%------------------------------------------------------------------------------
% File     : SWW470_3 : TPTP v8.2.0. Released v5.3.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 200, 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_tff_l200 [Bla11]

% Status   : Theorem
% Rating   : 0.22 v8.2.0, 0.20 v8.1.0, 0.27 v7.5.0, 0.38 v7.4.0, 0.20 v7.3.0, 0.17 v7.1.0, 0.00 v6.4.0, 0.33 v6.3.0, 0.00 v6.1.0, 0.67 v6.0.0, 0.40 v5.5.0, 0.25 v5.4.0, 0.67 v5.3.0
% Syntax   : Number of formulae    : 2281 ( 466 unt; 917 typ;   0 def)
%            Number of atoms       : 3118 (1143 equ)
%            Maximal formula atoms :    9 (   1 avg)
%            Number of connectives : 2091 ( 337   ~;  66   |; 137   &)
%                                         ( 203 <=>;1348  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   5 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of types       :  315 ( 314 usr)
%            Number of type conns  :  709 ( 372   >; 337   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  602 ( 602 usr; 231 con; 0-3 aty)
%            Number of variables   : 3804 (3724   !;  80   ?;3804   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 17:22:03
%------------------------------------------------------------------------------
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tff(sy_c_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_I_203,type,
    cOMBS_902588832ol_nat: fun_fu1071091151_a_nat ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_int_bool_bool: fun_fu587508068t_bool ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__Nat__Onat_000tc__Nat__Onat,type,
    cOMBS_int_nat_nat: fun_fu1650807462nt_nat ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Ot,type,
    cOMBS_443177668ol_nat: fun_fu2087062900nt_nat ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__N,type,
    cOMBS_1999073191ol_nat: fun_fu1989253562nt_nat ).

tff(sy_c_COMBS_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__N,type,
    cOMBS_1063588099ol_nat: fun_fu776947314nt_nat ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_nat_bool_bool: fun_fu1901564496t_bool ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat,type,
    cOMBS_nat_nat_nat: fun_fu1382375754at_nat ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Ot,type,
    cOMBS_1514622496ol_nat: fun_fu1446087120at_nat ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__N,type,
    cOMBS_2106653443ol_nat: fun_fu1081900566at_nat ).

tff(sy_c_Com_Ocom_OAss,type,
    ass: ( vname * fun_state_nat ) > com ).

tff(sy_c_Com_Ocom_OSKIP,type,
    skip: com ).

tff(sy_c_Com_Ocom_OSemi,type,
    semi: ( com * com ) > com ).

tff(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
    finite_card_int: fun_fun_int_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
    finite_card_nat: fun_fun_nat_bool_nat ).

tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____wdcyoqcnvy__Ot,type,
    finite164030737iple_a: fun_Ho483360497iple_a > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____wdcyoqcnvy__Ot_204,type,
    finite1617795063a_bool: fun_Ho112928087a_bool > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__Int__Oint,type,
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tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_,type,
    finite1321096241t_bool: fun_in719570932t_bool > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__Nat__Onat,type,
    finite2130160977at_nat: fun_nat_fun_nat_nat > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_,type,
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tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otrip,type,
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tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otrip_205,type,
    finite1306023740a_bool: fun_Ho112928087a_bool > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000tc__Int__Oint,type,
    finite1704255308nt_int: fun_int_fun_int_int > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc,type,
    finite58652534t_bool: fun_in719570932t_bool > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__Nat__Onat,type,
    finite1860950092at_nat: fun_nat_fun_nat_nat > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc,type,
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tff(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_,type,
    finite217552220iple_a: fun_fu2026518849l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
    finite_finite_int: fun_fu819804446l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
    finite_finite_nat: fun_fu425979586l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otrip,type,
    finite1832883692a_bool: fun_fu1342882871l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    finite1395289673t_bool: fun_fu1540142484l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    finite2012431853t_bool: fun_fu814369080l_bool ).

tff(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__,type,
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tff(sy_c_Finite__Set_Ofold1Set_000tc__Int__Oint,type,
    finite_fold1Set_int: ( fun_int_fun_int_int * fun_int_bool ) > fun_int_bool ).

tff(sy_c_Finite__Set_Ofold1Set_000tc__Nat__Onat,type,
    finite_fold1Set_nat: ( fun_nat_fun_nat_nat * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    finite2092228073iple_a: fun_Ho483360497iple_a > fun_fu253747899iple_a ).

tff(sy_c_Finite__Set_Ofold1_000tc__Int__Oint,type,
    finite_fold1_int: fun_int_fun_int_int > fun_fun_int_bool_int ).

tff(sy_c_Finite__Set_Ofold1_000tc__Nat__Onat,type,
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tff(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_,type,
    finite2089344904iple_a: ( fun_Ho483360497iple_a * hoare_2076525193iple_a ) > fun_fu253747899iple_a ).

tff(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J__206,type,
    finite718386368a_bool: ( fun_Ho112928087a_bool * fun_Ho882550911a_bool ) > fun_fu1663341283a_bool ).

tff(sy_c_Finite__Set_Ofold_000tc__Int__Oint_000tc__Int__Oint,type,
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tff(sy_c_Finite__Set_Ofold_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Oboo,type,
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tff(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__Nat__Onat,type,
    finite_fold_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_fun_nat_bool_nat ).

tff(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Oboo,type,
    finite1578363458t_bool: ( fun_na35521688t_bool * fun_nat_bool ) > fun_fu821463397t_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,type,
    finite974267142iple_a: fun_Ho483360497iple_a > fun_Ho112928087a_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_207,type,
    finite655745986a_bool: fun_Ho112928087a_bool > fun_fu2104275287l_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Int__Oint_000tc__Int__Oint,type,
    finite772772422nt_int: fun_int_fun_int_int > fun_in719570932t_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HO,type,
    finite1008073724t_bool: fun_in719570932t_bool > fun_fu1284533364l_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__Nat__Onat,type,
    finite929467206at_nat: fun_nat_fun_nat_nat > fun_na35521688t_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HO,type,
    finite758726980t_bool: fun_na35521688t_bool > fun_fu1216549656l_bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_,type,
    finite232328454iple_a: ( fun_Ho483360497iple_a * fun_fu253747899iple_a ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,type,
    finite1626084323ne_int: ( fun_int_fun_int_int * fun_fun_int_bool_int ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
    finite988810631ne_nat: ( fun_nat_fun_nat_nat * fun_fun_nat_bool_nat ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____wdcyoqcnvy__Ot,type,
    finite1659845635iple_a: ( fun_Ho483360497iple_a * fun_fu253747899iple_a ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,type,
    finite1432773856em_int: ( fun_int_fun_int_int * fun_fun_int_bool_int ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
    finite795500164em_nat: ( fun_nat_fun_nat_nat * fun_fun_nat_bool_nat ) > bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__HOL__Obool,type,
    minus_minus_bool: fun_bo1549164019l_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
    minus_minus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnv,type,
    minus_1649626414a_bool: fun_fu2051904533a_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    minus_1449998731t_bool: fun_fu1075512626t_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    minus_2067140911t_bool: fun_fu915923926t_bool ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
    one_one_int: int ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
    one_one_nat: nat ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
    plus_plus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
    plus_plus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
    times_times_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
    times_times_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__Int__Oint,type,
    uminus_uminus_int: fun_int_int ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
    zero_zero_int: int ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
    zero_zero_nat: nat ).

tff(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    the_Ho1456473914iple_a: fun_fu253747899iple_a ).

tff(sy_c_HOL_OThe_000tc__Int__Oint,type,
    the_int: fun_fun_int_bool_int ).

tff(sy_c_HOL_OThe_000tc__Nat__Onat,type,
    the_nat: fun_fun_nat_bool_nat ).

tff(sy_c_HOL_OThe_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_,type,
    the_fu972267150a_bool: fun_fu1819779801a_bool ).

tff(sy_c_HOL_OThe_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    the_fun_int_bool: fun_fu1421432211t_bool ).

tff(sy_c_HOL_OThe_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    the_fun_nat_bool: fun_fu496092379t_bool ).

tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Ohoare__derivs_000t__a,type,
    hoare_572465367rivs_a: fun_Ho882550911a_bool > fun_fu2026518849l_bool ).

tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple_Otriple_000t__a,type,
    hoare_1876769875iple_a: ( fun_a_fun_state_bool * com * fun_a_fun_state_bool ) > hoare_2076525193iple_a ).

tff(sy_c_If_000tc__Nat__Onat,type,
    if_nat: bool > fun_nat_fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
    semiri1621563631at_int: fun_nat_int ).

tff(sy_c_Natural_Oupdate,type,
    update: fun_st1816159843_state ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,type,
    bot_bot_bool: bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy,type,
    bot_bo214064617a_bool: fun_Ho882550911a_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    bot_bot_fun_int_bool: fun_int_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    bot_bot_fun_nat_bool: fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
    ord_less_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le1912455174t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le382113706t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__HOL__Obool,type,
    ord_less_eq_bool: fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Hoare____Mirabelle____wdcyo,type,
    ord_le940607413a_bool: fun_fu1789210355l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le951220754t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_000tc__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omin_000tc__Int__Oint,type,
    ord_min_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otrip,type,
    partia428461311iple_a: hoare_2076525193iple_a > fun_fu253747899iple_a ).

tff(sy_c_Partial__Function_Oflat__lub_000tc__Int__Oint,type,
    partial_flat_lub_int: int > fun_fun_int_bool_int ).

tff(sy_c_Partial__Function_Oflat__lub_000tc__Nat__Onat,type,
    partial_flat_lub_nat: nat > fun_fun_nat_bool_nat ).

tff(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
    power_power_int: fun_int_fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Int__Oint,type,
    ord_at875362053st_int: int > fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Nat__Onat,type,
    ord_at238088361st_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OgreaterThanAtMost_000tc__Nat__Onat,type,
    ord_gr375877188st_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OlessThan_000tc__Int__Oint,type,
    ord_lessThan_int: fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OlessThan_000tc__Nat__Onat,type,
    ord_lessThan_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    collec977865374iple_a: fun_fu1663341283a_bool ).

tff(sy_c_Set_OCollect_000tc__Int__Oint,type,
    collect_int: fun_fu554825757t_bool ).

tff(sy_c_Set_OCollect_000tc__Nat__Onat,type,
    collect_nat: fun_fu821463397t_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__,type,
    collec416005674a_bool: fun_fu1354972029l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    collect_fun_int_bool: fun_fu1070826941l_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    collect_fun_nat_bool: fun_fu616551101l_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__,type,
    image_1416816325iple_a: ( fun_Ho1518252029iple_a * fun_Ho882550911a_bool ) > fun_Ho882550911a_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc___208,type,
    image_268182434_a_int: ( fun_Ho1026035802_a_int * fun_Ho882550911a_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc___209,type,
    image_1778392390_a_nat: ( fun_Ho864122878_a_nat * fun_Ho882550911a_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otripl,type,
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tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Int__Oint,type,
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tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Nat__Onat,type,
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tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otripl,type,
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tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
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tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    insert1582918915iple_a: fun_Ho112928087a_bool ).

tff(sy_c_Set_Oinsert_000tc__Int__Oint,type,
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tff(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
    insert_nat: fun_na35521688t_bool ).

tff(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    the_el2146174248iple_a: fun_fu253747899iple_a ).

tff(sy_c_Set_Othe__elem_000tc__Int__Oint,type,
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tff(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
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tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: fun_bo1549164019l_bool ).

tff(sy_c_fdisj,type,
    fdisj: fun_bo1549164019l_bool ).

tff(sy_c_fequal_000tc__Com__Ostate,type,
    fequal_state: fun_st1506752259e_bool ).

tff(sy_c_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    fequal1589671816iple_a: fun_Ho943022881a_bool ).

tff(sy_c_fequal_000tc__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_000tc__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_Mt,type,
    fequal291249344a_bool: fun_fu1789210355l_bool ).

tff(sy_c_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    fequal_fun_int_bool: fun_fu1151772339l_bool ).

tff(sy_c_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    fequal_fun_nat_bool: fun_fu1217155507l_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1549164019l_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J,type,
    hAPP_a2036067514e_bool: ( fun_a_fun_state_bool * x_a ) > fun_state_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__HOL_,type,
    hAPP_a849909144l_bool: ( fun_a_1632297036l_bool * x_a ) > fun_st2063251938l_bool ).

tff(sy_c_hAPP_000t__a_000tc__fun_Itc__HOL__Obool_Mtc__fun_Itc__Com__Ostate_Mtc__HOL_,type,
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tff(sy_c_hAPP_000t__a_000tc__fun_Itc__fun_Itc__Com__Ostate_Mtc__Com__Ostate_J_Mtc__f,type,
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tff(sy_c_hAPP_000t__a_000tc__fun_Itc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_Mtc__fu,type,
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tff(sy_c_hAPP_000tc__Com__Ostate_000tc__Com__Ostate,type,
    hAPP_state_state: ( fun_state_state * state ) > state ).

tff(sy_c_hAPP_000tc__Com__Ostate_000tc__HOL__Obool,type,
    hAPP_state_bool: ( fun_state_bool * state ) > bool ).

tff(sy_c_hAPP_000tc__Com__Ostate_000tc__Nat__Onat,type,
    hAPP_state_nat: ( fun_state_nat * state ) > nat ).

tff(sy_c_hAPP_000tc__Com__Ostate_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J,type,
    hAPP_s1806633685e_bool: ( fun_st1506752259e_bool * state ) > fun_state_bool ).

tff(sy_c_hAPP_000tc__Com__Ostate_000tc__fun_Itc__Com__Ovname_Mtc__fun_Itc__Nat__Onat,type,
    hAPP_s712361723_state: ( fun_st1816159843_state * state ) > fun_vn774578976_state ).

tff(sy_c_hAPP_000tc__Com__Ostate_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Com__Ostate_000tc__fun_Itc__Nat__Onat_Mtc__Com__Ostate_J,type,
    hAPP_s1892499976_state: ( fun_st2072884720_state * state ) > fun_nat_state ).

tff(sy_c_hAPP_000tc__Com__Ovname_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Nat__Onat,type,
    hAPP_v365393659_state: ( fun_vn299216483_state * vname ) > fun_st2072884720_state ).

tff(sy_c_hAPP_000tc__Com__Ovname_000tc__fun_Itc__Nat__Onat_Mtc__Com__Ostate_J,type,
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tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
    hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL_,type,
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tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Ot,type,
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tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__HOL__O,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__Hoare_,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__Int__O,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__Nat__O,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_210,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_211,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_212,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_213,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_214,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_215,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_216,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_217,type,
    hAPP_H1566092493l_bool: ( fun_Ho1910759733l_bool * hoare_2076525193iple_a ) > fun_fu2026518849l_bool ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_218,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_219,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_220,type,
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tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_000tc__fun_It_221,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,type,
    hAPP_int_bool: ( fun_int_bool * int ) > bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint,type,
    hAPP_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Nat__Onat,type,
    hAPP_int_nat: ( fun_int_nat * int ) > nat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otr,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otr_222,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otr_223,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J,type,
    hAPP_int_fun_int_nat: ( fun_int_fun_int_nat * int ) > fun_int_nat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__Int__Oint_J,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoq,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J__224,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J__225,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_M,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Com__Ostate,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otr,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otr_226,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
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tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_J_0_335,type,
    hAPP_f1408247010at_nat: ( fun_fu1382375754at_nat * fun_nat_fun_nat_nat ) > fun_fu325663101at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_J_0_336,type,
    hAPP_f1731313045at_nat: ( fun_fu1799328765at_nat * fun_nat_fun_nat_nat ) > fun_fu2064863920at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Hoare____Mirabelle___,type,
    hAPP_f1173305809at_nat: ( fun_fu55165875at_nat * fun_na228779103ol_nat ) > fun_fu662538351at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Hoare____Mirabelle____337,type,
    hAPP_f1323480046at_nat: ( fun_fu1446087120at_nat * fun_na228779103ol_nat ) > fun_fu546267020at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Hoare____Mirabelle____338,type,
    hAPP_f928984021_a_nat: ( fun_fu1480131645_a_nat * fun_na1588060912_a_nat ) > fun_fu1472438000_a_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__O,type,
    hAPP_f630508183at_nat: ( fun_fu1373519097at_nat * fun_na1402637250ol_nat ) > fun_fu2071915986at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__O_339,type,
    hAPP_f1750007732at_nat: ( fun_fu1081900566at_nat * fun_na1402637250ol_nat ) > fun_fu1553240943at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Int__Oint_Mtc__Nat__O,type,
    hAPP_f1633513941nt_nat: ( fun_fu1251161661nt_nat * fun_na266550704nt_nat ) > fun_fu330349488nt_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__O,type,
    hAPP_f1722879237t_bool: ( fun_fu80187629t_bool * fun_na1436237685l_bool ) > fun_fu821463397t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__O_340,type,
    hAPP_f688831301t_bool: ( fun_fu1538037805t_bool * fun_na35521688t_bool ) > fun_fu429110920t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_Mtc__fun_Itc__C,type,
    hAPP_f1178339559l_bool: ( fun_fu1340893257l_bool * fun_fu1658206819l_bool ) > fun_fu2118559873l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_Mtc__fun_Itc__f,type,
    hAPP_f635443597e_bool: ( fun_fu817723125e_bool * fun_fu2127278875e_bool ) > fun_fu278206227e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__HOL,type,
    hAPP_f1006724181e_bool: ( fun_fu1860059197e_bool * fun_fu1219323149e_bool ) > fun_fu1591723597e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__HOL_341,type,
    hAPP_f375255701e_bool: ( fun_fu1631777789e_bool * fun_fu1047394976e_bool ) > fun_fu2008829792e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It_,type,
    hAPP_f1490793353l_bool: ( fun_fu1342882871l_bool * fun_fu2026518849l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__342,type,
    hAPP_f1758315691a_bool: ( fun_fu1819779801a_bool * fun_fu2026518849l_bool ) > fun_Ho882550911a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__343,type,
    hAPP_f354682389l_bool: ( fun_fu1354972029l_bool * fun_fu2026518849l_bool ) > fun_fu2026518849l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__344,type,
    hAPP_f1827400025a_bool: ( fun_fu870629313a_bool * fun_fu2026518849l_bool ) > fun_fu445247621a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__345,type,
    hAPP_f228445643at_nat: ( fun_fu898837677at_nat * fun_fu328701628ol_nat ) > fun_fu546267020at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__346,type,
    hAPP_f967504915a_bool: ( fun_fu600978683a_bool * fun_fu1663341283a_bool ) > fun_fu466270817a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__347,type,
    hAPP_f1882853211a_bool: ( fun_fu1912089667a_bool * fun_fu1663341283a_bool ) > fun_fu1381619113a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__348,type,
    hAPP_f246259945l_bool: ( fun_fu1236051915l_bool * fun_fu1789210355l_bool ) > fun_fu1789210355l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__349,type,
    hAPP_f86610733a_bool: ( fun_fu1595031695a_bool * fun_fu2051904533a_bool ) > fun_fu2051904533a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__350,type,
    hAPP_f300022271a_bool: ( fun_fu743040865a_bool * fun_fu2051904533a_bool ) > fun_fu914297575a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__351,type,
    hAPP_f534494591ol_nat: ( fun_fu1739618017ol_nat * fun_fu827349075ol_nat ) > fun_fu2076872489ol_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__352,type,
    hAPP_f2033836159ol_nat: ( fun_fu774707681ol_nat * fun_fu827349075ol_nat ) > fun_fu1428730409ol_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    hAPP_f215623910l_bool: ( fun_fu1540142484l_bool * fun_fu819804446l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_353,type,
    hAPP_f631471077t_bool: ( fun_fu1421432211t_bool * fun_fu819804446l_bool ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_354,type,
    hAPP_f1948010709l_bool: ( fun_fu1070826941l_bool * fun_fu819804446l_bool ) > fun_fu819804446l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_355,type,
    hAPP_f423804115t_bool: ( fun_fu1671602619t_bool * fun_fu819804446l_bool ) > fun_fu1221449756t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__Nat__Onat_J_,type,
    hAPP_f296893331_a_nat: ( fun_fu2017547765_a_nat * fun_fun_int_bool_nat ) > fun_fu602188273_a_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__Nat__Onat_J__356,type,
    hAPP_f2100446809nt_nat: ( fun_fu1795303867nt_nat * fun_fun_int_bool_nat ) > fun_fu1396640951nt_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__Nat__Onat_J__357,type,
    hAPP_f109563153at_nat: ( fun_fu1888474227at_nat * fun_fun_int_bool_nat ) > fun_fu1553240943at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__Int,type,
    hAPP_f1871962969t_bool: ( fun_fu763112513t_bool * fun_fu554825757t_bool ) > fun_fu945366625t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__Int_358,type,
    hAPP_f654702867t_bool: ( fun_fu1521371387t_bool * fun_fu554825757t_bool ) > fun_fu257678363t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__Int_359,type,
    hAPP_f1986088027t_bool: ( fun_fu788822083t_bool * fun_fu554825757t_bool ) > fun_fu1589063523t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun,type,
    hAPP_f1399552105l_bool: ( fun_fu1276998475l_bool * fun_fu1151772339l_bool ) > fun_fu1151772339l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun_360,type,
    hAPP_f194616807t_bool: ( fun_fu680577737t_bool * fun_fu1075512626t_bool ) > fun_fu1075512626t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun_361,type,
    hAPP_f1596157055t_bool: ( fun_fu873308769t_bool * fun_fu1075512626t_bool ) > fun_fu1881253066t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun_,type,
    hAPP_f1772247755ol_nat: ( fun_fu651819309ol_nat * fun_fu670292377ol_nat ) > fun_fu876640303ol_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun__362,type,
    hAPP_f1331458699ol_nat: ( fun_fu657063149ol_nat * fun_fu670292377ol_nat ) > fun_fu1243635695ol_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun__363,type,
    hAPP_f1632710027ol_nat: ( fun_fu342964205ol_nat * fun_fu670292377ol_nat ) > fun_fu1544887023ol_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    hAPP_f1637334154l_bool: ( fun_fu814369080l_bool * fun_fu425979586l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_364,type,
    hAPP_f582319405t_bool: ( fun_fu496092379t_bool * fun_fu425979586l_bool ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_365,type,
    hAPP_f1246832597l_bool: ( fun_fu616551101l_bool * fun_fu425979586l_bool ) > fun_fu425979586l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_366,type,
    hAPP_f1777703707t_bool: ( fun_fu1402718723t_bool * fun_fu425979586l_bool ) > fun_fu531872008t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__Nat__Onat_J_,type,
    hAPP_f456064075_a_nat: ( fun_fu1155082413_a_nat * fun_fun_nat_bool_nat ) > fun_fu197504845_a_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__Nat__Onat_J__367,type,
    hAPP_f1964560145nt_nat: ( fun_fu523748467nt_nat * fun_fun_nat_bool_nat ) > fun_fu1424009747nt_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__Nat,type,
    hAPP_f69420889t_bool: ( fun_fu2137156673t_bool * fun_fu821463397t_bool ) > fun_fu623690153t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__Nat_368,type,
    hAPP_f1410409747t_bool: ( fun_fu719081723t_bool * fun_fu821463397t_bool ) > fun_fu401661795t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun,type,
    hAPP_f643944041l_bool: ( fun_fu820922699l_bool * fun_fu1217155507l_bool ) > fun_fu1217155507l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun_369,type,
    hAPP_f1311642927t_bool: ( fun_fu257437713t_bool * fun_fu915923926t_bool ) > fun_fu915923926t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun_370,type,
    hAPP_f1115950719t_bool: ( fun_fu1738364513t_bool * fun_fu915923926t_bool ) > fun_fu2023981606t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun_,type,
    hAPP_f450378965_a_nat: ( fun_fu687095869_a_nat * fun_fu1437424253_a_nat ) > fun_fu927620157_a_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun__371,type,
    hAPP_f1701814485nt_nat: ( fun_fu504533053nt_nat * fun_fu773339453nt_nat ) > fun_fu1086979837nt_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun__372,type,
    hAPP_f153943611ol_nat: ( fun_fu1838203037ol_nat * fun_fu990052945ol_nat ) > fun_fu394467047ol_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun__373,type,
    hAPP_f1029311995ol_nat: ( fun_fu294108253ol_nat * fun_fu990052945ol_nat ) > fun_fu1416337063ol_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun__374,type,
    hAPP_f250842005at_nat: ( fun_fu477648637at_nat * fun_fu1179887677at_nat ) > fun_fu2033973437at_nat ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__fun_Itc__fun__375,type,
    hAPP_f1315855317at_nat: ( fun_fu633097021at_nat * fun_fu1179887677at_nat ) > fun_fu199354621at_nat ).

tff(sy_c_hBOOL,type,
    hBOOL: bool > $o ).

tff(sy_c_member_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    member276146780iple_a: fun_Ho1910759733l_bool ).

tff(sy_c_member_000tc__Int__Oint,type,
    member_int: fun_in938817141l_bool ).

tff(sy_c_member_000tc__Nat__Onat,type,
    member_nat: fun_na1436237685l_bool ).

tff(sy_v_G,type,
    g: fun_Ho882550911a_bool ).

tff(sy_v_P,type,
    p: fun_a_fun_state_bool ).

tff(sy_v_b,type,
    b: fun_state_bool ).

tff(sy_v_c,type,
    c: com ).

%----Relevant facts (1198)
tff(fact_0_empty,axiom,
    ! [Ga: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),bot_bo214064617a_bool)) ).

tff(fact_1_triple_Oinject,axiom,
    ! [Fun1_2: fun_a_fun_state_bool,Com_2: com,Fun2_2: fun_a_fun_state_bool,Fun1_1: fun_a_fun_state_bool,Com_1: com,Fun2_1: fun_a_fun_state_bool] :
      ( ( hoare_1876769875iple_a(Fun1_2,Com_2,Fun2_2) = hoare_1876769875iple_a(Fun1_1,Com_1,Fun2_1) )
    <=> ( ( Fun1_2 = Fun1_1 )
        & ( Com_2 = Com_1 )
        & ( Fun2_2 = Fun2_1 ) ) ) ).

tff(fact_2_cut,axiom,
    ! [Ga: fun_Ho882550911a_bool,G_1: fun_Ho882550911a_bool,Ts: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(G_1),Ts))
     => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),G_1))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts)) ) ) ).

tff(fact_3_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho882550911a_bool,Ga: fun_Ho882550911a_bool,T_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,T_2),bot_bo214064617a_bool)))
     => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,T_2),Ts))) ) ) ).

tff(fact_4_constant,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool,C_5: bool] :
      ( ( hBOOL(C_5)
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_1)),bot_bo214064617a_bool))) )
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,hAPP_f340725611e_bool(hAPP_f1006724181e_bool(cOMBB_1348041619bool_a,cOMBC_231445413l_bool),hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),Pa))),C_5),Ca,Q_1)),bot_bo214064617a_bool))) ) ).

tff(fact_5_escape,axiom,
    ! [Ga: fun_Ho882550911a_bool,Ca: com,Q_1: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool] :
      ( ! [Z_11: x_a,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
         => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_2)),Ca,hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_a2036067514e_bool(Q_1,Z_11)))),bot_bo214064617a_bool))) )
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_1)),bot_bo214064617a_bool))) ) ).

tff(fact_6_conseq2,axiom,
    ! [Q_1: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,Ca: com,Q_3: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_3)),bot_bo214064617a_bool)))
     => ( ! [Z_11: x_a,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_11),S_2))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_11),S_2)) )
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_1)),bot_bo214064617a_bool))) ) ) ).

tff(fact_7_conseq1,axiom,
    ! [Pa: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,P_2: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(P_2,Ca,Q_1)),bot_bo214064617a_bool)))
     => ( ! [Z_11: x_a,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_11),S_2)) )
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_1)),bot_bo214064617a_bool))) ) ) ).

tff(fact_8_conseq12,axiom,
    ! [Q_1: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,P_2: fun_a_fun_state_bool,Ca: com,Q_3: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(P_2,Ca,Q_3)),bot_bo214064617a_bool)))
     => ( ! [Z_11: x_a,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
           => ! [S_3: state] :
                ( ! [Z_12: x_a] :
                    ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_12),S_2))
                   => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_12),S_3)) )
               => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_11),S_3)) ) )
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_1)),bot_bo214064617a_bool))) ) ) ).

tff(fact_9_insertE,axiom,
    ! [A_13: nat,Ba: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)))
     => ( ( A_13 != Ba )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A)) ) ) ).

tff(fact_10_insertE,axiom,
    ! [A_13: hoare_2076525193iple_a,Ba: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),A)))
     => ( ( A_13 != Ba )
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A)) ) ) ).

tff(fact_11_insertE,axiom,
    ! [A_13: int,Ba: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),A)))
     => ( ( A_13 != Ba )
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A)) ) ) ).

tff(fact_12_insertCI,axiom,
    ! [Ba: nat,A_13: nat,B: fun_nat_bool] :
      ( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),B))
       => ( A_13 = Ba ) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).

tff(fact_13_insertCI,axiom,
    ! [Ba: hoare_2076525193iple_a,A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] :
      ( ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),B))
       => ( A_13 = Ba ) )
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),B))) ) ).

tff(fact_14_insertCI,axiom,
    ! [Ba: int,A_13: int,B: fun_int_bool] :
      ( ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),B))
       => ( A_13 = Ba ) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),B))) ) ).

tff(fact_15_emptyE,axiom,
    ! [A_13: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),bot_bo214064617a_bool)) ).

tff(fact_16_emptyE,axiom,
    ! [A_13: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),bot_bot_fun_nat_bool)) ).

tff(fact_17_emptyE,axiom,
    ! [A_13: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),bot_bot_fun_int_bool)) ).

tff(fact_18_singleton__conv2,axiom,
    ! [A_13: nat] : hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(fequal_nat,A_13)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) ).

tff(fact_19_singleton__conv2,axiom,
    ! [A_13: hoare_2076525193iple_a] : hAPP_f642208769a_bool(collec977865374iple_a,hAPP_H2077279219a_bool(fequal1589671816iple_a,A_13)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool) ).

tff(fact_20_singleton__conv2,axiom,
    ! [A_13: int] : hAPP_f1805168059t_bool(collect_int,hAPP_i1948725293t_bool(fequal_int,A_13)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) ).

tff(fact_21_singleton__conv,axiom,
    ! [A_13: nat] : hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) ).

tff(fact_22_singleton__conv,axiom,
    ! [A_13: hoare_2076525193iple_a] : hAPP_f642208769a_bool(collec977865374iple_a,hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_13)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool) ).

tff(fact_23_singleton__conv,axiom,
    ! [A_13: int] : hAPP_f1805168059t_bool(collect_int,hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) ).

tff(fact_24_Collect__conv__if2,axiom,
    ! [Pa: fun_nat_bool,A_13: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_13))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_13))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_13))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_13))),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_25_Collect__conv__if2,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a] :
      ( ( hBOOL(hAPP_H454055633a_bool(Pa,A_13))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_H2077279219a_bool(fequal1589671816iple_a,A_13))),Pa)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool) ) )
      & ( ~ hBOOL(hAPP_H454055633a_bool(Pa,A_13))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_H2077279219a_bool(fequal1589671816iple_a,A_13))),Pa)) = bot_bo214064617a_bool ) ) ) ).

tff(fact_26_Collect__conv__if2,axiom,
    ! [Pa: fun_int_bool,A_13: int] :
      ( ( hBOOL(hAPP_int_bool(Pa,A_13))
       => ( hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(fequal_int,A_13))),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) ) )
      & ( ~ hBOOL(hAPP_int_bool(Pa,A_13))
       => ( hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(fequal_int,A_13))),Pa)) = bot_bot_fun_int_bool ) ) ) ).

tff(fact_27_Collect__conv__if,axiom,
    ! [Pa: fun_nat_bool,A_13: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_13))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_13))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13))),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_28_Collect__conv__if,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a] :
      ( ( hBOOL(hAPP_H454055633a_bool(Pa,A_13))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_13))),Pa)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool) ) )
      & ( ~ hBOOL(hAPP_H454055633a_bool(Pa,A_13))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_13))),Pa)) = bot_bo214064617a_bool ) ) ) ).

tff(fact_29_Collect__conv__if,axiom,
    ! [Pa: fun_int_bool,A_13: int] :
      ( ( hBOOL(hAPP_int_bool(Pa,A_13))
       => ( hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13))),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) ) )
      & ( ~ hBOOL(hAPP_int_bool(Pa,A_13))
       => ( hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13))),Pa)) = bot_bot_fun_int_bool ) ) ) ).

tff(fact_30_equals0D,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( ( A = bot_bo214064617a_bool )
     => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A)) ) ).

tff(fact_31_equals0D,axiom,
    ! [A_13: nat,A: fun_nat_bool] :
      ( ( A = bot_bot_fun_nat_bool )
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A)) ) ).

tff(fact_32_equals0D,axiom,
    ! [A_13: int,A: fun_int_bool] :
      ( ( A = bot_bot_fun_int_bool )
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A)) ) ).

tff(fact_33_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho882550911a_bool] :
      ( ( hAPP_f642208769a_bool(collec977865374iple_a,Pa) = bot_bo214064617a_bool )
    <=> ! [X_2: hoare_2076525193iple_a] : ~ hBOOL(hAPP_H454055633a_bool(Pa,X_2)) ) ).

tff(fact_34_Collect__empty__eq,axiom,
    ! [Pa: fun_int_bool] :
      ( ( hAPP_f1805168059t_bool(collect_int,Pa) = bot_bot_fun_int_bool )
    <=> ! [X_2: int] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).

tff(fact_35_Collect__empty__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(collect_nat,Pa) = bot_bot_fun_nat_bool )
    <=> ! [X_2: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

tff(fact_36_empty__iff,axiom,
    ! [Ca: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),bot_bo214064617a_bool)) ).

tff(fact_37_empty__iff,axiom,
    ! [Ca: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),bot_bot_fun_nat_bool)) ).

tff(fact_38_empty__iff,axiom,
    ! [Ca: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),bot_bot_fun_int_bool)) ).

tff(fact_39_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho882550911a_bool] :
      ( ( bot_bo214064617a_bool = hAPP_f642208769a_bool(collec977865374iple_a,Pa) )
    <=> ! [X_2: hoare_2076525193iple_a] : ~ hBOOL(hAPP_H454055633a_bool(Pa,X_2)) ) ).

tff(fact_40_empty__Collect__eq,axiom,
    ! [Pa: fun_int_bool] :
      ( ( bot_bot_fun_int_bool = hAPP_f1805168059t_bool(collect_int,Pa) )
    <=> ! [X_2: int] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).

tff(fact_41_empty__Collect__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( bot_bot_fun_nat_bool = hAPP_f800510211t_bool(collect_nat,Pa) )
    <=> ! [X_2: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

tff(fact_42_ex__in__conv,axiom,
    ! [A: fun_Ho882550911a_bool] :
      ( ? [X_2: hoare_2076525193iple_a] : hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
    <=> ( A != bot_bo214064617a_bool ) ) ).

tff(fact_43_ex__in__conv,axiom,
    ! [A: fun_nat_bool] :
      ( ? [X_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> ( A != bot_bot_fun_nat_bool ) ) ).

tff(fact_44_ex__in__conv,axiom,
    ! [A: fun_int_bool] :
      ( ? [X_2: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
    <=> ( A != bot_bot_fun_int_bool ) ) ).

tff(fact_45_all__not__in__conv,axiom,
    ! [A: fun_Ho882550911a_bool] :
      ( ! [X_2: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
    <=> ( A = bot_bo214064617a_bool ) ) ).

tff(fact_46_all__not__in__conv,axiom,
    ! [A: fun_nat_bool] :
      ( ! [X_2: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_47_all__not__in__conv,axiom,
    ! [A: fun_int_bool] :
      ( ! [X_2: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
    <=> ( A = bot_bot_fun_int_bool ) ) ).

tff(fact_48_empty__def,axiom,
    bot_bo214064617a_bool = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_b1299643615a_bool(cOMBK_861658160iple_a,fFalse)) ).

tff(fact_49_empty__def,axiom,
    bot_bot_fun_int_bool = hAPP_f1805168059t_bool(collect_int,hAPP_b396694332t_bool(cOMBK_bool_int,fFalse)) ).

tff(fact_50_empty__def,axiom,
    bot_bot_fun_nat_bool = hAPP_f800510211t_bool(collect_nat,hAPP_b1013836512t_bool(cOMBK_bool_nat,fFalse)) ).

tff(fact_51_insert__absorb,axiom,
    ! [A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) = A ) ) ).

tff(fact_52_insert__absorb,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
     => ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A) = A ) ) ).

tff(fact_53_insert__absorb,axiom,
    ! [A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) = A ) ) ).

tff(fact_54_insertI2,axiom,
    ! [Ba: nat,A_13: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).

tff(fact_55_insertI2,axiom,
    ! [Ba: hoare_2076525193iple_a,A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),B))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),B))) ) ).

tff(fact_56_insertI2,axiom,
    ! [Ba: int,A_13: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),B))) ) ).

tff(fact_57_insert__ident,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_58_insert__ident,axiom,
    ! [B: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B))
       => ( ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_59_insert__ident,axiom,
    ! [B: fun_int_bool,X_1: int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_60_insert__code,axiom,
    ! [Y_1: nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A),X_1))
    <=> ( ( Y_1 = X_1 )
        | hBOOL(hAPP_nat_bool(A,X_1)) ) ) ).

tff(fact_61_insert__code,axiom,
    ! [Y_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_1),A),X_1))
    <=> ( ( Y_1 = X_1 )
        | hBOOL(hAPP_H454055633a_bool(A,X_1)) ) ) ).

tff(fact_62_insert__code,axiom,
    ! [Y_1: int,A: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),A),X_1))
    <=> ( ( Y_1 = X_1 )
        | hBOOL(hAPP_int_bool(A,X_1)) ) ) ).

tff(fact_63_insert__iff,axiom,
    ! [A_13: nat,Ba: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)))
    <=> ( ( A_13 = Ba )
        | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A)) ) ) ).

tff(fact_64_insert__iff,axiom,
    ! [A_13: hoare_2076525193iple_a,Ba: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),A)))
    <=> ( ( A_13 = Ba )
        | hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A)) ) ) ).

tff(fact_65_insert__iff,axiom,
    ! [A_13: int,Ba: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),A)))
    <=> ( ( A_13 = Ba )
        | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A)) ) ) ).

tff(fact_66_insert__commute,axiom,
    ! [X_1: nat,Y_1: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) ).

tff(fact_67_insert__commute,axiom,
    ! [X_1: hoare_2076525193iple_a,Y_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_1),A)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) ).

tff(fact_68_insert__commute,axiom,
    ! [X_1: int,Y_1: int,A: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) ).

tff(fact_69_insert__absorb2,axiom,
    ! [X_1: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) ).

tff(fact_70_insert__absorb2,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A) ).

tff(fact_71_insert__absorb2,axiom,
    ! [X_1: int,A: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A) ).

tff(fact_72_insert__Collect,axiom,
    ! [A_13: nat,Pa: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(collect_nat,Pa)) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fimplies),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13)))),Pa)) ).

tff(fact_73_insert__Collect,axiom,
    ! [A_13: hoare_2076525193iple_a,Pa: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),hAPP_f642208769a_bool(collec977865374iple_a,Pa)) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fimplies),hAPP_f642208769a_bool(hAPP_f629438041a_bool(cOMBB_361863589iple_a,fNot),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_13)))),Pa)) ).

tff(fact_74_insert__Collect,axiom,
    ! [A_13: int,Pa: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(collect_int,Pa)) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fimplies),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fNot),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13)))),Pa)) ).

tff(fact_75_insert__compr,axiom,
    ! [A_13: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B))) ).

tff(fact_76_insert__compr,axiom,
    ! [A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fdisj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_13))),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),B))) ).

tff(fact_77_insert__compr,axiom,
    ! [A_13: int,B: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fdisj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),B))) ).

tff(fact_78_insertI1,axiom,
    ! [A_13: nat,B: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B))) ).

tff(fact_79_insertI1,axiom,
    ! [A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B))) ).

tff(fact_80_insertI1,axiom,
    ! [A_13: int,B: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B))) ).

tff(fact_81_insert__compr__raw,axiom,
    ! [X_2: nat,Xa: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),Xa) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_2))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),Xa))) ).

tff(fact_82_insert__compr__raw,axiom,
    ! [X_2: hoare_2076525193iple_a,Xa: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),Xa) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fdisj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),X_2))),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),Xa))) ).

tff(fact_83_insert__compr__raw,axiom,
    ! [X_2: int,Xa: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),Xa) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fdisj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),X_2))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),Xa))) ).

tff(fact_84_singleton__inject,axiom,
    ! [A_13: nat,Ba: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool) )
     => ( A_13 = Ba ) ) ).

tff(fact_85_singleton__inject,axiom,
    ! [A_13: hoare_2076525193iple_a,Ba: hoare_2076525193iple_a] :
      ( ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),bot_bo214064617a_bool) )
     => ( A_13 = Ba ) ) ).

tff(fact_86_singleton__inject,axiom,
    ! [A_13: int,Ba: int] :
      ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool) )
     => ( A_13 = Ba ) ) ).

tff(fact_87_singletonE,axiom,
    ! [Ba: nat,A_13: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))
     => ( Ba = A_13 ) ) ).

tff(fact_88_singletonE,axiom,
    ! [Ba: hoare_2076525193iple_a,A_13: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)))
     => ( Ba = A_13 ) ) ).

tff(fact_89_singletonE,axiom,
    ! [Ba: int,A_13: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))
     => ( Ba = A_13 ) ) ).

tff(fact_90_doubleton__eq__iff,axiom,
    ! [A_13: nat,Ba: nat,Ca: nat,D_1: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,D_1),bot_bot_fun_nat_bool)) )
    <=> ( ( ( A_13 = Ca )
          & ( Ba = D_1 ) )
        | ( ( A_13 = D_1 )
          & ( Ba = Ca ) ) ) ) ).

tff(fact_91_doubleton__eq__iff,axiom,
    ! [A_13: hoare_2076525193iple_a,Ba: hoare_2076525193iple_a,Ca: hoare_2076525193iple_a,D_1: hoare_2076525193iple_a] :
      ( ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),bot_bo214064617a_bool)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ca),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,D_1),bot_bo214064617a_bool)) )
    <=> ( ( ( A_13 = Ca )
          & ( Ba = D_1 ) )
        | ( ( A_13 = D_1 )
          & ( Ba = Ca ) ) ) ) ).

tff(fact_92_doubleton__eq__iff,axiom,
    ! [A_13: int,Ba: int,Ca: int,D_1: int] :
      ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,D_1),bot_bot_fun_int_bool)) )
    <=> ( ( ( A_13 = Ca )
          & ( Ba = D_1 ) )
        | ( ( A_13 = D_1 )
          & ( Ba = Ca ) ) ) ) ).

tff(fact_93_singleton__iff,axiom,
    ! [Ba: nat,A_13: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))
    <=> ( Ba = A_13 ) ) ).

tff(fact_94_singleton__iff,axiom,
    ! [Ba: hoare_2076525193iple_a,A_13: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)))
    <=> ( Ba = A_13 ) ) ).

tff(fact_95_singleton__iff,axiom,
    ! [Ba: int,A_13: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))
    <=> ( Ba = A_13 ) ) ).

tff(fact_96_insert__not__empty,axiom,
    ! [A_13: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) != bot_bot_fun_nat_bool ).

tff(fact_97_insert__not__empty,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A) != bot_bo214064617a_bool ).

tff(fact_98_insert__not__empty,axiom,
    ! [A_13: int,A: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) != bot_bot_fun_int_bool ).

tff(fact_99_empty__not__insert,axiom,
    ! [A_13: nat,A: fun_nat_bool] : bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) ).

tff(fact_100_empty__not__insert,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] : bot_bo214064617a_bool != hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A) ).

tff(fact_101_empty__not__insert,axiom,
    ! [A_13: int,A: fun_int_bool] : bot_bot_fun_int_bool != hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) ).

tff(fact_102_the__elem__eq,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(the_elem_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ).

tff(fact_103_the__elem__eq,axiom,
    ! [X_1: hoare_2076525193iple_a] : hAPP_f2009966983iple_a(the_el2146174248iple_a,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)) = X_1 ).

tff(fact_104_the__elem__eq,axiom,
    ! [X_1: int] : hAPP_f1594865479ol_int(the_elem_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ).

tff(fact_105_bot__apply,axiom,
    ! [X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(bot_bo214064617a_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_106_bot__apply,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_107_bot__apply,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_108_bot__fun__def,axiom,
    ! [X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(bot_bo214064617a_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_109_bot__fun__def,axiom,
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_110_bot__fun__def,axiom,
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_111_hoare__derivs_OSkip,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool] : hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,skip,Pa)),bot_bo214064617a_bool))) ).

tff(fact_112_Comp,axiom,
    ! [D_1: com,R_1: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_1)),bot_bo214064617a_bool)))
     => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Q_1,D_1,R_1)),bot_bo214064617a_bool)))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,semi(Ca,D_1),R_1)),bot_bo214064617a_bool))) ) ) ).

tff(fact_113_triple_Oexhaust,axiom,
    ! [Y_1: hoare_2076525193iple_a] :
      ~ ! [Fun1: fun_a_fun_state_bool,Com: com,Fun2: fun_a_fun_state_bool] : Y_1 != hoare_1876769875iple_a(Fun1,Com,Fun2) ).

tff(fact_114_Set_Oset__insert,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ~ ! [B_35: fun_nat_bool] :
            ( ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_35) )
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_35)) ) ) ).

tff(fact_115_Set_Oset__insert,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ~ ! [B_35: fun_Ho882550911a_bool] :
            ( ( A = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),B_35) )
           => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B_35)) ) ) ).

tff(fact_116_Set_Oset__insert,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ~ ! [B_35: fun_int_bool] :
            ( ( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_35) )
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_35)) ) ) ).

tff(fact_117_mk__disjoint__insert,axiom,
    ! [A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
     => ? [B_35: fun_nat_bool] :
          ( ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B_35) )
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),B_35)) ) ) ).

tff(fact_118_mk__disjoint__insert,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
     => ? [B_35: fun_Ho882550911a_bool] :
          ( ( A = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B_35) )
          & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),B_35)) ) ) ).

tff(fact_119_mk__disjoint__insert,axiom,
    ! [A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
     => ? [B_35: fun_int_bool] :
          ( ( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B_35) )
          & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),B_35)) ) ) ).

tff(fact_120_equals0I,axiom,
    ! [A: fun_Ho882550911a_bool] :
      ( ! [Y_21: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Y_21),A))
     => ( A = bot_bo214064617a_bool ) ) ).

tff(fact_121_equals0I,axiom,
    ! [A: fun_nat_bool] :
      ( ! [Y_21: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_21),A))
     => ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_122_equals0I,axiom,
    ! [A: fun_int_bool] :
      ( ! [Y_21: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_21),A))
     => ( A = bot_bot_fun_int_bool ) ) ).

tff(fact_123_conseq,axiom,
    ! [Q_1: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,Ca: com,Pa: fun_a_fun_state_bool] :
      ( ! [Z_11: x_a,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_11),S_2))
         => ? [P_1: fun_a_fun_state_bool,Q_2: fun_a_fun_state_bool] :
              ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(P_1,Ca,Q_2)),bot_bo214064617a_bool)))
              & ! [S_3: state] :
                  ( ! [Z_12: x_a] :
                      ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_12),S_2))
                     => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_2,Z_12),S_3)) )
                 => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_11),S_3)) ) ) )
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(Pa,Ca,Q_1)),bot_bo214064617a_bool))) ) ).

tff(fact_124_nonempty__iff,axiom,
    ! [A: fun_nat_bool] :
      ( ( A != bot_bot_fun_nat_bool )
    <=> ? [X_2: nat,B_35: fun_nat_bool] :
          ( ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_35) )
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_35)) ) ) ).

tff(fact_125_nonempty__iff,axiom,
    ! [A: fun_Ho882550911a_bool] :
      ( ( A != bot_bo214064617a_bool )
    <=> ? [X_2: hoare_2076525193iple_a,B_35: fun_Ho882550911a_bool] :
          ( ( A = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),B_35) )
          & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),B_35)) ) ) ).

tff(fact_126_nonempty__iff,axiom,
    ! [A: fun_int_bool] :
      ( ( A != bot_bot_fun_int_bool )
    <=> ? [X_2: int,B_35: fun_int_bool] :
          ( ( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),B_35) )
          & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B_35)) ) ) ).

tff(fact_127_com_Osimps_I13_J,axiom,
    ! [Com1_2: com,Com2_2: com] : semi(Com1_2,Com2_2) != skip ).

tff(fact_128_com_Osimps_I12_J,axiom,
    ! [Com1_2: com,Com2_2: com] : skip != semi(Com1_2,Com2_2) ).

tff(fact_129_the__elem__def,axiom,
    ! [X_28: fun_nat_bool] : hAPP_f22106695ol_nat(the_elem_nat,X_28) = hAPP_f22106695ol_nat(the_nat,hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,hAPP_f103356543l_bool(fequal_fun_nat_bool,X_28)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool))) ).

tff(fact_130_the__elem__def,axiom,
    ! [X_28: fun_Ho882550911a_bool] : hAPP_f2009966983iple_a(the_el2146174248iple_a,X_28) = hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_f338738851a_bool(hAPP_f1827400025a_bool(cOMBB_1676209667iple_a,hAPP_f256727871l_bool(fequal291249344a_bool,X_28)),hAPP_f219434403a_bool(hAPP_f1824299331a_bool(cOMBC_175592824a_bool,insert1582918915iple_a),bot_bo214064617a_bool))) ).

tff(fact_131_the__elem__def,axiom,
    ! [X_28: fun_int_bool] : hAPP_f1594865479ol_int(the_elem_int,X_28) = hAPP_f1594865479ol_int(the_int,hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,hAPP_f284875647l_bool(fequal_fun_int_bool,X_28)),hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,insert_int),bot_bot_fun_int_bool))) ).

tff(fact_132_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 ) ) ) ).

tff(fact_133_fold1Set__sing,axiom,
    ! [F: fun_nat_fun_nat_nat,A_13: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)),Ba))
    <=> ( A_13 = Ba ) ) ).

tff(fact_134_fold1Set__sing,axiom,
    ! [F: fun_Ho483360497iple_a,A_13: hoare_2076525193iple_a,Ba: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(finite100168025iple_a(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)),Ba))
    <=> ( A_13 = Ba ) ) ).

tff(fact_135_fold1Set__sing,axiom,
    ! [F: fun_int_fun_int_int,A_13: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)),Ba))
    <=> ( A_13 = Ba ) ) ).

tff(fact_136_folding__one_Osingleton,axiom,
    ! [X_1: nat,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ) ) ).

tff(fact_137_folding__one_Osingleton,axiom,
    ! [X_1: hoare_2076525193iple_a,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)) = X_1 ) ) ).

tff(fact_138_folding__one_Osingleton,axiom,
    ! [X_1: int,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ) ) ).

tff(fact_139_bot__empty__eq,axiom,
    ! [X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(bot_bo214064617a_bool,X_2))
    <=> hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),bot_bo214064617a_bool)) ) ).

tff(fact_140_bot__empty__eq,axiom,
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),bot_bot_fun_nat_bool)) ) ).

tff(fact_141_bot__empty__eq,axiom,
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_2))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),bot_bot_fun_int_bool)) ) ).

tff(fact_142_Ass,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,X_28: vname,A_13: fun_state_nat] : hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),X_28)),A_13)),ass(X_28,A_13),Pa)),bot_bo214064617a_bool))) ).

tff(fact_143_fold1__singleton,axiom,
    ! [F: fun_nat_fun_nat_nat,A_13: nat] : hAPP_f22106695ol_nat(finite_fold1_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)) = A_13 ).

tff(fact_144_fold1__singleton,axiom,
    ! [F: fun_Ho483360497iple_a,A_13: hoare_2076525193iple_a] : hAPP_f2009966983iple_a(finite2092228073iple_a(F),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)) = A_13 ).

tff(fact_145_fold1__singleton,axiom,
    ! [F: fun_int_fun_int_int,A_13: int] : hAPP_f1594865479ol_int(finite_fold1_int(F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)) = A_13 ).

tff(fact_146_fold1__singleton__def,axiom,
    ! [A_13: nat,G: fun_fun_nat_bool_nat,F: fun_nat_fun_nat_nat] :
      ( ( G = finite_fold1_nat(F) )
     => ( hAPP_f22106695ol_nat(G,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)) = A_13 ) ) ).

tff(fact_147_fold1__singleton__def,axiom,
    ! [A_13: hoare_2076525193iple_a,G: fun_fu253747899iple_a,F: fun_Ho483360497iple_a] :
      ( ( G = finite2092228073iple_a(F) )
     => ( hAPP_f2009966983iple_a(G,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)) = A_13 ) ) ).

tff(fact_148_fold1__singleton__def,axiom,
    ! [A_13: int,G: fun_fun_int_bool_int,F: fun_int_fun_int_int] :
      ( ( G = finite_fold1_int(F) )
     => ( hAPP_f1594865479ol_int(G,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)) = A_13 ) ) ).

tff(fact_149_image__constant__conv,axiom,
    ! [Ca: nat,A: fun_Ho882550911a_bool] :
      ( ( ( A = bot_bo214064617a_bool )
       => ( image_1778392390_a_nat(hAPP_n1236213193_a_nat(cOMBK_1654814733iple_a,Ca),A) = bot_bot_fun_nat_bool ) )
      & ( ( A != bot_bo214064617a_bool )
       => ( image_1778392390_a_nat(hAPP_n1236213193_a_nat(cOMBK_1654814733iple_a,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_150_image__constant__conv,axiom,
    ! [Ca: nat,A: fun_int_bool] :
      ( ( ( A = bot_bot_fun_int_bool )
       => ( image_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,Ca),A) = bot_bot_fun_nat_bool ) )
      & ( ( A != bot_bot_fun_int_bool )
       => ( image_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_151_image__constant__conv,axiom,
    ! [Ca: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( ( ( A = bot_bo214064617a_bool )
       => ( image_1416816325iple_a(hAPP_H1417936777iple_a(cOMBK_788297294iple_a,Ca),A) = bot_bo214064617a_bool ) )
      & ( ( A != bot_bo214064617a_bool )
       => ( image_1416816325iple_a(hAPP_H1417936777iple_a(cOMBK_788297294iple_a,Ca),A) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ca),bot_bo214064617a_bool) ) ) ) ).

tff(fact_152_image__constant__conv,axiom,
    ! [Ca: int,A: fun_int_bool] :
      ( ( ( A = bot_bot_fun_int_bool )
       => ( image_int_int(hAPP_int_fun_int_int(cOMBK_int_int,Ca),A) = bot_bot_fun_int_bool ) )
      & ( ( A != bot_bot_fun_int_bool )
       => ( image_int_int(hAPP_int_fun_int_int(cOMBK_int_int,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ) ) ).

tff(fact_153_image__constant__conv,axiom,
    ! [Ca: nat,A: fun_nat_bool] :
      ( ( ( A = bot_bot_fun_nat_bool )
       => ( image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = bot_bot_fun_nat_bool ) )
      & ( ( A != bot_bot_fun_nat_bool )
       => ( image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_154_image__constant__conv,axiom,
    ! [Ca: int,A: fun_nat_bool] :
      ( ( ( A = bot_bot_fun_nat_bool )
       => ( image_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,Ca),A) = bot_bot_fun_int_bool ) )
      & ( ( A != bot_bot_fun_nat_bool )
       => ( image_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ) ) ).

tff(fact_155_image__eqI,axiom,
    ! [A: fun_Ho882550911a_bool,Ba: hoare_2076525193iple_a,F: fun_Ho1518252029iple_a,X_1: hoare_2076525193iple_a] :
      ( ( Ba = hAPP_H1789220885iple_a(F,X_1) )
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_1416816325iple_a(F,A))) ) ) ).

tff(fact_156_image__eqI,axiom,
    ! [A: fun_nat_bool,Ba: hoare_2076525193iple_a,F: fun_na1240513468iple_a,X_1: nat] :
      ( ( Ba = hAPP_n887873492iple_a(F,X_1) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_344477316iple_a(F,A))) ) ) ).

tff(fact_157_image__eqI,axiom,
    ! [A: fun_int_bool,Ba: hoare_2076525193iple_a,F: fun_in1219646560iple_a,X_1: int] :
      ( ( Ba = hAPP_i867006584iple_a(F,X_1) )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_323610408iple_a(F,A))) ) ) ).

tff(fact_158_image__eqI,axiom,
    ! [A: fun_Ho882550911a_bool,Ba: nat,F: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a] :
      ( ( Ba = hAPP_H174304918_a_nat(F,X_1) )
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1778392390_a_nat(F,A))) ) ) ).

tff(fact_159_image__eqI,axiom,
    ! [A: fun_int_bool,Ba: int,F: fun_int_int,X_1: int] :
      ( ( Ba = hAPP_int_int(F,X_1) )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_int_int(F,A))) ) ) ).

tff(fact_160_image__eqI,axiom,
    ! [A: fun_Ho882550911a_bool,Ba: int,F: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a] :
      ( ( Ba = hAPP_H811578610_a_int(F,X_1) )
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_268182434_a_int(F,A))) ) ) ).

tff(fact_161_image__eqI,axiom,
    ! [A: fun_nat_bool,Ba: nat,F: fun_nat_nat,X_1: nat] :
      ( ( Ba = hAPP_nat_nat(F,X_1) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F,A))) ) ) ).

tff(fact_162_image__eqI,axiom,
    ! [A: fun_nat_bool,Ba: int,F: fun_nat_int,X_1: nat] :
      ( ( Ba = hAPP_nat_int(F,X_1) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_nat_int(F,A))) ) ) ).

tff(fact_163_fold1__def,axiom,
    ! [F: fun_int_fun_int_int,A: fun_int_bool] : hAPP_f1594865479ol_int(finite_fold1_int(F),A) = hAPP_f1594865479ol_int(the_int,finite_fold1Set_int(F,A)) ).

tff(fact_164_fold1__def,axiom,
    ! [F: fun_Ho483360497iple_a,A: fun_Ho882550911a_bool] : hAPP_f2009966983iple_a(finite2092228073iple_a(F),A) = hAPP_f2009966983iple_a(the_Ho1456473914iple_a,finite100168025iple_a(F,A)) ).

tff(fact_165_fold1__def,axiom,
    ! [F: fun_nat_fun_nat_nat,A: fun_nat_bool] : hAPP_f22106695ol_nat(finite_fold1_nat(F),A) = hAPP_f22106695ol_nat(the_nat,finite_fold1Set_nat(F,A)) ).

tff(fact_166_com_Osimps_I1_J,axiom,
    ! [Vname_1: vname,Fun_1: fun_state_nat,Vname: vname,Fun: fun_state_nat] :
      ( ( ass(Vname_1,Fun_1) = ass(Vname,Fun) )
    <=> ( ( Vname_1 = Vname )
        & ( Fun_1 = Fun ) ) ) ).

tff(fact_167_rev__image__eqI,axiom,
    ! [Ba: hoare_2076525193iple_a,F: fun_Ho1518252029iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( ( Ba = hAPP_H1789220885iple_a(F,X_1) )
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_1416816325iple_a(F,A))) ) ) ).

tff(fact_168_rev__image__eqI,axiom,
    ! [Ba: hoare_2076525193iple_a,F: fun_na1240513468iple_a,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ( Ba = hAPP_n887873492iple_a(F,X_1) )
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_344477316iple_a(F,A))) ) ) ).

tff(fact_169_rev__image__eqI,axiom,
    ! [Ba: hoare_2076525193iple_a,F: fun_in1219646560iple_a,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( ( Ba = hAPP_i867006584iple_a(F,X_1) )
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_323610408iple_a(F,A))) ) ) ).

tff(fact_170_rev__image__eqI,axiom,
    ! [Ba: nat,F: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( ( Ba = hAPP_H174304918_a_nat(F,X_1) )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1778392390_a_nat(F,A))) ) ) ).

tff(fact_171_rev__image__eqI,axiom,
    ! [Ba: int,F: fun_int_int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( ( Ba = hAPP_int_int(F,X_1) )
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_int_int(F,A))) ) ) ).

tff(fact_172_rev__image__eqI,axiom,
    ! [Ba: int,F: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( ( Ba = hAPP_H811578610_a_int(F,X_1) )
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_268182434_a_int(F,A))) ) ) ).

tff(fact_173_rev__image__eqI,axiom,
    ! [Ba: nat,F: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ( Ba = hAPP_nat_nat(F,X_1) )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F,A))) ) ) ).

tff(fact_174_rev__image__eqI,axiom,
    ! [Ba: int,F: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ( Ba = hAPP_nat_int(F,X_1) )
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_nat_int(F,A))) ) ) ).

tff(fact_175_imageI,axiom,
    ! [F: fun_Ho1518252029iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_H1789220885iple_a(F,X_1)),image_1416816325iple_a(F,A))) ) ).

tff(fact_176_imageI,axiom,
    ! [F: fun_na1240513468iple_a,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_n887873492iple_a(F,X_1)),image_344477316iple_a(F,A))) ) ).

tff(fact_177_imageI,axiom,
    ! [F: fun_in1219646560iple_a,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_i867006584iple_a(F,X_1)),image_323610408iple_a(F,A))) ) ).

tff(fact_178_imageI,axiom,
    ! [F: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_H174304918_a_nat(F,X_1)),image_1778392390_a_nat(F,A))) ) ).

tff(fact_179_imageI,axiom,
    ! [F: fun_int_int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(F,X_1)),image_int_int(F,A))) ) ).

tff(fact_180_imageI,axiom,
    ! [F: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_H811578610_a_int(F,X_1)),image_268182434_a_int(F,A))) ) ).

tff(fact_181_imageI,axiom,
    ! [F: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(F,X_1)),image_nat_nat(F,A))) ) ).

tff(fact_182_imageI,axiom,
    ! [F: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A))) ) ).

tff(fact_183_image__iff,axiom,
    ! [Z_1: hoare_2076525193iple_a,F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Z_1),image_1416816325iple_a(F,A)))
    <=> ? [X_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
          & ( Z_1 = hAPP_H1789220885iple_a(F,X_2) ) ) ) ).

tff(fact_184_image__iff,axiom,
    ! [Z_1: int,F: fun_int_int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Z_1),image_int_int(F,A)))
    <=> ? [X_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
          & ( Z_1 = hAPP_int_int(F,X_2) ) ) ) ).

tff(fact_185_image__iff,axiom,
    ! [Z_1: nat,F: fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Z_1),image_nat_nat(F,A)))
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
          & ( Z_1 = hAPP_nat_nat(F,X_2) ) ) ) ).

tff(fact_186_image__iff,axiom,
    ! [Z_1: int,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Z_1),image_nat_int(F,A)))
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
          & ( Z_1 = hAPP_nat_int(F,X_2) ) ) ) ).

tff(fact_187_image__image,axiom,
    ! [F: fun_nat_int,G: fun_int_nat,A: fun_int_bool] : image_nat_int(F,image_int_nat(G,A)) = image_int_int(hAPP_f147134065nt_int(hAPP_f1673907925nt_int(cOMBB_nat_int_int,F),G),A) ).

tff(fact_188_image__image,axiom,
    ! [F: fun_int_int,G: fun_nat_int,A: fun_nat_bool] : image_int_int(F,image_nat_int(G,A)) = image_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,F),G),A) ).

tff(fact_189_image__ident,axiom,
    ! [Y_23: fun_int_bool] : image_int_int(cOMBI_int,Y_23) = Y_23 ).

tff(fact_190_image__ident,axiom,
    ! [Y_23: fun_Ho882550911a_bool] : image_1416816325iple_a(cOMBI_1928055558iple_a,Y_23) = Y_23 ).

tff(fact_191_image__ident,axiom,
    ! [Y_23: fun_nat_bool] : image_nat_nat(cOMBI_nat,Y_23) = Y_23 ).

tff(fact_192_com_Osimps_I24_J,axiom,
    ! [Vname_1: vname,Fun_1: fun_state_nat,Com1: com,Com2: com] : ass(Vname_1,Fun_1) != semi(Com1,Com2) ).

tff(fact_193_com_Osimps_I25_J,axiom,
    ! [Com1: com,Com2: com,Vname_1: vname,Fun_1: fun_state_nat] : semi(Com1,Com2) != ass(Vname_1,Fun_1) ).

tff(fact_194_com_Osimps_I8_J,axiom,
    ! [Vname: vname,Fun: fun_state_nat] : skip != ass(Vname,Fun) ).

tff(fact_195_mem__def,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
    <=> hBOOL(hAPP_H454055633a_bool(A,X_1)) ) ).

tff(fact_196_mem__def,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_nat_bool(A,X_1)) ) ).

tff(fact_197_mem__def,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
    <=> hBOOL(hAPP_int_bool(A,X_1)) ) ).

tff(fact_198_Collect__def,axiom,
    ! [Pa: fun_int_bool] : hAPP_f1805168059t_bool(collect_int,Pa) = Pa ).

tff(fact_199_Collect__def,axiom,
    ! [Pa: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(collec977865374iple_a,Pa) = Pa ).

tff(fact_200_Collect__def,axiom,
    ! [Pa: fun_nat_bool] : hAPP_f800510211t_bool(collect_nat,Pa) = Pa ).

tff(fact_201_com_Osimps_I9_J,axiom,
    ! [Vname: vname,Fun: fun_state_nat] : ass(Vname,Fun) != skip ).

tff(fact_202_image__is__empty,axiom,
    ! [F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool] :
      ( ( image_1416816325iple_a(F,A) = bot_bo214064617a_bool )
    <=> ( A = bot_bo214064617a_bool ) ) ).

tff(fact_203_image__is__empty,axiom,
    ! [F: fun_int_int,A: fun_int_bool] :
      ( ( image_int_int(F,A) = bot_bot_fun_int_bool )
    <=> ( A = bot_bot_fun_int_bool ) ) ).

tff(fact_204_image__is__empty,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool] :
      ( ( image_nat_nat(F,A) = bot_bot_fun_nat_bool )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_205_image__is__empty,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool] :
      ( ( image_nat_int(F,A) = bot_bot_fun_int_bool )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_206_image__empty,axiom,
    ! [F: fun_Ho1518252029iple_a] : image_1416816325iple_a(F,bot_bo214064617a_bool) = bot_bo214064617a_bool ).

tff(fact_207_image__empty,axiom,
    ! [F: fun_int_int] : image_int_int(F,bot_bot_fun_int_bool) = bot_bot_fun_int_bool ).

tff(fact_208_image__empty,axiom,
    ! [F: fun_nat_nat] : image_nat_nat(F,bot_bot_fun_nat_bool) = bot_bot_fun_nat_bool ).

tff(fact_209_image__empty,axiom,
    ! [F: fun_nat_int] : image_nat_int(F,bot_bot_fun_nat_bool) = bot_bot_fun_int_bool ).

tff(fact_210_empty__is__image,axiom,
    ! [F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool] :
      ( ( bot_bo214064617a_bool = image_1416816325iple_a(F,A) )
    <=> ( A = bot_bo214064617a_bool ) ) ).

tff(fact_211_empty__is__image,axiom,
    ! [F: fun_int_int,A: fun_int_bool] :
      ( ( bot_bot_fun_int_bool = image_int_int(F,A) )
    <=> ( A = bot_bot_fun_int_bool ) ) ).

tff(fact_212_empty__is__image,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool] :
      ( ( bot_bot_fun_nat_bool = image_nat_nat(F,A) )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_213_empty__is__image,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool] :
      ( ( bot_bot_fun_int_bool = image_nat_int(F,A) )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_214_insert__image,axiom,
    ! [F: fun_int_nat,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_int_nat(F,X_1)),image_int_nat(F,A)) = image_int_nat(F,A) ) ) ).

tff(fact_215_insert__image,axiom,
    ! [F: fun_Ho1518252029iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_H1789220885iple_a(F,X_1)),image_1416816325iple_a(F,A)) = image_1416816325iple_a(F,A) ) ) ).

tff(fact_216_insert__image,axiom,
    ! [F: fun_int_int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_int_int(F,X_1)),image_int_int(F,A)) = image_int_int(F,A) ) ) ).

tff(fact_217_insert__image,axiom,
    ! [F: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_H811578610_a_int(F,X_1)),image_268182434_a_int(F,A)) = image_268182434_a_int(F,A) ) ) ).

tff(fact_218_insert__image,axiom,
    ! [F: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_nat_nat(F,X_1)),image_nat_nat(F,A)) = image_nat_nat(F,A) ) ) ).

tff(fact_219_insert__image,axiom,
    ! [F: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A)) = image_nat_int(F,A) ) ) ).

tff(fact_220_image__insert,axiom,
    ! [F: fun_Ho864122878_a_nat,A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] : image_1778392390_a_nat(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_H174304918_a_nat(F,A_13)),image_1778392390_a_nat(F,B)) ).

tff(fact_221_image__insert,axiom,
    ! [F: fun_int_nat,A_13: int,B: fun_int_bool] : image_int_nat(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_int_nat(F,A_13)),image_int_nat(F,B)) ).

tff(fact_222_image__insert,axiom,
    ! [F: fun_Ho1518252029iple_a,A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] : image_1416816325iple_a(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_H1789220885iple_a(F,A_13)),image_1416816325iple_a(F,B)) ).

tff(fact_223_image__insert,axiom,
    ! [F: fun_na1240513468iple_a,A_13: nat,B: fun_nat_bool] : image_344477316iple_a(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_n887873492iple_a(F,A_13)),image_344477316iple_a(F,B)) ).

tff(fact_224_image__insert,axiom,
    ! [F: fun_int_int,A_13: int,B: fun_int_bool] : image_int_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_int_int(F,A_13)),image_int_int(F,B)) ).

tff(fact_225_image__insert,axiom,
    ! [F: fun_nat_nat,A_13: nat,B: fun_nat_bool] : image_nat_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_nat_nat(F,A_13)),image_nat_nat(F,B)) ).

tff(fact_226_image__insert,axiom,
    ! [F: fun_nat_int,A_13: nat,B: fun_nat_bool] : image_nat_int(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,A_13)),image_nat_int(F,B)) ).

tff(fact_227_fold1Set__nonempty,axiom,
    ! [F: fun_Ho483360497iple_a,A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(finite100168025iple_a(F,A),X_1))
     => ( A != bot_bo214064617a_bool ) ) ).

tff(fact_228_fold1Set__nonempty,axiom,
    ! [F: fun_int_fun_int_int,A: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,A),X_1))
     => ( A != bot_bot_fun_int_bool ) ) ).

tff(fact_229_fold1Set__nonempty,axiom,
    ! [F: fun_nat_fun_nat_nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,A),X_1))
     => ( A != bot_bot_fun_nat_bool ) ) ).

tff(fact_230_empty__fold1SetE,axiom,
    ! [F: fun_Ho483360497iple_a,X_1: hoare_2076525193iple_a] : ~ hBOOL(hAPP_H454055633a_bool(finite100168025iple_a(F,bot_bo214064617a_bool),X_1)) ).

tff(fact_231_empty__fold1SetE,axiom,
    ! [F: fun_int_fun_int_int,X_1: int] : ~ hBOOL(hAPP_int_bool(finite_fold1Set_int(F,bot_bot_fun_int_bool),X_1)) ).

tff(fact_232_empty__fold1SetE,axiom,
    ! [F: fun_nat_fun_nat_nat,X_1: nat] : ~ hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,bot_bot_fun_nat_bool),X_1)) ).

tff(fact_233_image__constant,axiom,
    ! [Ca: nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( image_1778392390_a_nat(hAPP_n1236213193_a_nat(cOMBK_1654814733iple_a,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).

tff(fact_234_image__constant,axiom,
    ! [Ca: nat,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( image_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).

tff(fact_235_image__constant,axiom,
    ! [Ca: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( image_1416816325iple_a(hAPP_H1417936777iple_a(cOMBK_788297294iple_a,Ca),A) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ca),bot_bo214064617a_bool) ) ) ).

tff(fact_236_image__constant,axiom,
    ! [Ca: int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( image_int_int(hAPP_int_fun_int_int(cOMBK_int_int,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ) ).

tff(fact_237_image__constant,axiom,
    ! [Ca: int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( image_268182434_a_int(hAPP_i1058170057_a_int(cOMBK_1633947825iple_a,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ) ).

tff(fact_238_image__constant,axiom,
    ! [Ca: nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).

tff(fact_239_image__constant,axiom,
    ! [Ca: int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( image_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,Ca),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ca),bot_bot_fun_int_bool) ) ) ).

tff(fact_240_imageE,axiom,
    ! [Ba: hoare_2076525193iple_a,F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_1416816325iple_a(F,A)))
     => ~ ! [X_2: hoare_2076525193iple_a] :
            ( ( Ba = hAPP_H1789220885iple_a(F,X_2) )
           => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A)) ) ) ).

tff(fact_241_imageE,axiom,
    ! [Ba: nat,F: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1778392390_a_nat(F,A)))
     => ~ ! [X_2: hoare_2076525193iple_a] :
            ( ( Ba = hAPP_H174304918_a_nat(F,X_2) )
           => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A)) ) ) ).

tff(fact_242_imageE,axiom,
    ! [Ba: int,F: fun_Ho1026035802_a_int,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_268182434_a_int(F,A)))
     => ~ ! [X_2: hoare_2076525193iple_a] :
            ( ( Ba = hAPP_H811578610_a_int(F,X_2) )
           => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A)) ) ) ).

tff(fact_243_imageE,axiom,
    ! [Ba: hoare_2076525193iple_a,F: fun_na1240513468iple_a,A: fun_nat_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_344477316iple_a(F,A)))
     => ~ ! [X_2: nat] :
            ( ( Ba = hAPP_n887873492iple_a(F,X_2) )
           => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).

tff(fact_244_imageE,axiom,
    ! [Ba: int,F: fun_int_int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_int_int(F,A)))
     => ~ ! [X_2: int] :
            ( ( Ba = hAPP_int_int(F,X_2) )
           => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A)) ) ) ).

tff(fact_245_imageE,axiom,
    ! [Ba: hoare_2076525193iple_a,F: fun_in1219646560iple_a,A: fun_int_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ba),image_323610408iple_a(F,A)))
     => ~ ! [X_2: int] :
            ( ( Ba = hAPP_i867006584iple_a(F,X_2) )
           => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A)) ) ) ).

tff(fact_246_imageE,axiom,
    ! [Ba: nat,F: fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F,A)))
     => ~ ! [X_2: nat] :
            ( ( Ba = hAPP_nat_nat(F,X_2) )
           => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).

tff(fact_247_imageE,axiom,
    ! [Ba: int,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),image_nat_int(F,A)))
     => ~ ! [X_2: nat] :
            ( ( Ba = hAPP_nat_int(F,X_2) )
           => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).

tff(fact_248_the__sym__eq__trivial,axiom,
    ! [X_1: int] : hAPP_f1594865479ol_int(the_int,hAPP_i1948725293t_bool(fequal_int,X_1)) = X_1 ).

tff(fact_249_the__sym__eq__trivial,axiom,
    ! [X_1: hoare_2076525193iple_a] : hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_H2077279219a_bool(fequal1589671816iple_a,X_1)) = X_1 ).

tff(fact_250_the__sym__eq__trivial,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(the_nat,hAPP_n1699378549t_bool(fequal_nat,X_1)) = X_1 ).

tff(fact_251_the__eq__trivial,axiom,
    ! [A_13: int] : hAPP_f1594865479ol_int(the_int,hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),A_13)) = A_13 ).

tff(fact_252_the__eq__trivial,axiom,
    ! [A_13: hoare_2076525193iple_a] : hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_13)) = A_13 ).

tff(fact_253_the__eq__trivial,axiom,
    ! [A_13: nat] : hAPP_f22106695ol_nat(the_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_13)) = A_13 ).

tff(fact_254_If__def,axiom,
    ! [X_1: int,Y_1: int,Pa: bool] :
      ( ( hBOOL(Pa)
       => ( X_1 = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),X_1)))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),Y_1)))) ) )
      & ( ~ hBOOL(Pa)
       => ( Y_1 = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),X_1)))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),Y_1)))) ) ) ) ).

tff(fact_255_If__def,axiom,
    ! [X_1: hoare_2076525193iple_a,Y_1: hoare_2076525193iple_a,Pa: bool] :
      ( ( hBOOL(Pa)
       => ( X_1 = hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f629438041a_bool(cOMBB_361863589iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),X_1)))),hAPP_f642208769a_bool(hAPP_f629438041a_bool(cOMBB_361863589iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),Y_1)))) ) )
      & ( ~ hBOOL(Pa)
       => ( Y_1 = hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f629438041a_bool(cOMBB_361863589iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),X_1)))),hAPP_f642208769a_bool(hAPP_f629438041a_bool(cOMBB_361863589iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),Y_1)))) ) ) ) ).

tff(fact_256_If__def,axiom,
    ! [X_1: nat,Y_1: nat,Pa: bool] :
      ( ( hBOOL(Pa)
       => ( X_1 = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_1)))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),Y_1)))) ) )
      & ( ~ hBOOL(Pa)
       => ( Y_1 = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_1)))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),Y_1)))) ) ) ) ).

tff(fact_257_image__cong,axiom,
    ! [F: fun_Ho1518252029iple_a,G: fun_Ho1518252029iple_a,M_3: fun_Ho882550911a_bool,N_3: fun_Ho882550911a_bool] :
      ( ( M_3 = N_3 )
     => ( ! [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),N_3))
           => ( hAPP_H1789220885iple_a(F,X_2) = hAPP_H1789220885iple_a(G,X_2) ) )
       => ( image_1416816325iple_a(F,M_3) = image_1416816325iple_a(G,N_3) ) ) ) ).

tff(fact_258_image__cong,axiom,
    ! [F: fun_int_int,G: fun_int_int,M_3: fun_int_bool,N_3: fun_int_bool] :
      ( ( M_3 = N_3 )
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),N_3))
           => ( hAPP_int_int(F,X_2) = hAPP_int_int(G,X_2) ) )
       => ( image_int_int(F,M_3) = image_int_int(G,N_3) ) ) ) ).

tff(fact_259_image__cong,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,M_3: fun_nat_bool,N_3: fun_nat_bool] :
      ( ( M_3 = N_3 )
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
           => ( hAPP_nat_nat(F,X_2) = hAPP_nat_nat(G,X_2) ) )
       => ( image_nat_nat(F,M_3) = image_nat_nat(G,N_3) ) ) ) ).

tff(fact_260_image__cong,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,M_3: fun_nat_bool,N_3: fun_nat_bool] :
      ( ( M_3 = N_3 )
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
           => ( hAPP_nat_int(F,X_2) = hAPP_nat_int(G,X_2) ) )
       => ( image_nat_int(F,M_3) = image_nat_int(G,N_3) ) ) ) ).

tff(fact_261_fold1Set_Ointros,axiom,
    ! [F: fun_nat_fun_nat_nat,A_13: nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),A_13),A),X_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)),X_1)) ) ) ).

tff(fact_262_fold1Set_Ointros,axiom,
    ! [F: fun_Ho483360497iple_a,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),A_13),A),X_1))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => hBOOL(hAPP_H454055633a_bool(finite100168025iple_a(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A)),X_1)) ) ) ).

tff(fact_263_fold1Set_Ointros,axiom,
    ! [F: fun_int_fun_int_int,A_13: int,A: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),A_13),A),X_1))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => hBOOL(hAPP_int_bool(finite_fold1Set_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)),X_1)) ) ) ).

tff(fact_264_the__equality,axiom,
    ! [Pa: fun_int_bool,A_13: int] :
      ( hBOOL(hAPP_int_bool(Pa,A_13))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(Pa,X_2))
           => ( X_2 = A_13 ) )
       => ( hAPP_f1594865479ol_int(the_int,Pa) = A_13 ) ) ) ).

tff(fact_265_the__equality,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(Pa,A_13))
     => ( ! [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_H454055633a_bool(Pa,X_2))
           => ( X_2 = A_13 ) )
       => ( hAPP_f2009966983iple_a(the_Ho1456473914iple_a,Pa) = A_13 ) ) ) ).

tff(fact_266_the__equality,axiom,
    ! [Pa: fun_nat_bool,A_13: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,A_13))
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,X_2))
           => ( X_2 = A_13 ) )
       => ( hAPP_f22106695ol_nat(the_nat,Pa) = A_13 ) ) ) ).

tff(fact_267_folding__one_Oinsert,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( ( A != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ) ) ).

tff(fact_268_folding__one_Oinsert,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( ( A != bot_bo214064617a_bool )
           => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(F_1,A)) ) ) ) ) ) ).

tff(fact_269_folding__one_Oinsert,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( ( A != bot_bot_fun_int_bool )
           => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ) ) ).

tff(fact_270_folding__one_Oeq__fold,axiom,
    ! [A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f2009966983iple_a(F_1,A) = hAPP_f2009966983iple_a(finite2092228073iple_a(F),A) ) ) ) ).

tff(fact_271_folding__one_Oeq__fold,axiom,
    ! [A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(F_1,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F),A) ) ) ) ).

tff(fact_272_folding__one_Oeq__fold,axiom,
    ! [A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1594865479ol_int(F_1,A) = hAPP_f1594865479ol_int(finite_fold1_int(F),A) ) ) ) ).

tff(fact_273_folding__one_Oclosed,axiom,
    ! [A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( ! [X_2: nat,Y_21: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_21)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_21),bot_bot_fun_nat_bool))))
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F_1,A)),A)) ) ) ) ) ).

tff(fact_274_folding__one_Oclosed,axiom,
    ! [A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ( A != bot_bo214064617a_bool )
         => ( ! [X_2: hoare_2076525193iple_a,Y_21: hoare_2076525193iple_a] : hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),Y_21)),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_21),bot_bo214064617a_bool))))
           => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_f2009966983iple_a(F_1,A)),A)) ) ) ) ) ).

tff(fact_275_folding__one_Oclosed,axiom,
    ! [A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( A != bot_bot_fun_int_bool )
         => ( ! [X_2: int,Y_21: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_21)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_21),bot_bot_fun_int_bool))))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(F_1,A)),A)) ) ) ) ) ).

tff(fact_276_finite__Collect__conjI,axiom,
    ! [Q_1: fun_Ho882550911a_bool,Pa: fun_Ho882550911a_bool] :
      ( ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Pa)))
        | hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Q_1))) )
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),Pa)),Q_1)))) ) ).

tff(fact_277_finite__Collect__conjI,axiom,
    ! [Q_1: fun_int_bool,Pa: fun_int_bool] :
      ( ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Pa)))
        | hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Q_1))) )
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),Pa)),Q_1)))) ) ).

tff(fact_278_finite__Collect__conjI,axiom,
    ! [Q_1: fun_nat_bool,Pa: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Pa)))
        | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Q_1))) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),Q_1)))) ) ).

tff(fact_279_finite_OemptyI,axiom,
    hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,bot_bo214064617a_bool)) ).

tff(fact_280_finite_OemptyI,axiom,
    hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).

tff(fact_281_finite_OemptyI,axiom,
    hBOOL(hAPP_f448129468l_bool(finite_finite_int,bot_bot_fun_int_bool)) ).

tff(fact_282_finite_OinsertI,axiom,
    ! [A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A))) ) ).

tff(fact_283_finite_OinsertI,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A))) ) ).

tff(fact_284_finite_OinsertI,axiom,
    ! [A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A))) ) ).

tff(fact_285_finite__imageI,axiom,
    ! [H: fun_Ho1518252029iple_a,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,image_1416816325iple_a(H,F_1))) ) ).

tff(fact_286_finite__imageI,axiom,
    ! [H: fun_na1240513468iple_a,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,image_344477316iple_a(H,F_1))) ) ).

tff(fact_287_finite__imageI,axiom,
    ! [H: fun_in1219646560iple_a,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,image_323610408iple_a(H,F_1))) ) ).

tff(fact_288_finite__imageI,axiom,
    ! [H: fun_Ho864122878_a_nat,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1778392390_a_nat(H,F_1))) ) ).

tff(fact_289_finite__imageI,axiom,
    ! [H: fun_int_int,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_int_int(H,F_1))) ) ).

tff(fact_290_finite__imageI,axiom,
    ! [H: fun_Ho1026035802_a_int,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_268182434_a_int(H,F_1))) ) ).

tff(fact_291_finite__imageI,axiom,
    ! [H: fun_nat_nat,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_nat_nat(H,F_1))) ) ).

tff(fact_292_finite__imageI,axiom,
    ! [H: fun_nat_int,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(H,F_1))) ) ).

tff(fact_293_finite__Collect__disjI,axiom,
    ! [Pa: fun_Ho882550911a_bool,Q_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fdisj),Pa)),Q_1))))
    <=> ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Pa)))
        & hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Q_1))) ) ) ).

tff(fact_294_finite__Collect__disjI,axiom,
    ! [Pa: fun_int_bool,Q_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fdisj),Pa)),Q_1))))
    <=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Pa)))
        & hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,Q_1))) ) ) ).

tff(fact_295_finite__Collect__disjI,axiom,
    ! [Pa: fun_nat_bool,Q_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),Pa)),Q_1))))
    <=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Pa)))
        & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Q_1))) ) ) ).

tff(fact_296_finite__insert,axiom,
    ! [A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ).

tff(fact_297_finite__insert,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A)))
    <=> hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A)) ) ).

tff(fact_298_finite__insert,axiom,
    ! [A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ).

tff(fact_299_fold__graph_OemptyI,axiom,
    ! [F: fun_Ho483360497iple_a,Z_1: hoare_2076525193iple_a] : hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),bot_bo214064617a_bool),Z_1)) ).

tff(fact_300_fold__graph_OemptyI,axiom,
    ! [F: fun_int_fun_int_int,Z_1: int] : hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),bot_bot_fun_int_bool),Z_1)) ).

tff(fact_301_fold__graph_OemptyI,axiom,
    ! [F: fun_nat_fun_nat_nat,Z_1: nat] : hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),bot_bot_fun_nat_bool),Z_1)) ).

tff(fact_302_empty__fold__graphE,axiom,
    ! [F: fun_Ho483360497iple_a,Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),bot_bo214064617a_bool),X_1))
     => ( X_1 = Z_1 ) ) ).

tff(fact_303_empty__fold__graphE,axiom,
    ! [F: fun_int_fun_int_int,Z_1: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),bot_bot_fun_int_bool),X_1))
     => ( X_1 = Z_1 ) ) ).

tff(fact_304_empty__fold__graphE,axiom,
    ! [F: fun_nat_fun_nat_nat,Z_1: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),bot_bot_fun_nat_bool),X_1))
     => ( X_1 = Z_1 ) ) ).

tff(fact_305_fold__graph_OinsertI,axiom,
    ! [F: fun_nat_fun_nat_nat,Z_1: nat,Y_1: nat,X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
       => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Y_1))) ) ) ).

tff(fact_306_fold__graph_OinsertI,axiom,
    ! [F: fun_Ho483360497iple_a,Z_1: hoare_2076525193iple_a,Y_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),Y_1))
       => hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)),hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),Y_1))) ) ) ).

tff(fact_307_fold__graph_OinsertI,axiom,
    ! [F: fun_int_fun_int_int,Z_1: int,Y_1: int,X_1: int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
       => hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Y_1))) ) ) ).

tff(fact_308_insert__fold1SetE,axiom,
    ! [F: fun_nat_fun_nat_nat,A_13: nat,X_28: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),X_28)),X_1))
     => ~ ! [A_12: nat,A_43: fun_nat_bool] :
            ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),X_28) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),A_43) )
           => ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),A_12),A_43),X_1))
             => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A_43)) ) ) ) ).

tff(fact_309_insert__fold1SetE,axiom,
    ! [F: fun_Ho483360497iple_a,A_13: hoare_2076525193iple_a,X_28: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(finite100168025iple_a(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),X_28)),X_1))
     => ~ ! [A_12: hoare_2076525193iple_a,A_43: fun_Ho882550911a_bool] :
            ( ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),X_28) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_12),A_43) )
           => ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),A_12),A_43),X_1))
             => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_12),A_43)) ) ) ) ).

tff(fact_310_insert__fold1SetE,axiom,
    ! [F: fun_int_fun_int_int,A_13: int,X_28: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),X_28)),X_1))
     => ~ ! [A_12: int,A_43: fun_int_bool] :
            ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),X_28) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),A_43) )
           => ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),A_12),A_43),X_1))
             => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A_43)) ) ) ) ).

tff(fact_311_finite__nonempty__imp__fold1Set,axiom,
    ! [F: fun_Ho483360497iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ( A != bot_bo214064617a_bool )
       => ? [X1: hoare_2076525193iple_a] : hBOOL(hAPP_H454055633a_bool(finite100168025iple_a(F,A),X1)) ) ) ).

tff(fact_312_finite__nonempty__imp__fold1Set,axiom,
    ! [F: fun_nat_fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ? [X1: nat] : hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,A),X1)) ) ) ).

tff(fact_313_finite__nonempty__imp__fold1Set,axiom,
    ! [F: fun_int_fun_int_int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ( A != bot_bot_fun_int_bool )
       => ? [X1: int] : hBOOL(hAPP_int_bool(finite_fold1Set_int(F,A),X1)) ) ) ).

tff(fact_314_finite_Osimps,axiom,
    ! [A_13: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_13))
    <=> ( ( A_13 = bot_bot_fun_nat_bool )
        | ? [A_43: fun_nat_bool,A_12: nat] :
            ( ( A_13 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),A_43) )
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_43)) ) ) ) ).

tff(fact_315_finite_Osimps,axiom,
    ! [A_13: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_13))
    <=> ( ( A_13 = bot_bo214064617a_bool )
        | ? [A_43: fun_Ho882550911a_bool,A_12: hoare_2076525193iple_a] :
            ( ( A_13 = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_12),A_43) )
            & hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_43)) ) ) ) ).

tff(fact_316_finite_Osimps,axiom,
    ! [A_13: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_13))
    <=> ( ( A_13 = bot_bot_fun_int_bool )
        | ? [A_43: fun_int_bool,A_12: int] :
            ( ( A_13 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),A_43) )
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_43)) ) ) ) ).

tff(fact_317_finite__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
       => ( ! [X_2: nat,F_2: fun_nat_bool] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
             => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ).

tff(fact_318_finite__induct,axiom,
    ! [Pa: fun_fu2026518849l_bool,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(Pa,bot_bo214064617a_bool))
       => ( ! [X_2: hoare_2076525193iple_a,F_2: fun_Ho882550911a_bool] :
              ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_2))
             => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),F_2))
               => ( hBOOL(hAPP_f500854239l_bool(Pa,F_2))
                 => hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f500854239l_bool(Pa,F_1)) ) ) ) ).

tff(fact_319_finite__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
       => ( ! [X_2: int,F_2: fun_int_bool] :
              ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
             => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
               => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                 => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ).

tff(fact_320_pigeonhole__infinite,axiom,
    ! [F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,image_1416816325iple_a(F,A)))
       => ? [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
            & ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A))),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,hAPP_f1075886601a_bool(hAPP_f256103387a_bool(cOMBB_135702095iple_a,fequal1589671816iple_a),F)),hAPP_H1789220885iple_a(F,X_2)))))) ) ) ) ).

tff(fact_321_pigeonhole__infinite,axiom,
    ! [F: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1778392390_a_nat(F,A)))
       => ? [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
            & ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A))),hAPP_n1335981428a_bool(hAPP_f1040149703a_bool(cOMBC_240829083t_bool,hAPP_f570970635t_bool(hAPP_f2096737119t_bool(cOMBB_2017898829iple_a,fequal_nat),F)),hAPP_H174304918_a_nat(F,X_2)))))) ) ) ) ).

tff(fact_322_pigeonhole__infinite,axiom,
    ! [F: fun_int_nat,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_n1082236369t_bool(hAPP_f175561985t_bool(cOMBC_int_nat_bool,hAPP_f876579787t_bool(hAPP_f2026117279t_bool(cOMBB_1437810218ol_int,fequal_nat),F)),hAPP_int_nat(F,X_2)))))) ) ) ) ).

tff(fact_323_pigeonhole__infinite,axiom,
    ! [F: fun_Ho1026035802_a_int,A: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_268182434_a_int(F,A)))
       => ? [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
            & ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A))),hAPP_i28602320a_bool(hAPP_f1732083583a_bool(cOMBC_1771170551t_bool,hAPP_f841024195t_bool(hAPP_f1444703951t_bool(cOMBB_2132063381iple_a,fequal_int),F)),hAPP_H811578610_a_int(F,X_2)))))) ) ) ) ).

tff(fact_324_pigeonhole__infinite,axiom,
    ! [F: fun_na1240513468iple_a,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,image_344477316iple_a(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_H904886772t_bool(hAPP_f1460432071t_bool(cOMBC_2097822937a_bool,hAPP_f1324734153a_bool(hAPP_f946563611a_bool(cOMBB_315605200ol_nat,fequal1589671816iple_a),F)),hAPP_n887873492iple_a(F,X_2)))))) ) ) ) ).

tff(fact_325_pigeonhole__infinite,axiom,
    ! [F: fun_int_int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_int_int(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,fequal_int),F)),hAPP_int_int(F,X_2)))))) ) ) ) ).

tff(fact_326_pigeonhole__infinite,axiom,
    ! [F: fun_in1219646560iple_a,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,image_323610408iple_a(F,A)))
       => ? [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_H287744592t_bool(hAPP_f510295935t_bool(cOMBC_1135948669a_bool,hAPP_f1362463689a_bool(hAPP_f438022427a_bool(cOMBB_952878892ol_int,fequal1589671816iple_a),F)),hAPP_i867006584iple_a(F,X_2)))))) ) ) ) ).

tff(fact_327_pigeonhole__infinite,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_nat_nat(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,fequal_nat),F)),hAPP_nat_nat(F,X_2)))))) ) ) ) ).

tff(fact_328_pigeonhole__infinite,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(F,A)))
       => ? [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_i418383825t_bool(hAPP_f482008321t_bool(cOMBC_nat_int_bool,hAPP_f1533130627t_bool(hAPP_f879109391t_bool(cOMBB_1628441366ol_nat,fequal_int),F)),hAPP_nat_int(F,X_2)))))) ) ) ) ).

tff(fact_329_finite__imp__fold__graph,axiom,
    ! [F: fun_Ho483360497iple_a,Z_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ? [X1: hoare_2076525193iple_a] : hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),X1)) ) ).

tff(fact_330_finite__imp__fold__graph,axiom,
    ! [F: fun_nat_fun_nat_nat,Z_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ? [X1: nat] : hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),X1)) ) ).

tff(fact_331_finite__imp__fold__graph,axiom,
    ! [F: fun_int_fun_int_int,Z_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ? [X1: int] : hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),X1)) ) ).

tff(fact_332_fold1Set_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A1: fun_nat_bool,A2: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F,A1),A2))
    <=> ? [A_12: nat,A_43: fun_nat_bool,X_2: nat] :
          ( ( A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),A_43) )
          & ( A2 = X_2 )
          & hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),A_12),A_43),X_2))
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A_43)) ) ) ).

tff(fact_333_fold1Set_Osimps,axiom,
    ! [F: fun_Ho483360497iple_a,A1: fun_Ho882550911a_bool,A2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(finite100168025iple_a(F,A1),A2))
    <=> ? [A_12: hoare_2076525193iple_a,A_43: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
          ( ( A1 = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_12),A_43) )
          & ( A2 = X_2 )
          & hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),A_12),A_43),X_2))
          & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_12),A_43)) ) ) ).

tff(fact_334_fold1Set_Osimps,axiom,
    ! [F: fun_int_fun_int_int,A1: fun_int_bool,A2: int] :
      ( hBOOL(hAPP_int_bool(finite_fold1Set_int(F,A1),A2))
    <=> ? [A_12: int,A_43: fun_int_bool,X_2: int] :
          ( ( A1 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),A_43) )
          & ( A2 = X_2 )
          & hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),A_12),A_43),X_2))
          & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A_43)) ) ) ).

tff(fact_335_theI,axiom,
    ! [Pa: fun_int_bool,A_13: int] :
      ( hBOOL(hAPP_int_bool(Pa,A_13))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(Pa,X_2))
           => ( X_2 = A_13 ) )
       => hBOOL(hAPP_int_bool(Pa,hAPP_f1594865479ol_int(the_int,Pa))) ) ) ).

tff(fact_336_theI,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(Pa,A_13))
     => ( ! [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_H454055633a_bool(Pa,X_2))
           => ( X_2 = A_13 ) )
       => hBOOL(hAPP_H454055633a_bool(Pa,hAPP_f2009966983iple_a(the_Ho1456473914iple_a,Pa))) ) ) ).

tff(fact_337_theI,axiom,
    ! [Pa: fun_nat_bool,A_13: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,A_13))
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,X_2))
           => ( X_2 = A_13 ) )
       => hBOOL(hAPP_nat_bool(Pa,hAPP_f22106695ol_nat(the_nat,Pa))) ) ) ).

tff(fact_338_the1__equality,axiom,
    ! [A_13: int,Pa: fun_int_bool] :
      ( ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(Pa,X_2))
          & ! [Y_21: int] :
              ( hBOOL(hAPP_int_bool(Pa,Y_21))
             => ( Y_21 = X_2 ) ) )
     => ( hBOOL(hAPP_int_bool(Pa,A_13))
       => ( hAPP_f1594865479ol_int(the_int,Pa) = A_13 ) ) ) ).

tff(fact_339_the1__equality,axiom,
    ! [A_13: hoare_2076525193iple_a,Pa: fun_Ho882550911a_bool] :
      ( ? [X_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_H454055633a_bool(Pa,X_2))
          & ! [Y_21: hoare_2076525193iple_a] :
              ( hBOOL(hAPP_H454055633a_bool(Pa,Y_21))
             => ( Y_21 = X_2 ) ) )
     => ( hBOOL(hAPP_H454055633a_bool(Pa,A_13))
       => ( hAPP_f2009966983iple_a(the_Ho1456473914iple_a,Pa) = A_13 ) ) ) ).

tff(fact_340_the1__equality,axiom,
    ! [A_13: nat,Pa: fun_nat_bool] :
      ( ? [X_2: nat] :
          ( hBOOL(hAPP_nat_bool(Pa,X_2))
          & ! [Y_21: nat] :
              ( hBOOL(hAPP_nat_bool(Pa,Y_21))
             => ( Y_21 = X_2 ) ) )
     => ( hBOOL(hAPP_nat_bool(Pa,A_13))
       => ( hAPP_f22106695ol_nat(the_nat,Pa) = A_13 ) ) ) ).

tff(fact_341_theI_H,axiom,
    ! [Pa: fun_int_bool] :
      ( ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(Pa,X_2))
          & ! [Y_21: int] :
              ( hBOOL(hAPP_int_bool(Pa,Y_21))
             => ( Y_21 = X_2 ) ) )
     => hBOOL(hAPP_int_bool(Pa,hAPP_f1594865479ol_int(the_int,Pa))) ) ).

tff(fact_342_theI_H,axiom,
    ! [Pa: fun_Ho882550911a_bool] :
      ( ? [X_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_H454055633a_bool(Pa,X_2))
          & ! [Y_21: hoare_2076525193iple_a] :
              ( hBOOL(hAPP_H454055633a_bool(Pa,Y_21))
             => ( Y_21 = X_2 ) ) )
     => hBOOL(hAPP_H454055633a_bool(Pa,hAPP_f2009966983iple_a(the_Ho1456473914iple_a,Pa))) ) ).

tff(fact_343_theI_H,axiom,
    ! [Pa: fun_nat_bool] :
      ( ? [X_2: nat] :
          ( hBOOL(hAPP_nat_bool(Pa,X_2))
          & ! [Y_21: nat] :
              ( hBOOL(hAPP_nat_bool(Pa,Y_21))
             => ( Y_21 = X_2 ) ) )
     => hBOOL(hAPP_nat_bool(Pa,hAPP_f22106695ol_nat(the_nat,Pa))) ) ).

tff(fact_344_fold__graph_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,Z_1: nat,A1: fun_nat_bool,A2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A1),A2))
    <=> ( ( ( A1 = bot_bot_fun_nat_bool )
          & ( A2 = Z_1 ) )
        | ? [X_2: nat,A_43: fun_nat_bool,Y_21: nat] :
            ( ( A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_43) )
            & ( A2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_21) )
            & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_43))
            & hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A_43),Y_21)) ) ) ) ).

tff(fact_345_fold__graph_Osimps,axiom,
    ! [F: fun_Ho483360497iple_a,Z_1: hoare_2076525193iple_a,A1: fun_Ho882550911a_bool,A2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A1),A2))
    <=> ( ( ( A1 = bot_bo214064617a_bool )
          & ( A2 = Z_1 ) )
        | ? [X_2: hoare_2076525193iple_a,A_43: fun_Ho882550911a_bool,Y_21: hoare_2076525193iple_a] :
            ( ( A1 = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_43) )
            & ( A2 = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),Y_21) )
            & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_43))
            & hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A_43),Y_21)) ) ) ) ).

tff(fact_346_fold__graph_Osimps,axiom,
    ! [F: fun_int_fun_int_int,Z_1: int,A1: fun_int_bool,A2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A1),A2))
    <=> ( ( ( A1 = bot_bot_fun_int_bool )
          & ( A2 = Z_1 ) )
        | ? [X_2: int,A_43: fun_int_bool,Y_21: int] :
            ( ( A1 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),A_43) )
            & ( A2 = hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_21) )
            & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_43))
            & hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A_43),Y_21)) ) ) ) ).

tff(fact_347_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ) ).

tff(fact_348_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ( A != bot_bo214064617a_bool )
         => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(F_1,A)) ) ) ) ) ).

tff(fact_349_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( A != bot_bot_fun_int_bool )
         => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ) ).

tff(fact_350_finite__ne__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( ( F_1 != bot_bot_fun_nat_bool )
       => ( ! [X_2: nat] : hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)))
         => ( ! [X_2: nat,F_2: fun_nat_bool] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( ( F_2 != bot_bot_fun_nat_bool )
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_351_finite__ne__induct,axiom,
    ! [Pa: fun_fu2026518849l_bool,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( ( F_1 != bot_bo214064617a_bool )
       => ( ! [X_2: hoare_2076525193iple_a] : hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),bot_bo214064617a_bool)))
         => ( ! [X_2: hoare_2076525193iple_a,F_2: fun_Ho882550911a_bool] :
                ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_2))
               => ( ( F_2 != bot_bo214064617a_bool )
                 => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),F_2))
                   => ( hBOOL(hAPP_f500854239l_bool(Pa,F_2))
                     => hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f500854239l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_352_finite__ne__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( ( F_1 != bot_bot_fun_int_bool )
       => ( ! [X_2: int] : hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),bot_bot_fun_int_bool)))
         => ( ! [X_2: int,F_2: fun_int_bool] :
                ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
               => ( ( F_2 != bot_bot_fun_int_bool )
                 => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
                   => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                     => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_353_semilattice__big_OF__eq,axiom,
    ! [A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(big_se82156639iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f2009966983iple_a(F_1,A) = hAPP_f2009966983iple_a(finite2092228073iple_a(F),A) ) ) ) ).

tff(fact_354_semilattice__big_OF__eq,axiom,
    ! [A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(big_se275732192ig_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(F_1,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F),A) ) ) ) ).

tff(fact_355_semilattice__big_OF__eq,axiom,
    ! [A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(big_se913005884ig_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1594865479ol_int(F_1,A) = hAPP_f1594865479ol_int(finite_fold1_int(F),A) ) ) ) ).

tff(fact_356_folding__one_Oremove,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F_1,A) = X_1 ) )
            & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F_1,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ) ).

tff(fact_357_folding__one_Oremove,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)) = bot_bo214064617a_bool )
             => ( hAPP_f2009966983iple_a(F_1,A) = X_1 ) )
            & ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)) != bot_bo214064617a_bool )
             => ( hAPP_f2009966983iple_a(F_1,A) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ) ) ) ).

tff(fact_358_folding__one_Oremove,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool )
             => ( hAPP_f1594865479ol_int(F_1,A) = X_1 ) )
            & ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool )
             => ( hAPP_f1594865479ol_int(F_1,A) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ) ).

tff(fact_359_folding__one_Oinsert__remove,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = X_1 ) )
          & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).

tff(fact_360_folding__one_Oinsert__remove,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)) = bot_bo214064617a_bool )
           => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = X_1 ) )
          & ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)) != bot_bo214064617a_bool )
           => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ) ) ).

tff(fact_361_folding__one_Oinsert__remove,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool )
           => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = X_1 ) )
          & ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool )
           => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ).

tff(fact_362_fold1__insert,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( ( A != bot_bot_fun_nat_bool )
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)) ) ) ) ) ).

tff(fact_363_fold1__insert,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( ( A != bot_bot_fun_int_bool )
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),A)) ) ) ) ) ).

tff(fact_364_DiffI,axiom,
    ! [B: fun_Ho882550911a_bool,Ca: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),B))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B))) ) ) ).

tff(fact_365_DiffI,axiom,
    ! [B: fun_nat_bool,Ca: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ) ).

tff(fact_366_DiffI,axiom,
    ! [B: fun_int_bool,Ca: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ) ).

tff(fact_367_DiffE,axiom,
    ! [Ca: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)))
     => ~ ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),A))
         => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),B)) ) ) ).

tff(fact_368_DiffE,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_369_DiffE,axiom,
    ! [Ca: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ) ).

tff(fact_370_finite__Diff,axiom,
    ! [B: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B))) ) ).

tff(fact_371_finite__Diff,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

tff(fact_372_finite__Diff,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).

tff(fact_373_folding__one__idem_Oidem,axiom,
    ! [X_1: int,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hAPP_int_int(hAPP_int_fun_int_int(F,X_1),X_1) = X_1 ) ) ).

tff(fact_374_folding__one__idem_Oidem,axiom,
    ! [X_1: hoare_2076525193iple_a,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),X_1) = X_1 ) ) ).

tff(fact_375_folding__one__idem_Oidem,axiom,
    ! [X_1: nat,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),X_1) = X_1 ) ) ).

tff(fact_376_set__diff__eq,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A))),hAPP_f642208769a_bool(hAPP_f629438041a_bool(cOMBB_361863589iple_a,fNot),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),B)))) ).

tff(fact_377_set__diff__eq,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B)))) ).

tff(fact_378_set__diff__eq,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),A))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fNot),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),B)))) ).

tff(fact_379_Diff__iff,axiom,
    ! [Ca: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)))
    <=> ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),A))
        & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),B)) ) ) ).

tff(fact_380_Diff__iff,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
        & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_381_Diff__iff,axiom,
    ! [Ca: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
        & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ) ).

tff(fact_382_Diff__idemp,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ).

tff(fact_383_Diff__idemp,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)),B) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B) ).

tff(fact_384_Diff__idemp,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ).

tff(fact_385_DiffD1,axiom,
    ! [Ca: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),A)) ) ).

tff(fact_386_DiffD1,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) ) ).

tff(fact_387_DiffD1,axiom,
    ! [Ca: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A)) ) ).

tff(fact_388_DiffD2,axiom,
    ! [Ca: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)))
     => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),B)) ) ).

tff(fact_389_DiffD2,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ).

tff(fact_390_DiffD2,axiom,
    ! [Ca: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ).

tff(fact_391_empty__Diff,axiom,
    ! [A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,bot_bo214064617a_bool),A) = bot_bo214064617a_bool ).

tff(fact_392_empty__Diff,axiom,
    ! [A: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,bot_bot_fun_int_bool),A) = bot_bot_fun_int_bool ).

tff(fact_393_empty__Diff,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A) = bot_bot_fun_nat_bool ).

tff(fact_394_Diff__empty,axiom,
    ! [A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),bot_bo214064617a_bool) = A ).

tff(fact_395_Diff__empty,axiom,
    ! [A: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),bot_bot_fun_int_bool) = A ).

tff(fact_396_Diff__empty,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),bot_bot_fun_nat_bool) = A ).

tff(fact_397_Diff__cancel,axiom,
    ! [A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),A) = bot_bo214064617a_bool ).

tff(fact_398_Diff__cancel,axiom,
    ! [A: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),A) = bot_bot_fun_int_bool ).

tff(fact_399_Diff__cancel,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),A) = bot_bot_fun_nat_bool ).

tff(fact_400_finite__Diff2,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)))
      <=> hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A)) ) ) ).

tff(fact_401_finite__Diff2,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
      <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_402_finite__Diff2,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
      <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

tff(fact_403_insert__Diff__if,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) ) ) ) ).

tff(fact_404_insert__Diff__if,axiom,
    ! [A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] :
      ( ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B))
       => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)),B) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B))
       => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)),B) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)) ) ) ) ).

tff(fact_405_insert__Diff__if,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) ) ) ) ).

tff(fact_406_insert__Diff1,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) ) ).

tff(fact_407_insert__Diff1,axiom,
    ! [A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B))
     => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)),B) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B) ) ) ).

tff(fact_408_insert__Diff1,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
     => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ) ) ).

tff(fact_409_fold__graph__permute__diff,axiom,
    ! [A_13: nat,Ba: nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),Ba),A),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A))
         => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),A_13),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))),X_1)) ) ) ) ).

tff(fact_410_fold__graph__permute__diff,axiom,
    ! [A_13: int,Ba: int,A: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),Ba),A),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),A))
         => hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),A_13),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))),X_1)) ) ) ) ).

tff(fact_411_Diff__insert,axiom,
    ! [A: fun_nat_bool,A_13: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)) ).

tff(fact_412_Diff__insert,axiom,
    ! [A: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)) ).

tff(fact_413_Diff__insert,axiom,
    ! [A: fun_int_bool,A_13: int,B: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)) ).

tff(fact_414_Diff__insert2,axiom,
    ! [A: fun_nat_bool,A_13: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))),B) ).

tff(fact_415_Diff__insert2,axiom,
    ! [A: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))),B) ).

tff(fact_416_Diff__insert2,axiom,
    ! [A: fun_int_bool,A_13: int,B: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))),B) ).

tff(fact_417_insert__Diff__single,axiom,
    ! [A_13: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A) ).

tff(fact_418_insert__Diff__single,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A) ).

tff(fact_419_insert__Diff__single,axiom,
    ! [A_13: int,A: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A) ).

tff(fact_420_Diff__insert__absorb,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = A ) ) ).

tff(fact_421_Diff__insert__absorb,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)) = A ) ) ).

tff(fact_422_Diff__insert__absorb,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = A ) ) ).

tff(fact_423_insert__Diff,axiom,
    ! [A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = A ) ) ).

tff(fact_424_insert__Diff,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
     => ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))) = A ) ) ).

tff(fact_425_insert__Diff,axiom,
    ! [A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = A ) ) ).

tff(fact_426_finite__Diff__insert,axiom,
    ! [A: fun_nat_bool,A_13: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B))))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

tff(fact_427_finite__Diff__insert,axiom,
    ! [A: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B))))
    <=> hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B))) ) ).

tff(fact_428_finite__Diff__insert,axiom,
    ! [A: fun_int_bool,A_13: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B))))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).

tff(fact_429_fold__graph__insert__swap,axiom,
    ! [Z_1: nat,Ba: nat,A: fun_nat_bool,Y_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),Ba),A),Y_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A))
       => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(times_times_nat),Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),Y_1))) ) ) ).

tff(fact_430_fold__graph__insert__swap,axiom,
    ! [Z_1: int,Ba: int,A: fun_int_bool,Y_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),Ba),A),Y_1))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),A))
       => hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(times_times_int),Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Y_1))) ) ) ).

tff(fact_431_folding__one__idem_Oin__idem,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(F_1,A)) = hAPP_f2009966983iple_a(F_1,A) ) ) ) ) ).

tff(fact_432_folding__one__idem_Oin__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) = hAPP_f22106695ol_nat(F_1,A) ) ) ) ) ).

tff(fact_433_folding__one__idem_Oin__idem,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) = hAPP_f1594865479ol_int(F_1,A) ) ) ) ) ).

tff(fact_434_fold1__in,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ! [X_2: nat,Y_21: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_21)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_21),bot_bot_fun_nat_bool))))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)),A)) ) ) ) ).

tff(fact_435_fold1__in,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ( A != bot_bot_fun_int_bool )
       => ( ! [X_2: int,Y_21: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_21)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_21),bot_bot_fun_int_bool))))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),A)),A)) ) ) ) ).

tff(fact_436_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_Ho882550911a_bool,H: fun_Ho1518252029iple_a,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( ! [X_2: hoare_2076525193iple_a,Y_21: hoare_2076525193iple_a] : hAPP_H1789220885iple_a(H,hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),Y_21)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,hAPP_H1789220885iple_a(H,X_2)),hAPP_H1789220885iple_a(H,Y_21))
       => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,N_3))
         => ( ( N_3 != bot_bo214064617a_bool )
           => ( hAPP_H1789220885iple_a(H,hAPP_f2009966983iple_a(F_1,N_3)) = hAPP_f2009966983iple_a(F_1,image_1416816325iple_a(H,N_3)) ) ) ) ) ) ).

tff(fact_437_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_int_bool,H: fun_int_int,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( ! [X_2: int,Y_21: int] : hAPP_int_int(H,hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_21)) = hAPP_int_int(hAPP_int_fun_int_int(F,hAPP_int_int(H,X_2)),hAPP_int_int(H,Y_21))
       => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,N_3))
         => ( ( N_3 != bot_bot_fun_int_bool )
           => ( hAPP_int_int(H,hAPP_f1594865479ol_int(F_1,N_3)) = hAPP_f1594865479ol_int(F_1,image_int_int(H,N_3)) ) ) ) ) ) ).

tff(fact_438_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_nat_bool,H: fun_nat_nat,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( ! [X_2: nat,Y_21: nat] : hAPP_nat_nat(H,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_nat_nat(H,X_2)),hAPP_nat_nat(H,Y_21))
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
         => ( ( N_3 != bot_bot_fun_nat_bool )
           => ( hAPP_nat_nat(H,hAPP_f22106695ol_nat(F_1,N_3)) = hAPP_f22106695ol_nat(F_1,image_nat_nat(H,N_3)) ) ) ) ) ) ).

tff(fact_439_finite__empty__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,A))
       => ( ! [A_12: nat,A_43: fun_nat_bool] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_43))
             => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A_43))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,A_43))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_43),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),bot_bot_fun_nat_bool)))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) ) ) ) ).

tff(fact_440_finite__empty__induct,axiom,
    ! [Pa: fun_fu2026518849l_bool,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(Pa,A))
       => ( ! [A_12: hoare_2076525193iple_a,A_43: fun_Ho882550911a_bool] :
              ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_43))
             => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_12),A_43))
               => ( hBOOL(hAPP_f500854239l_bool(Pa,A_43))
                 => hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_43),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_12),bot_bo214064617a_bool)))) ) ) )
         => hBOOL(hAPP_f500854239l_bool(Pa,bot_bo214064617a_bool)) ) ) ) ).

tff(fact_441_finite__empty__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(Pa,A))
       => ( ! [A_12: int,A_43: fun_int_bool] :
              ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_43))
             => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A_43))
               => ( hBOOL(hAPP_f448129468l_bool(Pa,A_43))
                 => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_43),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),bot_bot_fun_int_bool)))) ) ) )
         => hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool)) ) ) ) ).

tff(fact_442_minus__apply,axiom,
    ! [A: fun_int_bool,B: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B),X_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_int_bool(A,X_1)),hAPP_int_bool(B,X_1))) ) ).

tff(fact_443_minus__apply,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B),X_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_H454055633a_bool(A,X_1)),hAPP_H454055633a_bool(B,X_1))) ) ).

tff(fact_444_minus__apply,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B),X_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_nat_bool(A,X_1)),hAPP_nat_bool(B,X_1))) ) ).

tff(fact_445_fun__diff__def,axiom,
    ! [A: fun_int_bool,B: fun_int_bool,X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_int_bool(A,X_2)),hAPP_int_bool(B,X_2))) ) ).

tff(fact_446_fun__diff__def,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_H454055633a_bool(A,X_2)),hAPP_H454055633a_bool(B,X_2))) ) ).

tff(fact_447_fun__diff__def,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_nat_bool(A,X_2)),hAPP_nat_bool(B,X_2))) ) ).

tff(fact_448_comp__fun__idem__remove,axiom,
    hBOOL(finite1956789438t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,hAPP_f1311642927t_bool(cOMBC_646315179t_bool,minus_2067140911t_bool)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)))) ).

tff(fact_449_comp__fun__idem__remove,axiom,
    hBOOL(finite1306023740a_bool(hAPP_f102426163a_bool(hAPP_f300022271a_bool(cOMBB_2036667361iple_a,hAPP_f86610733a_bool(cOMBC_2005334440a_bool,minus_1649626414a_bool)),hAPP_f219434403a_bool(hAPP_f1824299331a_bool(cOMBC_175592824a_bool,insert1582918915iple_a),bot_bo214064617a_bool)))) ).

tff(fact_450_comp__fun__idem__remove,axiom,
    hBOOL(finite58652534t_bool(hAPP_f1468280982t_bool(hAPP_f1596157055t_bool(cOMBB_765314529ol_int,hAPP_f194616807t_bool(cOMBC_922884543t_bool,minus_1449998731t_bool)),hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,insert_int),bot_bot_fun_int_bool)))) ).

tff(fact_451_comp__fun__idem_Ofun__left__idem,axiom,
    ! [X_1: int,Z_1: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite58652534t_bool(F))
     => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1) ) ) ).

tff(fact_452_comp__fun__idem_Ofun__left__idem,axiom,
    ! [X_1: hoare_2076525193iple_a,Z_1: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1306023740a_bool(F))
     => ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),Z_1)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),Z_1) ) ) ).

tff(fact_453_comp__fun__idem_Ofun__left__idem,axiom,
    ! [X_1: nat,Z_1: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1956789438t_bool(F))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1) ) ) ).

tff(fact_454_comp__fun__idem__insert,axiom,
    hBOOL(finite1956789438t_bool(insert_nat)) ).

tff(fact_455_comp__fun__idem__insert,axiom,
    hBOOL(finite1306023740a_bool(insert1582918915iple_a)) ).

tff(fact_456_comp__fun__idem__insert,axiom,
    hBOOL(finite58652534t_bool(insert_int)) ).

tff(fact_457_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
    ! [A_13: nat,Z_1: nat,A: fun_nat_bool,Y_1: nat,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
         => ? [Y_22: nat] :
              ( ( Y_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,A_13),Y_22) )
              & hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))),Y_22)) ) ) ) ) ).

tff(fact_458_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
    ! [A_13: hoare_2076525193iple_a,Z_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,Y_1: hoare_2076525193iple_a,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),Y_1))
       => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
         => ? [Y_22: hoare_2076525193iple_a] :
              ( ( Y_1 = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,A_13),Y_22) )
              & hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))),Y_22)) ) ) ) ) ).

tff(fact_459_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
    ! [A_13: int,Z_1: int,A: fun_int_bool,Y_1: int,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
         => ? [Y_22: int] :
              ( ( Y_1 = hAPP_int_int(hAPP_int_fun_int_int(F,A_13),Y_22) )
              & hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))),Y_22)) ) ) ) ) ).

tff(fact_460_setsum__diff1,axiom,
    ! [F: fun_nat_int,A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
         => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A)),hAPP_nat_int(F,A_13)) ) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
         => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A) ) ) ) ) ).

tff(fact_461_setsum__diff1,axiom,
    ! [F: fun_Ho1026035802_a_int,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
         => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),A)),hAPP_H811578610_a_int(F,A_13)) ) )
        & ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
         => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))) = hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),A) ) ) ) ) ).

tff(fact_462_setsum__diff1,axiom,
    ! [F: fun_int_int,A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
         => ( hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A)),hAPP_int_int(F,A_13)) ) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
         => ( hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A) ) ) ) ) ).

tff(fact_463_setsum__diff1__ring,axiom,
    ! [F: fun_nat_int,A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A)),hAPP_nat_int(F,A_13)) ) ) ) ).

tff(fact_464_setsum__diff1__ring,axiom,
    ! [F: fun_Ho1026035802_a_int,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),A)),hAPP_H811578610_a_int(F,A_13)) ) ) ) ).

tff(fact_465_setsum__diff1__ring,axiom,
    ! [F: fun_int_int,A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => ( hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A)),hAPP_int_int(F,A_13)) ) ) ) ).

tff(fact_466_setprod_Oinsert__remove,axiom,
    ! [G: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).

tff(fact_467_setprod_Oinsert__remove,axiom,
    ! [G: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f190641160ol_nat(big_co1398215719_a_nat(G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H174304918_a_nat(G,X_1)),hAPP_f190641160ol_nat(big_co1398215719_a_nat(G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ).

tff(fact_468_setprod_Oinsert__remove,axiom,
    ! [G: fun_int_nat,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f957591787ol_nat(big_co911457418nt_nat(G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(big_co911457418nt_nat(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).

tff(fact_469_setprod_Oinsert__remove,axiom,
    ! [G: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).

tff(fact_470_setprod_Oinsert__remove,axiom,
    ! [G: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f827914852ol_int(big_co2035489411_a_int(G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H811578610_a_int(G,X_1)),hAPP_f827914852ol_int(big_co2035489411_a_int(G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ).

tff(fact_471_setprod_Oinsert__remove,axiom,
    ! [G: fun_int_int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f1594865479ol_int(big_co1548731110nt_int(G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(big_co1548731110nt_int(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).

tff(fact_472_setprod_Oremove,axiom,
    ! [G: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(big_co1705425894at_nat(G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_473_setprod_Oremove,axiom,
    ! [G: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f190641160ol_nat(big_co1398215719_a_nat(G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H174304918_a_nat(G,X_1)),hAPP_f190641160ol_nat(big_co1398215719_a_nat(G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_474_setprod_Oremove,axiom,
    ! [G: fun_int_nat,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f957591787ol_nat(big_co911457418nt_nat(G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(big_co911457418nt_nat(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_475_setprod_Oremove,axiom,
    ! [G: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f659380387ol_int(big_co195215938at_int(G),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_476_setprod_Oremove,axiom,
    ! [G: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f827914852ol_int(big_co2035489411_a_int(G),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H811578610_a_int(G,X_1)),hAPP_f827914852ol_int(big_co2035489411_a_int(G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_477_setprod_Oremove,axiom,
    ! [G: fun_int_int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f1594865479ol_int(big_co1548731110nt_int(G),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(big_co1548731110nt_int(G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_478_minus__fold__remove,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),A) = hAPP_f800510211t_bool(finite1578363458t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,hAPP_f1311642927t_bool(cOMBC_646315179t_bool,minus_2067140911t_bool)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)),B),A) ) ) ).

tff(fact_479_minus__fold__remove,axiom,
    ! [B: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,B),A) = hAPP_f642208769a_bool(finite718386368a_bool(hAPP_f102426163a_bool(hAPP_f300022271a_bool(cOMBB_2036667361iple_a,hAPP_f86610733a_bool(cOMBC_2005334440a_bool,minus_1649626414a_bool)),hAPP_f219434403a_bool(hAPP_f1824299331a_bool(cOMBC_175592824a_bool,insert1582918915iple_a),bot_bo214064617a_bool)),B),A) ) ) ).

tff(fact_480_minus__fold__remove,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B),A) = hAPP_f1805168059t_bool(finite1827710202t_bool(hAPP_f1468280982t_bool(hAPP_f1596157055t_bool(cOMBB_765314529ol_int,hAPP_f194616807t_bool(cOMBC_922884543t_bool,minus_1449998731t_bool)),hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,insert_int),bot_bot_fun_int_bool)),B),A) ) ) ).

tff(fact_481_comp__fun__commute_Ofold__fun__comm,axiom,
    ! [X_1: hoare_2076525193iple_a,Z_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),A)) = hAPP_f2009966983iple_a(finite2089344904iple_a(F,hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_482_comp__fun__commute_Ofold__fun__comm,axiom,
    ! [X_1: hoare_2076525193iple_a,Z_1: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1617795063a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),A)) = hAPP_f642208769a_bool(finite718386368a_bool(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_483_comp__fun__commute_Ofold__fun__comm,axiom,
    ! [X_1: nat,Z_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_484_comp__fun__commute_Ofold__fun__comm,axiom,
    ! [X_1: nat,Z_1: fun_nat_bool,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1071749497t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A)) = hAPP_f800510211t_bool(finite1578363458t_bool(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_485_comp__fun__commute_Ofold__fun__comm,axiom,
    ! [X_1: int,Z_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_486_comp__fun__commute_Ofold__fun__comm,axiom,
    ! [X_1: int,Z_1: fun_int_bool,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite1321096241t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A)) = hAPP_f1805168059t_bool(finite1827710202t_bool(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_487_comp__fun__commute_Ofun__left__comm,axiom,
    ! [X_1: int,Y_1: int,Z_1: int,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_int_int(hAPP_int_fun_int_int(F,Y_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(F,Y_1),hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)) ) ) ).

tff(fact_488_comp__fun__commute_Ofun__left__comm,axiom,
    ! [X_1: nat,Y_1: nat,Z_1: nat,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,Y_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,Y_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)) ) ) ).

tff(fact_489_setsum__commute,axiom,
    ! [F: fun_nat_fun_int_nat,B: fun_int_bool,A: fun_nat_bool] : hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),F)),B)),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),hAPP_f446977493at_nat(cOMBC_nat_int_nat,F))),A)),B) ).

tff(fact_490_setsum__commute,axiom,
    ! [F: fun_na841142961_a_nat,B: fun_Ho882550911a_bool,A: fun_nat_bool] : hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f479558075at_nat(hAPP_f1173305809at_nat(cOMBC_129115664ol_nat,hAPP_f1356625915ol_nat(hAPP_f2033836159ol_nat(cOMBB_1944586014at_nat,big_co922834230_a_nat),F)),B)),A) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f1047071419_a_nat(hAPP_f1217804815_a_nat(cOMBC_1664565648ol_nat,hAPP_f1296292793ol_nat(hAPP_f153943611ol_nat(cOMBB_636937375iple_a,big_co387207925at_nat),hAPP_f982543381at_nat(cOMBC_23578254_a_nat,F))),A)),B) ).

tff(fact_491_setsum__commute,axiom,
    ! [F: fun_int_fun_nat_nat,B: fun_nat_bool,A: fun_int_bool] : hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),F)),B)),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),hAPP_f879494613nt_nat(cOMBC_int_nat_nat,F))),A)),B) ).

tff(fact_492_setsum__commute,axiom,
    ! [F: fun_Ho1036508337at_nat,B: fun_nat_bool,A: fun_Ho882550911a_bool] : hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f1047071419_a_nat(hAPP_f1217804815_a_nat(cOMBC_1664565648ol_nat,hAPP_f1296292793ol_nat(hAPP_f153943611ol_nat(cOMBB_636937375iple_a,big_co387207925at_nat),F)),B)),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f479558075at_nat(hAPP_f1173305809at_nat(cOMBC_129115664ol_nat,hAPP_f1356625915ol_nat(hAPP_f2033836159ol_nat(cOMBB_1944586014at_nat,big_co922834230_a_nat),hAPP_f668391445_a_nat(cOMBC_1942990924at_nat,F))),A)),B) ).

tff(fact_493_comp__fun__commute_Ofold__equality,axiom,
    ! [Z_1: int,A: fun_int_bool,Y_1: int,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
       => ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A) = Y_1 ) ) ) ).

tff(fact_494_comp__fun__commute_Ofold__equality,axiom,
    ! [Z_1: nat,A: fun_nat_bool,Y_1: nat,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
       => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A) = Y_1 ) ) ) ).

tff(fact_495_comp__fun__commute_Ofold__equality,axiom,
    ! [Z_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,Y_1: hoare_2076525193iple_a,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),Y_1))
       => ( hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),A) = Y_1 ) ) ) ).

tff(fact_496_comp__fun__commute_Ofold__equality,axiom,
    ! [Z_1: fun_int_bool,A: fun_int_bool,Y_1: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite1321096241t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(hAPP_f1634429330l_bool(finite1008073724t_bool(F),Z_1),A),Y_1))
       => ( hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A) = Y_1 ) ) ) ).

tff(fact_497_comp__fun__commute_Ofold__equality,axiom,
    ! [Z_1: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1617795063a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(hAPP_f845487733l_bool(finite655745986a_bool(F),Z_1),A),Y_1))
       => ( hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),A) = Y_1 ) ) ) ).

tff(fact_498_comp__fun__commute_Ofold__equality,axiom,
    ! [Z_1: fun_nat_bool,A: fun_nat_bool,Y_1: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1071749497t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f1266913334l_bool(finite758726980t_bool(F),Z_1),A),Y_1))
       => ( hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A) = Y_1 ) ) ) ).

tff(fact_499_fold__empty,axiom,
    ! [F: fun_Ho483360497iple_a,Z_1: hoare_2076525193iple_a] : hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),bot_bo214064617a_bool) = Z_1 ).

tff(fact_500_fold__empty,axiom,
    ! [F: fun_Ho112928087a_bool,Z_1: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),bot_bo214064617a_bool) = Z_1 ).

tff(fact_501_fold__empty,axiom,
    ! [F: fun_int_fun_int_int,Z_1: int] : hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),bot_bot_fun_int_bool) = Z_1 ).

tff(fact_502_fold__empty,axiom,
    ! [F: fun_in719570932t_bool,Z_1: fun_int_bool] : hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),bot_bot_fun_int_bool) = Z_1 ).

tff(fact_503_fold__empty,axiom,
    ! [F: fun_nat_fun_nat_nat,Z_1: nat] : hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),bot_bot_fun_nat_bool) = Z_1 ).

tff(fact_504_fold__empty,axiom,
    ! [F: fun_na35521688t_bool,Z_1: fun_nat_bool] : hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),bot_bot_fun_nat_bool) = Z_1 ).

tff(fact_505_comp__fun__commute_Ofold__insert2,axiom,
    ! [Z_1: nat,X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A) ) ) ) ) ).

tff(fact_506_comp__fun__commute_Ofold__insert2,axiom,
    ! [Z_1: fun_nat_bool,X_1: nat,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1071749497t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(finite1578363458t_bool(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)),A) ) ) ) ) ).

tff(fact_507_comp__fun__commute_Ofold__insert2,axiom,
    ! [Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f2009966983iple_a(finite2089344904iple_a(F,hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),Z_1)),A) ) ) ) ) ).

tff(fact_508_comp__fun__commute_Ofold__insert2,axiom,
    ! [Z_1: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1617795063a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f642208769a_bool(finite718386368a_bool(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),Z_1)),A) ) ) ) ) ).

tff(fact_509_comp__fun__commute_Ofold__insert2,axiom,
    ! [Z_1: int,X_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A) ) ) ) ) ).

tff(fact_510_comp__fun__commute_Ofold__insert2,axiom,
    ! [Z_1: fun_int_bool,X_1: int,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite1321096241t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(finite1827710202t_bool(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)),A) ) ) ) ) ).

tff(fact_511_comp__fun__commute_Ofold__insert,axiom,
    ! [Z_1: nat,X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A)) ) ) ) ) ).

tff(fact_512_comp__fun__commute_Ofold__insert,axiom,
    ! [Z_1: fun_nat_bool,X_1: nat,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1071749497t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A)) ) ) ) ) ).

tff(fact_513_comp__fun__commute_Ofold__insert,axiom,
    ! [Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),A)) ) ) ) ) ).

tff(fact_514_comp__fun__commute_Ofold__insert,axiom,
    ! [Z_1: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1617795063a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),A)) ) ) ) ) ).

tff(fact_515_comp__fun__commute_Ofold__insert,axiom,
    ! [Z_1: int,X_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A)) ) ) ) ) ).

tff(fact_516_comp__fun__commute_Ofold__insert,axiom,
    ! [Z_1: fun_int_bool,X_1: int,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite1321096241t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A)) ) ) ) ) ).

tff(fact_517_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),A))) ) ) ).

tff(fact_518_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z_1: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1617795063a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(hAPP_f845487733l_bool(finite655745986a_bool(F),Z_1),A),hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),A))) ) ) ).

tff(fact_519_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A))) ) ) ).

tff(fact_520_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z_1: fun_nat_bool,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1071749497t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f1266913334l_bool(finite758726980t_bool(F),Z_1),A),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A))) ) ) ).

tff(fact_521_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A))) ) ) ).

tff(fact_522_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z_1: fun_int_bool,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite1321096241t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(hAPP_f1634429330l_bool(finite1008073724t_bool(F),Z_1),A),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A))) ) ) ).

tff(fact_523_setsum__product,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool,G: fun_int_nat,B: fun_int_bool] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),B)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),hAPP_f237327688nt_nat(hAPP_f1633513941nt_nat(cOMBC_1218031117nt_nat,hAPP_f2080483477nt_nat(hAPP_f1701814485nt_nat(cOMBB_737513486at_nat,cOMBB_nat_nat_int),hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,times_times_nat),F))),G))),B)),A) ).

tff(fact_524_setsum__product,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool,G: fun_Ho864122878_a_nat,B: fun_Ho882550911a_bool] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),B)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f479558075at_nat(hAPP_f1173305809at_nat(cOMBC_129115664ol_nat,hAPP_f1356625915ol_nat(hAPP_f2033836159ol_nat(cOMBB_1944586014at_nat,big_co922834230_a_nat),hAPP_f536288520_a_nat(hAPP_f928984021_a_nat(cOMBC_130242509_a_nat,hAPP_f935848405_a_nat(hAPP_f450378965_a_nat(cOMBB_1353751246at_nat,cOMBB_1877004557iple_a),hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,times_times_nat),F))),G))),B)),A) ).

tff(fact_525_setsum__product,axiom,
    ! [F: fun_int_nat,A: fun_int_bool,G: fun_nat_nat,B: fun_nat_bool] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B)) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),hAPP_f1169617132at_nat(hAPP_f701299925at_nat(cOMBC_1294079849at_nat,hAPP_f1033905301at_nat(hAPP_f1315855317at_nat(cOMBB_891709290at_int,cOMBB_nat_nat_nat),hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,times_times_nat),F))),G))),B)),A) ).

tff(fact_526_setsum__product,axiom,
    ! [F: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool,G: fun_nat_nat,B: fun_nat_bool] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B)) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f1047071419_a_nat(hAPP_f1217804815_a_nat(cOMBC_1664565648ol_nat,hAPP_f1296292793ol_nat(hAPP_f153943611ol_nat(cOMBB_636937375iple_a,big_co387207925at_nat),hAPP_f1016910345at_nat(hAPP_f1112019605at_nat(cOMBC_340603916at_nat,hAPP_f564893909at_nat(hAPP_f250842005at_nat(cOMBB_151870093iple_a,cOMBB_nat_nat_nat),hAPP_f731653896at_nat(hAPP_f648261589at_nat(cOMBB_1416330906iple_a,times_times_nat),F))),G))),B)),A) ).

tff(fact_527_setsum__left__distrib,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool,R_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),R_2) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,times_times_nat),F)),R_2)),A) ).

tff(fact_528_setsum__left__distrib,axiom,
    ! [F: fun_int_nat,A: fun_int_bool,R_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),R_2) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_nat_fun_int_nat(hAPP_f879494613nt_nat(cOMBC_int_nat_nat,hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,times_times_nat),F)),R_2)),A) ).

tff(fact_529_setsum__left__distrib,axiom,
    ! [F: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool,R_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A)),R_2) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_n1236213193_a_nat(hAPP_f668391445_a_nat(cOMBC_1942990924at_nat,hAPP_f731653896at_nat(hAPP_f648261589at_nat(cOMBB_1416330906iple_a,times_times_nat),F)),R_2)),A) ).

tff(fact_530_setsum__right__distrib,axiom,
    ! [R_2: nat,F: fun_nat_nat,A: fun_nat_bool] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_2),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,hAPP_nat_fun_nat_nat(times_times_nat,R_2)),F)),A) ).

tff(fact_531_setsum__right__distrib,axiom,
    ! [R_2: nat,F: fun_int_nat,A: fun_int_bool] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_2),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2132704789nt_nat(hAPP_f901215189nt_nat(cOMBB_nat_nat_int,hAPP_nat_fun_nat_nat(times_times_nat,R_2)),F)),A) ).

tff(fact_532_setsum__right__distrib,axiom,
    ! [R_2: nat,F: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_2),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A)) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f2075363285_a_nat(hAPP_f375421205_a_nat(cOMBB_1877004557iple_a,hAPP_nat_fun_nat_nat(times_times_nat,R_2)),F)),A) ).

tff(fact_533_comp__fun__commute,axiom,
    hBOOL(finite2130160977at_nat(times_times_nat)) ).

tff(fact_534_comp__fun__commute,axiom,
    hBOOL(finite1973466193nt_int(times_times_int)) ).

tff(fact_535_comp__fun__commute_Ofold__graph__determ,axiom,
    ! [Y_1: int,Z_1: int,A: fun_int_bool,X_1: int,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),X_1))
       => ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_1))
         => ( Y_1 = X_1 ) ) ) ) ).

tff(fact_536_comp__fun__commute_Ofold__graph__determ,axiom,
    ! [Y_1: nat,Z_1: nat,A: fun_nat_bool,X_1: nat,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),X_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_1))
         => ( Y_1 = X_1 ) ) ) ) ).

tff(fact_537_comp__fun__commute_Ofold__graph__determ,axiom,
    ! [Y_1: hoare_2076525193iple_a,Z_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),X_1))
       => ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),Y_1))
         => ( Y_1 = X_1 ) ) ) ) ).

tff(fact_538_setprod_Odistrib,axiom,
    ! [G: fun_Ho864122878_a_nat,H: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f190641160ol_nat(big_co1398215719_a_nat(hAPP_f2075363285_a_nat(hAPP_f1659339553_a_nat(cOMBS_885136444at_nat,hAPP_f731653896at_nat(hAPP_f648261589at_nat(cOMBB_1416330906iple_a,times_times_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_f190641160ol_nat(big_co1398215719_a_nat(G),A)),hAPP_f190641160ol_nat(big_co1398215719_a_nat(H),A)) ) ) ).

tff(fact_539_setprod_Odistrib,axiom,
    ! [G: fun_Ho1026035802_a_int,H: fun_Ho1026035802_a_int,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f827914852ol_int(big_co2035489411_a_int(hAPP_f1382137045_a_int(hAPP_f1031139617_a_int(cOMBS_728441660nt_int,hAPP_f534508204nt_int(hAPP_f1566620885nt_int(cOMBB_364359670iple_a,times_times_int),G)),H)),A) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_f827914852ol_int(big_co2035489411_a_int(G),A)),hAPP_f827914852ol_int(big_co2035489411_a_int(H),A)) ) ) ).

tff(fact_540_comp__fun__commute_Ofold__rec,axiom,
    ! [Z_1: nat,X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ).

tff(fact_541_comp__fun__commute_Ofold__rec,axiom,
    ! [Z_1: fun_nat_bool,X_1: nat,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1071749497t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ).

tff(fact_542_comp__fun__commute_Ofold__rec,axiom,
    ! [Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),A) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ) ).

tff(fact_543_comp__fun__commute_Ofold__rec,axiom,
    ! [Z_1: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1617795063a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),A) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ) ).

tff(fact_544_comp__fun__commute_Ofold__rec,axiom,
    ! [Z_1: int,X_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ).

tff(fact_545_comp__fun__commute_Ofold__rec,axiom,
    ! [Z_1: fun_int_bool,X_1: int,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite1321096241t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ).

tff(fact_546_comp__fun__commute_Ofold__insert__remove,axiom,
    ! [Z_1: nat,X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_547_comp__fun__commute_Ofold__insert__remove,axiom,
    ! [Z_1: fun_nat_bool,X_1: nat,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1071749497t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_548_comp__fun__commute_Ofold__insert__remove,axiom,
    ! [Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_549_comp__fun__commute_Ofold__insert__remove,axiom,
    ! [Z_1: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1617795063a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_550_comp__fun__commute_Ofold__insert__remove,axiom,
    ! [Z_1: int,X_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_551_comp__fun__commute_Ofold__insert__remove,axiom,
    ! [Z_1: fun_int_bool,X_1: int,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite1321096241t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_552_setprod__insert,axiom,
    ! [F: fun_nat_nat,A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f22106695ol_nat(big_co1705425894at_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(F,A_13)),hAPP_f22106695ol_nat(big_co1705425894at_nat(F),A)) ) ) ) ).

tff(fact_553_setprod__insert,axiom,
    ! [F: fun_Ho864122878_a_nat,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => ( hAPP_f190641160ol_nat(big_co1398215719_a_nat(F),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H174304918_a_nat(F,A_13)),hAPP_f190641160ol_nat(big_co1398215719_a_nat(F),A)) ) ) ) ).

tff(fact_554_setprod__insert,axiom,
    ! [F: fun_nat_int,A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f659380387ol_int(big_co195215938at_int(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(F,A_13)),hAPP_f659380387ol_int(big_co195215938at_int(F),A)) ) ) ) ).

tff(fact_555_setprod__insert,axiom,
    ! [F: fun_Ho1026035802_a_int,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => ( hAPP_f827914852ol_int(big_co2035489411_a_int(F),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H811578610_a_int(F,A_13)),hAPP_f827914852ol_int(big_co2035489411_a_int(F),A)) ) ) ) ).

tff(fact_556_setprod_Oinsert,axiom,
    ! [G: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(big_co1705425894at_nat(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(big_co1705425894at_nat(G),A)) ) ) ) ).

tff(fact_557_setprod_Oinsert,axiom,
    ! [G: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f190641160ol_nat(big_co1398215719_a_nat(G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_H174304918_a_nat(G,X_1)),hAPP_f190641160ol_nat(big_co1398215719_a_nat(G),A)) ) ) ) ).

tff(fact_558_setprod_Oinsert,axiom,
    ! [G: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f659380387ol_int(big_co195215938at_int(G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(big_co195215938at_int(G),A)) ) ) ) ).

tff(fact_559_setprod_Oinsert,axiom,
    ! [G: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f827914852ol_int(big_co2035489411_a_int(G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_H811578610_a_int(G,X_1)),hAPP_f827914852ol_int(big_co2035489411_a_int(G),A)) ) ) ) ).

tff(fact_560_setsum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A_13: nat,A: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_nat_nat(F,A_13)) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) ) ) ) ).

tff(fact_561_setsum__diff1__nat,axiom,
    ! [F: fun_Ho864122878_a_nat,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A)),hAPP_H174304918_a_nat(F,A_13)) ) )
      & ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool))) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A) ) ) ) ).

tff(fact_562_setsum__diff1__nat,axiom,
    ! [F: fun_int_nat,A_13: int,A: fun_int_bool] :
      ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_int_nat(F,A_13)) ) )
      & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool))) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) ) ) ) ).

tff(fact_563_fold__def,axiom,
    ! [F: fun_int_fun_int_int,Z_1: int,A: fun_int_bool] : hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A) = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A)) ).

tff(fact_564_fold__def,axiom,
    ! [F: fun_nat_fun_nat_nat,Z_1: nat,A: fun_nat_bool] : hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A) = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A)) ).

tff(fact_565_fold__def,axiom,
    ! [F: fun_Ho483360497iple_a,Z_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] : hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),A) = hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A)) ).

tff(fact_566_fold__def,axiom,
    ! [F: fun_in719570932t_bool,Z_1: fun_int_bool,A: fun_int_bool] : hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A) = hAPP_f631471077t_bool(the_fun_int_bool,hAPP_f284875647l_bool(hAPP_f1634429330l_bool(finite1008073724t_bool(F),Z_1),A)) ).

tff(fact_567_fold__def,axiom,
    ! [F: fun_Ho112928087a_bool,Z_1: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),A) = hAPP_f1758315691a_bool(the_fu972267150a_bool,hAPP_f256727871l_bool(hAPP_f845487733l_bool(finite655745986a_bool(F),Z_1),A)) ).

tff(fact_568_fold__def,axiom,
    ! [F: fun_na35521688t_bool,Z_1: fun_nat_bool,A: fun_nat_bool] : hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A) = hAPP_f582319405t_bool(the_fun_nat_bool,hAPP_f103356543l_bool(hAPP_f1266913334l_bool(finite758726980t_bool(F),Z_1),A)) ).

tff(fact_569_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z_1: nat,X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite1860950092at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_570_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z_1: fun_nat_bool,X_1: nat,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1956789438t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(finite1578363458t_bool(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_571_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite1532229644iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f2009966983iple_a(finite2089344904iple_a(F,hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_572_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z_1: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1306023740a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f642208769a_bool(finite718386368a_bool(F,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_573_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z_1: int,X_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1704255308nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_574_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z_1: fun_int_bool,X_1: int,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite58652534t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(finite1827710202t_bool(F,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),Z_1)),A) ) ) ) ).

tff(fact_575_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z_1: nat,X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite1860950092at_nat(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A)) ) ) ) ).

tff(fact_576_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z_1: fun_nat_bool,X_1: nat,A: fun_nat_bool,F: fun_na35521688t_bool] :
      ( hBOOL(finite1956789438t_bool(F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(F,X_1),hAPP_f800510211t_bool(finite1578363458t_bool(F,Z_1),A)) ) ) ) ).

tff(fact_577_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite1532229644iple_a(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),hAPP_f2009966983iple_a(finite2089344904iple_a(F,Z_1),A)) ) ) ) ).

tff(fact_578_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z_1: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho112928087a_bool] :
      ( hBOOL(finite1306023740a_bool(F))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(F,X_1),hAPP_f642208769a_bool(finite718386368a_bool(F,Z_1),A)) ) ) ) ).

tff(fact_579_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z_1: int,X_1: int,A: fun_int_bool,F: fun_int_fun_int_int] :
      ( hBOOL(finite1704255308nt_int(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A)) ) ) ) ).

tff(fact_580_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z_1: fun_int_bool,X_1: int,A: fun_int_bool,F: fun_in719570932t_bool] :
      ( hBOOL(finite58652534t_bool(F))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(F,X_1),hAPP_f1805168059t_bool(finite1827710202t_bool(F,Z_1),A)) ) ) ) ).

tff(fact_581_folding__one_Oeq__fold_H,axiom,
    ! [X_1: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,X_1),A) ) ) ) ) ).

tff(fact_582_folding__one_Oeq__fold_H,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_f2009966983iple_a(finite2089344904iple_a(F,X_1),A) ) ) ) ) ).

tff(fact_583_folding__one_Oeq__fold_H,axiom,
    ! [X_1: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,X_1),A) ) ) ) ) ).

tff(fact_584_folding__one__idem_Oeq__fold__idem_H,axiom,
    ! [A_13: nat,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,A_13),A) ) ) ) ).

tff(fact_585_folding__one__idem_Oeq__fold__idem_H,axiom,
    ! [A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),A)) = hAPP_f2009966983iple_a(finite2089344904iple_a(F,A_13),A) ) ) ) ).

tff(fact_586_folding__one__idem_Oeq__fold__idem_H,axiom,
    ! [A_13: int,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,A_13),A) ) ) ) ).

tff(fact_587_fold1__eq__fold,axiom,
    ! [A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(times_times_nat,A_13),A) ) ) ) ).

tff(fact_588_fold1__eq__fold,axiom,
    ! [A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => ( hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),A)) = hAPP_f1594865479ol_int(finite_fold_int_int(times_times_int,A_13),A) ) ) ) ).

tff(fact_589_comp__fun__commute_Ofold__graph__insertE,axiom,
    ! [Z_1: nat,X_1: nat,A: fun_nat_bool,V: nat,F: fun_nat_fun_nat_nat] :
      ( hBOOL(finite2130160977at_nat(F))
     => ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),V))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ~ ! [Y_21: nat] :
                ( ( V = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Y_21) )
               => ~ hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(finite929467206at_nat(F),Z_1),A),Y_21)) ) ) ) ) ).

tff(fact_590_comp__fun__commute_Ofold__graph__insertE,axiom,
    ! [Z_1: hoare_2076525193iple_a,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,V: hoare_2076525193iple_a,F: fun_Ho483360497iple_a] :
      ( hBOOL(finite164030737iple_a(F))
     => ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)),V))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => ~ ! [Y_21: hoare_2076525193iple_a] :
                ( ( V = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_1),Y_21) )
               => ~ hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(finite974267142iple_a(F),Z_1),A),Y_21)) ) ) ) ) ).

tff(fact_591_comp__fun__commute_Ofold__graph__insertE,axiom,
    ! [Z_1: int,X_1: int,A: fun_int_bool,V: int,F: fun_int_fun_int_int] :
      ( hBOOL(finite1973466193nt_int(F))
     => ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),V))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ~ ! [Y_21: int] :
                ( ( V = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Y_21) )
               => ~ hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(finite772772422nt_int(F),Z_1),A),Y_21)) ) ) ) ) ).

tff(fact_592_setsum__image__gen,axiom,
    ! [G: fun_Ho864122878_a_nat,F: fun_Ho864122878_a_nat,S_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,S_1))
     => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f420120536at_nat(hAPP_f228445643at_nat(cOMBB_605112827at_nat,hAPP_f1697440293ol_nat(big_co922834230_a_nat,G)),hAPP_f1869543111a_bool(hAPP_f1882853211a_bool(cOMBB_1211601634ol_nat,collec977865374iple_a),hAPP_f1869543111a_bool(hAPP_f1882853211a_bool(cOMBB_1211601634ol_nat,hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),S_1)))),hAPP_f1040149703a_bool(cOMBC_240829083t_bool,hAPP_f570970635t_bool(hAPP_f2096737119t_bool(cOMBB_2017898829iple_a,fequal_nat),F)))))),image_1778392390_a_nat(F,S_1)) ) ) ).

tff(fact_593_setsum__image__gen,axiom,
    ! [G: fun_int_nat,F: fun_int_nat,S_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
     => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f389300155at_nat(hAPP_f109563153at_nat(cOMBB_1381347800at_nat,hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G)),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,collect_int),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f175561985t_bool(cOMBC_int_nat_bool,hAPP_f876579787t_bool(hAPP_f2026117279t_bool(cOMBB_1437810218ol_int,fequal_nat),F)))))),image_int_nat(F,S_1)) ) ) ).

tff(fact_594_setsum__image__gen,axiom,
    ! [G: fun_nat_nat,F: fun_na1240513468iple_a,S_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
     => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),S_1) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f1753252377_a_nat(hAPP_f456064075_a_nat(cOMBB_1733887227iple_a,hAPP_f782000547ol_nat(big_co387207925at_nat,G)),hAPP_f631038663t_bool(hAPP_f69420889t_bool(cOMBB_1063904223iple_a,collect_nat),hAPP_f631038663t_bool(hAPP_f69420889t_bool(cOMBB_1063904223iple_a,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f1460432071t_bool(cOMBC_2097822937a_bool,hAPP_f1324734153a_bool(hAPP_f946563611a_bool(cOMBB_315605200ol_nat,fequal1589671816iple_a),F)))))),image_344477316iple_a(F,S_1)) ) ) ).

tff(fact_595_setsum__image__gen,axiom,
    ! [G: fun_int_nat,F: fun_int_int,S_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
     => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1134349059nt_nat(hAPP_f2100446809nt_nat(cOMBB_2018621492at_int,hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G)),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,collect_int),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f428220345t_bool(cOMBC_int_int_bool,hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,fequal_int),F)))))),image_int_int(F,S_1)) ) ) ).

tff(fact_596_setsum__image__gen,axiom,
    ! [G: fun_int_nat,F: fun_in1219646560iple_a,S_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
     => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),S_1) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f10452157_a_nat(hAPP_f296893331_a_nat(cOMBB_1983399767iple_a,hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G)),hAPP_f741913983t_bool(hAPP_f1871962969t_bool(cOMBB_349652263iple_a,collect_int),hAPP_f741913983t_bool(hAPP_f1871962969t_bool(cOMBB_349652263iple_a,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f510295935t_bool(cOMBC_1135948669a_bool,hAPP_f1362463689a_bool(hAPP_f438022427a_bool(cOMBB_952878892ol_int,fequal1589671816iple_a),F)))))),image_323610408iple_a(F,S_1)) ) ) ).

tff(fact_597_setsum__image__gen,axiom,
    ! [G: fun_nat_nat,F: fun_nat_int,S_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
     => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1161717855nt_nat(hAPP_f1964560145nt_nat(cOMBB_1125113304at_int,hAPP_f782000547ol_nat(big_co387207925at_nat,G)),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,collect_nat),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f482008321t_bool(cOMBC_nat_int_bool,hAPP_f1533130627t_bool(hAPP_f879109391t_bool(cOMBB_1628441366ol_nat,fequal_int),F)))))),image_nat_int(F,S_1)) ) ) ).

tff(fact_598_setsum_Ocong,axiom,
    ! [G: fun_Ho864122878_a_nat,H: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ! [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
           => ( hAPP_H174304918_a_nat(G,X_2) = hAPP_H174304918_a_nat(H,X_2) ) )
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),A) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,H),A) ) ) ) ).

tff(fact_599_setsum_Ocong,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
           => ( hAPP_nat_nat(G,X_2) = hAPP_nat_nat(H,X_2) ) )
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,H),A) ) ) ) ).

tff(fact_600_setsum_Ocong,axiom,
    ! [G: fun_int_nat,H: fun_int_nat,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
           => ( hAPP_int_nat(G,X_2) = hAPP_int_nat(H,X_2) ) )
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,H),A) ) ) ) ).

tff(fact_601_setsum__setsum__restrict,axiom,
    ! [F: fun_int_fun_nat_nat,R_1: fun_int_fun_nat_bool,T_1: fun_nat_bool,S_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,T_1))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1161717855nt_nat(hAPP_f369809936nt_nat(cOMBS_1063588099ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),F)),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,collect_nat),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),T_1)))),R_1)))),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f389300155at_nat(hAPP_f1750007732at_nat(cOMBS_2106653443ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),hAPP_f879494613nt_nat(cOMBC_int_nat_nat,F))),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,collect_int),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f175561985t_bool(cOMBC_int_nat_bool,R_1))))),T_1) ) ) ) ).

tff(fact_602_setsum__setsum__restrict,axiom,
    ! [F: fun_Ho1036508337at_nat,R_1: fun_Ho1349715618t_bool,T_1: fun_nat_bool,S_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,S_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,T_1))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f1753252377_a_nat(hAPP_f991915629_a_nat(cOMBS_902588832ol_nat,hAPP_f1296292793ol_nat(hAPP_f153943611ol_nat(cOMBB_636937375iple_a,big_co387207925at_nat),F)),hAPP_f631038663t_bool(hAPP_f69420889t_bool(cOMBB_1063904223iple_a,collect_nat),hAPP_f631038663t_bool(hAPP_f69420889t_bool(cOMBB_1063904223iple_a,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),T_1)))),R_1)))),S_1) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f420120536at_nat(hAPP_f1323480046at_nat(cOMBS_1514622496ol_nat,hAPP_f1356625915ol_nat(hAPP_f2033836159ol_nat(cOMBB_1944586014at_nat,big_co922834230_a_nat),hAPP_f668391445_a_nat(cOMBC_1942990924at_nat,F))),hAPP_f1869543111a_bool(hAPP_f1882853211a_bool(cOMBB_1211601634ol_nat,collec977865374iple_a),hAPP_f1869543111a_bool(hAPP_f1882853211a_bool(cOMBB_1211601634ol_nat,hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),S_1)))),hAPP_f1040149703a_bool(cOMBC_240829083t_bool,R_1))))),T_1) ) ) ) ).

tff(fact_603_setsum__setsum__restrict,axiom,
    ! [F: fun_Ho1488374613nt_nat,R_1: fun_Ho1608761598t_bool,T_1: fun_int_bool,S_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,S_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,T_1))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f10452157_a_nat(hAPP_f209886133_a_nat(cOMBS_1838073924ol_nat,hAPP_f1998933761ol_nat(hAPP_f1772247755ol_nat(cOMBB_1701789015iple_a,big_co1740723097nt_nat),F)),hAPP_f741913983t_bool(hAPP_f1871962969t_bool(cOMBB_349652263iple_a,collect_int),hAPP_f741913983t_bool(hAPP_f1871962969t_bool(cOMBB_349652263iple_a,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),T_1)))),R_1)))),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1503898400nt_nat(hAPP_f807099282nt_nat(cOMBS_443177668ol_nat,hAPP_f1914880763ol_nat(hAPP_f534494591ol_nat(cOMBB_434376058at_int,big_co922834230_a_nat),hAPP_f1096027413_a_nat(cOMBC_1149022448nt_nat,F))),hAPP_f1500465535a_bool(hAPP_f967504915a_bool(cOMBB_1848875326ol_int,collec977865374iple_a),hAPP_f1500465535a_bool(hAPP_f967504915a_bool(cOMBB_1848875326ol_int,hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),S_1)))),hAPP_f1732083583a_bool(cOMBC_1771170551t_bool,R_1))))),T_1) ) ) ) ).

tff(fact_604_setsum__setsum__restrict,axiom,
    ! [F: fun_nat_fun_int_nat,R_1: fun_nat_fun_int_bool,T_1: fun_int_bool,S_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,T_1))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f389300155at_nat(hAPP_f1750007732at_nat(cOMBS_2106653443ol_nat,hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,big_co1740723097nt_nat),F)),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,collect_int),hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),T_1)))),R_1)))),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1161717855nt_nat(hAPP_f369809936nt_nat(cOMBS_1063588099ol_nat,hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,big_co387207925at_nat),hAPP_f446977493at_nat(cOMBC_nat_int_nat,F))),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,collect_nat),hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f482008321t_bool(cOMBC_nat_int_bool,R_1))))),T_1) ) ) ) ).

tff(fact_605_setsum__setsum__restrict,axiom,
    ! [F: fun_na841142961_a_nat,R_1: fun_na1758826658a_bool,T_1: fun_Ho882550911a_bool,S_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,T_1))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f420120536at_nat(hAPP_f1323480046at_nat(cOMBS_1514622496ol_nat,hAPP_f1356625915ol_nat(hAPP_f2033836159ol_nat(cOMBB_1944586014at_nat,big_co922834230_a_nat),F)),hAPP_f1869543111a_bool(hAPP_f1882853211a_bool(cOMBB_1211601634ol_nat,collec977865374iple_a),hAPP_f1869543111a_bool(hAPP_f1882853211a_bool(cOMBB_1211601634ol_nat,hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),T_1)))),R_1)))),S_1) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f1753252377_a_nat(hAPP_f991915629_a_nat(cOMBS_902588832ol_nat,hAPP_f1296292793ol_nat(hAPP_f153943611ol_nat(cOMBB_636937375iple_a,big_co387207925at_nat),hAPP_f982543381at_nat(cOMBC_23578254_a_nat,F))),hAPP_f631038663t_bool(hAPP_f69420889t_bool(cOMBB_1063904223iple_a,collect_nat),hAPP_f631038663t_bool(hAPP_f69420889t_bool(cOMBB_1063904223iple_a,hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))),hAPP_f1460432071t_bool(cOMBC_2097822937a_bool,R_1))))),T_1) ) ) ) ).

tff(fact_606_setsum__setsum__restrict,axiom,
    ! [F: fun_int_fun_int_nat,R_1: fun_int_fun_int_bool,T_1: fun_int_bool,S_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,T_1))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1134349059nt_nat(hAPP_f446447448nt_nat(cOMBS_1999073191ol_nat,hAPP_f1548925761ol_nat(hAPP_f1331458699ol_nat(cOMBB_2091167284at_int,big_co1740723097nt_nat),F)),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,collect_int),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),T_1)))),R_1)))),S_1) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1134349059nt_nat(hAPP_f446447448nt_nat(cOMBS_1999073191ol_nat,hAPP_f1548925761ol_nat(hAPP_f1331458699ol_nat(cOMBB_2091167284at_int,big_co1740723097nt_nat),hAPP_f909851349nt_nat(cOMBC_int_int_nat,F))),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,collect_int),hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f428220345t_bool(cOMBC_int_int_bool,R_1))))),T_1) ) ) ) ).

tff(fact_607_setsum__setsum__restrict,axiom,
    ! [F: fun_in501186901_a_nat,R_1: fun_in451447550a_bool,T_1: fun_Ho882550911a_bool,S_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,S_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,T_1))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f1503898400nt_nat(hAPP_f807099282nt_nat(cOMBS_443177668ol_nat,hAPP_f1914880763ol_nat(hAPP_f534494591ol_nat(cOMBB_434376058at_int,big_co922834230_a_nat),F)),hAPP_f1500465535a_bool(hAPP_f967504915a_bool(cOMBB_1848875326ol_int,collec977865374iple_a),hAPP_f1500465535a_bool(hAPP_f967504915a_bool(cOMBB_1848875326ol_int,hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),T_1)))),R_1)))),S_1) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f10452157_a_nat(hAPP_f209886133_a_nat(cOMBS_1838073924ol_nat,hAPP_f1998933761ol_nat(hAPP_f1772247755ol_nat(cOMBB_1701789015iple_a,big_co1740723097nt_nat),hAPP_f287449365nt_nat(cOMBC_1499522794_a_nat,F))),hAPP_f741913983t_bool(hAPP_f1871962969t_bool(cOMBB_349652263iple_a,collect_int),hAPP_f741913983t_bool(hAPP_f1871962969t_bool(cOMBB_349652263iple_a,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))),hAPP_f510295935t_bool(cOMBC_1135948669a_bool,R_1))))),T_1) ) ) ) ).

tff(fact_608_setsum_Oremove,axiom,
    ! [G: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_609_setsum_Oremove,axiom,
    ! [G: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_610_setsum_Oremove,axiom,
    ! [G: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,G),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H811578610_a_int(G,X_1)),hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_611_setsum_Oremove,axiom,
    ! [G: fun_int_int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_612_setsum_Oremove,axiom,
    ! [G: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H174304918_a_nat(G,X_1)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_613_setsum_Oremove,axiom,
    ! [G: fun_int_nat,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_614_setsum__diff1_H,axiom,
    ! [F: fun_nat_nat,A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,A_13)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_615_setsum__diff1_H,axiom,
    ! [F: fun_nat_int,A_13: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),A))
       => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(F,A_13)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_616_setsum__diff1_H,axiom,
    ! [F: fun_Ho1026035802_a_int,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H811578610_a_int(F,A_13)),hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_617_setsum__diff1_H,axiom,
    ! [F: fun_int_int,A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => ( hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(F,A_13)),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_618_setsum__diff1_H,axiom,
    ! [F: fun_Ho864122878_a_nat,A_13: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),A))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H174304918_a_nat(F,A_13)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_619_setsum__diff1_H,axiom,
    ! [F: fun_int_nat,A_13: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),A))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(F,A_13)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_620_setsum_Oinsert__remove,axiom,
    ! [G: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).

tff(fact_621_setsum_Oinsert__remove,axiom,
    ! [G: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).

tff(fact_622_setsum_Oinsert__remove,axiom,
    ! [G: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H811578610_a_int(G,X_1)),hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ).

tff(fact_623_setsum_Oinsert__remove,axiom,
    ! [G: fun_int_int,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(G,X_1)),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).

tff(fact_624_setsum_Oinsert__remove,axiom,
    ! [G: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H174304918_a_nat(G,X_1)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))) ) ) ).

tff(fact_625_setsum_Oinsert__remove,axiom,
    ! [G: fun_int_nat,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).

tff(fact_626_setsum__addf,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A: fun_nat_bool] : hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),F)),G)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A)) ).

tff(fact_627_setsum__addf,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,A: fun_int_bool] : hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2132704789nt_nat(hAPP_f1399363134nt_nat(cOMBS_int_nat_nat,hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,plus_plus_nat),F)),G)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A)) ).

tff(fact_628_setsum__addf,axiom,
    ! [F: fun_Ho864122878_a_nat,G: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] : hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f2075363285_a_nat(hAPP_f1659339553_a_nat(cOMBS_885136444at_nat,hAPP_f731653896at_nat(hAPP_f648261589at_nat(cOMBB_1416330906iple_a,plus_plus_nat),F)),G)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),A)) ).

tff(fact_629_setsum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,H),A)) ) ) ).

tff(fact_630_setsum_Odistrib,axiom,
    ! [G: fun_Ho864122878_a_nat,H: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,hAPP_f2075363285_a_nat(hAPP_f1659339553_a_nat(cOMBS_885136444at_nat,hAPP_f731653896at_nat(hAPP_f648261589at_nat(cOMBB_1416330906iple_a,plus_plus_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),A)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,H),A)) ) ) ).

tff(fact_631_setsum_Odistrib,axiom,
    ! [G: fun_Ho1026035802_a_int,H: fun_Ho1026035802_a_int,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,hAPP_f1382137045_a_int(hAPP_f1031139617_a_int(cOMBS_728441660nt_int,hAPP_f534508204nt_int(hAPP_f1566620885nt_int(cOMBB_364359670iple_a,plus_plus_int),G)),H)),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,G),A)),hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,H),A)) ) ) ).

tff(fact_632_setsum_Odistrib,axiom,
    ! [G: fun_int_nat,H: fun_int_nat,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,hAPP_f2132704789nt_nat(hAPP_f1399363134nt_nat(cOMBS_int_nat_nat,hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,plus_plus_nat),G)),H)),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,H),A)) ) ) ).

tff(fact_633_setsum__insert,axiom,
    ! [F: fun_nat_nat,A_13: nat,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),F_1))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),F_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,A_13)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),F_1)) ) ) ) ).

tff(fact_634_setsum__insert,axiom,
    ! [F: fun_nat_int,A_13: nat,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_13),F_1))
       => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),F_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(F,A_13)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,F),F_1)) ) ) ) ).

tff(fact_635_setsum__insert,axiom,
    ! [F: fun_Ho1026035802_a_int,A_13: hoare_2076525193iple_a,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),F_1))
       => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),F_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H811578610_a_int(F,A_13)),hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,F),F_1)) ) ) ) ).

tff(fact_636_setsum__insert,axiom,
    ! [F: fun_Ho864122878_a_nat,A_13: hoare_2076525193iple_a,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_13),F_1))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),F_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H174304918_a_nat(F,A_13)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),F_1)) ) ) ) ).

tff(fact_637_setsum__insert,axiom,
    ! [F: fun_int_nat,A_13: int,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_13),F_1))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),F_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(F,A_13)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),F_1)) ) ) ) ).

tff(fact_638_setsum_Oinsert,axiom,
    ! [G: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(G,X_1)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A)) ) ) ) ).

tff(fact_639_setsum_Oinsert,axiom,
    ! [G: fun_nat_int,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(G,X_1)),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,G),A)) ) ) ) ).

tff(fact_640_setsum_Oinsert,axiom,
    ! [G: fun_Ho1026035802_a_int,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_H811578610_a_int(G,X_1)),hAPP_f827914852ol_int(hAPP_f1870360797ol_int(big_co1560107922_a_int,G),A)) ) ) ) ).

tff(fact_641_setsum_Oinsert,axiom,
    ! [G: fun_Ho864122878_a_nat,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_H174304918_a_nat(G,X_1)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),A)) ) ) ) ).

tff(fact_642_setsum_Oinsert,axiom,
    ! [G: fun_int_nat,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(G,X_1)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A)) ) ) ) ).

tff(fact_643_eq__add__iff2,axiom,
    ! [A_13: int,E: int,Ca: int,Ba: int,D_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1) )
    <=> ( Ca = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),A_13)),E)),D_1) ) ) ).

tff(fact_644_eq__add__iff1,axiom,
    ! [A_13: int,E: int,Ca: int,Ba: int,D_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1) )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba)),E)),Ca) = D_1 ) ) ).

tff(fact_645_diff__add__cancel,axiom,
    ! [A_42: int,B_34: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_42),B_34)),B_34) = A_42 ).

tff(fact_646_add__diff__cancel,axiom,
    ! [A_41: int,B_33: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_41),B_33)),B_33) = A_41 ).

tff(fact_647_crossproduct__eq,axiom,
    ! [W: nat,Y_1: nat,X_1: nat,Z_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W),Z_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_1)) )
    <=> ( ( W = X_1 )
        | ( Y_1 = Z_1 ) ) ) ).

tff(fact_648_crossproduct__eq,axiom,
    ! [W: int,Y_1: int,X_1: int,Z_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Y_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_1)) )
    <=> ( ( W = X_1 )
        | ( Y_1 = Z_1 ) ) ) ).

tff(fact_649_combine__common__factor,axiom,
    ! [A_40: nat,E_1: nat,B_32: nat,C_28: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_40),E_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_32),E_1)),C_28)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_40),B_32)),E_1)),C_28) ).

tff(fact_650_combine__common__factor,axiom,
    ! [A_40: int,E_1: int,B_32: int,C_28: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_40),E_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_32),E_1)),C_28)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_40),B_32)),E_1)),C_28) ).

tff(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_39: nat,M_4: nat,B_31: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_39),M_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_31),M_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_39),B_31)),M_4) ).

tff(fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_39: int,M_4: int,B_31: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_39),M_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_31),M_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_39),B_31)),M_4) ).

tff(fact_653_add__right__imp__eq,axiom,
    ! [B_30: nat,A_38: nat,C_27: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_30),A_38) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_27),A_38) )
     => ( B_30 = C_27 ) ) ).

tff(fact_654_add__right__imp__eq,axiom,
    ! [B_30: int,A_38: int,C_27: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_30),A_38) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_27),A_38) )
     => ( B_30 = C_27 ) ) ).

tff(fact_655_add__imp__eq,axiom,
    ! [A_37: nat,B_29: nat,C_26: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_37),B_29) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_37),C_26) )
     => ( B_29 = C_26 ) ) ).

tff(fact_656_add__imp__eq,axiom,
    ! [A_37: int,B_29: int,C_26: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_37),B_29) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_37),C_26) )
     => ( B_29 = C_26 ) ) ).

tff(fact_657_add__left__imp__eq,axiom,
    ! [A_36: nat,B_28: nat,C_25: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_36),B_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_36),C_25) )
     => ( B_28 = C_25 ) ) ).

tff(fact_658_add__left__imp__eq,axiom,
    ! [A_36: int,B_28: int,C_25: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_36),B_28) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_36),C_25) )
     => ( B_28 = C_25 ) ) ).

tff(fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_35: nat,B_27: nat,C_24: nat,D_6: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_35),B_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_24),D_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_35),C_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_27),D_6)) ).

tff(fact_660_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_35: int,B_27: int,C_24: int,D_6: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_35),B_27)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_24),D_6)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_35),C_24)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_27),D_6)) ).

tff(fact_661_add__right__cancel,axiom,
    ! [Ba: nat,A_13: nat,Ca: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),A_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),A_13) )
    <=> ( Ba = Ca ) ) ).

tff(fact_662_add__right__cancel,axiom,
    ! [Ba: int,A_13: int,Ca: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),A_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),A_13) )
    <=> ( Ba = Ca ) ) ).

tff(fact_663_add__left__cancel,axiom,
    ! [A_13: nat,Ba: nat,Ca: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),Ba) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),Ca) )
    <=> ( Ba = Ca ) ) ).

tff(fact_664_add__left__cancel,axiom,
    ! [A_13: int,Ba: int,Ca: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),Ba) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),Ca) )
    <=> ( Ba = Ca ) ) ).

tff(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_34: nat,B_26: nat,C_23: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_34),B_26)),C_23) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_34),C_23)),B_26) ).

tff(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_34: int,B_26: int,C_23: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_34),B_26)),C_23) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_34),C_23)),B_26) ).

tff(fact_667_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_33: nat,B_25: nat,C_22: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_33),B_25)),C_22) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_33),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_25),C_22)) ).

tff(fact_668_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_33: int,B_25: int,C_22: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_33),B_25)),C_22) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_33),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_25),C_22)) ).

tff(fact_669_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_32: nat,B_24: nat,C_21: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_32),B_24)),C_21) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_32),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_24),C_21)) ).

tff(fact_670_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_32: int,B_24: int,C_21: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_32),B_24)),C_21) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_32),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_24),C_21)) ).

tff(fact_671_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_31: nat,C_20: nat,D_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_31),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_20),D_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_31),C_20)),D_5) ).

tff(fact_672_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_31: int,C_20: int,D_5: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_31),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_20),D_5)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_31),C_20)),D_5) ).

tff(fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_30: nat,C_19: nat,D_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_19),D_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_19),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),D_4)) ).

tff(fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_30: int,C_19: int,D_4: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_19),D_4)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_19),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),D_4)) ).

tff(fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_29: nat,C_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_29),C_18) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_18),A_29) ).

tff(fact_676_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_29: int,C_18: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_29),C_18) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_18),A_29) ).

tff(fact_677_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ly_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),Ry_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Rx_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Ry_4)) ).

tff(fact_678_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ly_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),Ry_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Rx_6)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_4),Ry_4)) ).

tff(fact_679_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),Ry_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_680_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),Ry_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2))) ).

tff(fact_682_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2))) ).

tff(fact_683_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: nat,Ly_1: nat,Rx_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Ly_1)),Rx_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Rx_3)),Ly_1) ).

tff(fact_684_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: int,Ly_1: int,Rx_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_1)),Rx_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_1) ).

tff(fact_685_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: nat,Ly: nat,Rx_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),Ly)),Rx_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Rx_2)) ).

tff(fact_686_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: int,Ly: int,Rx_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),Ly)),Rx_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Rx_2)) ).

tff(fact_687_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_28: nat,B_23: nat,C_17: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_28),B_23)),C_17) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_28),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_23),C_17)) ).

tff(fact_688_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_28: int,B_23: int,C_17: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_28),B_23)),C_17) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_28),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_23),C_17)) ).

tff(fact_689_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: nat,Rx_1: nat,Ry_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Rx_1)),Ry_1) ).

tff(fact_690_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: int,Rx_1: int,Ry_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Rx_1)),Ry_1) ).

tff(fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: nat,Rx: nat,Ry: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ry)) ).

tff(fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: int,Rx: int,Ry: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ry)) ).

tff(fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_27: nat,B_22: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_27),B_22) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_22),A_27) ).

tff(fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_27: int,B_22: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_27),B_22) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_22),A_27) ).

tff(fact_695_diff__eq__diff__eq,axiom,
    ! [A_13: int,Ba: int,Ca: int,D_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ca),D_1) )
     => ( ( A_13 = Ba )
      <=> ( Ca = D_1 ) ) ) ).

tff(fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_27: nat,Y_20: nat,Z_10: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_20),Z_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_27),Y_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_27),Z_10)) ).

tff(fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_27: int,Y_20: int,Z_10: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_20),Z_10)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),Y_20)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_27),Z_10)) ).

tff(fact_698_crossproduct__noteq,axiom,
    ! [Ca: nat,D_1: nat,A_13: nat,Ba: nat] :
      ( ( ( A_13 != Ba )
        & ( Ca != D_1 ) )
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_13),Ca)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),D_1)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_13),D_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ba),Ca)) ) ) ).

tff(fact_699_crossproduct__noteq,axiom,
    ! [Ca: int,D_1: int,A_13: int,Ba: int] :
      ( ( ( A_13 != Ba )
        & ( Ca != D_1 ) )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),Ca)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),D_1)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),Ca)) ) ) ).

tff(fact_700_comm__semiring__class_Odistrib,axiom,
    ! [A_26: int,B_21: int,C_16: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_26),B_21)),C_16) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_26),C_16)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_21),C_16)) ).

tff(fact_701_comm__semiring__class_Odistrib,axiom,
    ! [A_26: nat,B_21: nat,C_16: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_26),B_21)),C_16) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_26),C_16)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_21),C_16)) ).

tff(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_25: int,B_20: int,C_15: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_25),B_20)),C_15) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_25),C_15)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_20),C_15)) ).

tff(fact_703_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_25: nat,B_20: nat,C_15: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_25),B_20)),C_15) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_25),C_15)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_20),C_15)) ).

tff(fact_704_setsum__cong2,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,A: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
         => ( hAPP_int_nat(F,X_2) = hAPP_int_nat(G,X_2) ) )
     => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),A) ) ) ).

tff(fact_705_setsum__cong2,axiom,
    ! [F: fun_Ho864122878_a_nat,G: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool] :
      ( ! [X_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
         => ( hAPP_H174304918_a_nat(F,X_2) = hAPP_H174304918_a_nat(G,X_2) ) )
     => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),A) ) ) ).

tff(fact_706_setsum__cong2,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => ( hAPP_nat_nat(F,X_2) = hAPP_nat_nat(G,X_2) ) )
     => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),A) ) ) ).

tff(fact_707_setsum__cong,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B))
           => ( hAPP_int_nat(F,X_2) = hAPP_int_nat(G,X_2) ) )
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),B) ) ) ) ).

tff(fact_708_setsum__cong,axiom,
    ! [F: fun_Ho864122878_a_nat,G: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( ( A = B )
     => ( ! [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),B))
           => ( hAPP_H174304918_a_nat(F,X_2) = hAPP_H174304918_a_nat(G,X_2) ) )
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),B) ) ) ) ).

tff(fact_709_setsum__cong,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))
           => ( hAPP_nat_nat(F,X_2) = hAPP_nat_nat(G,X_2) ) )
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B) ) ) ) ).

tff(fact_710_setsum_OF__cong,axiom,
    ! [H: fun_int_nat,G: fun_int_nat,A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B))
           => ( hAPP_int_nat(H,X_2) = hAPP_int_nat(G,X_2) ) )
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,H),A) = hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),B) ) ) ) ).

tff(fact_711_setsum_OF__cong,axiom,
    ! [H: fun_Ho864122878_a_nat,G: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( ( A = B )
     => ( ! [X_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),B))
           => ( hAPP_H174304918_a_nat(H,X_2) = hAPP_H174304918_a_nat(G,X_2) ) )
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,H),A) = hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),B) ) ) ) ).

tff(fact_712_setsum_OF__cong,axiom,
    ! [H: fun_nat_nat,G: fun_nat_nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))
           => ( hAPP_nat_nat(H,X_2) = hAPP_nat_nat(G,X_2) ) )
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,H),A) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),B) ) ) ) ).

tff(fact_713_single__Diff__lessThan,axiom,
    ! [K_1: nat] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,K_1),bot_bot_fun_nat_bool)),hAPP_n1699378549t_bool(ord_lessThan_nat,K_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,K_1),bot_bot_fun_nat_bool) ).

tff(fact_714_single__Diff__lessThan,axiom,
    ! [K_1: int] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,K_1),bot_bot_fun_int_bool)),hAPP_i1948725293t_bool(ord_lessThan_int,K_1)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,K_1),bot_bot_fun_int_bool) ).

tff(fact_715_setsum__diff__nat,axiom,
    ! [F: fun_Ho864122878_a_nat,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,B),A))
       => ( hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),A)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),B)) ) ) ) ).

tff(fact_716_setsum__diff__nat,axiom,
    ! [F: fun_int_nat,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
       => ( hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),A)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),B)) ) ) ) ).

tff(fact_717_setsum__diff__nat,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
       => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),A)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),B)) ) ) ) ).

tff(fact_718_order__refl,axiom,
    ! [X_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),X_1)) ).

tff(fact_719_order__refl,axiom,
    ! [X_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),X_1)) ).

tff(fact_720_order__refl,axiom,
    ! [X_26: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_26),X_26)) ).

tff(fact_721_order__refl,axiom,
    ! [X_26: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_26),X_26)) ).

tff(fact_722_equalityI,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
       => ( A = B ) ) ) ).

tff(fact_723_equalityI,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
       => ( A = B ) ) ) ).

tff(fact_724_subsetD,axiom,
    ! [Ca: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Ca),B)) ) ) ).

tff(fact_725_subsetD,axiom,
    ! [Ca: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B)) ) ) ).

tff(fact_726_subsetD,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_727_empty__subsetI,axiom,
    ! [A: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A)) ).

tff(fact_728_empty__subsetI,axiom,
    ! [A: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A)) ).

tff(fact_729_empty__subsetI,axiom,
    ! [A: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,bot_bo214064617a_bool),A)) ).

tff(fact_730_finite__Collect__subsets,axiom,
    ! [A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,ord_le940607413a_bool),A)))) ) ).

tff(fact_731_finite__Collect__subsets,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,hAPP_f1948010709l_bool(collect_fun_int_bool,hAPP_f284875647l_bool(hAPP_f1399552105l_bool(cOMBC_605892544l_bool,ord_le951220754t_bool),A)))) ) ).

tff(fact_732_finite__Collect__subsets,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,hAPP_f1246832597l_bool(collect_fun_nat_bool,hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,ord_le1568362934t_bool),A)))) ) ).

tff(fact_733_pred__subset__eq,axiom,
    ! [R_1: fun_Ho882550911a_bool,S_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),R_1)),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),S_1)))
    <=> hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,R_1),S_1)) ) ).

tff(fact_734_pred__subset__eq,axiom,
    ! [R_1: fun_int_bool,S_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),R_1)),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),S_1)))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,R_1),S_1)) ) ).

tff(fact_735_pred__subset__eq,axiom,
    ! [R_1: fun_nat_bool,S_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_1)))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,R_1),S_1)) ) ).

tff(fact_736_linorder__le__cases,axiom,
    ! [X_25: int,Y_19: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_25),Y_19))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_19),X_25)) ) ).

tff(fact_737_linorder__le__cases,axiom,
    ! [X_25: nat,Y_19: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_25),Y_19))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_19),X_25)) ) ).

tff(fact_738_le__funE,axiom,
    ! [X_1: int,F: fun_int_bool,G: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))) ) ).

tff(fact_739_le__funE,axiom,
    ! [X_1: nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1))) ) ).

tff(fact_740_xt1_I6_J,axiom,
    ! [Z_1: fun_int_bool,Y_1: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_1),Y_1))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_1),X_1)) ) ) ).

tff(fact_741_xt1_I6_J,axiom,
    ! [Z_1: fun_nat_bool,Y_1: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_1),Y_1))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_1),X_1)) ) ) ).

tff(fact_742_xt1_I6_J,axiom,
    ! [Z_9: int,Y_18: int,X_24: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_18),X_24))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_9),Y_18))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_9),X_24)) ) ) ).

tff(fact_743_xt1_I6_J,axiom,
    ! [Z_9: nat,Y_18: nat,X_24: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_18),X_24))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_9),Y_18))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_9),X_24)) ) ) ).

tff(fact_744_xt1_I5_J,axiom,
    ! [Y_1: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_745_xt1_I5_J,axiom,
    ! [Y_1: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_746_xt1_I5_J,axiom,
    ! [Y_17: int,X_23: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_17),X_23))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_23),Y_17))
       => ( X_23 = Y_17 ) ) ) ).

tff(fact_747_xt1_I5_J,axiom,
    ! [Y_17: nat,X_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_17),X_23))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_23),Y_17))
       => ( X_23 = Y_17 ) ) ) ).

tff(fact_748_order__trans,axiom,
    ! [Z_1: fun_int_bool,X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),Z_1))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Z_1)) ) ) ).

tff(fact_749_order__trans,axiom,
    ! [Z_1: fun_nat_bool,X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),Z_1))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Z_1)) ) ) ).

tff(fact_750_order__trans,axiom,
    ! [Z_8: int,X_22: int,Y_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_22),Y_16))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_16),Z_8))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_22),Z_8)) ) ) ).

tff(fact_751_order__trans,axiom,
    ! [Z_8: nat,X_22: nat,Y_16: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Y_16))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_16),Z_8))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Z_8)) ) ) ).

tff(fact_752_order__antisym,axiom,
    ! [X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_753_order__antisym,axiom,
    ! [X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
       => ( X_1 = Y_1 ) ) ) ).

tff(fact_754_order__antisym,axiom,
    ! [X_21: int,Y_15: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_21),Y_15))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_15),X_21))
       => ( X_21 = Y_15 ) ) ) ).

tff(fact_755_order__antisym,axiom,
    ! [X_21: nat,Y_15: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_21),Y_15))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_15),X_21))
       => ( X_21 = Y_15 ) ) ) ).

tff(fact_756_xt1_I4_J,axiom,
    ! [Ca: fun_int_bool,Ba: fun_int_bool,A_13: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),A_13))
     => ( ( Ba = Ca )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ca),A_13)) ) ) ).

tff(fact_757_xt1_I4_J,axiom,
    ! [Ca: fun_nat_bool,Ba: fun_nat_bool,A_13: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),A_13))
     => ( ( Ba = Ca )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ca),A_13)) ) ) ).

tff(fact_758_xt1_I4_J,axiom,
    ! [C_14: int,B_19: int,A_24: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),A_24))
     => ( ( B_19 = C_14 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),A_24)) ) ) ).

tff(fact_759_xt1_I4_J,axiom,
    ! [C_14: nat,B_19: nat,A_24: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),A_24))
     => ( ( B_19 = C_14 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),A_24)) ) ) ).

tff(fact_760_ord__le__eq__trans,axiom,
    ! [Ca: fun_int_bool,A_13: fun_int_bool,Ba: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),Ba))
     => ( ( Ba = Ca )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),Ca)) ) ) ).

tff(fact_761_ord__le__eq__trans,axiom,
    ! [Ca: fun_nat_bool,A_13: fun_nat_bool,Ba: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),Ba))
     => ( ( Ba = Ca )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),Ca)) ) ) ).

tff(fact_762_ord__le__eq__trans,axiom,
    ! [C_13: int,A_23: int,B_18: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_23),B_18))
     => ( ( B_18 = C_13 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_23),C_13)) ) ) ).

tff(fact_763_ord__le__eq__trans,axiom,
    ! [C_13: nat,A_23: nat,B_18: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_23),B_18))
     => ( ( B_18 = C_13 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_23),C_13)) ) ) ).

tff(fact_764_xt1_I3_J,axiom,
    ! [Ca: fun_int_bool,A_13: fun_int_bool,Ba: fun_int_bool] :
      ( ( A_13 = Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ca),Ba))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ca),A_13)) ) ) ).

tff(fact_765_xt1_I3_J,axiom,
    ! [Ca: fun_nat_bool,A_13: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( A_13 = Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ca),Ba))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ca),A_13)) ) ) ).

tff(fact_766_xt1_I3_J,axiom,
    ! [C_12: int,A_22: int,B_17: int] :
      ( ( A_22 = B_17 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),B_17))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),A_22)) ) ) ).

tff(fact_767_xt1_I3_J,axiom,
    ! [C_12: nat,A_22: nat,B_17: nat] :
      ( ( A_22 = B_17 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),B_17))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),A_22)) ) ) ).

tff(fact_768_ord__eq__le__trans,axiom,
    ! [Ca: fun_int_bool,A_13: fun_int_bool,Ba: fun_int_bool] :
      ( ( A_13 = Ba )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ba),Ca))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),Ca)) ) ) ).

tff(fact_769_ord__eq__le__trans,axiom,
    ! [Ca: fun_nat_bool,A_13: fun_nat_bool,Ba: fun_nat_bool] :
      ( ( A_13 = Ba )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ba),Ca))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),Ca)) ) ) ).

tff(fact_770_ord__eq__le__trans,axiom,
    ! [C_11: int,A_21: int,B_16: int] :
      ( ( A_21 = B_16 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_16),C_11))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),C_11)) ) ) ).

tff(fact_771_ord__eq__le__trans,axiom,
    ! [C_11: nat,A_21: nat,B_16: nat] :
      ( ( A_21 = B_16 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_16),C_11))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),C_11)) ) ) ).

tff(fact_772_order__antisym__conv,axiom,
    ! [Y_1: fun_int_bool,X_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_773_order__antisym__conv,axiom,
    ! [Y_1: fun_nat_bool,X_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_774_order__antisym__conv,axiom,
    ! [Y_1: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_775_order__antisym__conv,axiom,
    ! [Y_1: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
      <=> ( X_1 = Y_1 ) ) ) ).

tff(fact_776_le__funD,axiom,
    ! [X_1: int,F: fun_int_bool,G: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))) ) ).

tff(fact_777_le__funD,axiom,
    ! [X_1: nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1))) ) ).

tff(fact_778_order__eq__refl,axiom,
    ! [X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( ( X_1 = Y_1 )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1)) ) ).

tff(fact_779_order__eq__refl,axiom,
    ! [X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( ( X_1 = Y_1 )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1)) ) ).

tff(fact_780_order__eq__refl,axiom,
    ! [X_20: int,Y_14: int] :
      ( ( X_20 = Y_14 )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_20),Y_14)) ) ).

tff(fact_781_order__eq__refl,axiom,
    ! [X_20: nat,Y_14: nat] :
      ( ( X_20 = Y_14 )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_20),Y_14)) ) ).

tff(fact_782_order__eq__iff,axiom,
    ! [X_1: fun_int_bool,Y_1: fun_int_bool] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1)) ) ) ).

tff(fact_783_order__eq__iff,axiom,
    ! [X_1: fun_nat_bool,Y_1: fun_nat_bool] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1)) ) ) ).

tff(fact_784_order__eq__iff,axiom,
    ! [X_1: int,Y_1: int] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).

tff(fact_785_order__eq__iff,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( ( X_1 = Y_1 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).

tff(fact_786_linorder__linear,axiom,
    ! [X_19: int,Y_13: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_19),Y_13))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_13),X_19)) ) ).

tff(fact_787_linorder__linear,axiom,
    ! [X_19: nat,Y_13: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_19),Y_13))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_13),X_19)) ) ).

tff(fact_788_le__fun__def,axiom,
    ! [F: fun_int_bool,G: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
    <=> ! [X_2: int] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G,X_2))) ) ).

tff(fact_789_le__fun__def,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
    <=> ! [X_2: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G,X_2))) ) ).

tff(fact_790_equalityE,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
         => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).

tff(fact_791_equalityE,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
         => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).

tff(fact_792_subset__trans,axiom,
    ! [C_5: fun_int_bool,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_5))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_5)) ) ) ).

tff(fact_793_subset__trans,axiom,
    ! [C_5: fun_nat_bool,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_5))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_5)) ) ) ).

tff(fact_794_set__mp,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B)) ) ) ).

tff(fact_795_set__mp,axiom,
    ! [X_1: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

tff(fact_796_set__mp,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_797_set__rev__mp,axiom,
    ! [B: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B)) ) ) ).

tff(fact_798_set__rev__mp,axiom,
    ! [B: fun_int_bool,X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

tff(fact_799_set__rev__mp,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_800_in__mono,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B)) ) ) ).

tff(fact_801_in__mono,axiom,
    ! [X_1: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

tff(fact_802_in__mono,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_803_equalityD2,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ).

tff(fact_804_equalityD2,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ).

tff(fact_805_equalityD1,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).

tff(fact_806_equalityD1,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).

tff(fact_807_set__eq__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( ( A = B )
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).

tff(fact_808_set__eq__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).

tff(fact_809_subset__refl,axiom,
    ! [A: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),A)) ).

tff(fact_810_subset__refl,axiom,
    ! [A: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),A)) ).

tff(fact_811_predicate1D,axiom,
    ! [X_1: int,Pa: fun_int_bool,Q_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Pa),Q_1))
     => ( hBOOL(hAPP_int_bool(Pa,X_1))
       => hBOOL(hAPP_int_bool(Q_1,X_1)) ) ) ).

tff(fact_812_predicate1D,axiom,
    ! [X_1: nat,Pa: fun_nat_bool,Q_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Pa),Q_1))
     => ( hBOOL(hAPP_nat_bool(Pa,X_1))
       => hBOOL(hAPP_nat_bool(Q_1,X_1)) ) ) ).

tff(fact_813_rev__predicate1D,axiom,
    ! [Q_1: fun_int_bool,Pa: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(Pa,X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Pa),Q_1))
       => hBOOL(hAPP_int_bool(Q_1,X_1)) ) ) ).

tff(fact_814_rev__predicate1D,axiom,
    ! [Q_1: fun_nat_bool,Pa: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Pa),Q_1))
       => hBOOL(hAPP_nat_bool(Q_1,X_1)) ) ) ).

tff(fact_815_lessThan__subset__iff,axiom,
    ! [X_1: int,Y_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_i1948725293t_bool(ord_lessThan_int,X_1)),hAPP_i1948725293t_bool(ord_lessThan_int,Y_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1)) ) ).

tff(fact_816_lessThan__subset__iff,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(ord_lessThan_nat,X_1)),hAPP_n1699378549t_bool(ord_lessThan_nat,Y_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).

tff(fact_817_lessThan__eq__iff,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( ( hAPP_n1699378549t_bool(ord_lessThan_nat,X_1) = hAPP_n1699378549t_bool(ord_lessThan_nat,Y_1) )
    <=> ( X_1 = Y_1 ) ) ).

tff(fact_818_add__le__imp__le__left,axiom,
    ! [C_10: int,A_20: int,B_15: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_10),A_20)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_10),B_15)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_20),B_15)) ) ).

tff(fact_819_add__le__imp__le__left,axiom,
    ! [C_10: nat,A_20: nat,B_15: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_10),A_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_10),B_15)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_20),B_15)) ) ).

tff(fact_820_add__le__imp__le__right,axiom,
    ! [A_19: int,C_9: int,B_14: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_19),C_9)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_14),C_9)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),B_14)) ) ).

tff(fact_821_add__le__imp__le__right,axiom,
    ! [A_19: nat,C_9: nat,B_14: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_19),C_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_14),C_9)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_19),B_14)) ) ).

tff(fact_822_add__mono,axiom,
    ! [C_8: int,D_3: int,A_18: int,B_13: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_18),B_13))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_8),D_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_18),C_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_13),D_3))) ) ) ).

tff(fact_823_add__mono,axiom,
    ! [C_8: nat,D_3: nat,A_18: nat,B_13: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),B_13))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_8),D_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_18),C_8)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_13),D_3))) ) ) ).

tff(fact_824_add__left__mono,axiom,
    ! [C_7: int,A_17: int,B_12: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_17),B_12))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),A_17)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),B_12))) ) ).

tff(fact_825_add__left__mono,axiom,
    ! [C_7: nat,A_17: nat,B_12: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_17),B_12))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),A_17)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),B_12))) ) ).

tff(fact_826_add__right__mono,axiom,
    ! [C_6: int,A_16: int,B_11: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_16),B_11))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_16),C_6)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_11),C_6))) ) ).

tff(fact_827_add__right__mono,axiom,
    ! [C_6: nat,A_16: nat,B_11: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_16),B_11))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_16),C_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_11),C_6))) ) ).

tff(fact_828_add__le__cancel__left,axiom,
    ! [Ca: int,A_13: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),A_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),Ba)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),Ba)) ) ).

tff(fact_829_add__le__cancel__left,axiom,
    ! [Ca: nat,A_13: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),A_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),Ba)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),Ba)) ) ).

tff(fact_830_add__le__cancel__right,axiom,
    ! [A_13: int,Ca: int,Ba: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),Ca)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),Ba)) ) ).

tff(fact_831_add__le__cancel__right,axiom,
    ! [A_13: nat,Ca: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),Ca)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),Ca)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),Ba)) ) ).

tff(fact_832_le__bot,axiom,
    ! [A_13: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_13),bot_bot_bool))
     => ( hBOOL(A_13)
      <=> hBOOL(bot_bot_bool) ) ) ).

tff(fact_833_le__bot,axiom,
    ! [A_13: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),bot_bot_fun_nat_bool))
     => ( A_13 = bot_bot_fun_nat_bool ) ) ).

tff(fact_834_le__bot,axiom,
    ! [A_13: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),bot_bot_fun_int_bool))
     => ( A_13 = bot_bot_fun_int_bool ) ) ).

tff(fact_835_le__bot,axiom,
    ! [A_13: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A_13),bot_bo214064617a_bool))
     => ( A_13 = bot_bo214064617a_bool ) ) ).

tff(fact_836_le__bot,axiom,
    ! [A_15: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_15),bot_bot_nat))
     => ( A_15 = bot_bot_nat ) ) ).

tff(fact_837_bot__unique,axiom,
    ! [A_13: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_13),bot_bot_bool))
    <=> ( hBOOL(A_13)
      <=> hBOOL(bot_bot_bool) ) ) ).

tff(fact_838_bot__unique,axiom,
    ! [A_13: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_13),bot_bot_fun_nat_bool))
    <=> ( A_13 = bot_bot_fun_nat_bool ) ) ).

tff(fact_839_bot__unique,axiom,
    ! [A_13: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_13),bot_bot_fun_int_bool))
    <=> ( A_13 = bot_bot_fun_int_bool ) ) ).

tff(fact_840_bot__unique,axiom,
    ! [A_13: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A_13),bot_bo214064617a_bool))
    <=> ( A_13 = bot_bo214064617a_bool ) ) ).

tff(fact_841_bot__unique,axiom,
    ! [A_13: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),bot_bot_nat))
    <=> ( A_13 = bot_bot_nat ) ) ).

tff(fact_842_bot__least,axiom,
    ! [A_13: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,bot_bot_bool),A_13)) ).

tff(fact_843_bot__least,axiom,
    ! [A_13: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_13)) ).

tff(fact_844_bot__least,axiom,
    ! [A_13: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A_13)) ).

tff(fact_845_bot__least,axiom,
    ! [A_13: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,bot_bo214064617a_bool),A_13)) ).

tff(fact_846_bot__least,axiom,
    ! [A_14: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_14)) ).

tff(fact_847_diff__eq__diff__less__eq,axiom,
    ! [A_13: int,Ba: int,Ca: int,D_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ca),D_1) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),Ba))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),D_1)) ) ) ).

tff(fact_848_subset__empty,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),bot_bot_fun_nat_bool))
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_849_subset__empty,axiom,
    ! [A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),bot_bot_fun_int_bool))
    <=> ( A = bot_bot_fun_int_bool ) ) ).

tff(fact_850_subset__empty,axiom,
    ! [A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),bot_bo214064617a_bool))
    <=> ( A = bot_bo214064617a_bool ) ) ).

tff(fact_851_finite__subset,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B))
       => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A)) ) ) ).

tff(fact_852_finite__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

tff(fact_853_finite__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_854_rev__finite__subset,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
       => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A)) ) ) ).

tff(fact_855_rev__finite__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

tff(fact_856_rev__finite__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_857_insert__mono,axiom,
    ! [A_13: nat,C_5: fun_nat_bool,D_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_5),D_2))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),C_5)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),D_2))) ) ).

tff(fact_858_insert__mono,axiom,
    ! [A_13: int,C_5: fun_int_bool,D_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_5),D_2))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),C_5)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),D_2))) ) ).

tff(fact_859_insert__mono,axiom,
    ! [A_13: hoare_2076525193iple_a,C_5: fun_Ho882550911a_bool,D_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,C_5),D_2))
     => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),C_5)),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),D_2))) ) ).

tff(fact_860_subset__insertI2,axiom,
    ! [Ba: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).

tff(fact_861_subset__insertI2,axiom,
    ! [Ba: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),B))) ) ).

tff(fact_862_subset__insertI2,axiom,
    ! [Ba: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
     => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),B))) ) ).

tff(fact_863_subset__insert,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
      <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

tff(fact_864_subset__insert,axiom,
    ! [B: fun_int_bool,X_1: int,A: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).

tff(fact_865_subset__insert,axiom,
    ! [B: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),B)))
      <=> hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B)) ) ) ).

tff(fact_866_insert__subset,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

tff(fact_867_insert__subset,axiom,
    ! [X_1: int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).

tff(fact_868_insert__subset,axiom,
    ! [X_1: hoare_2076525193iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),A)),B))
    <=> ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),B))
        & hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B)) ) ) ).

tff(fact_869_subset__insertI,axiom,
    ! [B: fun_nat_bool,A_13: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_13),B))) ).

tff(fact_870_subset__insertI,axiom,
    ! [B: fun_int_bool,A_13: int] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_13),B))) ).

tff(fact_871_subset__insertI,axiom,
    ! [B: fun_Ho882550911a_bool,A_13: hoare_2076525193iple_a] : hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,B),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_13),B))) ).

tff(fact_872_image__mono,axiom,
    ! [F: fun_int_int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(F,A)),image_int_int(F,B))) ) ).

tff(fact_873_image__mono,axiom,
    ! [F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
     => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,image_1416816325iple_a(F,A)),image_1416816325iple_a(F,B))) ) ).

tff(fact_874_image__mono,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),image_nat_int(F,B))) ) ).

tff(fact_875_image__mono,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_nat_nat(F,A)),image_nat_nat(F,B))) ) ).

tff(fact_876_subset__image__iff,axiom,
    ! [B: fun_int_bool,F: fun_int_int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_int_int(F,A)))
    <=> ? [AA: fun_int_bool] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,AA),A))
          & ( B = image_int_int(F,AA) ) ) ) ).

tff(fact_877_subset__image__iff,axiom,
    ! [B: fun_Ho882550911a_bool,F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,B),image_1416816325iple_a(F,A)))
    <=> ? [AA: fun_Ho882550911a_bool] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,AA),A))
          & ( B = image_1416816325iple_a(F,AA) ) ) ) ).

tff(fact_878_subset__image__iff,axiom,
    ! [B: fun_int_bool,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
    <=> ? [AA: fun_nat_bool] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,AA),A))
          & ( B = image_nat_int(F,AA) ) ) ) ).

tff(fact_879_subset__image__iff,axiom,
    ! [B: fun_nat_bool,F: fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_nat_nat(F,A)))
    <=> ? [AA: fun_nat_bool] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,AA),A))
          & ( B = image_nat_nat(F,AA) ) ) ) ).

tff(fact_880_double__diff,axiom,
    ! [C_5: fun_int_bool,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_5))
       => ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_5),A)) = A ) ) ) ).

tff(fact_881_double__diff,axiom,
    ! [C_5: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,B),C_5))
       => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,B),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,C_5),A)) = A ) ) ) ).

tff(fact_882_double__diff,axiom,
    ! [C_5: fun_nat_bool,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_5))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_5),A)) = A ) ) ) ).

tff(fact_883_Diff__mono,axiom,
    ! [D_2: fun_int_bool,B: fun_int_bool,A: fun_int_bool,C_5: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_5))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,D_2),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_5),D_2))) ) ) ).

tff(fact_884_Diff__mono,axiom,
    ! [D_2: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool,C_5: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),C_5))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,D_2),B))
       => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,C_5),D_2))) ) ) ).

tff(fact_885_Diff__mono,axiom,
    ! [D_2: fun_nat_bool,B: fun_nat_bool,A: fun_nat_bool,C_5: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_5))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,D_2),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_5),D_2))) ) ) ).

tff(fact_886_Diff__subset,axiom,
    ! [A: fun_int_bool,B: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),A)) ).

tff(fact_887_Diff__subset,axiom,
    ! [A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)),A)) ).

tff(fact_888_Diff__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),A)) ).

tff(fact_889_weaken,axiom,
    ! [Ts: fun_Ho882550911a_bool,Ga: fun_Ho882550911a_bool,Ts_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts_1))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,Ts),Ts_1))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts)) ) ) ).

tff(fact_890_asm,axiom,
    ! [Ts: fun_Ho882550911a_bool,Ga: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,Ts),Ga))
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts)) ) ).

tff(fact_891_subset__singletonD,axiom,
    ! [A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))
     => ( ( A = bot_bot_fun_nat_bool )
        | ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_892_subset__singletonD,axiom,
    ! [A: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))
     => ( ( A = bot_bot_fun_int_bool )
        | ( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool) ) ) ) ).

tff(fact_893_subset__singletonD,axiom,
    ! [A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool)))
     => ( ( A = bot_bo214064617a_bool )
        | ( A = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool) ) ) ) ).

tff(fact_894_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_int_int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_int_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_895_finite__surj,axiom,
    ! [B: fun_Ho882550911a_bool,F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,B),image_1416816325iple_a(F,A)))
       => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B)) ) ) ).

tff(fact_896_finite__surj,axiom,
    ! [B: fun_int_bool,F: fun_nat_int,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

tff(fact_897_finite__surj,axiom,
    ! [B: fun_nat_bool,F: fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_nat_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

tff(fact_898_image__diff__subset,axiom,
    ! [F: fun_int_int,A: fun_int_bool,B: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,image_int_int(F,A)),image_int_int(F,B))),image_int_int(F,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))) ).

tff(fact_899_image__diff__subset,axiom,
    ! [F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,image_1416816325iple_a(F,A)),image_1416816325iple_a(F,B))),image_1416816325iple_a(F,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),B)))) ).

tff(fact_900_image__diff__subset,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool,B: fun_nat_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,image_nat_int(F,A)),image_nat_int(F,B))),image_nat_int(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ).

tff(fact_901_image__diff__subset,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool,B: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,image_nat_nat(F,A)),image_nat_nat(F,B))),image_nat_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ).

tff(fact_902_le__add__iff1,axiom,
    ! [A_13: int,E: int,Ca: int,Ba: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_13),Ba)),E)),Ca)),D_1)) ) ).

tff(fact_903_le__add__iff2,axiom,
    ! [A_13: int,E: int,Ca: int,Ba: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_13),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ba),E)),D_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ba),A_13)),E)),D_1))) ) ).

tff(fact_904_diff__single__insert,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B))) ) ) ).

tff(fact_905_diff__single__insert,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B))) ) ) ).

tff(fact_906_diff__single__insert,axiom,
    ! [A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool))),B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
       => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),B))) ) ) ).

tff(fact_907_subset__insert__iff,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
    <=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B)) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ) ).

tff(fact_908_subset__insert__iff,axiom,
    ! [A: fun_int_bool,X_1: int,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
    <=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B)) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ) ).

tff(fact_909_subset__insert__iff,axiom,
    ! [A: fun_Ho882550911a_bool,X_1: hoare_2076525193iple_a,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),B)))
    <=> ( ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_1),bot_bo214064617a_bool))),B)) )
        & ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_1),A))
         => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B)) ) ) ) ).

tff(fact_910_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A))
       => ( ( B != bot_bo214064617a_bool )
         => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,B),A))
           => ( hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,hAPP_f2009966983iple_a(F_1,B)),hAPP_f2009966983iple_a(F_1,A)) = hAPP_f2009966983iple_a(F_1,A) ) ) ) ) ) ).

tff(fact_911_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_int_bool,A: fun_int_bool,F: fun_int_fun_int_int,F_1: fun_fun_int_bool_int] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( B != bot_bot_fun_int_bool )
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
           => ( hAPP_int_int(hAPP_int_fun_int_int(F,hAPP_f1594865479ol_int(F_1,B)),hAPP_f1594865479ol_int(F_1,A)) = hAPP_f1594865479ol_int(F_1,A) ) ) ) ) ) ).

tff(fact_912_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( B != bot_bot_fun_nat_bool )
         => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,B)),hAPP_f22106695ol_nat(F_1,A)) = hAPP_f22106695ol_nat(F_1,A) ) ) ) ) ) ).

tff(fact_913_finite__subset__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,A: fun_nat_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F_1),A))
       => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
         => ( ! [A_12: nat,F_2: fun_nat_bool] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),A))
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_12),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_12),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_914_finite__subset__induct,axiom,
    ! [Pa: fun_fu819804446l_bool,A: fun_int_bool,F_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F_1),A))
       => ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
         => ( ! [A_12: int,F_2: fun_int_bool] :
                ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
               => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),A))
                 => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_12),F_2))
                   => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                     => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_12),F_2))) ) ) ) )
           => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_915_finite__subset__induct,axiom,
    ! [Pa: fun_fu2026518849l_bool,A: fun_Ho882550911a_bool,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,F_1),A))
       => ( hBOOL(hAPP_f500854239l_bool(Pa,bot_bo214064617a_bool))
         => ( ! [A_12: hoare_2076525193iple_a,F_2: fun_Ho882550911a_bool] :
                ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_2))
               => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_12),A))
                 => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_12),F_2))
                   => ( hBOOL(hAPP_f500854239l_bool(Pa,F_2))
                     => hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_12),F_2))) ) ) ) )
           => hBOOL(hAPP_f500854239l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_916_subsetI,axiom,
    ! [B: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool] :
      ( ! [X_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
         => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),B)) )
     => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),B)) ) ).

tff(fact_917_subsetI,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).

tff(fact_918_subsetI,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).

tff(fact_919_finite__subset__image,axiom,
    ! [F: fun_int_int,A: fun_int_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_int_int(F,A)))
       => ? [C_4: fun_int_bool] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_4),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_4))
            & ( B = image_int_int(F,C_4) ) ) ) ) ).

tff(fact_920_finite__subset__image,axiom,
    ! [F: fun_Ho1518252029iple_a,A: fun_Ho882550911a_bool,B: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,B),image_1416816325iple_a(F,A)))
       => ? [C_4: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,C_4),A))
            & hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,C_4))
            & ( B = image_1416816325iple_a(F,C_4) ) ) ) ) ).

tff(fact_921_finite__subset__image,axiom,
    ! [F: fun_nat_int,A: fun_nat_bool,B: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
       => ? [C_4: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_4),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_4))
            & ( B = image_nat_int(F,C_4) ) ) ) ) ).

tff(fact_922_finite__subset__image,axiom,
    ! [F: fun_nat_nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_nat_nat(F,A)))
       => ? [C_4: fun_nat_bool] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_4),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_4))
            & ( B = image_nat_nat(F,C_4) ) ) ) ) ).

tff(fact_923_setsum__mono,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,K_3: fun_int_bool] :
      ( ! [I_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,I_2),K_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(F,I_2)),hAPP_int_nat(G,I_2))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,F),K_3)),hAPP_f957591787ol_nat(hAPP_f1109019371ol_nat(big_co1740723097nt_nat,G),K_3))) ) ).

tff(fact_924_setsum__mono,axiom,
    ! [F: fun_Ho864122878_a_nat,G: fun_Ho864122878_a_nat,K_3: fun_Ho882550911a_bool] :
      ( ! [I_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,I_2),K_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_H174304918_a_nat(F,I_2)),hAPP_H174304918_a_nat(G,I_2))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,F),K_3)),hAPP_f190641160ol_nat(hAPP_f1697440293ol_nat(big_co922834230_a_nat,G),K_3))) ) ).

tff(fact_925_setsum__mono,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,K_3: fun_nat_bool] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,I_2),K_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(G,I_2))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),K_3)),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,G),K_3))) ) ).

tff(fact_926_image__subsetI,axiom,
    ! [F: fun_int_int,B: fun_int_bool,A: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(F,X_2)),B)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(F,A)),B)) ) ).

tff(fact_927_image__subsetI,axiom,
    ! [F: fun_Ho1518252029iple_a,B: fun_Ho882550911a_bool,A: fun_Ho882550911a_bool] :
      ( ! [X_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A))
         => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_H1789220885iple_a(F,X_2)),B)) )
     => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,image_1416816325iple_a(F,A)),B)) ) ).

tff(fact_928_image__subsetI,axiom,
    ! [F: fun_nat_int,B: fun_int_bool,A: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_2)),B)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),B)) ) ).

tff(fact_929_image__subsetI,axiom,
    ! [F: fun_nat_nat,B: fun_nat_bool,A: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(F,X_2)),B)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_nat_nat(F,A)),B)) ) ).

tff(fact_930_le__funI,axiom,
    ! [F: fun_int_bool,G: fun_int_bool] :
      ( ! [X_2: int] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G,X_2)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)) ) ).

tff(fact_931_le__funI,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool] :
      ( ! [X_2: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G,X_2)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G)) ) ).

tff(fact_932_flat__lub__def,axiom,
    ! [A: fun_int_bool,Ba: int] :
      ( ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)))
       => ( hAPP_f1594865479ol_int(partial_flat_lub_int(Ba),A) = Ba ) )
      & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)))
       => ( hAPP_f1594865479ol_int(partial_flat_lub_int(Ba),A) = hAPP_f1594865479ol_int(the_int,hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Ba),bot_bot_fun_int_bool)))) ) ) ) ).

tff(fact_933_flat__lub__def,axiom,
    ! [A: fun_Ho882550911a_bool,Ba: hoare_2076525193iple_a] :
      ( ( hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),bot_bo214064617a_bool)))
       => ( hAPP_f2009966983iple_a(partia428461311iple_a(Ba),A) = Ba ) )
      & ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),bot_bo214064617a_bool)))
       => ( hAPP_f2009966983iple_a(partia428461311iple_a(Ba),A) = hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Ba),bot_bo214064617a_bool)))) ) ) ) ).

tff(fact_934_flat__lub__def,axiom,
    ! [A: fun_nat_bool,Ba: nat] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))
       => ( hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A) = Ba ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))
       => ( hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A) = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_935_sum__diff__distrib,axiom,
    ! [N_1: nat,Q_1: fun_nat_nat,Pa: fun_nat_nat] :
      ( ! [X_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_1,X_2)),hAPP_nat_nat(Pa,X_2)))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Pa),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_1),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),Pa)),Q_1)),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1)) ) ) ).

tff(fact_936_fold1__antimono,axiom,
    ! [B: fun_int_bool,A: fun_int_bool] :
      ( ( A != bot_bot_fun_int_bool )
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f1594865479ol_int(finite_fold1_int(ord_min_int),B)),hAPP_f1594865479ol_int(finite_fold1_int(ord_min_int),A))) ) ) ) ).

tff(fact_937_fold1__antimono,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( ( A != bot_bot_fun_nat_bool )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_fold1_nat(ord_min_nat),B)),hAPP_f22106695ol_nat(finite_fold1_nat(ord_min_nat),A))) ) ) ) ).

tff(fact_938_predicate1I,axiom,
    ! [Q_1: fun_int_bool,Pa: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(Pa,X_2))
         => hBOOL(hAPP_int_bool(Q_1,X_2)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Pa),Q_1)) ) ).

tff(fact_939_predicate1I,axiom,
    ! [Q_1: fun_nat_bool,Pa: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_nat_bool(Pa,X_2))
         => hBOOL(hAPP_nat_bool(Q_1,X_2)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Pa),Q_1)) ) ).

tff(fact_940_Collect__mono,axiom,
    ! [Q_1: fun_int_bool,Pa: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(Pa,X_2))
         => hBOOL(hAPP_int_bool(Q_1,X_2)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(collect_int,Pa)),hAPP_f1805168059t_bool(collect_int,Q_1))) ) ).

tff(fact_941_Collect__mono,axiom,
    ! [Q_1: fun_Ho882550911a_bool,Pa: fun_Ho882550911a_bool] :
      ( ! [X_2: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_H454055633a_bool(Pa,X_2))
         => hBOOL(hAPP_H454055633a_bool(Q_1,X_2)) )
     => hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(ord_le940607413a_bool,hAPP_f642208769a_bool(collec977865374iple_a,Pa)),hAPP_f642208769a_bool(collec977865374iple_a,Q_1))) ) ).

tff(fact_942_Collect__mono,axiom,
    ! [Q_1: fun_nat_bool,Pa: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_nat_bool(Pa,X_2))
         => hBOOL(hAPP_nat_bool(Q_1,X_2)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(collect_nat,Pa)),hAPP_f800510211t_bool(collect_nat,Q_1))) ) ).

tff(fact_943_Min__antimono,axiom,
    ! [M_3: fun_int_bool,N_3: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,M_3),N_3))
     => ( ( M_3 != bot_bot_fun_int_bool )
       => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,N_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f1594865479ol_int(big_linorder_Min_int,N_3)),hAPP_f1594865479ol_int(big_linorder_Min_int,M_3))) ) ) ) ).

tff(fact_944_Min__antimono,axiom,
    ! [M_3: fun_nat_bool,N_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,M_3),N_3))
     => ( ( M_3 != bot_bot_fun_nat_bool )
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(big_linorder_Min_nat,N_3)),hAPP_f22106695ol_nat(big_linorder_Min_nat,M_3))) ) ) ) ).

tff(fact_945_Max__mono,axiom,
    ! [M_3: fun_int_bool,N_3: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,M_3),N_3))
     => ( ( M_3 != bot_bot_fun_int_bool )
       => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,N_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f1594865479ol_int(big_linorder_Max_int,M_3)),hAPP_f1594865479ol_int(big_linorder_Max_int,N_3))) ) ) ) ).

tff(fact_946_Max__mono,axiom,
    ! [M_3: fun_nat_bool,N_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,M_3),N_3))
     => ( ( M_3 != bot_bot_fun_nat_bool )
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(big_linorder_Max_nat,M_3)),hAPP_f22106695ol_nat(big_linorder_Max_nat,N_3))) ) ) ) ).

tff(fact_947_finite__lessThan,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K_1))) ).

tff(fact_948_finite__Collect__le__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1)))) ).

tff(fact_949_finite__nat__set__iff__bounded__le,axiom,
    ! [N_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
    <=> ? [M_1: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M_1)) ) ) ).

tff(fact_950_Min_Oin__idem,axiom,
    ! [X_1: int,A: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => ( hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_1),hAPP_f1594865479ol_int(big_linorder_Min_int,A)) = hAPP_f1594865479ol_int(big_linorder_Min_int,A) ) ) ) ).

tff(fact_951_Min_Oin__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),hAPP_f22106695ol_nat(big_linorder_Min_nat,A)) = hAPP_f22106695ol_nat(big_linorder_Min_nat,A) ) ) ) ).

tff(fact_952_Min_Oidem,axiom,
    ! [X_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_18),X_18) = X_18 ).

tff(fact_953_min__max_Oinf__assoc,axiom,
    ! [X_17: nat,Y_12: nat,Z_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_17),Y_12)),Z_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_12),Z_7)) ).

tff(fact_954_min__max_Oinf_Oassoc,axiom,
    ! [A_11: nat,B_10: nat,C_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_11),B_10)),C_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_10),C_3)) ).

tff(fact_955_min__max_Oinf__left__commute,axiom,
    ! [X_16: nat,Y_11: nat,Z_6: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_16),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_11),Z_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_16),Z_6)) ).

tff(fact_956_min__max_Oinf_Oleft__commute,axiom,
    ! [B_9: nat,A_10: nat,C_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_10),C_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_10),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_9),C_2)) ).

tff(fact_957_min__max_Oinf__left__idem,axiom,
    ! [X_15: nat,Y_10: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_15),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_15),Y_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_15),Y_10) ).

tff(fact_958_min__max_Oinf_Oleft__idem,axiom,
    ! [A_9: nat,B_8: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_9),B_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_9),B_8) ).

tff(fact_959_min__max_Oinf__commute,axiom,
    ! [X_14: nat,Y_9: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_14),Y_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_9),X_14) ).

tff(fact_960_min__max_Oinf_Ocommute,axiom,
    ! [A_8: nat,B_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_8),B_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_7),A_8) ).

tff(fact_961_min__max_Oinf_Oidem,axiom,
    ! [A_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_7),A_7) = A_7 ).

tff(fact_962_min__le__iff__disj,axiom,
    ! [X_1: nat,Y_1: nat,Z_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_1)),Z_1))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_1))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),Z_1)) ) ) ).

tff(fact_963_min__le__iff__disj,axiom,
    ! [X_1: int,Y_1: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_1),Y_1)),Z_1))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Z_1))
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),Z_1)) ) ) ).

tff(fact_964_min__max_Ole__infE,axiom,
    ! [X_13: nat,A_6: nat,B_6: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_6),B_6)))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),A_6))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),B_6)) ) ) ).

tff(fact_965_min__max_Ole__infE,axiom,
    ! [X_13: int,A_6: int,B_6: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_13),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_6),B_6)))
     => ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_13),A_6))
         => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_13),B_6)) ) ) ).

tff(fact_966_min__max_Oinf__mono,axiom,
    ! [B_5: nat,D: nat,A_5: nat,C_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),C_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_5),D))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_5),B_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,C_1),D))) ) ) ).

tff(fact_967_min__max_Oinf__mono,axiom,
    ! [B_5: int,D: int,A_5: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_5),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),D))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_5),B_5)),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,C_1),D))) ) ) ).

tff(fact_968_min__max_Oinf__greatest,axiom,
    ! [Z_5: nat,X_12: nat,Y_8: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_12),Y_8))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_12),Z_5))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_12),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_8),Z_5))) ) ) ).

tff(fact_969_min__max_Oinf__greatest,axiom,
    ! [Z_5: int,X_12: int,Y_8: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_12),Y_8))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_12),Z_5))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_12),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,Y_8),Z_5))) ) ) ).

tff(fact_970_min__max_Ole__infI,axiom,
    ! [B_4: nat,X_11: nat,A_4: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),A_4))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),B_4))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_4),B_4))) ) ) ).

tff(fact_971_min__max_Ole__infI,axiom,
    ! [B_4: int,X_11: int,A_4: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),A_4))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),B_4))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_4),B_4))) ) ) ).

tff(fact_972_min__max_Oinf__absorb2,axiom,
    ! [Y_7: nat,X_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_10))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_10),Y_7) = Y_7 ) ) ).

tff(fact_973_min__max_Oinf__absorb2,axiom,
    ! [Y_7: int,X_10: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),X_10))
     => ( hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_10),Y_7) = Y_7 ) ) ).

tff(fact_974_min__max_Oinf__absorb1,axiom,
    ! [X_9: nat,Y_6: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_9),Y_6))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_9),Y_6) = X_9 ) ) ).

tff(fact_975_min__max_Oinf__absorb1,axiom,
    ! [X_9: int,Y_6: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_9),Y_6))
     => ( hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_9),Y_6) = X_9 ) ) ).

tff(fact_976_min__max_Ole__infI2,axiom,
    ! [A_3: nat,B_3: nat,X_8: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_3),X_8))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_3),B_3)),X_8)) ) ).

tff(fact_977_min__max_Ole__infI2,axiom,
    ! [A_3: int,B_3: int,X_8: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_3),X_8))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_3),B_3)),X_8)) ) ).

tff(fact_978_min__max_Ole__infI1,axiom,
    ! [B_2: nat,A_2: nat,X_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),X_7))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_2),B_2)),X_7)) ) ).

tff(fact_979_min__max_Ole__infI1,axiom,
    ! [B_2: int,A_2: int,X_7: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),X_7))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,A_2),B_2)),X_7)) ) ).

tff(fact_980_min__max_Ole__inf__iff,axiom,
    ! [X_1: nat,Y_1: nat,Z_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_1),Z_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_1)) ) ) ).

tff(fact_981_min__max_Ole__inf__iff,axiom,
    ! [X_1: int,Y_1: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,Y_1),Z_1)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Z_1)) ) ) ).

tff(fact_982_min__max_Ole__iff__inf,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_1) = X_1 ) ) ).

tff(fact_983_min__max_Ole__iff__inf,axiom,
    ! [X_1: int,Y_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_1),Y_1) = X_1 ) ) ).

tff(fact_984_min__max_Oinf__le2,axiom,
    ! [X_6: nat,Y_5: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_6),Y_5)),Y_5)) ).

tff(fact_985_min__max_Oinf__le2,axiom,
    ! [X_6: int,Y_5: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_6),Y_5)),Y_5)) ).

tff(fact_986_min__max_Oinf__le1,axiom,
    ! [X_5: nat,Y_4: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_5),Y_4)),X_5)) ).

tff(fact_987_min__max_Oinf__le1,axiom,
    ! [X_5: int,Y_4: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_5),Y_4)),X_5)) ).

tff(fact_988_min__add__distrib__left,axiom,
    ! [X_4: int,Y_3: int,Z_4: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_4),Y_3)),Z_4) = hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_4),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_3),Z_4)) ).

tff(fact_989_min__add__distrib__left,axiom,
    ! [X_4: nat,Y_3: nat,Z_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_4),Y_3)),Z_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_4),Z_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_3),Z_4)) ).

tff(fact_990_min__diff__distrib__left,axiom,
    ! [X_3: int,Y_2: int,Z_3: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,X_3),Y_2)),Z_3) = hAPP_int_int(hAPP_int_fun_int_int(ord_min_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_3),Z_3)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),Z_3)) ).

tff(fact_991_finite__less__ub,axiom,
    ! [U: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F)),U)))) ) ).

tff(fact_992_le__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ) ) ).

tff(fact_993_mult__le__mono,axiom,
    ! [K: nat,L_1: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),L_1))) ) ) ).

tff(fact_994_mult__le__mono2,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

tff(fact_995_mult__le__mono1,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

tff(fact_996_le__cube,axiom,
    ! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M)))) ).

tff(fact_997_le__square,axiom,
    ! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M))) ).

tff(fact_998_le__refl,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).

tff(fact_999_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).

tff(fact_1000_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1001_le__trans,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K)) ) ) ).

tff(fact_1002_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

tff(fact_1003_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ) ).

tff(fact_1004_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1005_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ).

tff(fact_1006_add__le__mono,axiom,
    ! [K: nat,L_1: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).

tff(fact_1007_add__le__mono1,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

tff(fact_1008_trans__le__add2,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

tff(fact_1009_trans__le__add1,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

tff(fact_1010_nat__add__left__cancel__le,axiom,
    ! [K_1: nat,M_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ).

tff(fact_1011_le__iff__add,axiom,
    ! [M_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
    <=> ? [K_2: nat] : N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_2) ) ).

tff(fact_1012_le__add1,axiom,
    ! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

tff(fact_1013_le__add2,axiom,
    ! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

tff(fact_1014_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

tff(fact_1015_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),N)) ).

tff(fact_1016_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

tff(fact_1017_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

tff(fact_1018_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

tff(fact_1019_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

tff(fact_1020_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K)) ).

tff(fact_1021_diff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).

tff(fact_1022_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).

tff(fact_1023_diff__le__self,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

tff(fact_1024_diff__diff__right,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

tff(fact_1025_le__diff__conv,axiom,
    ! [J_1: nat,K_1: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_1)),I_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))) ) ).

tff(fact_1026_le__add__diff,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),K))) ) ).

tff(fact_1027_diff__le__mono2,axiom,
    ! [L_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M))) ) ).

tff(fact_1028_diff__le__mono,axiom,
    ! [L_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),L_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),L_1))) ) ).

tff(fact_1029_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

tff(fact_1030_le__add__diff__inverse,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

tff(fact_1031_add__diff__assoc,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

tff(fact_1032_le__diff__conv2,axiom,
    ! [I_1: nat,K_1: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J_1)) ) ) ).

tff(fact_1033_le__add__diff__inverse2,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = M ) ) ).

tff(fact_1034_le__imp__diff__is__add,axiom,
    ! [K_1: nat,I_1: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1) = K_1 )
      <=> ( J_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) ) ) ) ).

tff(fact_1035_diff__add__assoc,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

tff(fact_1036_add__diff__assoc2,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

tff(fact_1037_diff__add__assoc2,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

tff(fact_1038_eq__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1) )
        <=> ( M_2 = N_1 ) ) ) ) ).

tff(fact_1039_Nat_Odiff__diff__eq,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

tff(fact_1040_nat__eq__add__iff2,axiom,
    ! [U: nat,M_2: nat,N_1: nat,I_1: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1) )
      <=> ( M_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1)),U)),N_1) ) ) ) ).

tff(fact_1041_nat__diff__add__eq2,axiom,
    ! [U_1: nat,M: nat,N: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I)),U_1)),N)) ) ) ).

tff(fact_1042_nat__le__add__iff2,axiom,
    ! [U: nat,M_2: nat,N_1: nat,I_1: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1)),U)),N_1))) ) ) ).

tff(fact_1043_nat__eq__add__iff1,axiom,
    ! [U: nat,M_2: nat,N_1: nat,J_1: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1) )
      <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_1)),U)),M_2) = N_1 ) ) ) ).

tff(fact_1044_nat__diff__add__eq1,axiom,
    ! [U_1: nat,M: nat,N: nat,J: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),U_1)),M)),N) ) ) ).

tff(fact_1045_nat__mult__assoc,axiom,
    ! [M: nat,N: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K)) ).

tff(fact_1046_nat__mult__commute,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M) ).

tff(fact_1047_nat__add__right__cancel,axiom,
    ! [M_2: nat,K_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1) )
    <=> ( M_2 = N_1 ) ) ).

tff(fact_1048_nat__add__left__cancel,axiom,
    ! [K_1: nat,M_2: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1) )
    <=> ( M_2 = N_1 ) ) ).

tff(fact_1049_nat__add__assoc,axiom,
    ! [M: nat,N: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K)) ).

tff(fact_1050_nat__add__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

tff(fact_1051_nat__add__commute,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M) ).

tff(fact_1052_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),N)) ).

tff(fact_1053_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K)) ).

tff(fact_1054_left__add__mult__distrib,axiom,
    ! [I: nat,U_1: nat,J: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),U_1)),K) ).

tff(fact_1055_nat__le__add__iff1,axiom,
    ! [U: nat,M_2: nat,N_1: nat,J_1: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_1)),U)),M_2)),N_1)) ) ) ).

tff(fact_1056_termination__basic__simps_I3_J,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1057_termination__basic__simps_I4_J,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1058_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),M) ).

tff(fact_1059_card__lessThan,axiom,
    ! [U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U ).

tff(fact_1060_nat__mult__eq__1__iff,axiom,
    ! [M_2: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_1) = one_one_nat )
    <=> ( ( M_2 = one_one_nat )
        & ( N_1 = one_one_nat ) ) ) ).

tff(fact_1061_nat__mult__1__right,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

tff(fact_1062_nat__1__eq__mult__iff,axiom,
    ! [M_2: nat,N_1: nat] :
      ( ( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_1) )
    <=> ( ( M_2 = one_one_nat )
        & ( N_1 = one_one_nat ) ) ) ).

tff(fact_1063_nat__mult__1,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

tff(fact_1064_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N)),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).

tff(fact_1065_zadd__int,axiom,
    ! [M: nat,N: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

tff(fact_1066_zle__int,axiom,
    ! [M_2: nat,N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ).

tff(fact_1067_zmult__int,axiom,
    ! [M: nat,N: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) ).

tff(fact_1068_int__mult,axiom,
    ! [M: nat,N: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) ).

tff(fact_1069_int__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).

tff(fact_1070_zdiff__int,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

tff(fact_1071_negative__zle,axiom,
    ! [N: nat,M: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),hAPP_nat_int(semiri1621563631at_int,M))) ).

tff(fact_1072_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U))) ).

tff(fact_1073_finite__Collect__less__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1)))) ).

tff(fact_1074_zle__iff__zadd,axiom,
    ! [W: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1))
    <=> ? [N_2: nat] : Z_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),hAPP_nat_int(semiri1621563631at_int,N_2)) ) ).

tff(fact_1075_int__power,axiom,
    ! [M: nat,N: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) ).

tff(fact_1076_zpower__int,axiom,
    ! [M: nat,N: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) ).

tff(fact_1077_int__int__eq,axiom,
    ! [M_2: nat,N_1: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,M_2) = hAPP_nat_int(semiri1621563631at_int,N_1) )
    <=> ( M_2 = N_1 ) ) ).

tff(fact_1078_zmult__zminus,axiom,
    ! [Z: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z)),W_1) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),W_1)) ).

tff(fact_1079_diff__int__def__symmetric,axiom,
    ! [Z: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),hAPP_int_int(uminus_uminus_int,W_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z),W_1) ).

tff(fact_1080_diff__int__def,axiom,
    ! [Z: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z),W_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),hAPP_int_int(uminus_uminus_int,W_1)) ).

tff(fact_1081_zminus__zadd__distrib,axiom,
    ! [Z: int,W_1: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),W_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z)),hAPP_int_int(uminus_uminus_int,W_1)) ).

tff(fact_1082_zle__refl,axiom,
    ! [W_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),W_1)) ).

tff(fact_1083_zle__linear,axiom,
    ! [Z: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W_1))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z)) ) ).

tff(fact_1084_zle__trans,axiom,
    ! [K: int,I: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),K))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K)) ) ) ).

tff(fact_1085_zle__antisym,axiom,
    ! [Z: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z))
       => ( Z = W_1 ) ) ) ).

tff(fact_1086_zmult__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3)) ).

tff(fact_1087_zpower__zpower,axiom,
    ! [X: int,Y: nat,Z: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),Z) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z)) ).

tff(fact_1088_zmult__commute,axiom,
    ! [Z: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),W_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z) ).

tff(fact_1089_zdiff__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W_1)) ).

tff(fact_1090_zdiff__zmult__distrib2,axiom,
    ! [W_1: int,Z1: int,Z2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z2)) ).

tff(fact_1091_zadd__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) ).

tff(fact_1092_zadd__left__commute,axiom,
    ! [X: int,Y: int,Z: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z)) ).

tff(fact_1093_zadd__commute,axiom,
    ! [Z: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),W_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),Z) ).

tff(fact_1094_zpower__zadd__distrib,axiom,
    ! [X: int,Y: nat,Z: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z)) ).

tff(fact_1095_zadd__zmult__distrib2,axiom,
    ! [W_1: int,Z1: int,Z2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z2)) ).

tff(fact_1096_zadd__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W_1)) ).

tff(fact_1097_zadd__left__mono,axiom,
    ! [K: int,I: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),J))) ) ).

tff(fact_1098_zmult__1__right,axiom,
    ! [Z: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),one_one_int) = Z ).

tff(fact_1099_zmult__1,axiom,
    ! [Z: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z) = Z ).

tff(fact_1100_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).

tff(fact_1101_less__add__eq__less,axiom,
    ! [M: nat,N: nat,K: nat,L_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_1102_add__less__mono,axiom,
    ! [K: nat,L_1: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).

tff(fact_1103_add__less__mono1,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

tff(fact_1104_trans__less__add2,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

tff(fact_1105_trans__less__add1,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

tff(fact_1106_nat__add__left__cancel__less,axiom,
    ! [K_1: nat,M_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1)) ) ).

tff(fact_1107_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

tff(fact_1108_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

tff(fact_1109_nat__less__le,axiom,
    ! [M_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
        & ( M_2 != N_1 ) ) ) ).

tff(fact_1110_le__eq__less__or__eq,axiom,
    ! [M_2: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
        | ( M_2 = N_1 ) ) ) ).

tff(fact_1111_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1112_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_1113_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1114_less__imp__diff__less,axiom,
    ! [N: nat,J: nat,K: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).

tff(fact_1115_diff__less__mono2,axiom,
    ! [L_1: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M))) ) ) ).

tff(fact_1116_nat__less__cases,axiom,
    ! [Pa: fun_nat_fun_nat_bool,M_2: nat,N_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) )
     => ( ( ( M_2 = N_1 )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) )
       => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_2))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M_2)) ) ) ) ).

tff(fact_1117_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

tff(fact_1118_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

tff(fact_1119_less__irrefl__nat,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

tff(fact_1120_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).

tff(fact_1121_nat__neq__iff,axiom,
    ! [M_2: nat,N_1: nat] :
      ( ( M_2 != N_1 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_2)) ) ) ).

tff(fact_1122_less__not__refl,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

tff(fact_1123_termination__basic__simps_I5_J,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).

tff(fact_1124_termination__basic__simps_I2_J,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1125_termination__basic__simps_I1_J,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1126_image__add__atLeastAtMost,axiom,
    ! [K_1: nat,I_1: nat,J_1: nat] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J_1)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),K_1)) ).

tff(fact_1127_finite__nat__set__iff__bounded,axiom,
    ! [N_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
    <=> ? [M_1: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),M_1)) ) ) ).

tff(fact_1128_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun_nat_bool,I_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) ).

tff(fact_1129_card__Collect__less__nat,axiom,
    ! [N_1: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N_1))) = N_1 ).

tff(fact_1130_add__diff__inverse,axiom,
    ! [M: nat,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

tff(fact_1131_less__diff__conv,axiom,
    ! [I_1: nat,J_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J_1)) ) ).

tff(fact_1132_less__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1)) ) ) ) ).

tff(fact_1133_diff__less__mono,axiom,
    ! [C: nat,A_1: nat,B_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_1),C))) ) ) ).

tff(fact_1134_nat__less__add__iff1,axiom,
    ! [U: nat,M_2: nat,N_1: nat,J_1: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_1)),U)),M_2)),N_1)) ) ) ).

tff(fact_1135_nat__less__add__iff2,axiom,
    ! [U: nat,M_2: nat,N_1: nat,I_1: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1)),U)),N_1))) ) ) ).

tff(fact_1136_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M)))) ).

tff(fact_1137_zless__le,axiom,
    ! [Z_1: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),W))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
        & ( Z_1 != W ) ) ) ).

tff(fact_1138_zadd__strict__right__mono,axiom,
    ! [K: int,I: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J),K))) ) ).

tff(fact_1139_zminus__zminus,axiom,
    ! [Z: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z)) = Z ).

tff(fact_1140_zadd__zless__mono,axiom,
    ! [Z_2: int,Z: int,W_2: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_2),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),Z))) ) ) ).

tff(fact_1141_zless__int,axiom,
    ! [M_2: nat,N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1)) ) ).

tff(fact_1142_zless__add1__eq,axiom,
    ! [W: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),one_one_int)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1))
        | ( W = Z_1 ) ) ) ).

tff(fact_1143_atLeastAtMostPlus1__int__conv,axiom,
    ! [M_2: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),N_1)))
     => ( hAPP_i1948725293t_bool(ord_at875362053st_int(M_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),N_1)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),N_1)),hAPP_i1948725293t_bool(ord_at875362053st_int(M_2),N_1)) ) ) ).

tff(fact_1144_zless__imp__add1__zle,axiom,
    ! [W_1: int,Z: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z)) ) ).

tff(fact_1145_add1__zle__eq,axiom,
    ! [W: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1)) ) ).

tff(fact_1146_zle__add1__eq__le,axiom,
    ! [W: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) ) ).

tff(fact_1147_zle__diff1__eq,axiom,
    ! [W: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1)) ) ).

tff(fact_1148_int__le__induct,axiom,
    ! [Pa: fun_int_bool,I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
     => ( hBOOL(hAPP_int_bool(Pa,K_1))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(Pa,I_2))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).

tff(fact_1149_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U))) ).

tff(fact_1150_zless__linear,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
      | ( X = Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ).

tff(fact_1151_simp__from__to,axiom,
    ! [J_1: int,I_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),I_1))
       => ( hAPP_i1948725293t_bool(ord_at875362053st_int(I_1),J_1) = bot_bot_fun_int_bool ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),I_1))
       => ( hAPP_i1948725293t_bool(ord_at875362053st_int(I_1),J_1) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,I_1),hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_1),one_one_int)),J_1)) ) ) ) ).

tff(fact_1152_int__less__induct,axiom,
    ! [Pa: fun_int_bool,I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
     => ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),one_one_int)))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(Pa,I_2))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).

tff(fact_1153_transfer__nat__int__set__relations_I1_J,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))) ) ).

tff(fact_1154_transfer__nat__int__set__relations_I5_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

tff(fact_1155_transfer__nat__int__set__relations_I2_J,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_1)),image_nat_int(semiri1621563631at_int,A))) ) ).

tff(fact_1156_transfer__nat__int__set__relations_I3_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( A = B )
    <=> ( image_nat_int(semiri1621563631at_int,A) = image_nat_int(semiri1621563631at_int,B) ) ) ).

tff(fact_1157_int__if__cong,axiom,
    ! [X_1: nat,Y_1: nat,Pa: bool] :
      ( ( hBOOL(Pa)
       => ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) ) )
      & ( ~ hBOOL(Pa)
       => ( hAPP_nat_int(semiri1621563631at_int,Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) ) ) ) ).

tff(fact_1158_transfer__int__nat__relations_I1_J,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_1) )
    <=> ( X_1 = Y_1 ) ) ).

tff(fact_1159_transfer__nat__int__set__relations_I4_J,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

tff(fact_1160_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [A: fun_nat_bool] : hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A)) ).

tff(fact_1161_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
    bot_bot_fun_int_bool = image_nat_int(semiri1621563631at_int,bot_bot_fun_nat_bool) ).

tff(fact_1162_transfer__int__nat__relations_I2_J,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) ) ).

tff(fact_1163_transfer__int__nat__relations_I3_J,axiom,
    ! [X_1: nat,Y_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).

tff(fact_1164_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) ).

tff(fact_1165_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) ).

tff(fact_1166_transfer__int__nat__numerals_I2_J,axiom,
    one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).

tff(fact_1167_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
    ! [X: nat,N: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)) ).

tff(fact_1168_int__gr__induct,axiom,
    ! [Pa: fun_int_bool,K_1: int,I_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
     => ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int)))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2))
             => ( hBOOL(hAPP_int_bool(Pa,I_2))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).

tff(fact_1169_int__ge__induct,axiom,
    ! [Pa: fun_int_bool,K_1: int,I_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
     => ( hBOOL(hAPP_int_bool(Pa,K_1))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
             => ( hBOOL(hAPP_int_bool(Pa,I_2))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).

tff(fact_1170_mono__nat__linear__lb,axiom,
    ! [M_2: nat,K_1: nat,F: fun_nat_nat] :
      ( ! [M_1: nat,N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,N_2))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,M_2)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_1)))) ) ).

tff(fact_1171_bounded__nat__set__is__finite,axiom,
    ! [N_1: nat,N_3: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),N_1)) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3)) ) ).

tff(fact_1172_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M_1: nat,N_2: nat] : Z != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)) ).

tff(fact_1173_less__mono__imp__le__mono,axiom,
    ! [I_1: nat,J_1: nat,F: fun_nat_nat] :
      ( ! [I_2: nat,J_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_2))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_1))) ) ) ).

tff(fact_1174_int__induct,axiom,
    ! [I_1: int,Pa: fun_int_bool,K_1: int] :
      ( hBOOL(hAPP_int_bool(Pa,K_1))
     => ( ! [I_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
           => ( hBOOL(hAPP_int_bool(Pa,I_2))
             => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(Pa,I_2))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_1)) ) ) ) ).

tff(fact_1175_less__zeroE,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_1176_le0,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

tff(fact_1177_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr375877188st_nat(L),U))) ).

tff(fact_1178_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
    ! [N: nat,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).

tff(fact_1179_transfer__int__nat__quantifiers_I1_J,axiom,
    ! [Pa: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => hBOOL(hAPP_int_bool(Pa,X_2)) )
    <=> ! [X_2: nat] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_1180_transfer__int__nat__quantifiers_I2_J,axiom,
    ! [Pa: fun_int_bool] :
      ( ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_int_bool(Pa,X_2)) )
    <=> ? [X_2: nat] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_1181_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
    ! [Z: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z))) ).

tff(fact_1182_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

tff(fact_1183_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_1184_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_1185_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

tff(fact_1186_int__less__0__conv,axiom,
    ! [K: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K)),zero_zero_int)) ).

tff(fact_1187_zero__less__int__conv,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_1188_less__bin__lemma,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),L)),zero_zero_int)) ) ).

tff(fact_1189_zmult__zless__mono2,axiom,
    ! [K: int,I: int,J: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),J))) ) ) ).

tff(fact_1190_int__0__less__1,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).

tff(fact_1191_nat__zero__less__power__iff,axiom,
    ! [X_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1))
        | ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_1192_nat__power__less__imp__less,axiom,
    ! [M: nat,N: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_1193_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

tff(fact_1194_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

tff(fact_1195_less__nat__zero__code,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_1196_neq0__conv,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_1197_not__less0,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

%----Helper facts (165)
tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fNot,P))
      | ~ hBOOL(P) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fNot,P)) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(P) ) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(Q) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
      | hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ hBOOL(fFalse) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
      | ~ hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBI_1_1_COMBI_000tc__Int__Oint_U,axiom,
    ! [P: int] : hAPP_int_int(cOMBI_int,P) = P ).

tff(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
    ! [P: nat] : hAPP_nat_nat(cOMBI_nat,P) = P ).

tff(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
    ! [X: state,Y: state] :
      ( ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
    ! [X: state,Y: state] :
      ( ( X != Y )
      | hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: int,Q: int] : hAPP_int_int(hAPP_int_fun_int_int(cOMBK_int_int,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P: int,Q: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(cOMBK_int_nat,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Int__Oint_U,axiom,
    ! [P: nat,Q: int] : hAPP_int_nat(hAPP_nat_fun_int_nat(cOMBK_nat_int,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: nat,Q: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: bool,Q: int] : hAPP_int_bool(hAPP_b396694332t_bool(cOMBK_bool_int,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: bool,Q: nat] : hAPP_nat_bool(hAPP_b1013836512t_bool(cOMBK_bool_nat,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
    ! [P: bool,Q: state] : hAPP_state_bool(hAPP_b2019457360e_bool(cOMBK_bool_state,P),Q) = P ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_int_bool,Y: fun_int_bool] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_int_bool,Y: fun_int_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_nat_bool,Y: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_nat_bool,Y: fun_nat_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_int,Q: fun_nat_int,R: nat] : hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R) = hAPP_int_int(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: fun_nat_int,Q: fun_int_nat,R: int] : hAPP_int_int(hAPP_f147134065nt_int(hAPP_f1673907925nt_int(cOMBB_nat_int_int,P),Q),R) = hAPP_nat_int(P,hAPP_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Int__Oint_U,axiom,
    ! [P: fun_nat_nat,Q: fun_int_nat,R: int] : hAPP_int_nat(hAPP_f2132704789nt_nat(hAPP_f901215189nt_nat(cOMBB_nat_nat_int,P),Q),R) = hAPP_nat_nat(P,hAPP_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R) = hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_fun_int_nat,Q: int,R: int] : hAPP_int_nat(hAPP_int_fun_int_nat(hAPP_f909851349nt_nat(cOMBC_int_int_nat,P),Q),R) = hAPP_int_nat(hAPP_int_fun_int_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_fun_nat_nat,Q: nat,R: int] : hAPP_int_nat(hAPP_nat_fun_int_nat(hAPP_f879494613nt_nat(cOMBC_int_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_int_fun_nat_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_fun_int_nat,Q: int,R: nat] : hAPP_nat_nat(hAPP_int_fun_nat_nat(hAPP_f446977493at_nat(cOMBC_nat_int_nat,P),Q),R) = hAPP_int_nat(hAPP_nat_fun_int_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: nat,R: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_fun_nat_nat,Q: fun_int_nat,R: int] : hAPP_int_nat(hAPP_f2132704789nt_nat(hAPP_f1399363134nt_nat(cOMBS_int_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_int_fun_nat_nat(P,R),hAPP_int_nat(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_int_bool,Q: int,R: int] : hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_nat_bool,Q: nat,R: int] : hAPP_int_bool(hAPP_n1082236369t_bool(hAPP_f175561985t_bool(cOMBC_int_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_i418383825t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P: fun_nat_fun_int_bool,Q: int,R: nat] : hAPP_nat_bool(hAPP_i418383825t_bool(hAPP_f482008321t_bool(cOMBC_nat_int_bool,P),Q),R) = hAPP_int_bool(hAPP_n1082236369t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_bool_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_in531499254l_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).

tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a_U,axiom,
    ! [P: fun_state_bool,Q: x_a] : hAPP_a2036067514e_bool(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
    ! [P: fun_bool_bool,Q: fun_state_bool,R: state] : hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f2073279419e_bool(cOMBB_160679318_state,P),Q),R) = hAPP_bool_bool(P,hAPP_state_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st2063251938l_bool,Q: bool,R: state] : hAPP_state_bool(hAPP_b2019457360e_bool(hAPP_f167292325e_bool(cOMBC_231445413l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st2063251938l_bool,Q: fun_state_bool,R: state] : hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f644196280e_bool(cOMBS_1378840469l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),hAPP_state_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U,axiom,
    ! [P: fun_st2072884720_state,Q: fun_state_nat,R: state] : hAPP_state_state(hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,P),Q),R) = hAPP_nat_state(hAPP_s1892499976_state(P,R),hAPP_state_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Com__Ostate_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
    ! [P: fun_state_bool,Q: fun_state_state,R: state] : hAPP_state_bool(hAPP_f746301080e_bool(hAPP_f162060345e_bool(cOMBB_844853809_state,P),Q),R) = hAPP_state_bool(P,hAPP_state_state(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st1506752259e_bool,Q: state,R: state] : hAPP_state_bool(hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,P),Q),R) = hAPP_state_bool(hAPP_s1806633685e_bool(P,R),Q) ).

tff(help_COMBI_1_1_COMBI_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_U,axiom,
    ! [P: hoare_2076525193iple_a] : hAPP_H1789220885iple_a(cOMBI_1928055558iple_a,P) = P ).

tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_,axiom,
    ! [X: hoare_2076525193iple_a,Y: hoare_2076525193iple_a] :
      ( ~ hBOOL(hAPP_H454055633a_bool(hAPP_H2077279219a_bool(fequal1589671816iple_a,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_,axiom,
    ! [X: hoare_2076525193iple_a,Y: hoare_2076525193iple_a] :
      ( ( X != Y )
      | hBOOL(hAPP_H454055633a_bool(hAPP_H2077279219a_bool(fequal1589671816iple_a,X),Y)) ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__,axiom,
    ! [P: fun_a_998512028e_bool,Q: bool,R: x_a] : hAPP_a2036067514e_bool(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,P),Q),R) = hAPP_b2019457360e_bool(hAPP_a723219176e_bool(P,R),Q) ).

tff(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Hoare____Mirabelle____wdcyoqcnvy__O,axiom,
    ! [P: int,Q: hoare_2076525193iple_a] : hAPP_H811578610_a_int(hAPP_i1058170057_a_int(cOMBK_1633947825iple_a,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____wdcyoqcnvy__O,axiom,
    ! [P: nat,Q: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_n1236213193_a_nat(cOMBK_1654814733iple_a,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____wdcyoqcnvy__,axiom,
    ! [P: bool,Q: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_b1299643615a_bool(cOMBK_861658160iple_a,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_int_nat,R: int] : hAPP_int_fun_nat_nat(hAPP_f1463450952at_nat(hAPP_f1087393429at_nat(cOMBB_1601129847at_int,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_376,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_int_int,R: int] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__377,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_nat_int,R: nat] : hAPP_n1082236369t_bool(hAPP_f1533130627t_bool(hAPP_f879109391t_bool(cOMBB_1628441366ol_nat,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_int_nat,R: int] : hAPP_i418383825t_bool(hAPP_f876579787t_bool(hAPP_f2026117279t_bool(cOMBB_1437810218ol_int,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__378,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_nat_nat,R: nat] : hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Ona,axiom,
    ! [P: fun_fun_int_bool_nat,Q: fun_int_fun_int_bool,R: int] : hAPP_int_nat(hAPP_f1134349059nt_nat(hAPP_f2100446809nt_nat(cOMBB_2018621492at_int,P),Q),R) = hAPP_f957591787ol_nat(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Ona_379,axiom,
    ! [P: fun_fun_int_bool_nat,Q: fun_nat_fun_int_bool,R: nat] : hAPP_nat_nat(hAPP_f389300155at_nat(hAPP_f109563153at_nat(cOMBB_1381347800at_nat,P),Q),R) = hAPP_f957591787ol_nat(P,hAPP_n1082236369t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Ona,axiom,
    ! [P: fun_fun_nat_bool_nat,Q: fun_int_fun_nat_bool,R: int] : hAPP_int_nat(hAPP_f1161717855nt_nat(hAPP_f1964560145nt_nat(cOMBB_1125113304at_int,P),Q),R) = hAPP_f22106695ol_nat(P,hAPP_i418383825t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in1172847482ol_nat,Q: fun_nat_bool,R: int] : hAPP_int_nat(hAPP_f787214110nt_nat(hAPP_f261731407nt_nat(cOMBC_462113011ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_i1530422220ol_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1402637250ol_nat,Q: fun_int_bool,R: nat] : hAPP_nat_nat(hAPP_f1473602334at_nat(hAPP_f630508183at_nat(cOMBC_1505178355ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_n259507732ol_nat(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in1461553182ol_nat,Q: fun_int_fun_int_bool,R: int] : hAPP_int_nat(hAPP_f1134349059nt_nat(hAPP_f446447448nt_nat(cOMBS_1999073191ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_i318423664ol_nat(P,R),hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in1172847482ol_nat,Q: fun_int_fun_nat_bool,R: int] : hAPP_int_nat(hAPP_f1161717855nt_nat(hAPP_f369809936nt_nat(cOMBS_1063588099ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_i1530422220ol_nat(P,R),hAPP_i418383825t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1402637250ol_nat,Q: fun_nat_fun_int_bool,R: nat] : hAPP_nat_nat(hAPP_f389300155at_nat(hAPP_f1750007732at_nat(cOMBS_2106653443ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_n259507732ol_nat(P,R),hAPP_n1082236369t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu819804446l_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu425979586l_bool,Q: fun_nat_fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_n1699378549t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in938817141l_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_int_bool,R: int] : hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_380,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_381,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_state_bool,R: state] : hAPP_s58564346l_bool(hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_state_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ovname_000tc__fun_Itc__Nat__,axiom,
    ! [P: fun_st1816159843_state,Q: vname,R: state] : hAPP_s1892499976_state(hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,P),Q),R) = hAPP_v594194232_state(hAPP_s712361723_state(P,R),Q) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple,axiom,
    ! [X: fun_Ho882550911a_bool,Y: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(fequal291249344a_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple,axiom,
    ! [X: fun_Ho882550911a_bool,Y: fun_Ho882550911a_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(fequal291249344a_bool,X),Y)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Hoare____Mirabelle,axiom,
    ! [P: fun_nat_nat,Q: fun_Ho864122878_a_nat,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_f2075363285_a_nat(hAPP_f375421205_a_nat(cOMBB_1877004557iple_a,P),Q),R) = hAPP_nat_nat(P,hAPP_H174304918_a_nat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho1488374613nt_nat,Q: int,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_i896257133_a_nat(hAPP_f1096027413_a_nat(cOMBC_1149022448nt_nat,P),Q),R) = hAPP_int_nat(hAPP_H1774516717nt_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_382,axiom,
    ! [P: fun_Ho1036508337at_nat,Q: nat,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_n1236213193_a_nat(hAPP_f668391445_a_nat(cOMBC_1942990924at_nat,P),Q),R) = hAPP_nat_nat(hAPP_H421001545at_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Hoare____Mirabelle____wdcyoqcnvy__O,axiom,
    ! [P: fun_in501186901_a_nat,Q: hoare_2076525193iple_a,R: int] : hAPP_int_nat(hAPP_H1774516717nt_nat(hAPP_f287449365nt_nat(cOMBC_1499522794_a_nat,P),Q),R) = hAPP_H174304918_a_nat(hAPP_i896257133_a_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Hoare____Mirabelle____wdcyoqcnvy__O,axiom,
    ! [P: fun_na841142961_a_nat,Q: hoare_2076525193iple_a,R: nat] : hAPP_nat_nat(hAPP_H421001545at_nat(hAPP_f982543381at_nat(cOMBC_23578254_a_nat,P),Q),R) = hAPP_H174304918_a_nat(hAPP_n1236213193_a_nat(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho1977546161nt_int,Q: fun_Ho1026035802_a_int,R: hoare_2076525193iple_a] : hAPP_H811578610_a_int(hAPP_f1382137045_a_int(hAPP_f1031139617_a_int(cOMBS_728441660nt_int,P),Q),R) = hAPP_int_int(hAPP_H1936429641nt_int(P,R),hAPP_H811578610_a_int(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_383,axiom,
    ! [P: fun_Ho1036508337at_nat,Q: fun_Ho864122878_a_nat,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_f2075363285_a_nat(hAPP_f1659339553_a_nat(cOMBS_885136444at_nat,P),Q),R) = hAPP_nat_nat(hAPP_H421001545at_nat(P,R),hAPP_H174304918_a_nat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_384,axiom,
    ! [P: fun_Ho1608761598t_bool,Q: int,R: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_i28602320a_bool(hAPP_f1732083583a_bool(cOMBC_1771170551t_bool,P),Q),R) = hAPP_int_bool(hAPP_H287744592t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_385,axiom,
    ! [P: fun_Ho1349715618t_bool,Q: nat,R: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hAPP_f1040149703a_bool(cOMBC_240829083t_bool,P),Q),R) = hAPP_nat_bool(hAPP_H904886772t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Hoare____Mirabelle____wdcyoqcnvy__O_386,axiom,
    ! [P: fun_in451447550a_bool,Q: hoare_2076525193iple_a,R: int] : hAPP_int_bool(hAPP_H287744592t_bool(hAPP_f510295935t_bool(cOMBC_1135948669a_bool,P),Q),R) = hAPP_H454055633a_bool(hAPP_i28602320a_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Hoare____Mirabelle____wdcyoqcnvy__O_387,axiom,
    ! [P: fun_na1758826658a_bool,Q: hoare_2076525193iple_a,R: nat] : hAPP_nat_bool(hAPP_H904886772t_bool(hAPP_f1460432071t_bool(cOMBC_2097822937a_bool,P),Q),R) = hAPP_H454055633a_bool(hAPP_n1335981428a_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel,axiom,
    ! [P: fun_bool_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f629438041a_bool(cOMBB_361863589iple_a,P),Q),R) = hAPP_bool_bool(P,hAPP_H454055633a_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_388,axiom,
    ! [P: fun_Ho2037816211l_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H421140395l_bool(P,R),hAPP_H454055633a_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc,axiom,
    ! [P: fun_a_2117018159e_bool,Q: fun_state_bool,R: x_a] : hAPP_a2036067514e_bool(hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,P),Q),R) = hAPP_f1759915619e_bool(hAPP_a1200519163e_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__Com__Ostate_J_000t,axiom,
    ! [P: fun_a_340690546e_bool,Q: fun_state_state,R: x_a] : hAPP_a2036067514e_bool(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,P),Q),R) = hAPP_f746301080e_bool(hAPP_a1224971408e_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,axiom,
    ! [P: fun_in1791225684at_nat,Q: fun_nat_nat,R: int] : hAPP_int_fun_nat_nat(hAPP_f1169617132at_nat(hAPP_f701299925at_nat(cOMBC_1294079849at_nat,P),Q),R) = hAPP_f1914919701at_nat(hAPP_i1876697324at_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J,axiom,
    ! [P: fun_na266550704nt_nat,Q: fun_int_nat,R: nat] : hAPP_nat_fun_int_nat(hAPP_f237327688nt_nat(hAPP_f1633513941nt_nat(cOMBC_1218031117nt_nat,P),Q),R) = hAPP_f2132704789nt_nat(hAPP_n74706760nt_nat(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu554825757t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__389,axiom,
    ! [P: fun_fu554825757t_bool,Q: fun_nat_fun_int_bool,R: nat] : hAPP_n1082236369t_bool(hAPP_f2022049025t_bool(hAPP_f1986088027t_bool(cOMBB_2101530536ol_nat,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_n1082236369t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu821463397t_bool,Q: fun_int_fun_nat_bool,R: int] : hAPP_i418383825t_bool(hAPP_f783004929t_bool(hAPP_f1410409747t_bool(cOMBB_1746576572ol_int,P),Q),R) = hAPP_f800510211t_bool(P,hAPP_i418383825t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__390,axiom,
    ! [P: fun_in719570932t_bool,Q: fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__391,axiom,
    ! [P: fun_na35521688t_bool,Q: fun_nat_bool,R: nat] : hAPP_n1699378549t_bool(hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu1151772339l_bool,Q: fun_int_bool,R: fun_int_bool] : hAPP_f448129468l_bool(hAPP_f284875647l_bool(hAPP_f1399552105l_bool(cOMBC_605892544l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_f284875647l_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu1217155507l_bool,Q: fun_nat_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: hoare_2076525193iple_a,Q: hoare_2076525193iple_a] : hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(cOMBK_788297294iple_a,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
    ! [P: fun_int_fun_int_int,Q: fun_Ho1026035802_a_int,R: hoare_2076525193iple_a] : hAPP_H1936429641nt_int(hAPP_f534508204nt_int(hAPP_f1566620885nt_int(cOMBB_364359670iple_a,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_H811578610_a_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_392,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_Ho864122878_a_nat,R: hoare_2076525193iple_a] : hAPP_H421001545at_nat(hAPP_f731653896at_nat(hAPP_f648261589at_nat(cOMBB_1416330906iple_a,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_H174304918_a_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__393,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_Ho1026035802_a_int,R: hoare_2076525193iple_a] : hAPP_H287744592t_bool(hAPP_f841024195t_bool(hAPP_f1444703951t_bool(cOMBB_2132063381iple_a,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_H811578610_a_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__394,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_Ho864122878_a_nat,R: hoare_2076525193iple_a] : hAPP_H904886772t_bool(hAPP_f570970635t_bool(hAPP_f2096737119t_bool(cOMBB_2017898829iple_a,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_H174304918_a_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,axiom,
    ! [P: fun_fu328701628ol_nat,Q: fun_na1758826658a_bool,R: nat] : hAPP_nat_nat(hAPP_f420120536at_nat(hAPP_f228445643at_nat(cOMBB_605112827at_nat,P),Q),R) = hAPP_f190641160ol_nat(P,hAPP_n1335981428a_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Ona_395,axiom,
    ! [P: fun_fun_int_bool_nat,Q: fun_Ho1608761598t_bool,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_f10452157_a_nat(hAPP_f296893331_a_nat(cOMBB_1983399767iple_a,P),Q),R) = hAPP_f957591787ol_nat(P,hAPP_H287744592t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Ona_396,axiom,
    ! [P: fun_fun_nat_bool_nat,Q: fun_Ho1349715618t_bool,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_f1753252377_a_nat(hAPP_f456064075_a_nat(cOMBB_1733887227iple_a,P),Q),R) = hAPP_f22106695ol_nat(P,hAPP_H904886772t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_397,axiom,
    ! [P: fun_Ho1345928733ol_nat,Q: fun_nat_bool,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_f1047071419_a_nat(hAPP_f1217804815_a_nat(cOMBC_1664565648ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_H2118836847ol_nat(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcy,axiom,
    ! [P: fun_na228779103ol_nat,Q: fun_Ho882550911a_bool,R: nat] : hAPP_nat_nat(hAPP_f479558075at_nat(hAPP_f1173305809at_nat(cOMBC_129115664ol_nat,P),Q),R) = hAPP_f190641160ol_nat(hAPP_n370719537ol_nat(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_398,axiom,
    ! [P: fun_Ho1634634433ol_nat,Q: fun_Ho1608761598t_bool,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_f10452157_a_nat(hAPP_f209886133_a_nat(cOMBS_1838073924ol_nat,P),Q),R) = hAPP_f957591787ol_nat(hAPP_H906838291ol_nat(P,R),hAPP_H287744592t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_399,axiom,
    ! [P: fun_Ho1345928733ol_nat,Q: fun_Ho1349715618t_bool,R: hoare_2076525193iple_a] : hAPP_H174304918_a_nat(hAPP_f1753252377_a_nat(hAPP_f991915629_a_nat(cOMBS_902588832ol_nat,P),Q),R) = hAPP_f22106695ol_nat(hAPP_H2118836847ol_nat(P,R),hAPP_H904886772t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Hoare____Mirabelle____wdcy,axiom,
    ! [P: fun_in149899451ol_nat,Q: fun_in451447550a_bool,R: int] : hAPP_int_nat(hAPP_f1503898400nt_nat(hAPP_f807099282nt_nat(cOMBS_443177668ol_nat,P),Q),R) = hAPP_f190641160ol_nat(hAPP_i291839885ol_nat(P,R),hAPP_i28602320a_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcy,axiom,
    ! [P: fun_na228779103ol_nat,Q: fun_na1758826658a_bool,R: nat] : hAPP_nat_nat(hAPP_f420120536at_nat(hAPP_f1323480046at_nat(cOMBS_1514622496ol_nat,P),Q),R) = hAPP_f190641160ol_nat(hAPP_n370719537ol_nat(P,R),hAPP_n1335981428a_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_400,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : hAPP_H421140395l_bool(hAPP_f268025233l_bool(hAPP_f1261090107l_bool(cOMBB_345279363iple_a,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H454055633a_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P: fun_fu1658206819l_bool,Q: fun_a_fun_state_bool,R: x_a] : hAPP_a849909144l_bool(hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_a2036067514e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_000tc__fun_Itc__,axiom,
    ! [P: fun_fu670292377ol_nat,Q: fun_int_fun_int_nat,R: int] : hAPP_i318423664ol_nat(hAPP_f1548925761ol_nat(hAPP_f1331458699ol_nat(cOMBB_2091167284at_int,P),Q),R) = hAPP_f1109019371ol_nat(P,hAPP_int_fun_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_000tc__fun_Itc___401,axiom,
    ! [P: fun_fu670292377ol_nat,Q: fun_nat_fun_int_nat,R: nat] : hAPP_n259507732ol_nat(hAPP_f299305025ol_nat(hAPP_f1632710027ol_nat(cOMBB_1453893592at_nat,P),Q),R) = hAPP_f1109019371ol_nat(P,hAPP_nat_fun_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc__,axiom,
    ! [P: fun_fu990052945ol_nat,Q: fun_int_fun_nat_nat,R: int] : hAPP_i1530422220ol_nat(hAPP_f1501159417ol_nat(hAPP_f1029311995ol_nat(cOMBB_1562322300at_int,P),Q),R) = hAPP_f782000547ol_nat(P,hAPP_int_fun_nat_nat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_402,axiom,
    ! [P: fun_Ho943022881a_bool,Q: hoare_2076525193iple_a,R: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,P),Q),R) = hAPP_H454055633a_bool(hAPP_H2077279219a_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__403,axiom,
    ! [P: fun_fu1075512626t_bool,Q: fun_int_bool,R: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(hAPP_f194616807t_bool(cOMBC_922884543t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__404,axiom,
    ! [P: fun_fu915923926t_bool,Q: fun_nat_bool,R: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(hAPP_f1311642927t_bool(cOMBC_646315179t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_405,axiom,
    ! [P: fun_Ho755902321at_nat,Q: fun_nat_nat,R: hoare_2076525193iple_a] : hAPP_H421001545at_nat(hAPP_f1016910345at_nat(hAPP_f1112019605at_nat(cOMBC_340603916at_nat,P),Q),R) = hAPP_f1914919701at_nat(hAPP_H1504435465at_nat(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__406,axiom,
    ! [P: fun_fu554825757t_bool,Q: fun_Ho1608761598t_bool,R: hoare_2076525193iple_a] : hAPP_H287744592t_bool(hAPP_f741913983t_bool(hAPP_f1871962969t_bool(cOMBB_349652263iple_a,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_H287744592t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__407,axiom,
    ! [P: fun_fu821463397t_bool,Q: fun_Ho1349715618t_bool,R: hoare_2076525193iple_a] : hAPP_H904886772t_bool(hAPP_f631038663t_bool(hAPP_f69420889t_bool(cOMBB_1063904223iple_a,P),Q),R) = hAPP_f800510211t_bool(P,hAPP_H904886772t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H,axiom,
    ! [P: fun_fu1219323149e_bool,Q: fun_a_1632297036l_bool,R: x_a] : hAPP_a723219176e_bool(hAPP_f340725611e_bool(hAPP_f1006724181e_bool(cOMBB_1348041619bool_a,P),Q),R) = hAPP_f167292325e_bool(P,hAPP_a849909144l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___408,axiom,
    ! [P: fun_fu773339453nt_nat,Q: fun_nat_fun_nat_nat,R: nat] : hAPP_n74706760nt_nat(hAPP_f2080483477nt_nat(hAPP_f1701814485nt_nat(cOMBB_737513486at_nat,P),Q),R) = hAPP_f901215189nt_nat(P,hAPP_nat_fun_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___409,axiom,
    ! [P: fun_fu1179887677at_nat,Q: fun_int_fun_nat_nat,R: int] : hAPP_i1876697324at_nat(hAPP_f1033905301at_nat(hAPP_f1315855317at_nat(cOMBB_891709290at_int,P),Q),R) = hAPP_f1585078997at_nat(P,hAPP_int_fun_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_410,axiom,
    ! [P: fun_fu2127278875e_bool,Q: fun_a_fun_state_bool,R: x_a] : hAPP_a1224971408e_bool(hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,P),Q),R) = hAPP_f162060345e_bool(P,hAPP_a2036067514e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho943022881a_bool,Q: fun_in1219646560iple_a,R: int] : hAPP_i28602320a_bool(hAPP_f1362463689a_bool(hAPP_f438022427a_bool(cOMBB_952878892ol_int,P),Q),R) = hAPP_H2077279219a_bool(P,hAPP_i867006584iple_a(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_411,axiom,
    ! [P: fun_Ho943022881a_bool,Q: fun_na1240513468iple_a,R: nat] : hAPP_n1335981428a_bool(hAPP_f1324734153a_bool(hAPP_f946563611a_bool(cOMBB_315605200ol_nat,P),Q),R) = hAPP_H2077279219a_bool(P,hAPP_n887873492iple_a(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__412,axiom,
    ! [P: fun_fu1075512626t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1529485324t_bool(hAPP_f1468280982t_bool(hAPP_f1596157055t_bool(cOMBB_765314529ol_int,P),Q),R) = hAPP_f1223193598t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__413,axiom,
    ! [P: fun_fu915923926t_bool,Q: fun_nat_fun_nat_bool,R: nat] : hAPP_n1512601776t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,P),Q),R) = hAPP_f1730770594t_bool(P,hAPP_n1699378549t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_414,axiom,
    ! [P: fun_fu2026518849l_bool,Q: fun_Ho943022881a_bool,R: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_f338738851a_bool(hAPP_f1827400025a_bool(cOMBB_1676209667iple_a,P),Q),R) = hAPP_f500854239l_bool(P,hAPP_H2077279219a_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_415,axiom,
    ! [P: fun_Ho1910759733l_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,P),Q),R) = hAPP_f500854239l_bool(hAPP_H1566092493l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Nat__Onat_J_000tc__fun_Itc___416,axiom,
    ! [P: fun_fu670292377ol_nat,Q: fun_Ho1488374613nt_nat,R: hoare_2076525193iple_a] : hAPP_H906838291ol_nat(hAPP_f1998933761ol_nat(hAPP_f1772247755ol_nat(cOMBB_1701789015iple_a,P),Q),R) = hAPP_f1109019371ol_nat(P,hAPP_H1774516717nt_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___417,axiom,
    ! [P: fun_fu990052945ol_nat,Q: fun_Ho1036508337at_nat,R: hoare_2076525193iple_a] : hAPP_H2118836847ol_nat(hAPP_f1296292793ol_nat(hAPP_f153943611ol_nat(cOMBB_636937375iple_a,P),Q),R) = hAPP_f782000547ol_nat(P,hAPP_H421001545at_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_418,axiom,
    ! [P: fun_fu1047394976e_bool,Q: fun_a_1632297036l_bool,R: x_a] : hAPP_a1200519163e_bool(hAPP_f963367678e_bool(hAPP_f375255701e_bool(cOMBB_145932198bool_a,P),Q),R) = hAPP_f644196280e_bool(P,hAPP_a849909144l_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcy_419,axiom,
    ! [P: fun_na1588060912_a_nat,Q: fun_Ho864122878_a_nat,R: nat] : hAPP_n1236213193_a_nat(hAPP_f536288520_a_nat(hAPP_f928984021_a_nat(cOMBC_130242509_a_nat,P),Q),R) = hAPP_f2075363285_a_nat(hAPP_n226523272_a_nat(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_420,axiom,
    ! [P: fun_fu1663341283a_bool,Q: fun_in451447550a_bool,R: int] : hAPP_i28602320a_bool(hAPP_f1500465535a_bool(hAPP_f967504915a_bool(cOMBB_1848875326ol_int,P),Q),R) = hAPP_f642208769a_bool(P,hAPP_i28602320a_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_421,axiom,
    ! [P: fun_fu1663341283a_bool,Q: fun_na1758826658a_bool,R: nat] : hAPP_n1335981428a_bool(hAPP_f1869543111a_bool(hAPP_f1882853211a_bool(cOMBB_1211601634ol_nat,P),Q),R) = hAPP_f642208769a_bool(P,hAPP_n1335981428a_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,axiom,
    ! [P: fun_fu1789210355l_bool,Q: fun_Ho882550911a_bool,R: fun_Ho882550911a_bool] : hAPP_f500854239l_bool(hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,P),Q),R) = hAPP_f500854239l_bool(hAPP_f256727871l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___422,axiom,
    ! [P: fun_fu1179887677at_nat,Q: fun_Ho1036508337at_nat,R: hoare_2076525193iple_a] : hAPP_H1504435465at_nat(hAPP_f564893909at_nat(hAPP_f250842005at_nat(cOMBB_151870093iple_a,P),Q),R) = hAPP_f1585078997at_nat(P,hAPP_H421001545at_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_423,axiom,
    ! [P: fun_Ho943022881a_bool,Q: fun_Ho1518252029iple_a,R: hoare_2076525193iple_a] : hAPP_H2077279219a_bool(hAPP_f1075886601a_bool(hAPP_f256103387a_bool(cOMBB_135702095iple_a,P),Q),R) = hAPP_H2077279219a_bool(P,hAPP_H1789220885iple_a(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_424,axiom,
    ! [P: fun_fu827349075ol_nat,Q: fun_in501186901_a_nat,R: int] : hAPP_i291839885ol_nat(hAPP_f1914880763ol_nat(hAPP_f534494591ol_nat(cOMBB_434376058at_int,P),Q),R) = hAPP_f1697440293ol_nat(P,hAPP_i896257133_a_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_425,axiom,
    ! [P: fun_fu827349075ol_nat,Q: fun_na841142961_a_nat,R: nat] : hAPP_n370719537ol_nat(hAPP_f1356625915ol_nat(hAPP_f2033836159ol_nat(cOMBB_1944586014at_nat,P),Q),R) = hAPP_f1697440293ol_nat(P,hAPP_n1236213193_a_nat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_426,axiom,
    ! [P: fun_Ho112928087a_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : hAPP_H2077279219a_bool(hAPP_f219434403a_bool(hAPP_f1824299331a_bool(cOMBC_175592824a_bool,P),Q),R) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc___427,axiom,
    ! [P: fun_fu1437424253_a_nat,Q: fun_nat_fun_nat_nat,R: nat] : hAPP_n226523272_a_nat(hAPP_f935848405_a_nat(hAPP_f450378965_a_nat(cOMBB_1353751246at_nat,P),Q),R) = hAPP_f375421205_a_nat(P,hAPP_nat_fun_nat_nat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_428,axiom,
    ! [P: fun_fu2051904533a_bool,Q: fun_Ho882550911a_bool,R: fun_Ho882550911a_bool] : hAPP_f642208769a_bool(hAPP_f1418638817a_bool(hAPP_f86610733a_bool(cOMBC_2005334440a_bool,P),Q),R) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_429,axiom,
    ! [P: fun_fu2051904533a_bool,Q: fun_Ho943022881a_bool,R: hoare_2076525193iple_a] : hAPP_H1940471919a_bool(hAPP_f102426163a_bool(hAPP_f300022271a_bool(cOMBB_2036667361iple_a,P),Q),R) = hAPP_f1418638817a_bool(P,hAPP_H2077279219a_bool(Q,R)) ).

%----Conjectures (1)
tff(conj_0,conjecture,
    hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(g),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hoare_1876769875iple_a(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_b2019457360e_bool(cOMBK_bool_state,fFalse)),c,hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,hAPP_f963367678e_bool(hAPP_f375255701e_bool(cOMBB_145932198bool_a,cOMBS_1378840469l_bool),hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),p))),hAPP_f1759915619e_bool(hAPP_f2073279419e_bool(cOMBB_160679318_state,fNot),b)))),bot_bo214064617a_bool))) ).

%------------------------------------------------------------------------------