_eq_nat @ N_1 @ M_1 )
=> ( P @ ( suc @ M_1 ) ) )
=> ( P @ M_2 ) ) ) ).
thf(fact_1112_finite__nat__set__iff__bounded__le,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M_1: nat] :
! [X: nat] :
( ( member_nat @ X @ N_2 )
=> ( ord_less_eq_nat @ X @ M_1 ) ) ) ).
thf(fact_1113_finite__less__ub,axiom,
! [U: nat,F: nat > nat] :
( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
thf(fact_1114_termination__basic__simps_I3_J,axiom,
! [Z: nat,X_1: nat,Y: nat] :
( ( ord_less_eq_nat @ X_1 @ Y )
=> ( ord_less_eq_nat @ X_1 @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_1115_termination__basic__simps_I4_J,axiom,
! [Y: nat,X_1: nat,Z: nat] :
( ( ord_less_eq_nat @ X_1 @ Z )
=> ( ord_less_eq_nat @ X_1 @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_1116_less__zeroE,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ zero_zero_nat ) ).
thf(fact_1117_lessI,axiom,
! [N_1: nat] : ( ord_less_nat @ N_1 @ ( suc @ N_1 ) ) ).
thf(fact_1118_Suc__mono,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_1 ) ) ) ).
thf(fact_1119_finite__Collect__less__nat,axiom,
! [K_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K_1 ) ) ) ).
thf(fact_1120_zero__less__Suc,axiom,
! [N_1: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N_1 ) ) ).
thf(fact_1121_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( P @ K ) @ ( ord_less_nat @ K @ I_1 ) ) ) ) ).
thf(fact_1122_finite__nat__set__iff__bounded,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M_1: nat] :
! [X: nat] :
( ( member_nat @ X @ N_2 )
=> ( ord_less_nat @ X @ M_1 ) ) ) ).
thf(fact_1123_less__or__eq__imp__le,axiom,
! [M: nat,N_1: nat] :
( ( ( ord_less_nat @ M @ N_1 )
| ( M = N_1 ) )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1124_le__neq__implies__less,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( M != N_1 )
=> ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_1125_less__imp__le__nat,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1126_le__eq__less__or__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
<=> ( ( ord_less_nat @ M @ N_1 )
| ( M = N_1 ) ) ) ).
thf(fact_1127_nat__less__le,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
<=> ( ( ord_less_eq_nat @ M @ N_1 )
& ( M != N_1 ) ) ) ).
thf(fact_1128_Suc__le__lessD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N_1 )
=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1129_le__less__Suc__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ( ord_less_nat @ N_1 @ ( suc @ M ) )
<=> ( N_1 = M ) ) ) ).
thf(fact_1130_Suc__leI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N_1 ) ) ).
thf(fact_1131_le__imp__less__Suc,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ M @ N_1 )
=> ( ord_less_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_1132_Suc__le__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N_1 )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1133_less__Suc__eq__le,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
<=> ( ord_less_eq_nat @ M @ N_1 ) ) ).
thf(fact_1134_less__eq__Suc__le,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ N_1 @ M )
<=> ( ord_less_eq_nat @ ( suc @ N_1 ) @ M ) ) ).
thf(fact_1135_less__diff__iff,axiom,
! [N_1: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N_1 @ K_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ) ) ).
thf(fact_1136_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
thf(fact_1137_not__add__less1,axiom,
! [I_1: nat,J_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ I_1 ) ).
thf(fact_1138_not__add__less2,axiom,
! [J_1: nat,I_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J_1 @ I_1 ) @ I_1 ) ).
thf(fact_1139_nat__add__left__cancel__less,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1140_trans__less__add1,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ J_1 @ M ) ) ) ).
thf(fact_1141_trans__less__add2,axiom,
! [M: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ M @ J_1 ) ) ) ).
thf(fact_1142_add__less__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K_1 ) @ ( plus_plus_nat @ J_1 @ K_1 ) ) ) ).
thf(fact_1143_add__less__mono,axiom,
! [K_1: nat,L: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ K_1 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K_1 ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_1144_less__add__eq__less,axiom,
! [M: nat,N_1: nat,K_1: nat,L: nat] :
( ( ord_less_nat @ K_1 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K_1 @ N_1 ) )
=> ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_1145_add__lessD1,axiom,
! [I_1: nat,J_1: nat,K_1: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J_1 ) @ K_1 )
=> ( ord_less_nat @ I_1 @ K_1 ) ) ).
thf(fact_1146_diff__less__mono2,axiom,
! [L: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N_1 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_1147_less__imp__diff__less,axiom,
! [N_1: nat,J_1: nat,K_1: nat] :
( ( ord_less_nat @ J_1 @ K_1 )
=> ( ord_less_nat @ ( minus_minus_nat @ J_1 @ N_1 ) @ K_1 ) ) ).
thf(fact_1148_not__less0,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ zero_zero_nat ) ).
thf(fact_1149_neq0__conv,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ).
thf(fact_1150_less__nat__zero__code,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ zero_zero_nat ) ).
thf(fact_1151_gr__implies__not0,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( N_1 != zero_zero_nat ) ) ).
thf(fact_1152_gr0I,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ).
thf(fact_1153_not__less__eq,axiom,
! [M: nat,N_1: nat] :
( ~ ( ord_less_nat @ M @ N_1 )
<=> ( ord_less_nat @ N_1 @ ( suc @ M ) ) ) ).
thf(fact_1154_less__Suc__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
<=> ( ( ord_less_nat @ M @ N_1 )
| ( M = N_1 ) ) ) ).
thf(fact_1155_Suc__less__eq,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1156_not__less__less__Suc__eq,axiom,
! [N_1: nat,M: nat] :
( ~ ( ord_less_nat @ N_1 @ M )
=> ( ( ord_less_nat @ N_1 @ ( suc @ M ) )
<=> ( N_1 = M ) ) ) ).
thf(fact_1157_less__antisym,axiom,
! [N_1: nat,M: nat] :
( ~ ( ord_less_nat @ N_1 @ M )
=> ( ( ord_less_nat @ N_1 @ ( suc @ M ) )
=> ( M = N_1 ) ) ) ).
thf(fact_1158_less__SucI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ord_less_nat @ M @ ( suc @ N_1 ) ) ) ).
thf(fact_1159_Suc__lessI,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
=> ( ( ( suc @ M )
!= N_1 )
=> ( ord_less_nat @ ( suc @ M ) @ N_1 ) ) ) ).
thf(fact_1160_less__trans__Suc,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ J_1 @ K_1 )
=> ( ord_less_nat @ ( suc @ I_1 ) @ K_1 ) ) ) ).
thf(fact_1161_less__SucE,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
=> ( ~ ( ord_less_nat @ M @ N_1 )
=> ( M = N_1 ) ) ) ).
thf(fact_1162_Suc__lessD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N_1 )
=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1163_Suc__less__SucD,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_1 ) )
=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1164_less__not__refl,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ N_1 ) ).
thf(fact_1165_nat__neq__iff,axiom,
! [M: nat,N_1: nat] :
( ( M != N_1 )
<=> ( ( ord_less_nat @ M @ N_1 )
| ( ord_less_nat @ N_1 @ M ) ) ) ).
thf(fact_1166_linorder__neqE__nat,axiom,
! [X_1: nat,Y: nat] :
( ( X_1 != Y )
=> ( ~ ( ord_less_nat @ X_1 @ Y )
=> ( ord_less_nat @ Y @ X_1 ) ) ) ).
thf(fact_1167_less__irrefl__nat,axiom,
! [N_1: nat] :
~ ( ord_less_nat @ N_1 @ N_1 ) ).
thf(fact_1168_less__not__refl2,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ N_1 @ M )
=> ( M != N_1 ) ) ).
thf(fact_1169_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_1170_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N_1: nat] :
( ( ( ord_less_nat @ M @ N_1 )
=> ( P @ N_1 @ M ) )
=> ( ( ( M = N_1 )
=> ( P @ N_1 @ M ) )
=> ( ( ( ord_less_nat @ N_1 @ M )
=> ( P @ N_1 @ M ) )
=> ( P @ N_1 @ M ) ) ) ) ).
thf(fact_1171_less__diff__conv,axiom,
! [I_1: nat,J_1: nat,K_1: nat] :
( ( ord_less_nat @ I_1 @ ( minus_minus_nat @ J_1 @ K_1 ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K_1 ) @ J_1 ) ) ).
thf(fact_1172_add__diff__inverse,axiom,
! [M: nat,N_1: nat] :
( ~ ( ord_less_nat @ M @ N_1 )
=> ( ( plus_plus_nat @ N_1 @ ( minus_minus_nat @ M @ N_1 ) )
= M ) ) ).
thf(fact_1173_Suc__mult__less__cancel1,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K_1 ) @ M ) @ ( times_times_nat @ ( suc @ K_1 ) @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1174_diff__less__Suc,axiom,
! [M: nat,N_1: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N_1 ) @ ( suc @ M ) ) ).
thf(fact_1175_nat__0__less__mult__iff,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ) ).
thf(fact_1176_mult__less__cancel1,axiom,
! [K_1: nat,M: nat,N_1: nat] :
( ( ord_less_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_1177_mult__less__cancel2,axiom,
! [M: nat,K_1: nat,N_1: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N_1 @ K_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_1178_mult__less__mono1,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_nat @ ( times_times_nat @ I_1 @ K_1 ) @ ( times_times_nat @ J_1 @ K_1 ) ) ) ) ).
thf(fact_1179_mult__less__mono2,axiom,
! [K_1: nat,I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_nat @ ( times_times_nat @ K_1 @ I_1 ) @ ( times_times_nat @ K_1 @ J_1 ) ) ) ) ).
thf(fact_1180_nat__mult__eq__cancel1,axiom,
! [M: nat,N_1: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N_1 ) )
<=> ( M = N_1 ) ) ) ).
thf(fact_1181_nat__mult__less__cancel1,axiom,
! [M: nat,N_1: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ord_less_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N_1 ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ) ).
thf(fact_1182_diff__less,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N_1 ) @ M ) ) ) ).
thf(fact_1183_zero__less__diff,axiom,
! [N_1: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N_1 @ M ) )
<=> ( ord_less_nat @ M @ N_1 ) ) ).
thf(fact_1184_less__add__Suc1,axiom,
! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ I_1 @ M ) ) ) ).
thf(fact_1185_less__add__Suc2,axiom,
! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ M @ I_1 ) ) ) ).
thf(fact_1186_less__iff__Suc__add,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ N_1 )
<=> ? [K: nat] :
( N_1
= ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_1187_add__gr__0,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N_1 ) ) ) ).
thf(fact_1188_gr0__conv__Suc,axiom,
! [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
<=> ? [M_1: nat] :
( N_1
= ( suc @ M_1 ) ) ) ).
thf(fact_1189_less__Suc0,axiom,
! [N_1: nat] :
( ( ord_less_nat @ N_1 @ ( suc @ zero_zero_nat ) )
<=> ( N_1 = zero_zero_nat ) ) ).
thf(fact_1190_less__Suc__eq__0__disj,axiom,
! [M: nat,N_1: nat] :
( ( ord_less_nat @ M @ ( suc @ N_1 ) )
<=> ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N_1 ) ) ) ) ).
thf(fact_1191_card__Collect__less__nat,axiom,
! [N_1: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N_1 ) ) )
= N_1 ) ).
thf(fact_1192_less__eq__Suc__le__raw,axiom,
! [X: nat] :
( ( ord_less_nat @ X )
= ( ord_less_eq_nat @ ( suc @ X ) ) ) ).
thf(fact_1193_termination__basic__simps_I1_J,axiom,
! [Z: nat,X_1: nat,Y: nat] :
( ( ord_less_nat @ X_1 @ Y )
=> ( ord_less_nat @ X_1 @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_1194_termination__basic__simps_I2_J,axiom,
! [Y: nat,X_1: nat,Z: nat] :
( ( ord_less_nat @ X_1 @ Z )
=> ( ord_less_nat @ X_1 @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_1195_termination__basic__simps_I5_J,axiom,
! [X_1: nat,Y: nat] :
( ( ord_less_nat @ X_1 @ Y )
=> ( ord_less_eq_nat @ X_1 @ Y ) ) ).
%----Helper facts (12)
thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X_1: nat,Y: nat] :
( ~ ( fequal_nat @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X_1: nat,Y: nat] :
( ( X_1 != Y )
| ( fequal_nat @ X_1 @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ~ ( fequal_pname @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ( X_1 != Y )
| ( fequal_pname @ X_1 @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
! [X_1: state,Y: state] :
( ~ ( fequal_state @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
! [X_1: state,Y: state] :
( ( X_1 != Y )
| ( fequal_state @ X_1 @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_,axiom,
! [X_1: hoare_2091234717iple_a,Y: hoare_2091234717iple_a] :
( ~ ( fequal1604381340iple_a @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_,axiom,
! [X_1: hoare_2091234717iple_a,Y: hoare_2091234717iple_a] :
( ( X_1 != Y )
| ( fequal1604381340iple_a @ X_1 @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com,axiom,
! [X_1: hoare_1708887482_state,Y: hoare_1708887482_state] :
( ~ ( fequal224822779_state @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com,axiom,
! [X_1: hoare_1708887482_state,Y: hoare_1708887482_state] :
( ( X_1 != Y )
| ( fequal224822779_state @ X_1 @ Y ) ) ).
thf(help_fequal_1_1_fequal_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It,axiom,
! [X_1: hoare_2091234717iple_a > $o,Y: hoare_2091234717iple_a > $o] :
( ~ ( fequal845167073le_a_o @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It,axiom,
! [X_1: hoare_2091234717iple_a > $o,Y: hoare_2091234717iple_a > $o] :
( ( X_1 != Y )
| ( fequal845167073le_a_o @ X_1 @ Y ) ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [N: nat] :
( ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X
@ ( semila1052848428le_a_o @ g
@ ( image_231808478iple_a
@ ^ [Pn: pname] : ( hoare_657976383iple_a @ ( p @ Pn ) @ ( body @ Pn ) @ ( q @ Pn ) )
@ procs ) ) )
=> ( hoare_1421888935alid_a @ N @ X ) )
=> ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X
@ ( image_231808478iple_a
@ ^ [Pn: pname] : ( hoare_657976383iple_a @ ( p @ Pn ) @ ( the_com @ ( body_1 @ Pn ) ) @ ( q @ Pn ) )
@ procs ) )
=> ( hoare_1421888935alid_a @ N @ X ) ) ) ).
thf(conj_1,conjecture,
( ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X @ g )
=> ( hoare_1421888935alid_a @ n @ X ) )
=> ! [X: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X
@ ( image_231808478iple_a
@ ^ [Pn: pname] : ( hoare_657976383iple_a @ ( p @ Pn ) @ ( body @ Pn ) @ ( q @ Pn ) )
@ procs ) )
=> ( hoare_1421888935alid_a @ n @ X ) ) ) ).
%------------------------------------------------------------------------------