TPTP Problem File: SWW472_3.p
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%------------------------------------------------------------------------------
% File : SWW472_3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 327, 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_l327 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.5.0, 0.75 v5.4.0, 1.00 v5.3.0
% Syntax : Number of formulae : 1483 ( 336 unt; 240 typ; 0 def)
% Number of atoms : 3094 ( 888 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2235 ( 384 ~; 81 |; 124 &)
% ( 266 <=>;1380 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 61 ( 60 usr)
% Number of type conns : 185 ( 110 >; 75 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 179 ( 179 usr; 70 con; 0-3 aty)
% Number of variables : 3310 (3236 !; 74 ?;3310 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:40:21
%------------------------------------------------------------------------------
%----Should-be-implicit typings (60)
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%----Explicit typings (180)
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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_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_Hoare__Mirabelle__wdcyoqcnvy_OMGT,type,
hoare_Mirabelle_MGT: com > hoare_363350478_state ).
tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Ohoare__derivs_000tc__Com__Ostate,type,
hoare_95103338_state: fun_Ho320143120e_bool > fun_fu1996308102l_bool ).
tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Ohoare__valids_000tc__Com__Ostate,type,
hoare_501514600_state: fun_Ho320143120e_bool > fun_fu1996308102l_bool ).
tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple_Otriple_000tc__Com__Ostate,type,
hoare_608330478_state: ( fun_st1506752259e_bool * com * fun_st1506752259e_bool ) > hoare_363350478_state ).
tff(sy_c_If_000tc__Nat__Onat,type,
if_nat: bool > fun_nat_fun_nat_nat ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: fun_int_int ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: fun_int_nat ).
tff(sy_c_Nat_OSuc,type,
suc: fun_nat_nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
semiri1621563631at_int: fun_nat_int ).
tff(sy_c_Nat_Osize__class_Osize_000tc__Com__Ocom,type,
size_size_com: com > nat ).
tff(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: fun_int_fun_int_int ).
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_bo39439654e_bool: fun_Ho320143120e_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__HOL__Obool,type,
ord_less_bool: fun_bo1549164019l_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__Hoare____Mirabelle____wdcyoqcnv,type,
ord_le1392391526e_bool: fun_fu2036837167l_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_le704992346e_bool: fun_fu2036837167l_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,
ord_le1568362934t_bool: fun_fu1217155507l_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle,type,
ord_le2031828156l_bool: fun_fu1575689779l_bool ).
tff(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
ord_min_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Predicate_OPowp_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,type,
powp_H991500296_state: fun_Ho320143120e_bool > fun_fu1996308102l_bool ).
tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ost,type,
collec223185785_state: fun_fu1066574909e_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_Oinsert_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Osta,type,
insert1330351828_state: fun_Ho1500656352e_bool ).
tff(sy_c_Set_Oinsert_000tc__Int__Oint,type,
insert_int: fun_in719570932t_bool ).
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_Itc__Com__O,type,
the_el1666657391_state: fun_fu700438391_state ).
tff(sy_c_Set_Othe__elem_000tc__Int__Oint,type,
the_elem_int: fun_fun_int_bool_int ).
tff(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
the_elem_nat: fun_fun_nat_bool_nat ).
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__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J,type,
fequal1026769423_state: fun_Ho424838195e_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_fimplies,type,
fimplies: fun_bo1549164019l_bool ).
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__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J,type,
hAPP_s1806633685e_bool: ( fun_st1506752259e_bool * state ) > fun_state_bool ).
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_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_00,type,
hAPP_H1645464494e_bool: ( fun_Ho320143120e_bool * hoare_363350478_state ) > bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_00_024,type,
hAPP_H1982017439_state: ( fun_Ho901937921_state * hoare_363350478_state ) > hoare_363350478_state ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_00_025,type,
hAPP_H1173243534l_bool: ( fun_Ho544647106l_bool * hoare_363350478_state ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_00_026,type,
hAPP_H705653759e_bool: ( fun_Ho424838195e_bool * hoare_363350478_state ) > fun_Ho320143120e_bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_00_027,type,
hAPP_H230141040_state: ( fun_Ho31865764_state * hoare_363350478_state ) > fun_Ho901937921_state ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_00_028,type,
hAPP_H40373975l_bool: ( fun_Ho177530809l_bool * hoare_363350478_state ) > fun_fu1996308102l_bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_00_029,type,
hAPP_H122293036e_bool: ( fun_Ho1500656352e_bool * hoare_363350478_state ) > fun_fu1066574909e_bool ).
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__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,
hAPP_i68813070l_bool: ( fun_in531499254l_bool * int ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otr,type,
hAPP_i1871177343e_bool: ( fun_in255859559e_bool * int ) > fun_Ho320143120e_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
hAPP_int_fun_int_int: ( fun_int_fun_int_int * int ) > fun_int_int ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_,type,
hAPP_i2112223885l_bool: ( fun_in938817141l_bool * int ) > fun_fu819804446l_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J__030,type,
hAPP_i1529485324t_bool: ( fun_in719570932t_bool * int ) > fun_fu554825757t_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
hAPP_nat_int: ( fun_nat_int * nat ) > int ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_n1006566506l_bool: ( fun_na1469252690l_bool * nat ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otr,type,
hAPP_n753050587e_bool: ( fun_na1285216451e_bool * nat ) > fun_Ho320143120e_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
hAPP_n1699378549t_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
hAPP_nat_fun_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
hAPP_n215258509l_bool: ( fun_na1436237685l_bool * nat ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J__031,type,
hAPP_n1512601776t_bool: ( fun_na35521688t_bool * nat ) > fun_fu821463397t_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os,type,
hAPP_f568432088l_bool: ( fun_fu1996308102l_bool * fun_Ho320143120e_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os_032,type,
hAPP_f1242401993_state: ( fun_fu700438391_state * fun_Ho320143120e_bool ) > hoare_363350478_state ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os_033,type,
hAPP_f195773483ol_int: ( fun_fu807176339ol_int * fun_Ho320143120e_bool ) > int ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os_034,type,
hAPP_f1705983439ol_nat: ( fun_fu645263415ol_nat * fun_Ho320143120e_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os_035,type,
hAPP_f1749453397e_bool: ( fun_fu1066574909e_bool * fun_Ho320143120e_bool ) > fun_Ho320143120e_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os_036,type,
hAPP_f1422032445l_bool: ( fun_fu1212692075l_bool * fun_Ho320143120e_bool ) > fun_Ho544647106l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os_037,type,
hAPP_f726209793l_bool: ( fun_fu2036837167l_bool * fun_Ho320143120e_bool ) > fun_fu1996308102l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Os_038,type,
hAPP_f1043393922e_bool: ( fun_fu531943146e_bool * fun_Ho320143120e_bool ) > fun_fu1066574909e_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_f448129468l_bool: ( fun_fu819804446l_bool * fun_int_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Int__Oint,type,
hAPP_f1594865479ol_int: ( fun_fun_int_bool_int * fun_int_bool ) > int ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
hAPP_f957591787ol_nat: ( fun_fun_int_bool_nat * fun_int_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oint_,type,
hAPP_f1805168059t_bool: ( fun_fu554825757t_bool * fun_int_bool ) > fun_int_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__Int__Oint__039,type,
hAPP_f2144054103l_bool: ( fun_fu610077323l_bool * fun_int_bool ) > fun_in531499254l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__I,type,
hAPP_f284875647l_bool: ( fun_fu1151772339l_bool * fun_int_bool ) > fun_fu819804446l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__I_040,type,
hAPP_f1223193598t_bool: ( fun_fu1075512626t_bool * fun_int_bool ) > fun_fu554825757t_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_f54304608l_bool: ( fun_fu425979586l_bool * fun_nat_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
hAPP_f22106695ol_nat: ( fun_fun_nat_bool_nat * fun_nat_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat_,type,
hAPP_f800510211t_bool: ( fun_fu821463397t_bool * fun_nat_bool ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat__041,type,
hAPP_f1146629647l_bool: ( fun_fu2104074051l_bool * fun_nat_bool ) > fun_na1469252690l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__N,type,
hAPP_f103356543l_bool: ( fun_fu1217155507l_bool * fun_nat_bool ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__N_042,type,
hAPP_f1730770594t_bool: ( fun_fu915923926t_bool * fun_nat_bool ) > fun_fu821463397t_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc,type,
hAPP_f1287348838l_bool: ( fun_fu1972704968l_bool * fun_fu1996308102l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc_043,type,
hAPP_f746351743l_bool: ( fun_fu1575689779l_bool * fun_fu1996308102l_bool ) > fun_fu1972704968l_bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_c_member_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J,type,
member1253905979_state: fun_Ho177530809l_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_P,type,
p: fun_st1506752259e_bool ).
tff(sy_v_Q,type,
q: fun_st1506752259e_bool ).
tff(sy_v_c,type,
c: com ).
%----Relevant facts (1200)
tff(fact_0_empty,axiom,
! [G_1: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),bot_bo39439654e_bool)) ).
tff(fact_1_triple_Oinject,axiom,
! [Fun1_2: fun_st1506752259e_bool,Com_1: com,Fun2_2: fun_st1506752259e_bool,Fun1_1: fun_st1506752259e_bool,Com: com,Fun2_1: fun_st1506752259e_bool] :
( ( hoare_608330478_state(Fun1_2,Com_1,Fun2_2) = hoare_608330478_state(Fun1_1,Com,Fun2_1) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_1 = Com )
& ( Fun2_2 = Fun2_1 ) ) ) ).
tff(fact_2_hoare__sound,axiom,
! [G_1: fun_Ho320143120e_bool,Ts: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts))
=> hBOOL(hAPP_f568432088l_bool(hoare_501514600_state(G_1),Ts)) ) ).
tff(fact_3_cut,axiom,
! [G_1: fun_Ho320143120e_bool,G_2: fun_Ho320143120e_bool,Ts: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_2),Ts))
=> ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),G_2))
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts)) ) ) ).
tff(fact_4_hoare__derivs_Oinsert,axiom,
! [Ts: fun_Ho320143120e_bool,G_1: fun_Ho320143120e_bool,T_1: hoare_363350478_state] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),bot_bo39439654e_bool)))
=> ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts))
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),Ts))) ) ) ).
tff(fact_5_derivs__insertD,axiom,
! [G_1: fun_Ho320143120e_bool,T_1: hoare_363350478_state,Ts: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),Ts)))
=> ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),bot_bo39439654e_bool)))
& hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts)) ) ) ).
tff(fact_6_conseq2,axiom,
! [Qa: fun_st1506752259e_bool,G_1: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool,Ca: com,Q_4: fun_st1506752259e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,Ca,Q_4)),bot_bo39439654e_bool)))
=> ( ! [Z_13: state,S_1: state] :
( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_4,Z_13),S_1))
=> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_13),S_1)) )
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,Ca,Qa)),bot_bo39439654e_bool))) ) ) ).
tff(fact_7_conseq1,axiom,
! [Pa: fun_st1506752259e_bool,G_1: fun_Ho320143120e_bool,P_1: fun_st1506752259e_bool,Ca: com,Qa: fun_st1506752259e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(P_1,Ca,Qa)),bot_bo39439654e_bool)))
=> ( ! [Z_13: state,S_1: state] :
( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_13),S_1))
=> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_13),S_1)) )
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,Ca,Qa)),bot_bo39439654e_bool))) ) ) ).
tff(fact_8_insertE,axiom,
! [A_1: int,B_2: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),A_2)))
=> ( ( A_1 != B_2 )
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2)) ) ) ).
tff(fact_9_insertE,axiom,
! [A_1: nat,B_2: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),A_2)))
=> ( ( A_1 != B_2 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2)) ) ) ).
tff(fact_10_insertE,axiom,
! [A_1: hoare_363350478_state,B_2: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_2),A_2)))
=> ( ( A_1 != B_2 )
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2)) ) ) ).
tff(fact_11_insertCI,axiom,
! [B_2: int,A_1: int,B_29: fun_int_bool] :
( ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),B_29))
=> ( A_1 = B_2 ) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),B_29))) ) ).
tff(fact_12_insertCI,axiom,
! [B_2: nat,A_1: nat,B_29: fun_nat_bool] :
( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B_29))
=> ( A_1 = B_2 ) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),B_29))) ) ).
tff(fact_13_insertCI,axiom,
! [B_2: hoare_363350478_state,A_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),B_29))
=> ( A_1 = B_2 ) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_2),B_29))) ) ).
tff(fact_14_conseq12,axiom,
! [Qa: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool,G_1: fun_Ho320143120e_bool,P_1: fun_st1506752259e_bool,Ca: com,Q_4: fun_st1506752259e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(P_1,Ca,Q_4)),bot_bo39439654e_bool)))
=> ( ! [Z_13: state,S_1: state] :
( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_13),S_1))
=> ! [S_2: state] :
( ! [Z_14: state] :
( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_14),S_1))
=> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_4,Z_14),S_2)) )
=> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_13),S_2)) ) )
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,Ca,Qa)),bot_bo39439654e_bool))) ) ) ).
tff(fact_15_emptyE,axiom,
! [A_1: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),bot_bot_fun_int_bool)) ).
tff(fact_16_emptyE,axiom,
! [A_1: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),bot_bot_fun_nat_bool)) ).
tff(fact_17_emptyE,axiom,
! [A_1: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),bot_bo39439654e_bool)) ).
tff(fact_18_empty__not__insert,axiom,
! [A_1: int,A_2: fun_int_bool] : bot_bot_fun_int_bool != hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2) ).
tff(fact_19_empty__not__insert,axiom,
! [A_1: nat,A_2: fun_nat_bool] : bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2) ).
tff(fact_20_empty__not__insert,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] : bot_bo39439654e_bool != hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2) ).
tff(fact_21_insert__not__empty,axiom,
! [A_1: int,A_2: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2) != bot_bot_fun_int_bool ).
tff(fact_22_insert__not__empty,axiom,
! [A_1: nat,A_2: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2) != bot_bot_fun_nat_bool ).
tff(fact_23_insert__not__empty,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2) != bot_bo39439654e_bool ).
tff(fact_24_singleton__iff,axiom,
! [B_2: int,A_1: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool)))
<=> ( B_2 = A_1 ) ) ).
tff(fact_25_singleton__iff,axiom,
! [B_2: nat,A_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))
<=> ( B_2 = A_1 ) ) ).
tff(fact_26_singleton__iff,axiom,
! [B_2: hoare_363350478_state,A_1: hoare_363350478_state] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool)))
<=> ( B_2 = A_1 ) ) ).
tff(fact_27_doubleton__eq__iff,axiom,
! [A_1: int,B_2: int,Ca: int,D_1: int] :
( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),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_1 = Ca )
& ( B_2 = D_1 ) )
| ( ( A_1 = D_1 )
& ( B_2 = Ca ) ) ) ) ).
tff(fact_28_doubleton__eq__iff,axiom,
! [A_1: nat,B_2: nat,Ca: nat,D_1: nat] :
( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),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_1 = Ca )
& ( B_2 = D_1 ) )
| ( ( A_1 = D_1 )
& ( B_2 = Ca ) ) ) ) ).
tff(fact_29_doubleton__eq__iff,axiom,
! [A_1: hoare_363350478_state,B_2: hoare_363350478_state,Ca: hoare_363350478_state,D_1: hoare_363350478_state] :
( ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_2),bot_bo39439654e_bool)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Ca),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,D_1),bot_bo39439654e_bool)) )
<=> ( ( ( A_1 = Ca )
& ( B_2 = D_1 ) )
| ( ( A_1 = D_1 )
& ( B_2 = Ca ) ) ) ) ).
tff(fact_30_equals0D,axiom,
! [A_1: int,A_2: fun_int_bool] :
( ( A_2 = bot_bot_fun_int_bool )
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2)) ) ).
tff(fact_31_equals0D,axiom,
! [A_1: nat,A_2: fun_nat_bool] :
( ( A_2 = bot_bot_fun_nat_bool )
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2)) ) ).
tff(fact_32_equals0D,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( ( A_2 = bot_bo39439654e_bool )
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2)) ) ).
tff(fact_33_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_34_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_35_Collect__empty__eq,axiom,
! [Pa: fun_Ho320143120e_bool] :
( ( hAPP_f1749453397e_bool(collec223185785_state,Pa) = bot_bo39439654e_bool )
<=> ! [X_2: hoare_363350478_state] : ~ hBOOL(hAPP_H1645464494e_bool(Pa,X_2)) ) ).
tff(fact_36_empty__iff,axiom,
! [Ca: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),bot_bot_fun_int_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: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),bot_bo39439654e_bool)) ).
tff(fact_39_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_40_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_41_empty__Collect__eq,axiom,
! [Pa: fun_Ho320143120e_bool] :
( ( bot_bo39439654e_bool = hAPP_f1749453397e_bool(collec223185785_state,Pa) )
<=> ! [X_2: hoare_363350478_state] : ~ hBOOL(hAPP_H1645464494e_bool(Pa,X_2)) ) ).
tff(fact_42_ex__in__conv,axiom,
! [A_2: fun_int_bool] :
( ? [X_2: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_2))
<=> ( A_2 != bot_bot_fun_int_bool ) ) ).
tff(fact_43_ex__in__conv,axiom,
! [A_2: fun_nat_bool] :
( ? [X_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_2))
<=> ( A_2 != bot_bot_fun_nat_bool ) ) ).
tff(fact_44_ex__in__conv,axiom,
! [A_2: fun_Ho320143120e_bool] :
( ? [X_2: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),A_2))
<=> ( A_2 != bot_bo39439654e_bool ) ) ).
tff(fact_45_all__not__in__conv,axiom,
! [A_2: fun_int_bool] :
( ! [X_2: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_2))
<=> ( A_2 = bot_bot_fun_int_bool ) ) ).
tff(fact_46_all__not__in__conv,axiom,
! [A_2: fun_nat_bool] :
( ! [X_2: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_2))
<=> ( A_2 = bot_bot_fun_nat_bool ) ) ).
tff(fact_47_all__not__in__conv,axiom,
! [A_2: fun_Ho320143120e_bool] :
( ! [X_2: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),A_2))
<=> ( A_2 = bot_bo39439654e_bool ) ) ).
tff(fact_48_empty__def,axiom,
bot_bot_fun_int_bool = hAPP_f1805168059t_bool(collect_int,cOMBK_bool_int(fFalse)) ).
tff(fact_49_empty__def,axiom,
bot_bot_fun_nat_bool = hAPP_f800510211t_bool(collect_nat,cOMBK_bool_nat(fFalse)) ).
tff(fact_50_empty__def,axiom,
bot_bo39439654e_bool = hAPP_f1749453397e_bool(collec223185785_state,cOMBK_574827367_state(fFalse)) ).
tff(fact_51_insert__absorb,axiom,
! [A_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2) = A_2 ) ) ).
tff(fact_52_insert__absorb,axiom,
! [A_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2) = A_2 ) ) ).
tff(fact_53_insert__absorb,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2))
=> ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2) = A_2 ) ) ).
tff(fact_54_insertI2,axiom,
! [B_2: int,A_1: int,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),B_29))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),B_29))) ) ).
tff(fact_55_insertI2,axiom,
! [B_2: nat,A_1: nat,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B_29))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),B_29))) ) ).
tff(fact_56_insertI2,axiom,
! [B_2: hoare_363350478_state,A_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),B_29))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_2),B_29))) ) ).
tff(fact_57_insert__ident,axiom,
! [B_29: fun_int_bool,X_1: int,A_2: fun_int_bool] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29))
=> ( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_29) )
<=> ( A_2 = B_29 ) ) ) ) ).
tff(fact_58_insert__ident,axiom,
! [B_29: fun_nat_bool,X_1: nat,A_2: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29))
=> ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_29) )
<=> ( A_2 = B_29 ) ) ) ) ).
tff(fact_59_insert__ident,axiom,
! [B_29: fun_Ho320143120e_bool,X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29))
=> ( ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),B_29) )
<=> ( A_2 = B_29 ) ) ) ) ).
tff(fact_60_insert__code,axiom,
! [Y_1: int,A_2: fun_int_bool,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),A_2),X_1))
<=> ( ( Y_1 = X_1 )
| hBOOL(hAPP_int_bool(A_2,X_1)) ) ) ).
tff(fact_61_insert__code,axiom,
! [Y_1: nat,A_2: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A_2),X_1))
<=> ( ( Y_1 = X_1 )
| hBOOL(hAPP_nat_bool(A_2,X_1)) ) ) ).
tff(fact_62_insert__code,axiom,
! [Y_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_1),A_2),X_1))
<=> ( ( Y_1 = X_1 )
| hBOOL(hAPP_H1645464494e_bool(A_2,X_1)) ) ) ).
tff(fact_63_insert__iff,axiom,
! [A_1: int,B_2: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),A_2)))
<=> ( ( A_1 = B_2 )
| hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2)) ) ) ).
tff(fact_64_insert__iff,axiom,
! [A_1: nat,B_2: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),A_2)))
<=> ( ( A_1 = B_2 )
| hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2)) ) ) ).
tff(fact_65_insert__iff,axiom,
! [A_1: hoare_363350478_state,B_2: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_2),A_2)))
<=> ( ( A_1 = B_2 )
| hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2)) ) ) ).
tff(fact_66_insert__commute,axiom,
! [X_1: int,Y_1: int,A_2: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),A_2)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) ).
tff(fact_67_insert__commute,axiom,
! [X_1: nat,Y_1: nat,A_2: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) ).
tff(fact_68_insert__commute,axiom,
! [X_1: hoare_363350478_state,Y_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_1),A_2)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) ).
tff(fact_69_insert__absorb2,axiom,
! [X_1: int,A_2: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2) ).
tff(fact_70_insert__absorb2,axiom,
! [X_1: nat,A_2: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2) ).
tff(fact_71_insert__absorb2,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2) ).
tff(fact_72_insert__Collect,axiom,
! [A_1: int,Pa: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),hAPP_f1805168059t_bool(collect_int,Pa)) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(cOMBS_int_bool_bool(hAPP_f2144054103l_bool(cOMBB_1652995168ol_int(fimplies),hAPP_f1805168059t_bool(cOMBB_bool_bool_int(fNot),hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_1)))),Pa)) ).
tff(fact_73_insert__Collect,axiom,
! [A_1: nat,Pa: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(collect_nat,Pa)) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(cOMBS_nat_bool_bool(hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fimplies),hAPP_f800510211t_bool(cOMBB_bool_bool_nat(fNot),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_1)))),Pa)) ).
tff(fact_74_insert__Collect,axiom,
! [A_1: hoare_363350478_state,Pa: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),hAPP_f1749453397e_bool(collec223185785_state,Pa)) = hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(cOMBS_56025283l_bool(hAPP_f1422032445l_bool(cOMBB_877416532_state(fimplies),hAPP_f1749453397e_bool(cOMBB_2059354034_state(fNot),hAPP_H705653759e_bool(cOMBC_13083556e_bool(fequal1026769423_state),A_1)))),Pa)) ).
tff(fact_75_insert__compr,axiom,
! [A_1: int,B_29: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_29) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(cOMBS_int_bool_bool(hAPP_f2144054103l_bool(cOMBB_1652995168ol_int(fdisj),hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_1))),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),B_29))) ).
tff(fact_76_insert__compr,axiom,
! [A_1: nat,B_29: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_29) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(cOMBS_nat_bool_bool(hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fdisj),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_1))),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B_29))) ).
tff(fact_77_insert__compr,axiom,
! [A_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_29) = hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(cOMBS_56025283l_bool(hAPP_f1422032445l_bool(cOMBB_877416532_state(fdisj),hAPP_H705653759e_bool(cOMBC_13083556e_bool(fequal1026769423_state),A_1))),hAPP_f1749453397e_bool(cOMBC_1415895650l_bool(member1253905979_state),B_29))) ).
tff(fact_78_insertI1,axiom,
! [A_1: int,B_29: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_29))) ).
tff(fact_79_insertI1,axiom,
! [A_1: nat,B_29: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_29))) ).
tff(fact_80_insertI1,axiom,
! [A_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_29))) ).
tff(fact_81_singleton__inject,axiom,
! [A_1: int,B_2: int] :
( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),bot_bot_fun_int_bool) )
=> ( A_1 = B_2 ) ) ).
tff(fact_82_singleton__inject,axiom,
! [A_1: nat,B_2: nat] :
( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),bot_bot_fun_nat_bool) )
=> ( A_1 = B_2 ) ) ).
tff(fact_83_singleton__inject,axiom,
! [A_1: hoare_363350478_state,B_2: hoare_363350478_state] :
( ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_2),bot_bo39439654e_bool) )
=> ( A_1 = B_2 ) ) ).
tff(fact_84_singletonE,axiom,
! [B_2: int,A_1: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool)))
=> ( B_2 = A_1 ) ) ).
tff(fact_85_singletonE,axiom,
! [B_2: nat,A_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))
=> ( B_2 = A_1 ) ) ).
tff(fact_86_singletonE,axiom,
! [B_2: hoare_363350478_state,A_1: hoare_363350478_state] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool)))
=> ( B_2 = A_1 ) ) ).
tff(fact_87_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_88_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_89_the__elem__eq,axiom,
! [X_1: hoare_363350478_state] : hAPP_f1242401993_state(the_el1666657391_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)) = X_1 ).
tff(fact_90_bot__apply,axiom,
! [X_1: int] :
( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_91_bot__apply,axiom,
! [X_1: nat] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_92_bot__apply,axiom,
! [X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(bot_bo39439654e_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_93_bot__fun__def,axiom,
! [X_2: int] :
( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_2))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_94_bot__fun__def,axiom,
! [X_2: nat] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_95_bot__fun__def,axiom,
! [X_2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(bot_bo39439654e_bool,X_2))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_96_hoare__derivs_OSkip,axiom,
! [G_1: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool] : hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,skip,Pa)),bot_bo39439654e_bool))) ).
tff(fact_97_Comp,axiom,
! [D_1: com,R_3: fun_st1506752259e_bool,G_1: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool,Ca: com,Qa: fun_st1506752259e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,Ca,Qa)),bot_bo39439654e_bool)))
=> ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Qa,D_1,R_3)),bot_bo39439654e_bool)))
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,semi(Ca,D_1),R_3)),bot_bo39439654e_bool))) ) ) ).
tff(fact_98_triple_Oexhaust,axiom,
! [Y_1: hoare_363350478_state] :
~ ! [Fun1: fun_st1506752259e_bool,Com_2: com,Fun2: fun_st1506752259e_bool] : Y_1 != hoare_608330478_state(Fun1,Com_2,Fun2) ).
tff(fact_99_Set_Oset__insert,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ~ ! [B_46: fun_int_bool] :
( ( A_2 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_46) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_46)) ) ) ).
tff(fact_100_Set_Oset__insert,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ~ ! [B_46: fun_nat_bool] :
( ( A_2 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_46) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_46)) ) ) ).
tff(fact_101_Set_Oset__insert,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ~ ! [B_46: fun_Ho320143120e_bool] :
( ( A_2 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),B_46) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_46)) ) ) ).
tff(fact_102_mk__disjoint__insert,axiom,
! [A_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> ? [B_46: fun_int_bool] :
( ( A_2 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_46) )
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),B_46)) ) ) ).
tff(fact_103_mk__disjoint__insert,axiom,
! [A_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> ? [B_46: fun_nat_bool] :
( ( A_2 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_46) )
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B_46)) ) ) ).
tff(fact_104_mk__disjoint__insert,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2))
=> ? [B_46: fun_Ho320143120e_bool] :
( ( A_2 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_46) )
& ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),B_46)) ) ) ).
tff(fact_105_equals0I,axiom,
! [A_2: fun_int_bool] :
( ! [Y_5: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_5),A_2))
=> ( A_2 = bot_bot_fun_int_bool ) ) ).
tff(fact_106_equals0I,axiom,
! [A_2: fun_nat_bool] :
( ! [Y_5: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_5),A_2))
=> ( A_2 = bot_bot_fun_nat_bool ) ) ).
tff(fact_107_equals0I,axiom,
! [A_2: fun_Ho320143120e_bool] :
( ! [Y_5: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Y_5),A_2))
=> ( A_2 = bot_bo39439654e_bool ) ) ).
tff(fact_108_conseq,axiom,
! [Qa: fun_st1506752259e_bool,G_1: fun_Ho320143120e_bool,Ca: com,Pa: fun_st1506752259e_bool] :
( ! [Z_13: state,S_1: state] :
( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_13),S_1))
=> ? [P_2: fun_st1506752259e_bool,Q_3: fun_st1506752259e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(P_2,Ca,Q_3)),bot_bo39439654e_bool)))
& ! [S_2: state] :
( ! [Z_14: state] :
( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_2,Z_14),S_1))
=> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_3,Z_14),S_2)) )
=> hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_13),S_2)) ) ) )
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(Pa,Ca,Qa)),bot_bo39439654e_bool))) ) ).
tff(fact_109_nonempty__iff,axiom,
! [A_2: fun_int_bool] :
( ( A_2 != bot_bot_fun_int_bool )
<=> ? [X_2: int,B_46: fun_int_bool] :
( ( A_2 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),B_46) )
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B_46)) ) ) ).
tff(fact_110_nonempty__iff,axiom,
! [A_2: fun_nat_bool] :
( ( A_2 != bot_bot_fun_nat_bool )
<=> ? [X_2: nat,B_46: fun_nat_bool] :
( ( A_2 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_46) )
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_46)) ) ) ).
tff(fact_111_nonempty__iff,axiom,
! [A_2: fun_Ho320143120e_bool] :
( ( A_2 != bot_bo39439654e_bool )
<=> ? [X_2: hoare_363350478_state,B_46: fun_Ho320143120e_bool] :
( ( A_2 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),B_46) )
& ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),B_46)) ) ) ).
tff(fact_112_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_113_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_114_bot__empty__eq,axiom,
! [X_2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(bot_bo39439654e_bool,X_2))
<=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),bot_bo39439654e_bool)) ) ).
tff(fact_115_com_Osimps_I13_J,axiom,
! [Com1_3: com,Com2_3: com] : semi(Com1_3,Com2_3) != skip ).
tff(fact_116_com_Osimps_I12_J,axiom,
! [Com1_3: com,Com2_3: com] : skip != semi(Com1_3,Com2_3) ).
tff(fact_117_com_Osimps_I3_J,axiom,
! [Com1: com,Com2: com,Com1_1: com,Com2_1: com] :
( ( semi(Com1,Com2) = semi(Com1_1,Com2_1) )
<=> ( ( Com1 = Com1_1 )
& ( Com2 = Com2_1 ) ) ) ).
tff(fact_118_fold1Set__sing,axiom,
! [F: fun_int_fun_int_int,A_1: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite_fold1Set_int(F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool)),B_2))
<=> ( A_1 = B_2 ) ) ).
tff(fact_119_fold1Set__sing,axiom,
! [F: fun_nat_fun_nat_nat,A_1: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite_fold1Set_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)),B_2))
<=> ( A_1 = B_2 ) ) ).
tff(fact_120_fold1Set__sing,axiom,
! [F: fun_Ho31865764_state,A_1: hoare_363350478_state,B_2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1652217598_state(F),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool)),B_2))
<=> ( A_1 = B_2 ) ) ).
tff(fact_121_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_122_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_123_folding__one_Osingleton,axiom,
! [X_1: hoare_363350478_state,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite269920017_state(F,F_1))
=> ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)) = X_1 ) ) ).
tff(fact_124_fold1__singleton,axiom,
! [F: fun_int_fun_int_int,A_1: int] : hAPP_f1594865479ol_int(finite_fold1_int(F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool)) = A_1 ).
tff(fact_125_fold1__singleton,axiom,
! [F: fun_nat_fun_nat_nat,A_1: nat] : hAPP_f22106695ol_nat(finite_fold1_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)) = A_1 ).
tff(fact_126_fold1__singleton,axiom,
! [F: fun_Ho31865764_state,A_1: hoare_363350478_state] : hAPP_f1242401993_state(finite841610862_state(F),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool)) = A_1 ).
tff(fact_127_fold1__singleton__def,axiom,
! [A_1: 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_1),bot_bot_fun_int_bool)) = A_1 ) ) ).
tff(fact_128_fold1__singleton__def,axiom,
! [A_1: 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_1),bot_bot_fun_nat_bool)) = A_1 ) ) ).
tff(fact_129_fold1__singleton__def,axiom,
! [A_1: hoare_363350478_state,G: fun_fu700438391_state,F: fun_Ho31865764_state] :
( ( G = finite841610862_state(F) )
=> ( hAPP_f1242401993_state(G,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool)) = A_1 ) ) ).
tff(fact_130_empty__fold1SetE,axiom,
! [F: fun_int_fun_int_int,X_1: int] : ~ hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite_fold1Set_int(F),bot_bot_fun_int_bool),X_1)) ).
tff(fact_131_empty__fold1SetE,axiom,
! [F: fun_nat_fun_nat_nat,X_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite_fold1Set_nat(F),bot_bot_fun_nat_bool),X_1)) ).
tff(fact_132_empty__fold1SetE,axiom,
! [F: fun_Ho31865764_state,X_1: hoare_363350478_state] : ~ hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1652217598_state(F),bot_bo39439654e_bool),X_1)) ).
tff(fact_133_fold1Set__nonempty,axiom,
! [F: fun_int_fun_int_int,A_2: fun_int_bool,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite_fold1Set_int(F),A_2),X_1))
=> ( A_2 != bot_bot_fun_int_bool ) ) ).
tff(fact_134_fold1Set__nonempty,axiom,
! [F: fun_nat_fun_nat_nat,A_2: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite_fold1Set_nat(F),A_2),X_1))
=> ( A_2 != bot_bot_fun_nat_bool ) ) ).
tff(fact_135_fold1Set__nonempty,axiom,
! [F: fun_Ho31865764_state,A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1652217598_state(F),A_2),X_1))
=> ( A_2 != bot_bo39439654e_bool ) ) ).
tff(fact_136_subset__singletonD,axiom,
! [A_2: fun_int_bool,X_1: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))
=> ( ( A_2 = bot_bot_fun_int_bool )
| ( A_2 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool) ) ) ) ).
tff(fact_137_subset__singletonD,axiom,
! [A_2: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))
=> ( ( A_2 = bot_bot_fun_nat_bool )
| ( A_2 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool) ) ) ) ).
tff(fact_138_subset__singletonD,axiom,
! [A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))
=> ( ( A_2 = bot_bo39439654e_bool )
| ( A_2 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool) ) ) ) ).
tff(fact_139_order__refl,axiom,
! [X_1: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),X_1)) ).
tff(fact_140_order__refl,axiom,
! [X_36: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_36),X_36)) ).
tff(fact_141_order__refl,axiom,
! [X_36: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_36),X_36)) ).
tff(fact_142_equalityI,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),A_2))
=> ( A_2 = B_29 ) ) ) ).
tff(fact_143_subsetD,axiom,
! [Ca: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),B_29)) ) ) ).
tff(fact_144_subsetD,axiom,
! [Ca: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B_29)) ) ) ).
tff(fact_145_subsetD,axiom,
! [Ca: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B_29)) ) ) ).
tff(fact_146_empty__subsetI,axiom,
! [A_2: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A_2)) ).
tff(fact_147_empty__subsetI,axiom,
! [A_2: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_2)) ).
tff(fact_148_empty__subsetI,axiom,
! [A_2: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,bot_bo39439654e_bool),A_2)) ).
tff(fact_149_linorder__le__cases,axiom,
! [X_35: nat,Y_34: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_35),Y_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_34),X_35)) ) ).
tff(fact_150_linorder__le__cases,axiom,
! [X_35: int,Y_34: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_35),Y_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_34),X_35)) ) ).
tff(fact_151_le__funE,axiom,
! [X_1: hoare_363350478_state,F: fun_Ho320143120e_bool,G: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1645464494e_bool(F,X_1)),hAPP_H1645464494e_bool(G,X_1))) ) ).
tff(fact_152_xt1_I6_J,axiom,
! [Z_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Z_1),Y_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Z_1),X_1)) ) ) ).
tff(fact_153_xt1_I6_J,axiom,
! [Z_12: nat,Y_33: nat,X_34: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_33),X_34))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_12),Y_33))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_12),X_34)) ) ) ).
tff(fact_154_xt1_I6_J,axiom,
! [Z_12: int,Y_33: int,X_34: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_33),X_34))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_12),Y_33))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_12),X_34)) ) ) ).
tff(fact_155_xt1_I5_J,axiom,
! [Y_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
=> ( X_1 = Y_1 ) ) ) ).
tff(fact_156_xt1_I5_J,axiom,
! [Y_32: nat,X_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_32),X_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_33),Y_32))
=> ( X_33 = Y_32 ) ) ) ).
tff(fact_157_xt1_I5_J,axiom,
! [Y_32: int,X_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_32),X_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_33),Y_32))
=> ( X_33 = Y_32 ) ) ) ).
tff(fact_158_order__trans,axiom,
! [Z_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),Z_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Z_1)) ) ) ).
tff(fact_159_order__trans,axiom,
! [Z_11: nat,X_32: nat,Y_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_32),Y_31))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_31),Z_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_32),Z_11)) ) ) ).
tff(fact_160_order__trans,axiom,
! [Z_11: int,X_32: int,Y_31: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_32),Y_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_31),Z_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_32),Z_11)) ) ) ).
tff(fact_161_order__antisym,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),X_1))
=> ( X_1 = Y_1 ) ) ) ).
tff(fact_162_order__antisym,axiom,
! [X_31: nat,Y_30: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_31),Y_30))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_30),X_31))
=> ( X_31 = Y_30 ) ) ) ).
tff(fact_163_order__antisym,axiom,
! [X_31: int,Y_30: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_31),Y_30))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_30),X_31))
=> ( X_31 = Y_30 ) ) ) ).
tff(fact_164_xt1_I4_J,axiom,
! [Ca: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),A_1))
=> ( ( B_2 = Ca )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),A_1)) ) ) ).
tff(fact_165_xt1_I4_J,axiom,
! [C_35: nat,B_45: nat,A_61: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_45),A_61))
=> ( ( B_45 = C_35 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_35),A_61)) ) ) ).
tff(fact_166_xt1_I4_J,axiom,
! [C_35: int,B_45: int,A_61: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_45),A_61))
=> ( ( B_45 = C_35 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_35),A_61)) ) ) ).
tff(fact_167_ord__le__eq__trans,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),B_2))
=> ( ( B_2 = Ca )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),Ca)) ) ) ).
tff(fact_168_ord__le__eq__trans,axiom,
! [C_34: nat,A_60: nat,B_44: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_60),B_44))
=> ( ( B_44 = C_34 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_60),C_34)) ) ) ).
tff(fact_169_ord__le__eq__trans,axiom,
! [C_34: int,A_60: int,B_44: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_60),B_44))
=> ( ( B_44 = C_34 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_60),C_34)) ) ) ).
tff(fact_170_xt1_I3_J,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( ( A_1 = B_2 )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),B_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),A_1)) ) ) ).
tff(fact_171_xt1_I3_J,axiom,
! [C_33: nat,A_59: nat,B_43: nat] :
( ( A_59 = B_43 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_33),B_43))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_33),A_59)) ) ) ).
tff(fact_172_xt1_I3_J,axiom,
! [C_33: int,A_59: int,B_43: int] :
( ( A_59 = B_43 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_33),B_43))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_33),A_59)) ) ) ).
tff(fact_173_ord__eq__le__trans,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( ( A_1 = B_2 )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),Ca))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),Ca)) ) ) ).
tff(fact_174_ord__eq__le__trans,axiom,
! [C_32: nat,A_58: nat,B_42: nat] :
( ( A_58 = B_42 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_42),C_32))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_58),C_32)) ) ) ).
tff(fact_175_ord__eq__le__trans,axiom,
! [C_32: int,A_58: int,B_42: int] :
( ( A_58 = B_42 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_42),C_32))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_58),C_32)) ) ) ).
tff(fact_176_predicate1D,axiom,
! [X_1: hoare_363350478_state,Pa: fun_Ho320143120e_bool,Qa: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Pa),Qa))
=> ( hBOOL(hAPP_H1645464494e_bool(Pa,X_1))
=> hBOOL(hAPP_H1645464494e_bool(Qa,X_1)) ) ) ).
tff(fact_177_mem__def,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
<=> hBOOL(hAPP_H1645464494e_bool(A_2,X_1)) ) ).
tff(fact_178_mem__def,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
<=> hBOOL(hAPP_int_bool(A_2,X_1)) ) ).
tff(fact_179_mem__def,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
<=> hBOOL(hAPP_nat_bool(A_2,X_1)) ) ).
tff(fact_180_Collect__def,axiom,
! [Pa: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(collec223185785_state,Pa) = Pa ).
tff(fact_181_order__antisym__conv,axiom,
! [Y_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_182_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_183_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_184_le__funD,axiom,
! [X_1: hoare_363350478_state,F: fun_Ho320143120e_bool,G: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1645464494e_bool(F,X_1)),hAPP_H1645464494e_bool(G,X_1))) ) ).
tff(fact_185_order__eq__refl,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( ( X_1 = Y_1 )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1)) ) ).
tff(fact_186_order__eq__refl,axiom,
! [X_30: nat,Y_29: nat] :
( ( X_30 = Y_29 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_29)) ) ).
tff(fact_187_order__eq__refl,axiom,
! [X_30: int,Y_29: int] :
( ( X_30 = Y_29 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Y_29)) ) ).
tff(fact_188_rev__predicate1D,axiom,
! [Qa: fun_Ho320143120e_bool,Pa: fun_Ho320143120e_bool,X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(Pa,X_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Pa),Qa))
=> hBOOL(hAPP_H1645464494e_bool(Qa,X_1)) ) ) ).
tff(fact_189_order__eq__iff,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( ( X_1 = Y_1 )
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
& hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),X_1)) ) ) ).
tff(fact_190_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_191_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_192_linorder__linear,axiom,
! [X_29: nat,Y_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_28))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_28),X_29)) ) ).
tff(fact_193_linorder__linear,axiom,
! [X_29: int,Y_28: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_29),Y_28))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_28),X_29)) ) ).
tff(fact_194_le__fun__def,axiom,
! [F: fun_Ho320143120e_bool,G: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,F),G))
<=> ! [X_2: hoare_363350478_state] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1645464494e_bool(F,X_2)),hAPP_H1645464494e_bool(G,X_2))) ) ).
tff(fact_195_equalityE,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( ( A_2 = B_29 )
=> ~ ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),A_2)) ) ) ).
tff(fact_196_subset__trans,axiom,
! [C_26: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),C_26))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),C_26)) ) ) ).
tff(fact_197_set__mp,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29)) ) ) ).
tff(fact_198_set__mp,axiom,
! [X_1: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29)) ) ) ).
tff(fact_199_set__mp,axiom,
! [X_1: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29)) ) ) ).
tff(fact_200_set__rev__mp,axiom,
! [B_29: fun_Ho320143120e_bool,X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29)) ) ) ).
tff(fact_201_set__rev__mp,axiom,
! [B_29: fun_int_bool,X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29)) ) ) ).
tff(fact_202_set__rev__mp,axiom,
! [B_29: fun_nat_bool,X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29)) ) ) ).
tff(fact_203_in__mono,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29)) ) ) ).
tff(fact_204_in__mono,axiom,
! [X_1: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29)) ) ) ).
tff(fact_205_in__mono,axiom,
! [X_1: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29)) ) ) ).
tff(fact_206_equalityD2,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( ( A_2 = B_29 )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),A_2)) ) ).
tff(fact_207_equalityD1,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( ( A_2 = B_29 )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29)) ) ).
tff(fact_208_set__eq__subset,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( ( A_2 = B_29 )
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
& hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),A_2)) ) ) ).
tff(fact_209_subset__refl,axiom,
! [A_2: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),A_2)) ).
tff(fact_210_le__bot,axiom,
! [A_1: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),bot_bot_fun_int_bool))
=> ( A_1 = bot_bot_fun_int_bool ) ) ).
tff(fact_211_le__bot,axiom,
! [A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),bot_bot_fun_nat_bool))
=> ( A_1 = bot_bot_fun_nat_bool ) ) ).
tff(fact_212_le__bot,axiom,
! [A_1: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_1),bot_bot_bool))
=> ( hBOOL(A_1)
<=> hBOOL(bot_bot_bool) ) ) ).
tff(fact_213_le__bot,axiom,
! [A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),bot_bo39439654e_bool))
=> ( A_1 = bot_bo39439654e_bool ) ) ).
tff(fact_214_le__bot,axiom,
! [A_57: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_57),bot_bot_nat))
=> ( A_57 = bot_bot_nat ) ) ).
tff(fact_215_bot__unique,axiom,
! [A_1: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),bot_bot_fun_int_bool))
<=> ( A_1 = bot_bot_fun_int_bool ) ) ).
tff(fact_216_bot__unique,axiom,
! [A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),bot_bot_fun_nat_bool))
<=> ( A_1 = bot_bot_fun_nat_bool ) ) ).
tff(fact_217_bot__unique,axiom,
! [A_1: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_1),bot_bot_bool))
<=> ( hBOOL(A_1)
<=> hBOOL(bot_bot_bool) ) ) ).
tff(fact_218_bot__unique,axiom,
! [A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),bot_bo39439654e_bool))
<=> ( A_1 = bot_bo39439654e_bool ) ) ).
tff(fact_219_bot__unique,axiom,
! [A_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),bot_bot_nat))
<=> ( A_1 = bot_bot_nat ) ) ).
tff(fact_220_bot__least,axiom,
! [A_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A_1)) ).
tff(fact_221_bot__least,axiom,
! [A_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_1)) ).
tff(fact_222_bot__least,axiom,
! [A_1: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,bot_bot_bool),A_1)) ).
tff(fact_223_bot__least,axiom,
! [A_1: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,bot_bo39439654e_bool),A_1)) ).
tff(fact_224_bot__least,axiom,
! [A_56: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_56)) ).
tff(fact_225_subset__empty,axiom,
! [A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),bot_bot_fun_int_bool))
<=> ( A_2 = bot_bot_fun_int_bool ) ) ).
tff(fact_226_subset__empty,axiom,
! [A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),bot_bot_fun_nat_bool))
<=> ( A_2 = bot_bot_fun_nat_bool ) ) ).
tff(fact_227_subset__empty,axiom,
! [A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),bot_bo39439654e_bool))
<=> ( A_2 = bot_bo39439654e_bool ) ) ).
tff(fact_228_insert__mono,axiom,
! [A_1: int,C_26: fun_int_bool,D_10: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_26),D_10))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),C_26)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),D_10))) ) ).
tff(fact_229_insert__mono,axiom,
! [A_1: nat,C_26: fun_nat_bool,D_10: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_26),D_10))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),C_26)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),D_10))) ) ).
tff(fact_230_insert__mono,axiom,
! [A_1: hoare_363350478_state,C_26: fun_Ho320143120e_bool,D_10: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,C_26),D_10))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),C_26)),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),D_10))) ) ).
tff(fact_231_subset__insertI2,axiom,
! [B_2: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),B_29))) ) ).
tff(fact_232_subset__insertI2,axiom,
! [B_2: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),B_29))) ) ).
tff(fact_233_subset__insertI2,axiom,
! [B_2: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_2),B_29))) ) ).
tff(fact_234_subset__insert,axiom,
! [B_29: fun_int_bool,X_1: int,A_2: fun_int_bool] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_29)))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29)) ) ) ).
tff(fact_235_subset__insert,axiom,
! [B_29: fun_nat_bool,X_1: nat,A_2: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_29)))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29)) ) ) ).
tff(fact_236_subset__insert,axiom,
! [B_29: fun_Ho320143120e_bool,X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),B_29)))
<=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29)) ) ) ).
tff(fact_237_insert__subset,axiom,
! [X_1: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)),B_29))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29))
& hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29)) ) ) ).
tff(fact_238_insert__subset,axiom,
! [X_1: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)),B_29))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29)) ) ) ).
tff(fact_239_insert__subset,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)),B_29))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29))
& hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29)) ) ) ).
tff(fact_240_subset__insertI,axiom,
! [B_29: fun_int_bool,A_1: int] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_29),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_29))) ).
tff(fact_241_subset__insertI,axiom,
! [B_29: fun_nat_bool,A_1: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_29),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_29))) ).
tff(fact_242_subset__insertI,axiom,
! [B_29: fun_Ho320143120e_bool,A_1: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_29))) ).
tff(fact_243_thin,axiom,
! [G_1: fun_Ho320143120e_bool,G_2: fun_Ho320143120e_bool,Ts: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_2),Ts))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,G_2),G_1))
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts)) ) ) ).
tff(fact_244_weaken,axiom,
! [Ts: fun_Ho320143120e_bool,G_1: fun_Ho320143120e_bool,Ts_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ts),Ts_1))
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts)) ) ) ).
tff(fact_245_asm,axiom,
! [Ts: fun_Ho320143120e_bool,G_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ts),G_1))
=> hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts)) ) ).
tff(fact_246_subsetI,axiom,
! [B_29: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool] :
( ! [X_2: hoare_363350478_state] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),B_29)) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29)) ) ).
tff(fact_247_subsetI,axiom,
! [B_29: fun_int_bool,A_2: fun_int_bool] :
( ! [X_2: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B_29)) )
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29)) ) ).
tff(fact_248_subsetI,axiom,
! [B_29: fun_nat_bool,A_2: fun_nat_bool] :
( ! [X_2: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_29)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29)) ) ).
tff(fact_249_le__funI,axiom,
! [F: fun_Ho320143120e_bool,G: fun_Ho320143120e_bool] :
( ! [X_2: hoare_363350478_state] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1645464494e_bool(F,X_2)),hAPP_H1645464494e_bool(G,X_2)))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,F),G)) ) ).
tff(fact_250_predicate1I,axiom,
! [Qa: fun_Ho320143120e_bool,Pa: fun_Ho320143120e_bool] :
( ! [X_2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(Pa,X_2))
=> hBOOL(hAPP_H1645464494e_bool(Qa,X_2)) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Pa),Qa)) ) ).
tff(fact_251_Collect__mono,axiom,
! [Qa: fun_Ho320143120e_bool,Pa: fun_Ho320143120e_bool] :
( ! [X_2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(Pa,X_2))
=> hBOOL(hAPP_H1645464494e_bool(Qa,X_2)) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(collec223185785_state,Pa)),hAPP_f1749453397e_bool(collec223185785_state,Qa))) ) ).
tff(fact_252_xt2,axiom,
! [Ca: fun_Ho320143120e_bool,F: fun_fu645263415ol_nat,B_2: fun_Ho320143120e_bool,A_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,B_2)),A_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),B_2))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_5),X_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,Y_5)),hAPP_f1705983439ol_nat(F,X_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,Ca)),A_1)) ) ) ) ).
tff(fact_253_xt2,axiom,
! [Ca: fun_Ho320143120e_bool,F: fun_fu807176339ol_int,B_2: fun_Ho320143120e_bool,A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,B_2)),A_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),B_2))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_5),X_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,Y_5)),hAPP_f195773483ol_int(F,X_2))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,Ca)),A_1)) ) ) ) ).
tff(fact_254_xt2,axiom,
! [Ca: nat,F: fun_na1285216451e_bool,B_2: nat,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,B_2)),A_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ca),B_2))
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,Y_5)),hAPP_n753050587e_bool(F,X_2))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,Ca)),A_1)) ) ) ) ).
tff(fact_255_xt2,axiom,
! [Ca: int,F: fun_in255859559e_bool,B_2: int,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,B_2)),A_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),B_2))
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,Y_5)),hAPP_i1871177343e_bool(F,X_2))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,Ca)),A_1)) ) ) ) ).
tff(fact_256_xt1_I15_J,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,F: fun_fu1066574909e_bool,B_2: fun_Ho320143120e_bool] :
( ( A_1 = hAPP_f1749453397e_bool(F,B_2) )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),B_2))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_5),X_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(F,Y_5)),hAPP_f1749453397e_bool(F,X_2))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(F,Ca)),A_1)) ) ) ) ).
tff(fact_257_xt1_I15_J,axiom,
! [Ca: nat,A_1: nat,F: fun_nat_nat,B_2: nat] :
( ( A_1 = hAPP_nat_nat(F,B_2) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ca),B_2))
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Y_5)),hAPP_nat_nat(F,X_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Ca)),A_1)) ) ) ) ).
tff(fact_258_xt1_I15_J,axiom,
! [Ca: int,A_1: int,F: fun_int_int,B_2: int] :
( ( A_1 = hAPP_int_int(F,B_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),B_2))
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(F,Y_5)),hAPP_int_int(F,X_2))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(F,Ca)),A_1)) ) ) ) ).
tff(fact_259_order__subst1,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: nat,F: fun_fu645263415ol_nat,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),hAPP_f1705983439ol_nat(F,B_2)))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),Ca))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,X_2)),hAPP_f1705983439ol_nat(F,Y_5))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),hAPP_f1705983439ol_nat(F,Ca))) ) ) ) ).
tff(fact_260_order__subst1,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: int,F: fun_fu807176339ol_int,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),hAPP_f195773483ol_int(F,B_2)))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),Ca))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,X_2)),hAPP_f195773483ol_int(F,Y_5))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),hAPP_f195773483ol_int(F,Ca))) ) ) ) ).
tff(fact_261_order__subst1,axiom,
! [Ca: nat,A_1: fun_Ho320143120e_bool,F: fun_na1285216451e_bool,B_2: nat] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),hAPP_n753050587e_bool(F,B_2)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),Ca))
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,X_2)),hAPP_n753050587e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),hAPP_n753050587e_bool(F,Ca))) ) ) ) ).
tff(fact_262_order__subst1,axiom,
! [Ca: int,A_1: fun_Ho320143120e_bool,F: fun_in255859559e_bool,B_2: int] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),hAPP_i1871177343e_bool(F,B_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),Ca))
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,X_2)),hAPP_i1871177343e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),hAPP_i1871177343e_bool(F,Ca))) ) ) ) ).
tff(fact_263_ord__eq__le__subst,axiom,
! [Ca: nat,A_1: fun_Ho320143120e_bool,F: fun_na1285216451e_bool,B_2: nat] :
( ( A_1 = hAPP_n753050587e_bool(F,B_2) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),Ca))
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,X_2)),hAPP_n753050587e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),hAPP_n753050587e_bool(F,Ca))) ) ) ) ).
tff(fact_264_ord__eq__le__subst,axiom,
! [Ca: int,A_1: fun_Ho320143120e_bool,F: fun_in255859559e_bool,B_2: int] :
( ( A_1 = hAPP_i1871177343e_bool(F,B_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),Ca))
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,X_2)),hAPP_i1871177343e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),hAPP_i1871177343e_bool(F,Ca))) ) ) ) ).
tff(fact_265_ord__eq__le__subst,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: nat,F: fun_fu645263415ol_nat,B_2: fun_Ho320143120e_bool] :
( ( A_1 = hAPP_f1705983439ol_nat(F,B_2) )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),Ca))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,X_2)),hAPP_f1705983439ol_nat(F,Y_5))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),hAPP_f1705983439ol_nat(F,Ca))) ) ) ) ).
tff(fact_266_ord__eq__le__subst,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: int,F: fun_fu807176339ol_int,B_2: fun_Ho320143120e_bool] :
( ( A_1 = hAPP_f195773483ol_int(F,B_2) )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),Ca))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,X_2)),hAPP_f195773483ol_int(F,Y_5))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),hAPP_f195773483ol_int(F,Ca))) ) ) ) ).
tff(fact_267_order__subst2,axiom,
! [F: fun_na1285216451e_bool,Ca: fun_Ho320143120e_bool,A_1: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,B_2)),Ca))
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,X_2)),hAPP_n753050587e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,A_1)),Ca)) ) ) ) ).
tff(fact_268_order__subst2,axiom,
! [F: fun_in255859559e_bool,Ca: fun_Ho320143120e_bool,A_1: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,B_2)),Ca))
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,X_2)),hAPP_i1871177343e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,A_1)),Ca)) ) ) ) ).
tff(fact_269_order__subst2,axiom,
! [F: fun_fu645263415ol_nat,Ca: nat,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),B_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,B_2)),Ca))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,X_2)),hAPP_f1705983439ol_nat(F,Y_5))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,A_1)),Ca)) ) ) ) ).
tff(fact_270_order__subst2,axiom,
! [F: fun_fu807176339ol_int,Ca: int,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,B_2)),Ca))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,X_2)),hAPP_f195773483ol_int(F,Y_5))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,A_1)),Ca)) ) ) ) ).
tff(fact_271_ord__le__eq__subst,axiom,
! [F: fun_na1285216451e_bool,Ca: fun_Ho320143120e_bool,A_1: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_2))
=> ( ( hAPP_n753050587e_bool(F,B_2) = Ca )
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,X_2)),hAPP_n753050587e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,A_1)),Ca)) ) ) ) ).
tff(fact_272_ord__le__eq__subst,axiom,
! [F: fun_in255859559e_bool,Ca: fun_Ho320143120e_bool,A_1: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_2))
=> ( ( hAPP_i1871177343e_bool(F,B_2) = Ca )
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_5))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,X_2)),hAPP_i1871177343e_bool(F,Y_5))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,A_1)),Ca)) ) ) ) ).
tff(fact_273_ord__le__eq__subst,axiom,
! [F: fun_fu645263415ol_nat,Ca: nat,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),B_2))
=> ( ( hAPP_f1705983439ol_nat(F,B_2) = Ca )
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,X_2)),hAPP_f1705983439ol_nat(F,Y_5))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,A_1)),Ca)) ) ) ) ).
tff(fact_274_ord__le__eq__subst,axiom,
! [F: fun_fu807176339ol_int,Ca: int,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),B_2))
=> ( ( hAPP_f195773483ol_int(F,B_2) = Ca )
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_2),Y_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,X_2)),hAPP_f195773483ol_int(F,Y_5))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,A_1)),Ca)) ) ) ) ).
tff(fact_275_xt3,axiom,
! [Ca: fun_Ho320143120e_bool,F: fun_na1285216451e_bool,B_2: nat,A_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),A_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),hAPP_n753050587e_bool(F,B_2)))
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_n753050587e_bool(F,Y_5)),hAPP_n753050587e_bool(F,X_2))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),hAPP_n753050587e_bool(F,A_1))) ) ) ) ).
tff(fact_276_xt3,axiom,
! [Ca: fun_Ho320143120e_bool,F: fun_in255859559e_bool,B_2: int,A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),hAPP_i1871177343e_bool(F,B_2)))
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_i1871177343e_bool(F,Y_5)),hAPP_i1871177343e_bool(F,X_2))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),hAPP_i1871177343e_bool(F,A_1))) ) ) ) ).
tff(fact_277_xt3,axiom,
! [Ca: nat,F: fun_fu645263415ol_nat,B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),A_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ca),hAPP_f1705983439ol_nat(F,B_2)))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_5),X_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(F,Y_5)),hAPP_f1705983439ol_nat(F,X_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ca),hAPP_f1705983439ol_nat(F,A_1))) ) ) ) ).
tff(fact_278_xt3,axiom,
! [Ca: int,F: fun_fu807176339ol_int,B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),A_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),hAPP_f195773483ol_int(F,B_2)))
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_5),X_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_f195773483ol_int(F,Y_5)),hAPP_f195773483ol_int(F,X_2))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),hAPP_f195773483ol_int(F,A_1))) ) ) ) ).
tff(fact_279_xt1_I16_J,axiom,
! [F: fun_fu1066574909e_bool,Ca: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),A_1))
=> ( ( hAPP_f1749453397e_bool(F,B_2) = Ca )
=> ( ! [X_2: fun_Ho320143120e_bool,Y_5: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_5),X_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(F,Y_5)),hAPP_f1749453397e_bool(F,X_2))) )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Ca),hAPP_f1749453397e_bool(F,A_1))) ) ) ) ).
tff(fact_280_xt1_I16_J,axiom,
! [F: fun_nat_nat,Ca: nat,B_2: nat,A_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),A_1))
=> ( ( hAPP_nat_nat(F,B_2) = Ca )
=> ( ! [X_2: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Y_5)),hAPP_nat_nat(F,X_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ca),hAPP_nat_nat(F,A_1))) ) ) ) ).
tff(fact_281_xt1_I16_J,axiom,
! [F: fun_int_int,Ca: int,B_2: int,A_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_1))
=> ( ( hAPP_int_int(F,B_2) = Ca )
=> ( ! [X_2: int,Y_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(F,Y_5)),hAPP_int_int(F,X_2))) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),hAPP_int_int(F,A_1))) ) ) ) ).
tff(fact_282_fold1Set_Ointros,axiom,
! [F: fun_int_fun_int_int,A_1: int,A_2: fun_int_bool,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,A_1),A_2),X_1))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite_fold1Set_int(F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2)),X_1)) ) ) ).
tff(fact_283_fold1Set_Ointros,axiom,
! [F: fun_nat_fun_nat_nat,A_1: nat,A_2: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,A_1),A_2),X_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite_fold1Set_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2)),X_1)) ) ) ).
tff(fact_284_fold1Set_Ointros,axiom,
! [F: fun_Ho31865764_state,A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,A_1),A_2),X_1))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2))
=> hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1652217598_state(F),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2)),X_1)) ) ) ).
tff(fact_285_folding__one_Oinsert,axiom,
! [X_1: int,A_2: 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_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( ( A_2 != bot_bot_fun_int_bool )
=> ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A_2)) ) ) ) ) ) ).
tff(fact_286_folding__one_Oinsert,axiom,
! [X_1: nat,A_2: 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_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( ( A_2 != bot_bot_fun_nat_bool )
=> ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A_2)) ) ) ) ) ) ).
tff(fact_287_folding__one_Oinsert,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite269920017_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( ( A_2 != bot_bo39439654e_bool )
=> ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),hAPP_f1242401993_state(F_1,A_2)) ) ) ) ) ) ).
tff(fact_288_Powp__mono,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> hBOOL(hAPP_f1287348838l_bool(hAPP_f746351743l_bool(ord_le2031828156l_bool,powp_H991500296_state(A_2)),powp_H991500296_state(B_29))) ) ).
tff(fact_289_subset__insert__iff,axiom,
! [A_2: fun_int_bool,X_1: int,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_29)))
<=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B_29)) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29)) ) ) ) ).
tff(fact_290_subset__insert__iff,axiom,
! [A_2: fun_nat_bool,X_1: nat,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_29)))
<=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B_29)) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29)) ) ) ) ).
tff(fact_291_subset__insert__iff,axiom,
! [A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),B_29)))
<=> ( ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool))),B_29)) )
& ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29)) ) ) ) ).
tff(fact_292_diff__single__insert,axiom,
! [A_2: fun_int_bool,X_1: int,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_29))) ) ) ).
tff(fact_293_diff__single__insert,axiom,
! [A_2: fun_nat_bool,X_1: nat,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_29))) ) ) ).
tff(fact_294_diff__single__insert,axiom,
! [A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool))),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),B_29))) ) ) ).
tff(fact_295_folding__one_Oeq__fold,axiom,
! [A_2: 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_2))
=> ( hAPP_f1594865479ol_int(F_1,A_2) = hAPP_f1594865479ol_int(finite_fold1_int(F),A_2) ) ) ) ).
tff(fact_296_folding__one_Oeq__fold,axiom,
! [A_2: 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_2))
=> ( hAPP_f22106695ol_nat(F_1,A_2) = hAPP_f22106695ol_nat(finite_fold1_nat(F),A_2) ) ) ) ).
tff(fact_297_folding__one_Oeq__fold,axiom,
! [A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite269920017_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hAPP_f1242401993_state(F_1,A_2) = hAPP_f1242401993_state(finite841610862_state(F),A_2) ) ) ) ).
tff(fact_298_folding__one_Oclosed,axiom,
! [A_2: 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_2))
=> ( ( A_2 != bot_bot_fun_int_bool )
=> ( ! [X_2: int,Y_5: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_5)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_5),bot_bot_fun_int_bool))))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(F_1,A_2)),A_2)) ) ) ) ) ).
tff(fact_299_folding__one_Oclosed,axiom,
! [A_2: 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_2))
=> ( ( A_2 != bot_bot_fun_nat_bool )
=> ( ! [X_2: nat,Y_5: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_5)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_5),bot_bot_fun_nat_bool))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F_1,A_2)),A_2)) ) ) ) ) ).
tff(fact_300_folding__one_Oclosed,axiom,
! [A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite269920017_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ( A_2 != bot_bo39439654e_bool )
=> ( ! [X_2: hoare_363350478_state,Y_5: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,hAPP_H1982017439_state(hAPP_H230141040_state(F,X_2),Y_5)),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_5),bot_bo39439654e_bool))))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,hAPP_f1242401993_state(F_1,A_2)),A_2)) ) ) ) ) ).
tff(fact_301_DiffE,axiom,
! [Ca: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)))
=> ~ ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),B_29)) ) ) ).
tff(fact_302_DiffE,axiom,
! [Ca: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)))
=> ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B_29)) ) ) ).
tff(fact_303_DiffE,axiom,
! [Ca: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)))
=> ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B_29)) ) ) ).
tff(fact_304_DiffI,axiom,
! [B_29: fun_Ho320143120e_bool,Ca: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),A_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),B_29))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29))) ) ) ).
tff(fact_305_DiffI,axiom,
! [B_29: fun_int_bool,Ca: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B_29))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29))) ) ) ).
tff(fact_306_DiffI,axiom,
! [B_29: fun_nat_bool,Ca: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B_29))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29))) ) ) ).
tff(fact_307_finite_OemptyI,axiom,
hBOOL(hAPP_f448129468l_bool(finite_finite_int,bot_bot_fun_int_bool)) ).
tff(fact_308_finite_OemptyI,axiom,
hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).
tff(fact_309_finite_OemptyI,axiom,
hBOOL(hAPP_f568432088l_bool(finite280062779_state,bot_bo39439654e_bool)) ).
tff(fact_310_finite_OinsertI,axiom,
! [A_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2))) ) ).
tff(fact_311_finite_OinsertI,axiom,
! [A_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2))) ) ).
tff(fact_312_finite_OinsertI,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2))) ) ).
tff(fact_313_finite__Diff,axiom,
! [B_29: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29))) ) ).
tff(fact_314_finite__Diff,axiom,
! [B_29: fun_int_bool,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29))) ) ).
tff(fact_315_finite__Diff,axiom,
! [B_29: fun_nat_bool,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29))) ) ).
tff(fact_316_finite__Diff__insert,axiom,
! [A_2: fun_int_bool,A_1: int,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_29))))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29))) ) ).
tff(fact_317_finite__Diff__insert,axiom,
! [A_2: fun_nat_bool,A_1: nat,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_29))))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29))) ) ).
tff(fact_318_finite__Diff__insert,axiom,
! [A_2: fun_Ho320143120e_bool,A_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_29))))
<=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29))) ) ).
tff(fact_319_finite__Diff2,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)))
<=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2)) ) ) ).
tff(fact_320_finite__Diff2,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2)) ) ) ).
tff(fact_321_finite__Diff2,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2)) ) ) ).
tff(fact_322_DiffD2,axiom,
! [Ca: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)))
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),B_29)) ) ).
tff(fact_323_DiffD2,axiom,
! [Ca: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B_29)) ) ).
tff(fact_324_DiffD2,axiom,
! [Ca: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B_29)) ) ).
tff(fact_325_DiffD1,axiom,
! [Ca: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),A_2)) ) ).
tff(fact_326_DiffD1,axiom,
! [Ca: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A_2)) ) ).
tff(fact_327_DiffD1,axiom,
! [Ca: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A_2)) ) ).
tff(fact_328_Diff__idemp,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)),B_29) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29) ).
tff(fact_329_Diff__idemp,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)),B_29) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29) ).
tff(fact_330_Diff__idemp,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)),B_29) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29) ).
tff(fact_331_Diff__iff,axiom,
! [Ca: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),A_2))
& ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),B_29)) ) ) ).
tff(fact_332_Diff__iff,axiom,
! [Ca: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A_2))
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B_29)) ) ) ).
tff(fact_333_Diff__iff,axiom,
! [Ca: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A_2))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B_29)) ) ) ).
tff(fact_334_set__diff__eq,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29) = hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(cOMBS_56025283l_bool(hAPP_f1422032445l_bool(cOMBB_877416532_state(fconj),hAPP_f1749453397e_bool(cOMBC_1415895650l_bool(member1253905979_state),A_2))),hAPP_f1749453397e_bool(cOMBB_2059354034_state(fNot),hAPP_f1749453397e_bool(cOMBC_1415895650l_bool(member1253905979_state),B_29)))) ).
tff(fact_335_set__diff__eq,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29) = hAPP_f1805168059t_bool(collect_int,hAPP_f1805168059t_bool(cOMBS_int_bool_bool(hAPP_f2144054103l_bool(cOMBB_1652995168ol_int(fconj),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A_2))),hAPP_f1805168059t_bool(cOMBB_bool_bool_int(fNot),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),B_29)))) ).
tff(fact_336_set__diff__eq,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(cOMBS_nat_bool_bool(hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A_2))),hAPP_f800510211t_bool(cOMBB_bool_bool_nat(fNot),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B_29)))) ).
tff(fact_337_Diff__cancel,axiom,
! [A_2: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),A_2) = bot_bot_fun_int_bool ).
tff(fact_338_Diff__cancel,axiom,
! [A_2: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),A_2) = bot_bot_fun_nat_bool ).
tff(fact_339_Diff__cancel,axiom,
! [A_2: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),A_2) = bot_bo39439654e_bool ).
tff(fact_340_Diff__empty,axiom,
! [A_2: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),bot_bot_fun_int_bool) = A_2 ).
tff(fact_341_Diff__empty,axiom,
! [A_2: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),bot_bot_fun_nat_bool) = A_2 ).
tff(fact_342_Diff__empty,axiom,
! [A_2: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),bot_bo39439654e_bool) = A_2 ).
tff(fact_343_empty__Diff,axiom,
! [A_2: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,bot_bot_fun_int_bool),A_2) = bot_bot_fun_int_bool ).
tff(fact_344_empty__Diff,axiom,
! [A_2: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A_2) = bot_bot_fun_nat_bool ).
tff(fact_345_empty__Diff,axiom,
! [A_2: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,bot_bo39439654e_bool),A_2) = bot_bo39439654e_bool ).
tff(fact_346_insert__Diff1,axiom,
! [A_2: fun_int_bool,X_1: int,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29))
=> ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)),B_29) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29) ) ) ).
tff(fact_347_insert__Diff1,axiom,
! [A_2: fun_nat_bool,X_1: nat,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29))
=> ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)),B_29) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29) ) ) ).
tff(fact_348_insert__Diff1,axiom,
! [A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29))
=> ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)),B_29) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29) ) ) ).
tff(fact_349_insert__Diff__if,axiom,
! [A_2: fun_int_bool,X_1: int,B_29: fun_int_bool] :
( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29))
=> ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)),B_29) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29) ) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29))
=> ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)),B_29) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)) ) ) ) ).
tff(fact_350_insert__Diff__if,axiom,
! [A_2: fun_nat_bool,X_1: nat,B_29: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29))
=> ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)),B_29) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29))
=> ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)),B_29) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)) ) ) ) ).
tff(fact_351_insert__Diff__if,axiom,
! [A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29))
=> ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)),B_29) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29) ) )
& ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29))
=> ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)),B_29) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)) ) ) ) ).
tff(fact_352_double__diff,axiom,
! [C_26: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),C_26))
=> ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,B_29),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,C_26),A_2)) = A_2 ) ) ) ).
tff(fact_353_double__diff,axiom,
! [C_26: fun_int_bool,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_29),C_26))
=> ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B_29),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_26),A_2)) = A_2 ) ) ) ).
tff(fact_354_double__diff,axiom,
! [C_26: fun_nat_bool,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_29),C_26))
=> ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B_29),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_26),A_2)) = A_2 ) ) ) ).
tff(fact_355_Diff__mono,axiom,
! [D_10: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,C_26: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),C_26))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,D_10),B_29))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,C_26),D_10))) ) ) ).
tff(fact_356_Diff__mono,axiom,
! [D_10: fun_int_bool,B_29: fun_int_bool,A_2: fun_int_bool,C_26: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),C_26))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,D_10),B_29))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_26),D_10))) ) ) ).
tff(fact_357_Diff__mono,axiom,
! [D_10: fun_nat_bool,B_29: fun_nat_bool,A_2: fun_nat_bool,C_26: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),C_26))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,D_10),B_29))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_26),D_10))) ) ) ).
tff(fact_358_Diff__subset,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)),A_2)) ).
tff(fact_359_Diff__subset,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)),A_2)) ).
tff(fact_360_Diff__subset,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)),A_2)) ).
tff(fact_361_finite__insert,axiom,
! [A_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2)))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2)) ) ).
tff(fact_362_finite__insert,axiom,
! [A_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2)))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2)) ) ).
tff(fact_363_finite__insert,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2)))
<=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2)) ) ).
tff(fact_364_finite__subset,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2)) ) ) ).
tff(fact_365_finite__subset,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2)) ) ) ).
tff(fact_366_finite__subset,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2)) ) ) ).
tff(fact_367_rev__finite__subset,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2)) ) ) ).
tff(fact_368_rev__finite__subset,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2)) ) ) ).
tff(fact_369_rev__finite__subset,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2)) ) ) ).
tff(fact_370_empty__fold__graphE,axiom,
! [F: fun_int_fun_int_int,Z_1: int,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),bot_bot_fun_int_bool),X_1))
=> ( X_1 = Z_1 ) ) ).
tff(fact_371_empty__fold__graphE,axiom,
! [F: fun_nat_fun_nat_nat,Z_1: nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),bot_bot_fun_nat_bool),X_1))
=> ( X_1 = Z_1 ) ) ).
tff(fact_372_empty__fold__graphE,axiom,
! [F: fun_Ho31865764_state,Z_1: hoare_363350478_state,X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),bot_bo39439654e_bool),X_1))
=> ( X_1 = Z_1 ) ) ).
tff(fact_373_fold__graph_OemptyI,axiom,
! [F: fun_int_fun_int_int,Z_1: int] : hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),bot_bot_fun_int_bool),Z_1)) ).
tff(fact_374_fold__graph_OemptyI,axiom,
! [F: fun_nat_fun_nat_nat,Z_1: nat] : hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),bot_bot_fun_nat_bool),Z_1)) ).
tff(fact_375_fold__graph_OemptyI,axiom,
! [F: fun_Ho31865764_state,Z_1: hoare_363350478_state] : hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),bot_bo39439654e_bool),Z_1)) ).
tff(fact_376_fold__graph_OinsertI,axiom,
! [F: fun_int_fun_int_int,Z_1: int,Y_1: int,X_1: int,A_2: fun_int_bool] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),Y_1))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)),hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Y_1))) ) ) ).
tff(fact_377_fold__graph_OinsertI,axiom,
! [F: fun_nat_fun_nat_nat,Z_1: nat,Y_1: nat,X_1: nat,A_2: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),Y_1))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Y_1))) ) ) ).
tff(fact_378_fold__graph_OinsertI,axiom,
! [F: fun_Ho31865764_state,Z_1: hoare_363350478_state,Y_1: hoare_363350478_state,X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),Y_1))
=> hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)),hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),Y_1))) ) ) ).
tff(fact_379_folding__one_Oremove,axiom,
! [X_1: int,A_2: 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_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),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_2) = X_1 ) )
& ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),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_2) = 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_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ) ).
tff(fact_380_folding__one_Oremove,axiom,
! [X_1: nat,A_2: 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_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),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_2) = X_1 ) )
& ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),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_2) = 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_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ) ).
tff(fact_381_folding__one_Oremove,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite269920017_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)) = bot_bo39439654e_bool )
=> ( hAPP_f1242401993_state(F_1,A_2) = X_1 ) )
& ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)) != bot_bo39439654e_bool )
=> ( hAPP_f1242401993_state(F_1,A_2) = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))) ) ) ) ) ) ) ).
tff(fact_382_folding__one_Oinsert__remove,axiom,
! [X_1: int,A_2: 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_2))
=> ( ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),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_2)) = X_1 ) )
& ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),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_2)) = 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_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ).
tff(fact_383_folding__one_Oinsert__remove,axiom,
! [X_1: nat,A_2: 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_2))
=> ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),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_2)) = X_1 ) )
& ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),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_2)) = 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_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).
tff(fact_384_folding__one_Oinsert__remove,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite269920017_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)) = bot_bo39439654e_bool )
=> ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = X_1 ) )
& ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)) != bot_bo39439654e_bool )
=> ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))) ) ) ) ) ) ).
tff(fact_385_insert__Diff,axiom,
! [A_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool))) = A_2 ) ) ).
tff(fact_386_insert__Diff,axiom,
! [A_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))) = A_2 ) ) ).
tff(fact_387_insert__Diff,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2))
=> ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool))) = A_2 ) ) ).
tff(fact_388_Diff__insert__absorb,axiom,
! [X_1: int,A_2: fun_int_bool] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = A_2 ) ) ).
tff(fact_389_Diff__insert__absorb,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = A_2 ) ) ).
tff(fact_390_Diff__insert__absorb,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)) = A_2 ) ) ).
tff(fact_391_insert__Diff__single,axiom,
! [A_1: int,A_2: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool))) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2) ).
tff(fact_392_insert__Diff__single,axiom,
! [A_1: nat,A_2: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2) ).
tff(fact_393_insert__Diff__single,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool))) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2) ).
tff(fact_394_Diff__insert2,axiom,
! [A_2: fun_int_bool,A_1: int,B_29: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_29)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool))),B_29) ).
tff(fact_395_Diff__insert2,axiom,
! [A_2: fun_nat_bool,A_1: nat,B_29: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_29)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))),B_29) ).
tff(fact_396_Diff__insert2,axiom,
! [A_2: fun_Ho320143120e_bool,A_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_29)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool))),B_29) ).
tff(fact_397_Diff__insert,axiom,
! [A_2: fun_int_bool,A_1: int,B_29: fun_int_bool] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_29)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool)) ).
tff(fact_398_Diff__insert,axiom,
! [A_2: fun_nat_bool,A_1: nat,B_29: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_29)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)) ).
tff(fact_399_Diff__insert,axiom,
! [A_2: fun_Ho320143120e_bool,A_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] : hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_29)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool)) ).
tff(fact_400_finite__empty__induct,axiom,
! [Pa: fun_fu819804446l_bool,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,A_2))
=> ( ! [A_55: int,A_42: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_42))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_55),A_42))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,A_42))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_42),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_55),bot_bot_fun_int_bool)))) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool)) ) ) ) ).
tff(fact_401_finite__empty__induct,axiom,
! [Pa: fun_fu425979586l_bool,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,A_2))
=> ( ! [A_55: nat,A_42: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_42))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_55),A_42))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,A_42))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_42),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_55),bot_bot_fun_nat_bool)))) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) ) ) ) ).
tff(fact_402_finite__empty__induct,axiom,
! [Pa: fun_fu1996308102l_bool,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(Pa,A_2))
=> ( ! [A_55: hoare_363350478_state,A_42: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_42))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_55),A_42))
=> ( hBOOL(hAPP_f568432088l_bool(Pa,A_42))
=> hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_42),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_55),bot_bo39439654e_bool)))) ) ) )
=> hBOOL(hAPP_f568432088l_bool(Pa,bot_bo39439654e_bool)) ) ) ) ).
tff(fact_403_finite__subset__induct,axiom,
! [Pa: fun_fu819804446l_bool,A_2: 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_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
=> ( ! [A_55: 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_55),A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_55),F_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_55),F_2))) ) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_404_finite__subset__induct,axiom,
! [Pa: fun_fu425979586l_bool,A_2: 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_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
=> ( ! [A_55: 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_55),A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_55),F_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_55),F_2))) ) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_405_finite__subset__induct,axiom,
! [Pa: fun_fu1996308102l_bool,A_2: fun_Ho320143120e_bool,F_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,F_1),A_2))
=> ( hBOOL(hAPP_f568432088l_bool(Pa,bot_bo39439654e_bool))
=> ( ! [A_55: hoare_363350478_state,F_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_55),A_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_55),F_2))
=> ( hBOOL(hAPP_f568432088l_bool(Pa,F_2))
=> hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_55),F_2))) ) ) ) )
=> hBOOL(hAPP_f568432088l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_406_insert__fold1SetE,axiom,
! [F: fun_int_fun_int_int,A_1: int,X_28: fun_int_bool,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite_fold1Set_int(F),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),X_28)),X_1))
=> ~ ! [A_55: int,A_42: fun_int_bool] :
( ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),X_28) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_55),A_42) )
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,A_55),A_42),X_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_55),A_42)) ) ) ) ).
tff(fact_407_insert__fold1SetE,axiom,
! [F: fun_nat_fun_nat_nat,A_1: nat,X_28: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite_fold1Set_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),X_28)),X_1))
=> ~ ! [A_55: nat,A_42: fun_nat_bool] :
( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),X_28) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_55),A_42) )
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,A_55),A_42),X_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_55),A_42)) ) ) ) ).
tff(fact_408_insert__fold1SetE,axiom,
! [F: fun_Ho31865764_state,A_1: hoare_363350478_state,X_28: fun_Ho320143120e_bool,X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1652217598_state(F),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),X_28)),X_1))
=> ~ ! [A_55: hoare_363350478_state,A_42: fun_Ho320143120e_bool] :
( ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),X_28) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_55),A_42) )
=> ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,A_55),A_42),X_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_55),A_42)) ) ) ) ).
tff(fact_409_finite__nonempty__imp__fold1Set,axiom,
! [F: fun_int_fun_int_int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ( A_2 != bot_bot_fun_int_bool )
=> ? [X1: int] : hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite_fold1Set_int(F),A_2),X1)) ) ) ).
tff(fact_410_finite__nonempty__imp__fold1Set,axiom,
! [F: fun_nat_fun_nat_nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ( A_2 != bot_bot_fun_nat_bool )
=> ? [X1: nat] : hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite_fold1Set_nat(F),A_2),X1)) ) ) ).
tff(fact_411_finite__nonempty__imp__fold1Set,axiom,
! [F: fun_Ho31865764_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ( A_2 != bot_bo39439654e_bool )
=> ? [X1: hoare_363350478_state] : hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1652217598_state(F),A_2),X1)) ) ) ).
tff(fact_412_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_413_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_414_finite__induct,axiom,
! [Pa: fun_fu1996308102l_bool,F_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(Pa,bot_bo39439654e_bool))
=> ( ! [X_2: hoare_363350478_state,F_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),F_2))
=> ( hBOOL(hAPP_f568432088l_bool(Pa,F_2))
=> hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),F_2))) ) ) )
=> hBOOL(hAPP_f568432088l_bool(Pa,F_1)) ) ) ) ).
tff(fact_415_finite_Osimps,axiom,
! [A_1: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1))
<=> ( ( A_1 = bot_bot_fun_int_bool )
| ? [A_42: fun_int_bool,A_55: int] :
( ( A_1 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_55),A_42) )
& hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_42)) ) ) ) ).
tff(fact_416_finite_Osimps,axiom,
! [A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
<=> ( ( A_1 = bot_bot_fun_nat_bool )
| ? [A_42: fun_nat_bool,A_55: nat] :
( ( A_1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_55),A_42) )
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_42)) ) ) ) ).
tff(fact_417_finite_Osimps,axiom,
! [A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
<=> ( ( A_1 = bot_bo39439654e_bool )
| ? [A_42: fun_Ho320143120e_bool,A_55: hoare_363350478_state] :
( ( A_1 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_55),A_42) )
& hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_42)) ) ) ) ).
tff(fact_418_finite__imp__fold__graph,axiom,
! [F: fun_int_fun_int_int,Z_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ? [X1: int] : hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),X1)) ) ).
tff(fact_419_finite__imp__fold__graph,axiom,
! [F: fun_nat_fun_nat_nat,Z_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ? [X1: nat] : hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),X1)) ) ).
tff(fact_420_finite__imp__fold__graph,axiom,
! [F: fun_Ho31865764_state,Z_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ? [X1: hoare_363350478_state] : hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),X1)) ) ).
tff(fact_421_fold1Set_Osimps,axiom,
! [F: fun_int_fun_int_int,A1: fun_int_bool,A2: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite_fold1Set_int(F),A1),A2))
<=> ? [A_55: int,A_42: fun_int_bool,X_2: int] :
( ( A1 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_55),A_42) )
& ( A2 = X_2 )
& hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,A_55),A_42),X_2))
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_55),A_42)) ) ) ).
tff(fact_422_fold1Set_Osimps,axiom,
! [F: fun_nat_fun_nat_nat,A1: fun_nat_bool,A2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite_fold1Set_nat(F),A1),A2))
<=> ? [A_55: nat,A_42: fun_nat_bool,X_2: nat] :
( ( A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_55),A_42) )
& ( A2 = X_2 )
& hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,A_55),A_42),X_2))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_55),A_42)) ) ) ).
tff(fact_423_fold1Set_Osimps,axiom,
! [F: fun_Ho31865764_state,A1: fun_Ho320143120e_bool,A2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1652217598_state(F),A1),A2))
<=> ? [A_55: hoare_363350478_state,A_42: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
( ( A1 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_55),A_42) )
& ( A2 = X_2 )
& hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,A_55),A_42),X_2))
& ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_55),A_42)) ) ) ).
tff(fact_424_minus__apply,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool,X_1: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29),X_1))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_H1645464494e_bool(A_2,X_1)),hAPP_H1645464494e_bool(B_29,X_1))) ) ).
tff(fact_425_minus__apply,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29),X_1))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_int_bool(A_2,X_1)),hAPP_int_bool(B_29,X_1))) ) ).
tff(fact_426_minus__apply,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29),X_1))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_nat_bool(A_2,X_1)),hAPP_nat_bool(B_29,X_1))) ) ).
tff(fact_427_fun__diff__def,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29),X_2))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_H1645464494e_bool(A_2,X_2)),hAPP_H1645464494e_bool(B_29,X_2))) ) ).
tff(fact_428_fun__diff__def,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool,X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29),X_2))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_int_bool(A_2,X_2)),hAPP_int_bool(B_29,X_2))) ) ).
tff(fact_429_fun__diff__def,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool,X_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29),X_2))
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_nat_bool(A_2,X_2)),hAPP_nat_bool(B_29,X_2))) ) ).
tff(fact_430_diff__eq__diff__less__eq,axiom,
! [A_1: int,B_2: int,Ca: int,D_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_2) = 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_1),B_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ca),D_1)) ) ) ).
tff(fact_431_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(finite772772422nt_int(F,Z_1),A1),A2))
<=> ( ( ( A1 = bot_bot_fun_int_bool )
& ( A2 = Z_1 ) )
| ? [X_2: int,A_42: fun_int_bool,Y_5: int] :
( ( A1 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),A_42) )
& ( A2 = hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_5) )
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_42))
& hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_42),Y_5)) ) ) ) ).
tff(fact_432_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(finite929467206at_nat(F,Z_1),A1),A2))
<=> ( ( ( A1 = bot_bot_fun_nat_bool )
& ( A2 = Z_1 ) )
| ? [X_2: nat,A_42: fun_nat_bool,Y_5: nat] :
( ( A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_42) )
& ( A2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_5) )
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_42))
& hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_42),Y_5)) ) ) ) ).
tff(fact_433_fold__graph_Osimps,axiom,
! [F: fun_Ho31865764_state,Z_1: hoare_363350478_state,A1: fun_Ho320143120e_bool,A2: hoare_363350478_state] :
( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A1),A2))
<=> ( ( ( A1 = bot_bo39439654e_bool )
& ( A2 = Z_1 ) )
| ? [X_2: hoare_363350478_state,A_42: fun_Ho320143120e_bool,Y_5: hoare_363350478_state] :
( ( A1 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_42) )
& ( A2 = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_2),Y_5) )
& ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),A_42))
& hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_42),Y_5)) ) ) ) ).
tff(fact_434_folding__one__idem_Osubset__idem,axiom,
! [B_29: fun_int_bool,A_2: 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_2))
=> ( ( B_29 != bot_bot_fun_int_bool )
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_29),A_2))
=> ( hAPP_int_int(hAPP_int_fun_int_int(F,hAPP_f1594865479ol_int(F_1,B_29)),hAPP_f1594865479ol_int(F_1,A_2)) = hAPP_f1594865479ol_int(F_1,A_2) ) ) ) ) ) ).
tff(fact_435_folding__one__idem_Osubset__idem,axiom,
! [B_29: fun_nat_bool,A_2: 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_2))
=> ( ( B_29 != bot_bot_fun_nat_bool )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_29),A_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,B_29)),hAPP_f22106695ol_nat(F_1,A_2)) = hAPP_f22106695ol_nat(F_1,A_2) ) ) ) ) ) ).
tff(fact_436_folding__one__idem_Osubset__idem,axiom,
! [B_29: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite2031572_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ( B_29 != bot_bo39439654e_bool )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),A_2))
=> ( hAPP_H1982017439_state(hAPP_H230141040_state(F,hAPP_f1242401993_state(F_1,B_29)),hAPP_f1242401993_state(F_1,A_2)) = hAPP_f1242401993_state(F_1,A_2) ) ) ) ) ) ).
tff(fact_437_folding__one__idem_Oinsert__idem,axiom,
! [X_1: int,A_2: 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_2))
=> ( ( A_2 != bot_bot_fun_int_bool )
=> ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A_2)) ) ) ) ) ).
tff(fact_438_folding__one__idem_Oinsert__idem,axiom,
! [X_1: nat,A_2: 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_2))
=> ( ( A_2 != bot_bot_fun_nat_bool )
=> ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A_2)) ) ) ) ) ).
tff(fact_439_folding__one__idem_Oinsert__idem,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite2031572_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ( A_2 != bot_bo39439654e_bool )
=> ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),hAPP_f1242401993_state(F_1,A_2)) ) ) ) ) ).
tff(fact_440_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_441_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_442_finite__ne__induct,axiom,
! [Pa: fun_fu1996308102l_bool,F_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_1))
=> ( ( F_1 != bot_bo39439654e_bool )
=> ( ! [X_2: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),bot_bo39439654e_bool)))
=> ( ! [X_2: hoare_363350478_state,F_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_2))
=> ( ( F_2 != bot_bo39439654e_bool )
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),F_2))
=> ( hBOOL(hAPP_f568432088l_bool(Pa,F_2))
=> hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),F_2))) ) ) ) )
=> hBOOL(hAPP_f568432088l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_443_folding__one__idem_Oidem,axiom,
! [X_1: hoare_363350478_state,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite2031572_state(F,F_1))
=> ( hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),X_1) = X_1 ) ) ).
tff(fact_444_folding__one__idem_Oin__idem,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite2031572_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),hAPP_f1242401993_state(F_1,A_2)) = hAPP_f1242401993_state(F_1,A_2) ) ) ) ) ).
tff(fact_445_folding__one__idem_Oin__idem,axiom,
! [X_1: int,A_2: 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_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A_2)) = hAPP_f1594865479ol_int(F_1,A_2) ) ) ) ) ).
tff(fact_446_folding__one__idem_Oin__idem,axiom,
! [X_1: nat,A_2: 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_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A_2)) = hAPP_f22106695ol_nat(F_1,A_2) ) ) ) ) ).
tff(fact_447_diff__eq__diff__eq,axiom,
! [A_1: int,B_2: int,Ca: int,D_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ca),D_1) )
=> ( ( A_1 = B_2 )
<=> ( Ca = D_1 ) ) ) ).
tff(fact_448_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_1: int,Z_1: int,A_2: fun_int_bool,Y_1: int,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),Y_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> ? [Y_27: int] :
( ( Y_1 = hAPP_int_int(hAPP_int_fun_int_int(F,A_1),Y_27) )
& hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool))),Y_27)) ) ) ) ) ).
tff(fact_449_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_1: nat,Z_1: nat,A_2: fun_nat_bool,Y_1: nat,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),Y_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> ? [Y_27: nat] :
( ( Y_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,A_1),Y_27) )
& hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))),Y_27)) ) ) ) ) ).
tff(fact_450_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_1: hoare_363350478_state,Z_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,Y_1: hoare_363350478_state,F: fun_Ho31865764_state] :
( hBOOL(finite980784795_state(F))
=> ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2))
=> ? [Y_27: hoare_363350478_state] :
( ( Y_1 = hAPP_H1982017439_state(hAPP_H230141040_state(F,A_1),Y_27) )
& hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),bot_bo39439654e_bool))),Y_27)) ) ) ) ) ).
tff(fact_451_semilattice__big_OF__eq,axiom,
! [A_2: 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_2))
=> ( hAPP_f1594865479ol_int(F_1,A_2) = hAPP_f1594865479ol_int(finite_fold1_int(F),A_2) ) ) ) ).
tff(fact_452_semilattice__big_OF__eq,axiom,
! [A_2: 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_2))
=> ( hAPP_f22106695ol_nat(F_1,A_2) = hAPP_f22106695ol_nat(finite_fold1_nat(F),A_2) ) ) ) ).
tff(fact_453_semilattice__big_OF__eq,axiom,
! [A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(big_se798579832_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hAPP_f1242401993_state(F_1,A_2) = hAPP_f1242401993_state(finite841610862_state(F),A_2) ) ) ) ).
tff(fact_454_fold__graph__permute__diff,axiom,
! [A_1: nat,B_2: nat,A_2: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(times_times_nat,B_2),A_2),X_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_2),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(times_times_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))),X_1)) ) ) ) ).
tff(fact_455_fold__graph__permute__diff,axiom,
! [A_1: int,B_2: int,A_2: fun_int_bool,X_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(times_times_int,B_2),A_2),X_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_2),A_2))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(times_times_int,A_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),bot_bot_fun_int_bool)))),X_1)) ) ) ) ).
tff(fact_456_fold1__insert,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( ( A_2 != bot_bot_fun_nat_bool )
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A_2)) ) ) ) ) ).
tff(fact_457_fold1__insert,axiom,
! [X_1: int,A_2: fun_int_bool] :
( ( A_2 != bot_bot_fun_int_bool )
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),A_2)) ) ) ) ) ).
tff(fact_458_psubset__insert__iff,axiom,
! [A_2: fun_int_bool,X_1: int,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B_29)))
<=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_2),B_29)) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B_29))
=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B_29)) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29)) ) ) ) ) ) ).
tff(fact_459_psubset__insert__iff,axiom,
! [A_2: fun_nat_bool,X_1: nat,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_29)))
<=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_2),B_29)) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_29))
=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B_29)) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29)) ) ) ) ) ) ).
tff(fact_460_psubset__insert__iff,axiom,
! [A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),B_29)))
<=> ( ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29)) )
& ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),B_29))
=> ( ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool))),B_29)) )
& ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29)) ) ) ) ) ) ).
tff(fact_461_card__Diff1__le,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))),hAPP_f957591787ol_nat(finite_card_int,A_2))) ) ).
tff(fact_462_card__Diff1__le,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))),hAPP_f22106695ol_nat(finite_card_nat,A_2))) ) ).
tff(fact_463_card__Diff1__le,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))),hAPP_f1705983439ol_nat(finite987718268_state,A_2))) ) ).
tff(fact_464_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_54: nat,B_41: nat,C_31: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_54),B_41)),C_31) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_41),C_31)) ).
tff(fact_465_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_54: int,B_41: int,C_31: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_54),B_41)),C_31) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_54),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_41),C_31)) ).
tff(fact_466_psubsetD,axiom,
! [Ca: hoare_363350478_state,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),A_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Ca),B_29)) ) ) ).
tff(fact_467_psubsetD,axiom,
! [Ca: int,A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),A_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ca),B_29)) ) ) ).
tff(fact_468_psubsetD,axiom,
! [Ca: nat,A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_2),B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B_29)) ) ) ).
tff(fact_469_psubset__trans,axiom,
! [C_26: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_29),C_26))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),C_26)) ) ) ).
tff(fact_470_comp__fun__commute,axiom,
hBOOL(finite2130160977at_nat(times_times_nat)) ).
tff(fact_471_comp__fun__commute,axiom,
hBOOL(finite1973466193nt_int(times_times_int)) ).
tff(fact_472_order__less__irrefl,axiom,
! [X_1: fun_Ho320143120e_bool] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),X_1)) ).
tff(fact_473_order__less__irrefl,axiom,
! [X_27: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_27),X_27)) ).
tff(fact_474_order__less__irrefl,axiom,
! [X_27: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_27),X_27)) ).
tff(fact_475_linorder__neq__iff,axiom,
! [X_1: nat,Y_1: nat] :
( ( X_1 != Y_1 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1)) ) ) ).
tff(fact_476_linorder__neq__iff,axiom,
! [X_1: int,Y_1: int] :
( ( X_1 != Y_1 )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ) ).
tff(fact_477_not__less__iff__gr__or__eq,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_478_not__less__iff__gr__or__eq,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_479_linorder__less__linear,axiom,
! [X_26: nat,Y_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_26),Y_26))
| ( X_26 = Y_26 )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_26),X_26)) ) ).
tff(fact_480_linorder__less__linear,axiom,
! [X_26: int,Y_26: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_26),Y_26))
| ( X_26 = Y_26 )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_26),X_26)) ) ).
tff(fact_481_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_482_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_483_linorder__antisym__conv3,axiom,
! [Y_1: nat,X_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_484_linorder__antisym__conv3,axiom,
! [Y_1: int,X_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_485_linorder__neqE,axiom,
! [X_25: nat,Y_25: nat] :
( ( X_25 != Y_25 )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_25),Y_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_25),X_25)) ) ) ).
tff(fact_486_linorder__neqE,axiom,
! [X_25: int,Y_25: int] :
( ( X_25 != Y_25 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_25),Y_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_25),X_25)) ) ) ).
tff(fact_487_less__imp__neq,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ( X_1 != Y_1 ) ) ).
tff(fact_488_less__imp__neq,axiom,
! [X_24: nat,Y_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_24),Y_24))
=> ( X_24 != Y_24 ) ) ).
tff(fact_489_less__imp__neq,axiom,
! [X_24: int,Y_24: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_24),Y_24))
=> ( X_24 != Y_24 ) ) ).
tff(fact_490_order__less__not__sym,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),X_1)) ) ).
tff(fact_491_order__less__not__sym,axiom,
! [X_23: nat,Y_23: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_23),Y_23))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_23),X_23)) ) ).
tff(fact_492_order__less__not__sym,axiom,
! [X_23: int,Y_23: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_23),Y_23))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_23),X_23)) ) ).
tff(fact_493_order__less__imp__not__less,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),X_1)) ) ).
tff(fact_494_order__less__imp__not__less,axiom,
! [X_22: nat,Y_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Y_22))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_22),X_22)) ) ).
tff(fact_495_order__less__imp__not__less,axiom,
! [X_22: int,Y_22: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Y_22))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_22),X_22)) ) ).
tff(fact_496_order__less__imp__not__eq,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ( X_1 != Y_1 ) ) ).
tff(fact_497_order__less__imp__not__eq,axiom,
! [X_21: nat,Y_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_21))
=> ( X_21 != Y_21 ) ) ).
tff(fact_498_order__less__imp__not__eq,axiom,
! [X_21: int,Y_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_21))
=> ( X_21 != Y_21 ) ) ).
tff(fact_499_order__less__imp__not__eq2,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ( Y_1 != X_1 ) ) ).
tff(fact_500_order__less__imp__not__eq2,axiom,
! [X_20: nat,Y_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_20))
=> ( Y_20 != X_20 ) ) ).
tff(fact_501_order__less__imp__not__eq2,axiom,
! [X_20: int,Y_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_20),Y_20))
=> ( Y_20 != X_20 ) ) ).
tff(fact_502_order__less__imp__triv,axiom,
! [Pa: bool,X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),X_1))
=> hBOOL(Pa) ) ) ).
tff(fact_503_order__less__imp__triv,axiom,
! [Pa: bool,X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))
=> hBOOL(Pa) ) ) ).
tff(fact_504_order__less__imp__triv,axiom,
! [Pa: bool,X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1))
=> hBOOL(Pa) ) ) ).
tff(fact_505_order__less__asym_H,axiom,
! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),B_2))
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_2),A_1)) ) ).
tff(fact_506_order__less__asym_H,axiom,
! [A_53: nat,B_40: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_53),B_40))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_40),A_53)) ) ).
tff(fact_507_order__less__asym_H,axiom,
! [A_53: int,B_40: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_53),B_40))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_40),A_53)) ) ).
tff(fact_508_xt1_I9_J,axiom,
! [B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_2),A_1))
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),B_2)) ) ).
tff(fact_509_xt1_I9_J,axiom,
! [B_39: nat,A_52: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_39),A_52))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_52),B_39)) ) ).
tff(fact_510_xt1_I9_J,axiom,
! [B_39: int,A_52: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_39),A_52))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_52),B_39)) ) ).
tff(fact_511_ord__eq__less__trans,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( ( A_1 = B_2 )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_2),Ca))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),Ca)) ) ) ).
tff(fact_512_ord__eq__less__trans,axiom,
! [C_30: nat,A_51: nat,B_38: nat] :
( ( A_51 = B_38 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_38),C_30))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_51),C_30)) ) ) ).
tff(fact_513_ord__eq__less__trans,axiom,
! [C_30: int,A_51: int,B_38: int] :
( ( A_51 = B_38 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_38),C_30))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_51),C_30)) ) ) ).
tff(fact_514_xt1_I1_J,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( ( A_1 = B_2 )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Ca),B_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Ca),A_1)) ) ) ).
tff(fact_515_xt1_I1_J,axiom,
! [C_29: nat,A_50: nat,B_37: nat] :
( ( A_50 = B_37 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_29),B_37))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_29),A_50)) ) ) ).
tff(fact_516_xt1_I1_J,axiom,
! [C_29: int,A_50: int,B_37: int] :
( ( A_50 = B_37 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_29),B_37))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_29),A_50)) ) ) ).
tff(fact_517_ord__less__eq__trans,axiom,
! [Ca: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),B_2))
=> ( ( B_2 = Ca )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),Ca)) ) ) ).
tff(fact_518_ord__less__eq__trans,axiom,
! [C_28: nat,A_49: nat,B_36: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_49),B_36))
=> ( ( B_36 = C_28 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_49),C_28)) ) ) ).
tff(fact_519_ord__less__eq__trans,axiom,
! [C_28: int,A_49: int,B_36: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_49),B_36))
=> ( ( B_36 = C_28 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_49),C_28)) ) ) ).
tff(fact_520_xt1_I2_J,axiom,
! [Ca: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_2),A_1))
=> ( ( B_2 = Ca )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Ca),A_1)) ) ) ).
tff(fact_521_xt1_I2_J,axiom,
! [C_27: nat,B_35: nat,A_48: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_35),A_48))
=> ( ( B_35 = C_27 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_27),A_48)) ) ) ).
tff(fact_522_xt1_I2_J,axiom,
! [C_27: int,B_35: int,A_48: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_35),A_48))
=> ( ( B_35 = C_27 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_27),A_48)) ) ) ).
tff(fact_523_order__less__trans,axiom,
! [Z_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),Z_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Z_1)) ) ) ).
tff(fact_524_order__less__trans,axiom,
! [Z_10: nat,X_19: nat,Y_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Y_19))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_19),Z_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Z_10)) ) ) ).
tff(fact_525_order__less__trans,axiom,
! [Z_10: int,X_19: int,Y_19: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Y_19))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_19),Z_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Z_10)) ) ) ).
tff(fact_526_xt1_I10_J,axiom,
! [Z_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Z_1),Y_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Z_1),X_1)) ) ) ).
tff(fact_527_xt1_I10_J,axiom,
! [Z_9: nat,Y_18: nat,X_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_18),X_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_9),Y_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_9),X_18)) ) ) ).
tff(fact_528_xt1_I10_J,axiom,
! [Z_9: int,Y_18: int,X_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_18),X_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_9),Y_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_9),X_18)) ) ) ).
tff(fact_529_order__less__asym,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),X_1)) ) ).
tff(fact_530_order__less__asym,axiom,
! [X_17: nat,Y_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_17),Y_17))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_17),X_17)) ) ).
tff(fact_531_order__less__asym,axiom,
! [X_17: int,Y_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_17),Y_17))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_17),X_17)) ) ).
tff(fact_532_linorder__cases,axiom,
! [X_16: nat,Y_16: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_16),Y_16))
=> ( ( X_16 != Y_16 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_16),X_16)) ) ) ).
tff(fact_533_linorder__cases,axiom,
! [X_16: int,Y_16: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_16),Y_16))
=> ( ( X_16 != Y_16 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_16),X_16)) ) ) ).
tff(fact_534_less__fun__def,axiom,
! [F: fun_Ho320143120e_bool,G: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,F),G))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,F),G))
& ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,G),F)) ) ) ).
tff(fact_535_linorder__not__less,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ).
tff(fact_536_linorder__not__less,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ).
tff(fact_537_linorder__not__le,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1)) ) ).
tff(fact_538_linorder__not__le,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) ) ).
tff(fact_539_linorder__le__less__linear,axiom,
! [X_15: nat,Y_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),Y_15))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_15),X_15)) ) ).
tff(fact_540_linorder__le__less__linear,axiom,
! [X_15: int,Y_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_15),Y_15))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_15),X_15)) ) ).
tff(fact_541_order__less__le,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
& ( X_1 != Y_1 ) ) ) ).
tff(fact_542_order__less__le,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
& ( X_1 != Y_1 ) ) ) ).
tff(fact_543_order__less__le,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
& ( X_1 != Y_1 ) ) ) ).
tff(fact_544_less__le__not__le,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
& ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),X_1)) ) ) ).
tff(fact_545_less__le__not__le,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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_546_less__le__not__le,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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_547_order__le__less,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_548_order__le__less,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_549_order__le__less,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_550_leI,axiom,
! [X_14: nat,Y_14: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_14),Y_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_14),X_14)) ) ).
tff(fact_551_leI,axiom,
! [X_14: int,Y_14: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_14),Y_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_14),X_14)) ) ).
tff(fact_552_not__leE,axiom,
! [Y_13: nat,X_13: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_13),X_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_13)) ) ).
tff(fact_553_not__leE,axiom,
! [Y_13: int,X_13: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_13),X_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_13),Y_13)) ) ).
tff(fact_554_linorder__antisym__conv1,axiom,
! [X_1: nat,Y_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_555_linorder__antisym__conv1,axiom,
! [X_1: int,Y_1: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_556_order__neq__le__trans,axiom,
! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( ( A_1 != B_2 )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),B_2))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),B_2)) ) ) ).
tff(fact_557_order__neq__le__trans,axiom,
! [A_47: nat,B_34: nat] :
( ( A_47 != B_34 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_47),B_34))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_47),B_34)) ) ) ).
tff(fact_558_order__neq__le__trans,axiom,
! [A_47: int,B_34: int] :
( ( A_47 != B_34 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_47),B_34))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_47),B_34)) ) ) ).
tff(fact_559_xt1_I12_J,axiom,
! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( ( A_1 != B_2 )
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),A_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_2),A_1)) ) ) ).
tff(fact_560_xt1_I12_J,axiom,
! [A_46: nat,B_33: nat] :
( ( A_46 != B_33 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_33),A_46))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_33),A_46)) ) ) ).
tff(fact_561_xt1_I12_J,axiom,
! [A_46: int,B_33: int] :
( ( A_46 != B_33 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_33),A_46))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_33),A_46)) ) ) ).
tff(fact_562_leD,axiom,
! [Y_12: nat,X_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_12),X_12))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_12),Y_12)) ) ).
tff(fact_563_leD,axiom,
! [Y_12: int,X_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_12),X_12))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_12),Y_12)) ) ).
tff(fact_564_order__less__imp__le,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1)) ) ).
tff(fact_565_order__less__imp__le,axiom,
! [X_11: nat,Y_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_11),Y_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),Y_11)) ) ).
tff(fact_566_order__less__imp__le,axiom,
! [X_11: int,Y_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_11),Y_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),Y_11)) ) ).
tff(fact_567_linorder__antisym__conv2,axiom,
! [X_1: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_568_linorder__antisym__conv2,axiom,
! [X_1: int,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1))
<=> ( X_1 = Y_1 ) ) ) ).
tff(fact_569_order__le__imp__less__or__eq,axiom,
! [X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
| ( X_1 = Y_1 ) ) ) ).
tff(fact_570_order__le__imp__less__or__eq,axiom,
! [X_10: nat,Y_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_10),Y_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_10),Y_10))
| ( X_10 = Y_10 ) ) ) ).
tff(fact_571_order__le__imp__less__or__eq,axiom,
! [X_10: int,Y_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_10),Y_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_10),Y_10))
| ( X_10 = Y_10 ) ) ) ).
tff(fact_572_order__le__neq__trans,axiom,
! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_1),B_2))
=> ( ( A_1 != B_2 )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),B_2)) ) ) ).
tff(fact_573_order__le__neq__trans,axiom,
! [A_45: nat,B_32: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_45),B_32))
=> ( ( A_45 != B_32 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_45),B_32)) ) ) ).
tff(fact_574_order__le__neq__trans,axiom,
! [A_45: int,B_32: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_45),B_32))
=> ( ( A_45 != B_32 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_45),B_32)) ) ) ).
tff(fact_575_xt1_I11_J,axiom,
! [B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_2),A_1))
=> ( ( A_1 != B_2 )
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_2),A_1)) ) ) ).
tff(fact_576_xt1_I11_J,axiom,
! [B_31: nat,A_44: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_31),A_44))
=> ( ( A_44 != B_31 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_31),A_44)) ) ) ).
tff(fact_577_xt1_I11_J,axiom,
! [B_31: int,A_44: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_31),A_44))
=> ( ( A_44 != B_31 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_31),A_44)) ) ) ).
tff(fact_578_order__less__le__trans,axiom,
! [Z_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),Z_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Z_1)) ) ) ).
tff(fact_579_order__less__le__trans,axiom,
! [Z_8: nat,X_9: nat,Y_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_9),Z_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Z_8)) ) ) ).
tff(fact_580_order__less__le__trans,axiom,
! [Z_8: int,X_9: int,Y_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_9),Z_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Z_8)) ) ) ).
tff(fact_581_xt1_I7_J,axiom,
! [Z_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Z_1),Y_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Z_1),X_1)) ) ) ).
tff(fact_582_xt1_I7_J,axiom,
! [Z_7: nat,Y_8: nat,X_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_8),X_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_7),Y_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_7),X_8)) ) ) ).
tff(fact_583_xt1_I7_J,axiom,
! [Z_7: int,Y_8: int,X_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_8),X_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_7),Y_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_7),X_8)) ) ) ).
tff(fact_584_order__le__less__trans,axiom,
! [Z_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,X_1),Y_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Y_1),Z_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,X_1),Z_1)) ) ) ).
tff(fact_585_order__le__less__trans,axiom,
! [Z_6: nat,X_7: nat,Y_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),Y_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_7),Z_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_7),Z_6)) ) ) ).
tff(fact_586_order__le__less__trans,axiom,
! [Z_6: int,X_7: int,Y_7: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),Y_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_7),Z_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_7),Z_6)) ) ) ).
tff(fact_587_xt1_I8_J,axiom,
! [Z_1: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,X_1: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,Y_1),X_1))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Z_1),Y_1))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,Z_1),X_1)) ) ) ).
tff(fact_588_xt1_I8_J,axiom,
! [Z_5: nat,Y_6: nat,X_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_6),X_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_5),Y_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_5),X_6)) ) ) ).
tff(fact_589_xt1_I8_J,axiom,
! [Z_5: int,Y_6: int,X_6: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),X_6))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_5),Y_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_5),X_6)) ) ) ).
tff(fact_590_not__less__bot,axiom,
! [A_1: fun_int_bool] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),bot_bot_fun_int_bool)) ).
tff(fact_591_not__less__bot,axiom,
! [A_1: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),bot_bot_fun_nat_bool)) ).
tff(fact_592_not__less__bot,axiom,
! [A_1: bool] : ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_bool,A_1),bot_bot_bool)) ).
tff(fact_593_not__less__bot,axiom,
! [A_1: fun_Ho320143120e_bool] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_1),bot_bo39439654e_bool)) ).
tff(fact_594_not__less__bot,axiom,
! [A_43: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_43),bot_bot_nat)) ).
tff(fact_595_bot__less,axiom,
! [A_1: fun_int_bool] :
( ( A_1 != bot_bot_fun_int_bool )
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,bot_bot_fun_int_bool),A_1)) ) ).
tff(fact_596_bot__less,axiom,
! [A_1: fun_nat_bool] :
( ( A_1 != bot_bot_fun_nat_bool )
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,bot_bot_fun_nat_bool),A_1)) ) ).
tff(fact_597_bot__less,axiom,
! [A_1: bool] :
( ~ ( hBOOL(A_1)
<=> hBOOL(bot_bot_bool) )
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_bool,bot_bot_bool),A_1)) ) ).
tff(fact_598_bot__less,axiom,
! [A_1: fun_Ho320143120e_bool] :
( ( A_1 != bot_bo39439654e_bool )
<=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,bot_bo39439654e_bool),A_1)) ) ).
tff(fact_599_bot__less,axiom,
! [A_1: nat] :
( ( A_1 != bot_bot_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,bot_bot_nat),A_1)) ) ).
tff(fact_600_not__psubset__empty,axiom,
! [A_2: fun_int_bool] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_2),bot_bot_fun_int_bool)) ).
tff(fact_601_not__psubset__empty,axiom,
! [A_2: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_2),bot_bot_fun_nat_bool)) ).
tff(fact_602_not__psubset__empty,axiom,
! [A_2: fun_Ho320143120e_bool] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),bot_bo39439654e_bool)) ).
tff(fact_603_diff__eq__diff__less,axiom,
! [A_1: int,B_2: int,Ca: int,D_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ca),D_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ca),D_1)) ) ) ).
tff(fact_604_psubset__eq,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
& ( A_2 != B_29 ) ) ) ).
tff(fact_605_subset__iff__psubset__eq,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
<=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
| ( A_2 = B_29 ) ) ) ).
tff(fact_606_psubset__imp__subset,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29)) ) ).
tff(fact_607_psubset__subset__trans,axiom,
! [C_26: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),C_26))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),C_26)) ) ) ).
tff(fact_608_subset__psubset__trans,axiom,
! [C_26: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,B_29),C_26))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),C_26)) ) ) ).
tff(fact_609_card__insert__le,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)))) ) ).
tff(fact_610_card__insert__le,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)))) ) ).
tff(fact_611_card__insert__le,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)))) ) ).
tff(fact_612_card__mono,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),hAPP_f1705983439ol_nat(finite987718268_state,B_29))) ) ) ).
tff(fact_613_card__mono,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),hAPP_f957591787ol_nat(finite_card_int,B_29))) ) ) ).
tff(fact_614_card__mono,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),hAPP_f22106695ol_nat(finite_card_nat,B_29))) ) ) ).
tff(fact_615_card__seteq,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f1705983439ol_nat(finite987718268_state,B_29)),hAPP_f1705983439ol_nat(finite987718268_state,A_2)))
=> ( A_2 = B_29 ) ) ) ) ).
tff(fact_616_card__seteq,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,B_29)),hAPP_f957591787ol_nat(finite_card_int,A_2)))
=> ( A_2 = B_29 ) ) ) ) ).
tff(fact_617_card__seteq,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,B_29)),hAPP_f22106695ol_nat(finite_card_nat,A_2)))
=> ( A_2 = B_29 ) ) ) ) ).
tff(fact_618_diff__card__le__card__Diff,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),hAPP_f1705983439ol_nat(finite987718268_state,B_29))),hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)))) ) ).
tff(fact_619_diff__card__le__card__Diff,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),hAPP_f957591787ol_nat(finite_card_int,B_29))),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)))) ) ).
tff(fact_620_diff__card__le__card__Diff,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),hAPP_f22106695ol_nat(finite_card_nat,B_29))),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)))) ) ).
tff(fact_621_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_1: int,Z_1: int,A_2: fun_int_bool,X_1: int,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),X_1))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),Y_1))
=> ( Y_1 = X_1 ) ) ) ) ).
tff(fact_622_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_1: nat,Z_1: nat,A_2: fun_nat_bool,X_1: nat,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),Y_1))
=> ( Y_1 = X_1 ) ) ) ) ).
tff(fact_623_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_1: hoare_363350478_state,Z_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,X_1: hoare_363350478_state,F: fun_Ho31865764_state] :
( hBOOL(finite980784795_state(F))
=> ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),X_1))
=> ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),Y_1))
=> ( Y_1 = X_1 ) ) ) ) ).
tff(fact_624_fold__graph__insert__swap,axiom,
! [Z_1: int,B_2: int,A_2: fun_int_bool,Y_1: int] :
( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(times_times_int,B_2),A_2),Y_1))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_2),A_2))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(times_times_int,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_2),A_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Y_1))) ) ) ).
tff(fact_625_fold__graph__insert__swap,axiom,
! [Z_1: nat,B_2: nat,A_2: fun_nat_bool,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(times_times_nat,B_2),A_2),Y_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_2),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(times_times_nat,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_2),A_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),Y_1))) ) ) ).
tff(fact_626_card__Diff__subset,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,B_29),A_2))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),hAPP_f1705983439ol_nat(finite987718268_state,B_29)) ) ) ) ).
tff(fact_627_card__Diff__subset,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_29),A_2))
=> ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),hAPP_f957591787ol_nat(finite_card_int,B_29)) ) ) ) ).
tff(fact_628_card__Diff__subset,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_29),A_2))
=> ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),hAPP_f22106695ol_nat(finite_card_nat,B_29)) ) ) ) ).
tff(fact_629_fold1__in,axiom,
! [A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ( A_2 != bot_bot_fun_int_bool )
=> ( ! [X_2: int,Y_5: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_2),Y_5)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_5),bot_bot_fun_int_bool))))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),A_2)),A_2)) ) ) ) ).
tff(fact_630_fold1__in,axiom,
! [A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ( A_2 != bot_bot_fun_nat_bool )
=> ( ! [X_2: nat,Y_5: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_5)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_5),bot_bot_fun_nat_bool))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A_2)),A_2)) ) ) ) ).
tff(fact_631_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_1: int,X_1: int,A_2: fun_int_bool,V_2: int,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)),V_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ~ ! [Y_5: int] :
( ( V_2 = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Y_5) )
=> ~ hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),Y_5)) ) ) ) ) ).
tff(fact_632_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_1: nat,X_1: nat,A_2: fun_nat_bool,V_2: nat,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)),V_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ~ ! [Y_5: nat] :
( ( V_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Y_5) )
=> ~ hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),Y_5)) ) ) ) ) ).
tff(fact_633_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_1: hoare_363350478_state,X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,V_2: hoare_363350478_state,F: fun_Ho31865764_state] :
( hBOOL(finite980784795_state(F))
=> ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)),V_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ~ ! [Y_5: hoare_363350478_state] :
( ( V_2 = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_1),Y_5) )
=> ~ hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),Y_5)) ) ) ) ) ).
tff(fact_634_psubset__imp__ex__mem,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
=> ? [B_30: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_30),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,B_29),A_2))) ) ).
tff(fact_635_psubset__imp__ex__mem,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_2),B_29))
=> ? [B_30: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_30),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B_29),A_2))) ) ).
tff(fact_636_psubset__imp__ex__mem,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_2),B_29))
=> ? [B_30: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_30),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B_29),A_2))) ) ).
tff(fact_637_psubset__card__mono,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),hAPP_f1705983439ol_nat(finite987718268_state,B_29))) ) ) ).
tff(fact_638_psubset__card__mono,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_2),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),hAPP_f957591787ol_nat(finite_card_int,B_29))) ) ) ).
tff(fact_639_psubset__card__mono,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_2),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),hAPP_f22106695ol_nat(finite_card_nat,B_29))) ) ) ).
tff(fact_640_card__psubset,axiom,
! [A_2: fun_Ho320143120e_bool,B_29: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_29))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le704992346e_bool,A_2),B_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),hAPP_f1705983439ol_nat(finite987718268_state,B_29)))
=> hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(ord_le1392391526e_bool,A_2),B_29)) ) ) ) ).
tff(fact_641_card__psubset,axiom,
! [A_2: fun_int_bool,B_29: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B_29))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),hAPP_f957591787ol_nat(finite_card_int,B_29)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_2),B_29)) ) ) ) ).
tff(fact_642_card__psubset,axiom,
! [A_2: fun_nat_bool,B_29: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_29))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),hAPP_f22106695ol_nat(finite_card_nat,B_29)))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_2),B_29)) ) ) ) ).
tff(fact_643_card__Diff1__less,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))),hAPP_f957591787ol_nat(finite_card_int,A_2))) ) ) ).
tff(fact_644_card__Diff1__less,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))),hAPP_f22106695ol_nat(finite_card_nat,A_2))) ) ) ).
tff(fact_645_card__Diff1__less,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))),hAPP_f1705983439ol_nat(finite987718268_state,A_2))) ) ) ).
tff(fact_646_card__Diff2__less,axiom,
! [Y_1: int,X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_1),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_1),bot_bot_fun_int_bool)))),hAPP_f957591787ol_nat(finite_card_int,A_2))) ) ) ) ).
tff(fact_647_card__Diff2__less,axiom,
! [Y_1: nat,X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_1),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),bot_bot_fun_nat_bool)))),hAPP_f22106695ol_nat(finite_card_nat,A_2))) ) ) ) ).
tff(fact_648_card__Diff2__less,axiom,
! [Y_1: hoare_363350478_state,X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Y_1),A_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool))),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_1),bot_bo39439654e_bool)))),hAPP_f1705983439ol_nat(finite987718268_state,A_2))) ) ) ) ).
tff(fact_649_finite__linorder__max__induct,axiom,
! [Pa: fun_fu819804446l_bool,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
=> ( ! [B_30: int,A_42: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_42))
=> ( ! [X_2: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_42))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),B_30)) )
=> ( hBOOL(hAPP_f448129468l_bool(Pa,A_42))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_30),A_42))) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,A_2)) ) ) ) ).
tff(fact_650_finite__linorder__max__induct,axiom,
! [Pa: fun_fu425979586l_bool,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
=> ( ! [B_30: nat,A_42: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_42))
=> ( ! [X_2: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_42))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),B_30)) )
=> ( hBOOL(hAPP_f54304608l_bool(Pa,A_42))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_30),A_42))) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,A_2)) ) ) ) ).
tff(fact_651_diff__less__mono,axiom,
! [C_25: nat,A: nat,B_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_25),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),C_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_1),C_25))) ) ) ).
tff(fact_652_less__diff__iff,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))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ) ).
tff(fact_653_finite__linorder__min__induct,axiom,
! [Pa: fun_fu819804446l_bool,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
=> ( ! [B_30: int,A_42: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_42))
=> ( ! [X_2: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_42))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_30),X_2)) )
=> ( hBOOL(hAPP_f448129468l_bool(Pa,A_42))
=> hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_30),A_42))) ) ) )
=> hBOOL(hAPP_f448129468l_bool(Pa,A_2)) ) ) ) ).
tff(fact_654_finite__linorder__min__induct,axiom,
! [Pa: fun_fu425979586l_bool,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
=> ( ! [B_30: nat,A_42: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_42))
=> ( ! [X_2: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_42))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_30),X_2)) )
=> ( hBOOL(hAPP_f54304608l_bool(Pa,A_42))
=> hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_30),A_42))) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,A_2)) ) ) ) ).
tff(fact_655_le__diff__iff,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))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_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)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ) ).
tff(fact_656_Nat_Odiff__diff__eq,axiom,
! [N_1: nat,K_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
=> ( 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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ) ) ) ).
tff(fact_657_eq__diff__iff,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,M),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K) )
<=> ( M = N ) ) ) ) ).
tff(fact_658_less__not__refl,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_659_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) ) ) ).
tff(fact_660_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_661_less__irrefl__nat,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1)) ).
tff(fact_662_less__not__refl2,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
=> ( M_1 != N_1 ) ) ).
tff(fact_663_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_664_nat__less__cases,axiom,
! [Pa: fun_nat_fun_nat_bool,M: nat,N: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N),M)) )
=> ( ( ( M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N),M)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N),M)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N),M)) ) ) ) ).
tff(fact_665_mult__le__mono,axiom,
! [K_1: nat,L_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),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_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),L_1))) ) ) ).
tff(fact_666_le__antisym,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> ( M_1 = N_1 ) ) ) ).
tff(fact_667_le__trans,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1)) ) ) ).
tff(fact_668_mult__le__mono2,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),J))) ) ).
tff(fact_669_mult__le__mono1,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K_1))) ) ).
tff(fact_670_eq__imp__le,axiom,
! [M_1: nat,N_1: nat] :
( ( M_1 = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_671_nat__le__linear,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)) ) ).
tff(fact_672_le__cube,axiom,
! [M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1)))) ).
tff(fact_673_le__square,axiom,
! [M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1))) ).
tff(fact_674_le__refl,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
tff(fact_675_diff__mult__distrib,axiom,
! [M_1: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1)) ).
tff(fact_676_diff__mult__distrib2,axiom,
! [K_1: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)) ).
tff(fact_677_diff__commute,axiom,
! [I_1: nat,J: nat,K_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),K_1)),J) ).
tff(fact_678_nat__less__le,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))
& ( M != N ) ) ) ).
tff(fact_679_le__eq__less__or__eq,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_nat,M),N))
| ( M = N ) ) ) ).
tff(fact_680_less__imp__le__nat,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_681_le__neq__implies__less,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ( ( M_1 != N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ).
tff(fact_682_less__or__eq__imp__le,axiom,
! [M_1: nat,N_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
| ( M_1 = N_1 ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_683_diff__less__mono2,axiom,
! [L_1: nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),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_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M_1))) ) ) ).
tff(fact_684_less__imp__diff__less,axiom,
! [N_1: nat,J: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),N_1)),K_1)) ) ).
tff(fact_685_diff__le__self,axiom,
! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),M_1)) ).
tff(fact_686_diff__le__mono2,axiom,
! [L_1: nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_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,L_1),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M_1))) ) ).
tff(fact_687_diff__le__mono,axiom,
! [L_1: nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_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_1),L_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),L_1))) ) ).
tff(fact_688_diff__diff__cancel,axiom,
! [I_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),N_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I_1)) = I_1 ) ) ).
tff(fact_689_less__mono__imp__le__mono,axiom,
! [I_2: nat,J_2: nat,F: fun_nat_nat] :
( ! [I: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I)),hAPP_nat_nat(F,J_1))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_2))) ) ) ).
tff(fact_690_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_691_card__Diff__singleton__if,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),one_one_nat) ) )
& ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_f957591787ol_nat(finite_card_int,A_2) ) ) ) ) ).
tff(fact_692_card__Diff__singleton__if,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),one_one_nat) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_f22106695ol_nat(finite_card_nat,A_2) ) ) ) ) ).
tff(fact_693_card__Diff__singleton__if,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),one_one_nat) ) )
& ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool))) = hAPP_f1705983439ol_nat(finite987718268_state,A_2) ) ) ) ) ).
tff(fact_694_card__Diff__singleton,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A_2)),one_one_nat) ) ) ) ).
tff(fact_695_card__Diff__singleton,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A_2)),one_one_nat) ) ) ) ).
tff(fact_696_card__Diff__singleton,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f1705983439ol_nat(finite987718268_state,A_2)),one_one_nat) ) ) ) ).
tff(fact_697_card__Diff__insert,axiom,
! [B_29: fun_int_bool,A_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),B_29))
=> ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),B_29))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),B_29))),one_one_nat) ) ) ) ) ).
tff(fact_698_card__Diff__insert,axiom,
! [B_29: fun_nat_bool,A_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B_29))
=> ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_29))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),B_29))),one_one_nat) ) ) ) ) ).
tff(fact_699_card__Diff__insert,axiom,
! [B_29: fun_Ho320143120e_bool,A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),A_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_1),B_29))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),B_29))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),B_29))),one_one_nat) ) ) ) ) ).
tff(fact_700_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1: int,X_1: int,A_2: fun_int_bool,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = 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_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).
tff(fact_701_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_1: nat,X_1: nat,A_2: fun_nat_bool,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = 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_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).
tff(fact_702_nat__mult__assoc,axiom,
! [M_1: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1)) ).
tff(fact_703_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
tff(fact_704_nat__mult__commute,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M_1) ).
tff(fact_705_nat__mult__1__right,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),one_one_nat) = N_1 ).
tff(fact_706_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
tff(fact_707_nat__mult__1,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N_1) = N_1 ).
tff(fact_708_one__reorient,axiom,
! [X_1: int] :
( ( one_one_int = X_1 )
<=> ( X_1 = one_one_int ) ) ).
tff(fact_709_one__reorient,axiom,
! [X_1: nat] :
( ( one_one_nat = X_1 )
<=> ( X_1 = one_one_nat ) ) ).
tff(fact_710_mult__1__left,axiom,
! [A_41: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_41) = A_41 ).
tff(fact_711_mult__1__left,axiom,
! [A_41: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_41) = A_41 ).
tff(fact_712_mult__1,axiom,
! [A_40: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_40) = A_40 ).
tff(fact_713_mult__1,axiom,
! [A_40: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_40) = A_40 ).
tff(fact_714_mult__1__right,axiom,
! [A_39: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_39),one_one_int) = A_39 ).
tff(fact_715_mult__1__right,axiom,
! [A_39: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_39),one_one_nat) = A_39 ).
tff(fact_716_mult_Ocomm__neutral,axiom,
! [A_38: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_38),one_one_int) = A_38 ).
tff(fact_717_mult_Ocomm__neutral,axiom,
! [A_38: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_38),one_one_nat) = A_38 ).
tff(fact_718_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1: int,Z_1: int,A_2: fun_int_bool,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A_2)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A_2) ) ) ) ).
tff(fact_719_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_1: nat,Z_1: nat,A_2: fun_nat_bool,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A_2)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A_2) ) ) ) ).
tff(fact_720_comp__fun__commute_Ofold__equality,axiom,
! [Z_1: int,A_2: fun_int_bool,Y_1: int,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),Y_1))
=> ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A_2) = Y_1 ) ) ) ).
tff(fact_721_comp__fun__commute_Ofold__equality,axiom,
! [Z_1: nat,A_2: fun_nat_bool,Y_1: nat,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),Y_1))
=> ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A_2) = Y_1 ) ) ) ).
tff(fact_722_comp__fun__commute_Ofold__equality,axiom,
! [Z_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,Y_1: hoare_363350478_state,F: fun_Ho31865764_state] :
( hBOOL(finite980784795_state(F))
=> ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),Y_1))
=> ( hAPP_f1242401993_state(finite1620577938_state(F,Z_1),A_2) = Y_1 ) ) ) ).
tff(fact_723_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1: int,X_1: int,A_2: fun_int_bool,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,hAPP_int_int(hAPP_int_fun_int_int(F,X_1),Z_1)),A_2) ) ) ) ) ).
tff(fact_724_comp__fun__commute_Ofold__insert2,axiom,
! [Z_1: nat,X_1: nat,A_2: fun_nat_bool,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),Z_1)),A_2) ) ) ) ) ).
tff(fact_725_comp__fun__commute_Ofold__insert,axiom,
! [Z_1: int,X_1: int,A_2: fun_int_bool,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A_2)) ) ) ) ) ).
tff(fact_726_comp__fun__commute_Ofold__insert,axiom,
! [Z_1: nat,X_1: nat,A_2: fun_nat_bool,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A_2)) ) ) ) ) ).
tff(fact_727_folding__one_Oeq__fold_H,axiom,
! [X_1: int,A_2: 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_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,X_1),A_2) ) ) ) ) ).
tff(fact_728_folding__one_Oeq__fold_H,axiom,
! [X_1: nat,A_2: 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_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,X_1),A_2) ) ) ) ) ).
tff(fact_729_folding__one_Oeq__fold_H,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite269920017_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = hAPP_f1242401993_state(finite1620577938_state(F,X_1),A_2) ) ) ) ) ).
tff(fact_730_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_1: int,A_2: 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_2))
=> ( hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2)) = hAPP_f1594865479ol_int(finite_fold_int_int(F,A_1),A_2) ) ) ) ).
tff(fact_731_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_1: nat,A_2: 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_2))
=> ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F,A_1),A_2) ) ) ) ).
tff(fact_732_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
( hBOOL(finite2031572_state(F,F_1))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_1),A_2)) = hAPP_f1242401993_state(finite1620577938_state(F,A_1),A_2) ) ) ) ).
tff(fact_733_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1: int,A_2: fun_int_bool,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> hBOOL(hAPP_int_bool(hAPP_f1805168059t_bool(finite772772422nt_int(F,Z_1),A_2),hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A_2))) ) ) ).
tff(fact_734_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1: nat,A_2: fun_nat_bool,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(finite929467206at_nat(F,Z_1),A_2),hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A_2))) ) ) ).
tff(fact_735_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool,F: fun_Ho31865764_state] :
( hBOOL(finite980784795_state(F))
=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(finite1206205712_state(F,Z_1),A_2),hAPP_f1242401993_state(finite1620577938_state(F,Z_1),A_2))) ) ) ).
tff(fact_736_fold1__eq__fold,axiom,
! [A_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A_2))
=> ( hAPP_f1594865479ol_int(finite_fold1_int(times_times_int),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_1),A_2)) = hAPP_f1594865479ol_int(finite_fold_int_int(times_times_int,A_1),A_2) ) ) ) ).
tff(fact_737_fold1__eq__fold,axiom,
! [A_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A_2)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(times_times_nat,A_1),A_2) ) ) ) ).
tff(fact_738_comp__fun__commute_Ofold__rec,axiom,
! [Z_1: int,X_1: int,A_2: fun_int_bool,F: fun_int_fun_int_int] :
( hBOOL(finite1973466193nt_int(F))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f1594865479ol_int(finite_fold_int_int(F,Z_1),A_2) = 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_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ).
tff(fact_739_comp__fun__commute_Ofold__rec,axiom,
! [Z_1: nat,X_1: nat,A_2: fun_nat_bool,F: fun_nat_fun_nat_nat] :
( hBOOL(finite2130160977at_nat(F))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_fold_nat_nat(F,Z_1),A_2) = 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_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ).
tff(fact_740_less__1__mult,axiom,
! [N_3: int,M_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_8))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_8),N_3))) ) ) ).
tff(fact_741_less__1__mult,axiom,
! [N_3: nat,M_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_8),N_3))) ) ) ).
tff(fact_742_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_37: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_37) = A_37 ).
tff(fact_743_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_37: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_37) = A_37 ).
tff(fact_744_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_36: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_36),one_one_int) = A_36 ).
tff(fact_745_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_36: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_36),one_one_nat) = A_36 ).
tff(fact_746_card_Oremove,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f957591787ol_nat(finite_card_int,A_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ).
tff(fact_747_card_Oremove,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_card_nat,A_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ).
tff(fact_748_card_Oremove,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,A_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))) ) ) ) ).
tff(fact_749_card_Oinsert__remove,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).
tff(fact_750_card_Oinsert__remove,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).
tff(fact_751_card_Oinsert__remove,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))) ) ) ).
tff(fact_752_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_5: int,Y_4: int,Z_4: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_5),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_4),Z_4)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_5),Y_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_5),Z_4)) ).
tff(fact_753_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_5: nat,Y_4: nat,Z_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_4),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_5),Y_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_5),Z_4)) ).
tff(fact_754_crossproduct__noteq,axiom,
! [Ca: int,D_1: int,A_1: int,B_2: int] :
( ( ( A_1 != B_2 )
& ( 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_1),Ca)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),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_1),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Ca)) ) ) ).
tff(fact_755_crossproduct__noteq,axiom,
! [Ca: nat,D_1: nat,A_1: nat,B_2: nat] :
( ( ( A_1 != B_2 )
& ( 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_1),Ca)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),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_1),D_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Ca)) ) ) ).
tff(fact_756_comm__semiring__class_Odistrib,axiom,
! [A_35: int,B_28: int,C_24: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_35),B_28)),C_24) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_35),C_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_28),C_24)) ).
tff(fact_757_comm__semiring__class_Odistrib,axiom,
! [A_35: nat,B_28: nat,C_24: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_35),B_28)),C_24) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_35),C_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_28),C_24)) ).
tff(fact_758_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_34: int,B_27: int,C_23: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_34),B_27)),C_23) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_34),C_23)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_27),C_23)) ).
tff(fact_759_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_34: nat,B_27: nat,C_23: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_34),B_27)),C_23) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_34),C_23)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_27),C_23)) ).
tff(fact_760_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_33: int,M_7: int,B_26: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_33),M_7)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_26),M_7)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_33),B_26)),M_7) ).
tff(fact_761_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_33: nat,M_7: nat,B_26: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_33),M_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_26),M_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_33),B_26)),M_7) ).
tff(fact_762_combine__common__factor,axiom,
! [A_32: int,E_1: 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(times_times_int,A_32),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_25),E_1)),C_22)) = 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_32),B_25)),E_1)),C_22) ).
tff(fact_763_combine__common__factor,axiom,
! [A_32: nat,E_1: 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(times_times_nat,A_32),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_25),E_1)),C_22)) = 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_32),B_25)),E_1)),C_22) ).
tff(fact_764_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_765_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_766_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_31: int,C_21: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_31),C_21) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_21),A_31) ).
tff(fact_767_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_31: nat,C_21: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_31),C_21) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_21),A_31) ).
tff(fact_768_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_30: int,C_20: int,D_9: 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_20),D_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_20),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),D_9)) ).
tff(fact_769_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_30: nat,C_20: nat,D_9: 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_20),D_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_20),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),D_9)) ).
tff(fact_770_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_29: int,C_19: int,D_8: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_29),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_19),D_8)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_29),C_19)),D_8) ).
tff(fact_771_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_29: nat,C_19: nat,D_8: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_29),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_19),D_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_29),C_19)),D_8) ).
tff(fact_772_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_28: int,B_24: int,C_18: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_28),B_24)),C_18) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_28),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_24),C_18)) ).
tff(fact_773_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_28: nat,B_24: nat,C_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_28),B_24)),C_18) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_28),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_24),C_18)) ).
tff(fact_774_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_27: int,B_23: int,C_17: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_27),B_23)),C_17) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_27),C_17)),B_23) ).
tff(fact_775_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_27: nat,B_23: nat,C_17: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_27),B_23)),C_17) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_27),C_17)),B_23) ).
tff(fact_776_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_26: int,B_22: int,C_16: int,D_7: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_26),B_22)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_16),D_7)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_26),C_16)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_22),D_7)) ).
tff(fact_777_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_26: nat,B_22: nat,C_16: nat,D_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_26),B_22)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_16),D_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_26),C_16)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_22),D_7)) ).
tff(fact_778_add__right__imp__eq,axiom,
! [B_21: int,A_25: int,C_15: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_21),A_25) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_15),A_25) )
=> ( B_21 = C_15 ) ) ).
tff(fact_779_add__right__imp__eq,axiom,
! [B_21: nat,A_25: nat,C_15: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_21),A_25) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_15),A_25) )
=> ( B_21 = C_15 ) ) ).
tff(fact_780_add__imp__eq,axiom,
! [A_24: int,B_20: int,C_14: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_24),B_20) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_24),C_14) )
=> ( B_20 = C_14 ) ) ).
tff(fact_781_add__imp__eq,axiom,
! [A_24: nat,B_20: nat,C_14: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_24),B_20) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_24),C_14) )
=> ( B_20 = C_14 ) ) ).
tff(fact_782_add__left__imp__eq,axiom,
! [A_23: int,B_19: int,C_13: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_23),B_19) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_23),C_13) )
=> ( B_19 = C_13 ) ) ).
tff(fact_783_add__left__imp__eq,axiom,
! [A_23: nat,B_19: nat,C_13: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_23),B_19) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_23),C_13) )
=> ( B_19 = C_13 ) ) ).
tff(fact_784_add__right__cancel,axiom,
! [B_2: int,A_1: int,Ca: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),A_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),A_1) )
<=> ( B_2 = Ca ) ) ).
tff(fact_785_add__right__cancel,axiom,
! [B_2: nat,A_1: nat,Ca: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_2),A_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),A_1) )
<=> ( B_2 = Ca ) ) ).
tff(fact_786_add__left__cancel,axiom,
! [A_1: int,B_2: int,Ca: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),Ca) )
<=> ( B_2 = Ca ) ) ).
tff(fact_787_add__left__cancel,axiom,
! [A_1: nat,B_2: nat,Ca: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),B_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),Ca) )
<=> ( B_2 = Ca ) ) ).
tff(fact_788_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_22: int,B_18: int,C_12: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_22),B_18)),C_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_22),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_18),C_12)) ).
tff(fact_789_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_22: nat,B_18: nat,C_12: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_22),B_18)),C_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_22),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_18),C_12)) ).
tff(fact_790_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: 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) )
<=> ( M = N ) ) ).
tff(fact_791_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: 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) )
<=> ( M = N ) ) ).
tff(fact_792_nat__add__assoc,axiom,
! [M_1: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1)) ).
tff(fact_793_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_794_nat__add__commute,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1) ).
tff(fact_795_add__le__cancel__right,axiom,
! [A_1: int,Ca: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),Ca)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_2)) ) ).
tff(fact_796_add__le__cancel__right,axiom,
! [A_1: nat,Ca: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),Ca)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_2),Ca)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_2)) ) ).
tff(fact_797_add__le__cancel__left,axiom,
! [Ca: int,A_1: int,B_2: 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_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_2)) ) ).
tff(fact_798_add__le__cancel__left,axiom,
! [Ca: nat,A_1: nat,B_2: 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_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),B_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_2)) ) ).
tff(fact_799_add__right__mono,axiom,
! [C_11: int,A_21: int,B_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),B_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_21),C_11)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_17),C_11))) ) ).
tff(fact_800_add__right__mono,axiom,
! [C_11: nat,A_21: nat,B_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),B_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_21),C_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_17),C_11))) ) ).
tff(fact_801_add__left__mono,axiom,
! [C_10: int,A_20: int,B_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_20),B_16))
=> 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_16))) ) ).
tff(fact_802_add__left__mono,axiom,
! [C_10: nat,A_20: nat,B_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_20),B_16))
=> 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_16))) ) ).
tff(fact_803_add__mono,axiom,
! [C_9: int,D_6: int,A_19: int,B_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),B_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_9),D_6))
=> 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_15),D_6))) ) ) ).
tff(fact_804_add__mono,axiom,
! [C_9: nat,D_6: nat,A_19: nat,B_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_19),B_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_9),D_6))
=> 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_15),D_6))) ) ) ).
tff(fact_805_add__le__imp__le__right,axiom,
! [A_18: int,C_8: 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_18),C_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_14),C_8)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_18),B_14)) ) ).
tff(fact_806_add__le__imp__le__right,axiom,
! [A_18: nat,C_8: 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_18),C_8)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_14),C_8)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),B_14)) ) ).
tff(fact_807_add__le__imp__le__left,axiom,
! [C_7: int,A_17: int,B_13: int] :
( 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_13)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_17),B_13)) ) ).
tff(fact_808_add__le__imp__le__left,axiom,
! [C_7: nat,A_17: nat,B_13: nat] :
( 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_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_17),B_13)) ) ).
tff(fact_809_add__less__imp__less__left,axiom,
! [C_6: int,A_16: int,B_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),A_16)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),B_12)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_16),B_12)) ) ).
tff(fact_810_add__less__imp__less__left,axiom,
! [C_6: nat,A_16: nat,B_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),A_16)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),B_12)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_16),B_12)) ) ).
tff(fact_811_add__less__imp__less__right,axiom,
! [A_15: int,C_5: int,B_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_15),C_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_11),C_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_15),B_11)) ) ).
tff(fact_812_add__less__imp__less__right,axiom,
! [A_15: nat,C_5: nat,B_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_15),C_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_11),C_5)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_15),B_11)) ) ).
tff(fact_813_add__strict__mono,axiom,
! [C_4: int,D_5: int,A_14: int,B_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_14),B_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_4),D_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_14),C_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_10),D_5))) ) ) ).
tff(fact_814_add__strict__mono,axiom,
! [C_4: nat,D_5: nat,A_14: nat,B_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_14),B_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_4),D_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_14),C_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_10),D_5))) ) ) ).
tff(fact_815_add__strict__left__mono,axiom,
! [C_3: int,A_13: int,B_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_13),B_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_3),A_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_3),B_9))) ) ).
tff(fact_816_add__strict__left__mono,axiom,
! [C_3: nat,A_13: nat,B_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_13),B_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_3),A_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_3),B_9))) ) ).
tff(fact_817_add__strict__right__mono,axiom,
! [C_2: int,A_12: int,B_8: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_12),B_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_12),C_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_8),C_2))) ) ).
tff(fact_818_add__strict__right__mono,axiom,
! [C_2: nat,A_12: nat,B_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_12),B_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_12),C_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_8),C_2))) ) ).
tff(fact_819_add__less__cancel__left,axiom,
! [Ca: int,A_1: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),A_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ca),B_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_2)) ) ).
tff(fact_820_add__less__cancel__left,axiom,
! [Ca: nat,A_1: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),A_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ca),B_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_2)) ) ).
tff(fact_821_add__less__cancel__right,axiom,
! [A_1: int,Ca: int,B_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),Ca)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_2)) ) ).
tff(fact_822_add__less__cancel__right,axiom,
! [A_1: nat,Ca: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),Ca)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_2),Ca)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_2)) ) ).
tff(fact_823_add__diff__cancel,axiom,
! [A_11: int,B_7: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_11),B_7)),B_7) = A_11 ).
tff(fact_824_diff__add__cancel,axiom,
! [A_10: int,B_6: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_10),B_6)),B_6) = A_10 ).
tff(fact_825_add__lessD1,axiom,
! [I_1: nat,J: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1)) ) ).
tff(fact_826_less__add__eq__less,axiom,
! [M_1: nat,N_1: nat,K_1: nat,L_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L_1))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),L_1) = 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_1),N_1)) ) ) ).
tff(fact_827_add__less__mono,axiom,
! [K_1: nat,L_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L_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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).
tff(fact_828_add__less__mono1,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> 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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1))) ) ).
tff(fact_829_trans__less__add2,axiom,
! [M_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J))) ) ).
tff(fact_830_trans__less__add1,axiom,
! [M_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M_1))) ) ).
tff(fact_831_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_832_not__add__less2,axiom,
! [J: nat,I_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),I_1)) ).
tff(fact_833_not__add__less1,axiom,
! [I_1: 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_1),J)),I_1)) ).
tff(fact_834_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_835_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_836_le__add2,axiom,
! [N_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1))) ).
tff(fact_837_le__add1,axiom,
! [N_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1))) ).
tff(fact_838_le__iff__add,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
<=> ? [K_2: nat] : N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2) ) ).
tff(fact_839_nat__add__left__cancel__le,axiom,
! [K: nat,M: 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,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_840_trans__le__add1,axiom,
! [M_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M_1))) ) ).
tff(fact_841_trans__le__add2,axiom,
! [M_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J))) ) ).
tff(fact_842_add__le__mono1,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> 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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1))) ) ).
tff(fact_843_add__le__mono,axiom,
! [K_1: nat,L_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),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_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).
tff(fact_844_add__leD2,axiom,
! [M_1: nat,K_1: 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,M_1),K_1)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) ) ).
tff(fact_845_add__leD1,axiom,
! [M_1: nat,K_1: 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,M_1),K_1)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_846_add__leE,axiom,
! [M_1: nat,K_1: 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,M_1),K_1)),N_1))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) ) ) ).
tff(fact_847_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_848_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_849_diff__cancel2,axiom,
! [M_1: nat,K_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ).
tff(fact_850_diff__cancel,axiom,
! [K_1: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ).
tff(fact_851_diff__diff__left,axiom,
! [I_1: nat,J: nat,K_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).
tff(fact_852_diff__add__inverse,axiom,
! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1)),N_1) = M_1 ).
tff(fact_853_diff__add__inverse2,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),N_1) = M_1 ).
tff(fact_854_add__mult__distrib2,axiom,
! [K_1: nat,M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)) ).
tff(fact_855_add__mult__distrib,axiom,
! [M_1: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1)) ).
tff(fact_856_less__add__one,axiom,
! [A_9: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_9),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_9),one_one_int))) ).
tff(fact_857_less__add__one,axiom,
! [A_9: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_9),one_one_nat))) ).
tff(fact_858_eq__add__iff2,axiom,
! [A_1: int,E: int,Ca: int,B_2: 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_1),E)),Ca) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),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,B_2),A_1)),E)),D_1) ) ) ).
tff(fact_859_eq__add__iff1,axiom,
! [A_1: int,E: int,Ca: int,B_2: 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_1),E)),Ca) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),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_1),B_2)),E)),Ca) = D_1 ) ) ).
tff(fact_860_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_8: int,M_6: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_8),M_6)),M_6) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_8),one_one_int)),M_6) ).
tff(fact_861_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_8: nat,M_6: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_8),M_6)),M_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_8),one_one_nat)),M_6) ).
tff(fact_862_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_5: int,A_7: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_7),M_5)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_7),one_one_int)),M_5) ).
tff(fact_863_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_5: nat,A_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_7),M_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_7),one_one_nat)),M_5) ).
tff(fact_864_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_4),M_4) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),M_4) ).
tff(fact_865_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_4),M_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),M_4) ).
tff(fact_866_comp__fun__idem__insert,axiom,
hBOOL(finite58652534t_bool(insert_int)) ).
tff(fact_867_comp__fun__idem__insert,axiom,
hBOOL(finite1956789438t_bool(insert_nat)) ).
tff(fact_868_comp__fun__idem__insert,axiom,
hBOOL(finite377806280e_bool(insert1330351828_state)) ).
tff(fact_869_add__less__le__mono,axiom,
! [C_1: int,D_4: int,A_6: int,B_5: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_6),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),D_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_6),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_5),D_4))) ) ) ).
tff(fact_870_add__less__le__mono,axiom,
! [C_1: nat,D_4: nat,A_6: nat,B_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_6),B_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),D_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_6),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_5),D_4))) ) ) ).
tff(fact_871_add__le__less__mono,axiom,
! [C: int,D_3: int,A_5: int,B_4: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_5),B_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_5),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_4),D_3))) ) ) ).
tff(fact_872_add__le__less__mono,axiom,
! [C: nat,D_3: nat,A_5: nat,B_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),B_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C),D_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_5),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_4),D_3))) ) ) ).
tff(fact_873_le__add__iff2,axiom,
! [A_1: int,E: int,Ca: int,B_2: 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_1),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),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,B_2),A_1)),E)),D_1))) ) ).
tff(fact_874_le__add__iff1,axiom,
! [A_1: int,E: int,Ca: int,B_2: 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_1),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),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_1),B_2)),E)),Ca)),D_1)) ) ).
tff(fact_875_less__add__iff2,axiom,
! [A_1: int,E: int,Ca: int,B_2: int,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),E)),D_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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,B_2),A_1)),E)),D_1))) ) ).
tff(fact_876_less__add__iff1,axiom,
! [A_1: int,E: int,Ca: int,B_2: int,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),E)),Ca)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),E)),D_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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_1),B_2)),E)),Ca)),D_1)) ) ).
tff(fact_877_less__diff__conv,axiom,
! [I_2: nat,J_2: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)),J_2)) ) ).
tff(fact_878_add__diff__inverse,axiom,
! [M_1: nat,N_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) = M_1 ) ) ).
tff(fact_879_diff__add__assoc2,axiom,
! [I_1: nat,K_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),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_1)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1) ) ) ).
tff(fact_880_add__diff__assoc2,axiom,
! [I_1: nat,K_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),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_1)),I_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),K_1) ) ) ).
tff(fact_881_diff__add__assoc,axiom,
! [I_1: nat,K_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) ) ) ).
tff(fact_882_le__imp__diff__is__add,axiom,
! [K: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I_2) = K )
<=> ( J_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),I_2) ) ) ) ).
tff(fact_883_le__add__diff__inverse2,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),N_1) = M_1 ) ) ).
tff(fact_884_le__diff__conv2,axiom,
! [I_2: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)),J_2)) ) ) ).
tff(fact_885_add__diff__assoc,axiom,
! [I_1: nat,K_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1) ) ) ).
tff(fact_886_le__add__diff__inverse,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) = M_1 ) ) ).
tff(fact_887_le__add__diff,axiom,
! [M_1: nat,K_1: nat,N_1: nat] :
( 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,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1)),K_1))) ) ).
tff(fact_888_le__diff__conv,axiom,
! [J_2: nat,K: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K)),I_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K))) ) ).
tff(fact_889_diff__diff__right,axiom,
! [I_1: nat,K_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J) ) ) ).
tff(fact_890_linorder__neqE__linordered__idom,axiom,
! [X_4: int,Y_3: int] :
( ( X_4 != Y_3 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_4),Y_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_3),X_4)) ) ) ).
tff(fact_891_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_4: int,B_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_4),B_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_3),A_4) ).
tff(fact_892_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_4: nat,B_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_4),B_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_3),A_4) ).
tff(fact_893_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),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,Rx_6),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ry_4)) ).
tff(fact_894_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: nat,Rx_6: nat,Ry_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),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,Rx_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ry_4)) ).
tff(fact_895_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),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,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Rx_5)),Ry_3) ).
tff(fact_896_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: nat,Rx_5: nat,Ry_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Rx_5)),Ry_3) ).
tff(fact_897_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_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_4),Ly_4)),Rx_4) = 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_4),Rx_4)) ).
tff(fact_898_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: nat,Ly_4: nat,Rx_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_4),Ly_4)),Rx_4) = 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_4),Rx_4)) ).
tff(fact_899_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: 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_3)),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_3) ).
tff(fact_900_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_3: 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_3)),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_3) ).
tff(fact_901_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: 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_2),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),Ry_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_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),Ry_2))) ).
tff(fact_902_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: nat,Ly_2: nat,Rx_2: 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_2),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),Ry_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_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),Ry_2))) ).
tff(fact_903_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Ly_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,Rx_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),Ly_1)),Ry_1)) ).
tff(fact_904_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Ly_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,Rx_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),Ly_1)),Ry_1)) ).
tff(fact_905_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ly)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Rx)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Ry)) ).
tff(fact_906_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ly)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Rx)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Ry)) ).
tff(fact_907_card_Oinsert,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f957591787ol_nat(finite_card_int,A_2)) ) ) ) ).
tff(fact_908_card_Oinsert,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f22106695ol_nat(finite_card_nat,A_2)) ) ) ) ).
tff(fact_909_card_Oinsert,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),A_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f1705983439ol_nat(finite987718268_state,A_2)) ) ) ) ).
tff(fact_910_nat__less__add__iff1,axiom,
! [U_1: nat,M: nat,N: nat,J_2: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2))
=> ( 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_2),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_2),U_1)),N)))
<=> 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_2),J_2)),U_1)),M)),N)) ) ) ).
tff(fact_911_nat__less__add__iff2,axiom,
! [U_1: nat,M: nat,N: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( 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_2),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_2),U_1)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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_2),I_2)),U_1)),N))) ) ) ).
tff(fact_912_nat__le__add__iff1,axiom,
! [U_1: nat,M: nat,N: nat,J_2: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_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(hAPP_nat_fun_nat_nat(times_times_nat,I_2),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_2),U_1)),N)))
<=> 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_2),J_2)),U_1)),M)),N)) ) ) ).
tff(fact_913_nat__diff__add__eq1,axiom,
! [U_2: nat,M_1: nat,N_1: nat,J: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
=> ( 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_1),U_2)),M_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_2)),N_1)) = 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_1),J)),U_2)),M_1)),N_1) ) ) ).
tff(fact_914_nat__eq__add__iff1,axiom,
! [U_1: nat,M: nat,N: nat,J_2: nat,I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),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_2),U_1)),N) )
<=> ( 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_2),J_2)),U_1)),M) = N ) ) ) ).
tff(fact_915_left__add__mult__distrib,axiom,
! [I_1: nat,U_2: nat,J: nat,K_1: 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_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_2)),K_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(plus_plus_nat,I_1),J)),U_2)),K_1) ).
tff(fact_916_nat__eq__add__iff2,axiom,
! [U_1: nat,M: nat,N: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),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_2),U_1)),N) )
<=> ( 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_2),I_2)),U_1)),N) ) ) ) ).
tff(fact_917_nat__diff__add__eq2,axiom,
! [U_2: nat,M_1: nat,N_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),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_1),U_2)),M_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_2)),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_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,J),I_1)),U_2)),N_1)) ) ) ).
tff(fact_918_nat__le__add__iff2,axiom,
! [U_1: nat,M: nat,N: nat,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_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(hAPP_nat_fun_nat_nat(times_times_nat,I_2),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_2),U_1)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_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_2),I_2)),U_1)),N))) ) ) ).
tff(fact_919_mono__nat__linear__lb,axiom,
! [M: nat,K: nat,F: fun_nat_nat] :
( ! [M_2: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_2)),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)),K)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).
tff(fact_920_convex__bound__lt,axiom,
! [V_1: int,U: int,Y_2: int,X_3: int,A_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_1))
=> ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U),V_1) = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U),X_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_1),Y_2))),A_3)) ) ) ) ) ) ).
tff(fact_921_card__Suc__Diff1,axiom,
! [X_1: int,A_2: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_2))
=> ( hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) = hAPP_f957591787ol_nat(finite_card_int,A_2) ) ) ) ).
tff(fact_922_card__Suc__Diff1,axiom,
! [X_1: nat,A_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_2))
=> ( hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) = hAPP_f22106695ol_nat(finite_card_nat,A_2) ) ) ) ).
tff(fact_923_card__Suc__Diff1,axiom,
! [X_1: hoare_363350478_state,A_2: fun_Ho320143120e_bool] :
( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_1),A_2))
=> ( hAPP_nat_nat(suc,hAPP_f1705983439ol_nat(finite987718268_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_1),bot_bo39439654e_bool)))) = hAPP_f1705983439ol_nat(finite987718268_state,A_2) ) ) ) ).
tff(fact_924_less__zeroE,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_925_le0,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_926_zero__less__Suc,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_1))) ).
tff(fact_927_Suc__mono,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_928_lessI,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,N_1))) ).
tff(fact_929_add__is__1,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_930_one__is__add,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) )
<=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_931_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: 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) )
<=> ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
tff(fact_932_not__less__eq,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_nat,N),hAPP_nat_nat(suc,M))) ) ).
tff(fact_933_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| ( M = N ) ) ) ).
tff(fact_934_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_935_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
<=> ( N = M ) ) ) ).
tff(fact_936_less__antisym,axiom,
! [N_1: nat,M_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M_1)))
=> ( M_1 = N_1 ) ) ) ).
tff(fact_937_less__SucI,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_938_Suc__lessI,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> ( ( hAPP_nat_nat(suc,M_1) != N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N_1)) ) ) ).
tff(fact_939_less__trans__Suc,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_1)),K_1)) ) ) ).
tff(fact_940_less__SucE,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> ( M_1 = N_1 ) ) ) ).
tff(fact_941_Suc__lessD,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_942_Suc__less__SucD,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_943_add__Suc__right,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) ).
tff(fact_944_add__Suc,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) ).
tff(fact_945_add__Suc__shift,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N_1)) ).
tff(fact_946_Suc__leD,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_947_le__SucE,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N_1)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ( M_1 = hAPP_nat_nat(suc,N_1) ) ) ) ).
tff(fact_948_le__SucI,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_949_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),hAPP_nat_nat(suc,M)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
tff(fact_950_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| ( M = hAPP_nat_nat(suc,N) ) ) ) ).
tff(fact_951_not__less__eq__eq,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,hAPP_nat_nat(suc,N)),M)) ) ).
tff(fact_952_Suc__n__not__le__n,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1)) ).
tff(fact_953_diff__Suc__Suc,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) ).
tff(fact_954_Suc__diff__diff,axiom,
! [M_1: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N_1)),hAPP_nat_nat(suc,K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),K_1) ).
tff(fact_955_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),N) )
<=> ( M = N ) ) ).
tff(fact_956_not__less0,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_957_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
tff(fact_958_less__nat__zero__code,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat)) ).
tff(fact_959_gr__implies__not0,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> ( N_1 != zero_zero_nat ) ) ).
tff(fact_960_gr0I,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_961_plus__nat_Oadd__0,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N_1) = N_1 ).
tff(fact_962_Nat_Oadd__0__right,axiom,
! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),zero_zero_nat) = M_1 ).
tff(fact_963_add__is__0,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
tff(fact_964_add__eq__self__zero,axiom,
! [M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = M_1 )
=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_965_le__0__eq,axiom,
! [N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),zero_zero_nat))
<=> ( N = zero_zero_nat ) ) ).
tff(fact_966_less__eq__nat_Osimps_I1_J,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1)) ).
tff(fact_967_diff__0__eq__0,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_968_minus__nat_Odiff__0,axiom,
! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),zero_zero_nat) = M_1 ).
tff(fact_969_diff__self__eq__0,axiom,
! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),M_1) = zero_zero_nat ).
tff(fact_970_diffs0__imp__equal,axiom,
! [M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) = zero_zero_nat )
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1) = zero_zero_nat )
=> ( M_1 = N_1 ) ) ) ).
tff(fact_971_mult__cancel2,axiom,
! [M: nat,K: nat,N: 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) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
tff(fact_972_mult__cancel1,axiom,
! [K: nat,M: nat,N: 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) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
tff(fact_973_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
tff(fact_974_mult__0__right,axiom,
! [M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),zero_zero_nat) = zero_zero_nat ).
tff(fact_975_mult__0,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_976_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(suc,zero_zero_nat) )
<=> ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
tff(fact_977_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
tff(fact_978_Suc__neq__Zero,axiom,
! [M_1: nat] : hAPP_nat_nat(suc,M_1) != zero_zero_nat ).
tff(fact_979_Zero__neq__Suc,axiom,
! [M_1: nat] : zero_zero_nat != hAPP_nat_nat(suc,M_1) ).
tff(fact_980_nat_Oinject,axiom,
! [Nat_4: nat,Nat_3: nat] :
( ( hAPP_nat_nat(suc,Nat_4) = hAPP_nat_nat(suc,Nat_3) )
<=> ( Nat_4 = Nat_3 ) ) ).
tff(fact_981_Suc__n__not__n,axiom,
! [N_1: nat] : hAPP_nat_nat(suc,N_1) != N_1 ).
tff(fact_982_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] : hAPP_nat_nat(suc,Nat_2) != zero_zero_nat ).
tff(fact_983_Suc__not__Zero,axiom,
! [M_1: nat] : hAPP_nat_nat(suc,M_1) != zero_zero_nat ).
tff(fact_984_n__not__Suc__n,axiom,
! [N_1: nat] : N_1 != hAPP_nat_nat(suc,N_1) ).
tff(fact_985_nat_Osimps_I2_J,axiom,
! [Nat_1: nat] : zero_zero_nat != hAPP_nat_nat(suc,Nat_1) ).
tff(fact_986_Zero__not__Suc,axiom,
! [M_1: nat] : zero_zero_nat != hAPP_nat_nat(suc,M_1) ).
tff(fact_987_gr0__conv__Suc,axiom,
! [N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
<=> ? [M_2: nat] : N = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_988_less__Suc0,axiom,
! [N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,zero_zero_nat)))
<=> ( N = zero_zero_nat ) ) ).
tff(fact_989_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
<=> ( ( M = zero_zero_nat )
| ? [J_1: nat] :
( ( M = hAPP_nat_nat(suc,J_1) )
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N)) ) ) ) ).
tff(fact_990_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) ) ) ).
tff(fact_991_one__less__mult,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1))) ) ) ).
tff(fact_992_n__less__n__mult__m,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),M_1))) ) ) ).
tff(fact_993_n__less__m__mult__n,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1))) ) ) ).
tff(fact_994_Suc__pred,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))) = N_1 ) ) ).
tff(fact_995_diff__Suc__less,axiom,
! [I_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,I_1))),N_1)) ) ).
tff(fact_996_One__nat__def,axiom,
one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).
tff(fact_997_Suc__diff__1,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) = N_1 ) ) ).
tff(fact_998_less__add__Suc1,axiom,
! [I_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),M_1)))) ).
tff(fact_999_less__add__Suc2,axiom,
! [I_1: nat,M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),I_1)))) ).
tff(fact_1000_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
<=> ? [K_2: nat] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)) ) ).
tff(fact_1001_Suc__le__lessD,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_1002_le__less__Suc__eq,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_nat,N),hAPP_nat_nat(suc,M)))
<=> ( N = M ) ) ) ).
tff(fact_1003_Suc__leI,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1)) ) ).
tff(fact_1004_le__imp__less__Suc,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_1005_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_1006_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_1007_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M)) ) ).
tff(fact_1008_diff__less__Suc,axiom,
! [M_1: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),hAPP_nat_nat(suc,M_1))) ).
tff(fact_1009_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_1010_mult__Suc,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) ).
tff(fact_1011_mult__Suc__right,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) ).
tff(fact_1012_Suc__diff__le,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) ) ) ).
tff(fact_1013_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_1014_diff__Suc__1,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_1)),one_one_nat) = N_1 ).
tff(fact_1015_add__gr__0,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
tff(fact_1016_zero__less__diff,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_1017_diff__less,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)),M_1)) ) ) ).
tff(fact_1018_mult__less__mono2,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),J))) ) ) ).
tff(fact_1019_mult__less__mono1,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K_1))) ) ) ).
tff(fact_1020_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_1021_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_1022_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
tff(fact_1023_nat__mult__less__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_1024_nat__mult__eq__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( ( 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) )
<=> ( M = N ) ) ) ).
tff(fact_1025_diff__add__0,axiom,
! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),M_1)) = zero_zero_nat ).
tff(fact_1026_diff__is__0__eq_H,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) = zero_zero_nat ) ) ).
tff(fact_1027_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_1028_mult__eq__self__implies__10,axiom,
! [M_1: nat,N_1: nat] :
( ( M_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) )
=> ( ( N_1 = one_one_nat )
| ( M_1 = zero_zero_nat ) ) ) ).
tff(fact_1029_diff__Suc__diff__eq1,axiom,
! [M_1: nat,K_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),hAPP_nat_nat(suc,J)) ) ) ).
tff(fact_1030_diff__Suc__diff__eq2,axiom,
! [M_1: nat,K_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))),M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_1)) ) ) ).
tff(fact_1031_nat__diff__split,axiom,
! [Pa: fun_nat_bool,A_1: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),B_2)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_2))
=> hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) )
& ! [D_2: nat] :
( ( A_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_2),D_2) )
=> hBOOL(hAPP_nat_bool(Pa,D_2)) ) ) ) ).
tff(fact_1032_nat__diff__split__asm,axiom,
! [Pa: fun_nat_bool,A_1: nat,B_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),B_2)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_2))
& ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) )
| ? [D_2: nat] :
( ( A_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_2),D_2) )
& ~ hBOOL(hAPP_nat_bool(Pa,D_2)) ) ) ) ).
tff(fact_1033_mult__le__cancel2,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(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).
tff(fact_1034_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_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)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).
tff(fact_1035_nat__mult__le__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_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)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).
tff(fact_1036_ex__least__nat__less,axiom,
! [N: nat,Pa: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(Pa,N))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),N))
& ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))
=> ~ hBOOL(hAPP_nat_bool(Pa,I)) )
& hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),one_one_nat))) ) ) ) ).
tff(fact_1037_mult__eq__if,axiom,
! [N_1: nat,M_1: nat] :
( ( ( M_1 = zero_zero_nat )
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = zero_zero_nat ) )
& ( ( M_1 != zero_zero_nat )
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N_1)) ) ) ) ).
tff(fact_1038_add__eq__if,axiom,
! [N_1: nat,M_1: nat] :
( ( ( M_1 = zero_zero_nat )
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = N_1 ) )
& ( ( M_1 != zero_zero_nat )
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N_1)) ) ) ) ).
tff(fact_1039_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
tff(fact_1040_Suc__eq__plus1__left,axiom,
! [N_1: nat] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_1) ).
tff(fact_1041_Suc__eq__plus1,axiom,
! [N_1: nat] : hAPP_nat_nat(suc,N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat) ).
tff(fact_1042_diff__Suc__eq__diff__pred,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N_1) ).
tff(fact_1043_Suc__pred_H,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) ) ) ).
tff(fact_1044_inc__induct,axiom,
! [Pa: fun_nat_bool,I_2: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2))
=> ( hBOOL(hAPP_nat_bool(Pa,J_2))
=> ( ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,I)))
=> hBOOL(hAPP_nat_bool(Pa,I)) ) )
=> hBOOL(hAPP_nat_bool(Pa,I_2)) ) ) ) ).
tff(fact_1045_ex__least__nat__le,axiom,
! [N: nat,Pa: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(Pa,N))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N))
& ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2))
=> ~ hBOOL(hAPP_nat_bool(Pa,I)) )
& hBOOL(hAPP_nat_bool(Pa,K_2)) ) ) ) ).
tff(fact_1046_less__imp__Suc__add,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> ? [K_2: nat] : N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)) ) ).
tff(fact_1047_less__imp__add__positive,axiom,
! [I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
& ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_2) = J ) ) ) ).
tff(fact_1048_gr0__implies__Suc,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ? [M_2: nat] : N_1 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_1049_zero__induct__lemma,axiom,
! [I_2: nat,Pa: fun_nat_bool,K: nat] :
( hBOOL(hAPP_nat_bool(Pa,K))
=> ( ! [N_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_2)))
=> hBOOL(hAPP_nat_bool(Pa,N_2)) )
=> hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K),I_2))) ) ) ).
tff(fact_1050_Suc__le__D,axiom,
! [N_1: nat,M_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_3))
=> ? [M_2: nat] : M_3 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_1051_lessE,axiom,
! [I_1: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1))
=> ( ( K_1 != hAPP_nat_nat(suc,I_1) )
=> ~ ! [J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> ( K_1 != hAPP_nat_nat(suc,J_1) ) ) ) ) ).
tff(fact_1052_Suc__lessE,axiom,
! [I_1: nat,K_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_1)),K_1))
=> ~ ! [J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1))
=> ( K_1 != hAPP_nat_nat(suc,J_1) ) ) ) ).
tff(fact_1053_not0__implies__Suc,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
=> ? [M_2: nat] : N_1 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_1054_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] : Y != hAPP_nat_nat(suc,Nat) ) ).
tff(fact_1055_zero__induct,axiom,
! [Pa: fun_nat_bool,K: nat] :
( hBOOL(hAPP_nat_bool(Pa,K))
=> ( ! [N_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_2)))
=> hBOOL(hAPP_nat_bool(Pa,N_2)) )
=> hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) ) ) ).
tff(fact_1056_nat__induct,axiom,
! [N: nat,Pa: fun_nat_bool] :
( hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
=> ( ! [N_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,N_2))
=> hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_2))) )
=> hBOOL(hAPP_nat_bool(Pa,N)) ) ) ).
tff(fact_1057_com_Osize_I4_J,axiom,
! [Com1_2: com,Com2_2: com] : com_size(semi(Com1_2,Com2_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,com_size(Com1_2)),com_size(Com2_2))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1058_com_Osize_I12_J,axiom,
! [Com1_2: com,Com2_2: com] : size_size_com(semi(Com1_2,Com2_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_size_com(Com1_2)),size_size_com(Com2_2))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1059_com_Osize_I9_J,axiom,
size_size_com(skip) = zero_zero_nat ).
tff(fact_1060_com_Osize_I1_J,axiom,
com_size(skip) = zero_zero_nat ).
tff(fact_1061_com_Osize_I5_J,axiom,
! [Fun_1: fun_state_bool,Com1: com,Com2: com] : com_size(cond(Fun_1,Com1,Com2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,com_size(Com1)),com_size(Com2))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1062_com_Osimps_I45_J,axiom,
! [Fun: fun_state_bool,Com1_1: com,Com2_1: com,Com1: com,Com2: com] : cond(Fun,Com1_1,Com2_1) != semi(Com1,Com2) ).
tff(fact_1063_com_Osimps_I44_J,axiom,
! [Com1: com,Com2: com,Fun: fun_state_bool,Com1_1: com,Com2_1: com] : semi(Com1,Com2) != cond(Fun,Com1_1,Com2_1) ).
tff(fact_1064_com_Osimps_I15_J,axiom,
! [Fun: fun_state_bool,Com1_1: com,Com2_1: com] : cond(Fun,Com1_1,Com2_1) != skip ).
tff(fact_1065_com_Osimps_I14_J,axiom,
! [Fun: fun_state_bool,Com1_1: com,Com2_1: com] : skip != cond(Fun,Com1_1,Com2_1) ).
tff(fact_1066_com_Osimps_I4_J,axiom,
! [Fun_1: fun_state_bool,Com1: com,Com2: com,Fun: fun_state_bool,Com1_1: com,Com2_1: com] :
( ( cond(Fun_1,Com1,Com2) = cond(Fun,Com1_1,Com2_1) )
<=> ( ( Fun_1 = Fun )
& ( Com1 = Com1_1 )
& ( Com2 = Com2_1 ) ) ) ).
tff(fact_1067_com_Osize_I13_J,axiom,
! [Fun_1: fun_state_bool,Com1: com,Com2: com] : size_size_com(cond(Fun_1,Com1,Com2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_size_com(Com1)),size_size_com(Com2))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1068_com_Osize_I11_J,axiom,
! [Loc: loc,Fun_1: fun_state_nat,Com_1: com] : size_size_com(local(Loc,Fun_1,Com_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_size_com(Com_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1069_com_Osize_I3_J,axiom,
! [Loc: loc,Fun_1: fun_state_nat,Com_1: com] : com_size(local(Loc,Fun_1,Com_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,com_size(Com_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1070_com_Osimps_I36_J,axiom,
! [Loc: loc,Fun_1: fun_state_nat,Com_1: com,Fun: fun_state_bool,Com1_1: com,Com2_1: com] : local(Loc,Fun_1,Com_1) != cond(Fun,Com1_1,Com2_1) ).
tff(fact_1071_com_Osimps_I37_J,axiom,
! [Fun: fun_state_bool,Com1_1: com,Com2_1: com,Loc: loc,Fun_1: fun_state_nat,Com_1: com] : cond(Fun,Com1_1,Com2_1) != local(Loc,Fun_1,Com_1) ).
tff(fact_1072_com_Osimps_I2_J,axiom,
! [Loc: loc,Fun_1: fun_state_nat,Com_1: com,Loc_1: loc,Fun: fun_state_nat,Com: com] :
( ( local(Loc,Fun_1,Com_1) = local(Loc_1,Fun,Com) )
<=> ( ( Loc = Loc_1 )
& ( Fun_1 = Fun )
& ( Com_1 = Com ) ) ) ).
tff(fact_1073_com_Osimps_I10_J,axiom,
! [Loc_1: loc,Fun: fun_state_nat,Com: com] : skip != local(Loc_1,Fun,Com) ).
tff(fact_1074_com_Osimps_I11_J,axiom,
! [Loc_1: loc,Fun: fun_state_nat,Com: com] : local(Loc_1,Fun,Com) != skip ).
tff(fact_1075_com_Osimps_I35_J,axiom,
! [Com1_1: com,Com2_1: com,Loc: loc,Fun_1: fun_state_nat,Com_1: com] : semi(Com1_1,Com2_1) != local(Loc,Fun_1,Com_1) ).
tff(fact_1076_com_Osimps_I34_J,axiom,
! [Loc: loc,Fun_1: fun_state_nat,Com_1: com,Com1_1: com,Com2_1: com] : local(Loc,Fun_1,Com_1) != semi(Com1_1,Com2_1) ).
tff(fact_1077_com_Osize_I6_J,axiom,
! [Fun_1: fun_state_bool,Com_1: com] : com_size(while(Fun_1,Com_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,com_size(Com_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1078_com_Osize_I14_J,axiom,
! [Fun_1: fun_state_bool,Com_1: com] : size_size_com(while(Fun_1,Com_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_size_com(Com_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_1079_com_Osimps_I39_J,axiom,
! [Fun: fun_state_bool,Com: com,Loc: loc,Fun_1: fun_state_nat,Com_1: com] : while(Fun,Com) != local(Loc,Fun_1,Com_1) ).
tff(fact_1080_com_Osimps_I38_J,axiom,
! [Loc: loc,Fun_1: fun_state_nat,Com_1: com,Fun: fun_state_bool,Com: com] : local(Loc,Fun_1,Com_1) != while(Fun,Com) ).
tff(fact_1081_com_Osimps_I47_J,axiom,
! [Fun: fun_state_bool,Com: com,Com1: com,Com2: com] : while(Fun,Com) != semi(Com1,Com2) ).
tff(fact_1082_com_Osimps_I46_J,axiom,
! [Com1: com,Com2: com,Fun: fun_state_bool,Com: com] : semi(Com1,Com2) != while(Fun,Com) ).
tff(fact_1083_com_Osimps_I17_J,axiom,
! [Fun: fun_state_bool,Com: com] : while(Fun,Com) != skip ).
tff(fact_1084_com_Osimps_I16_J,axiom,
! [Fun: fun_state_bool,Com: com] : skip != while(Fun,Com) ).
tff(fact_1085_com_Osimps_I5_J,axiom,
! [Fun_1: fun_state_bool,Com_1: com,Fun: fun_state_bool,Com: com] :
( ( while(Fun_1,Com_1) = while(Fun,Com) )
<=> ( ( Fun_1 = Fun )
& ( Com_1 = Com ) ) ) ).
tff(fact_1086_com_Osimps_I53_J,axiom,
! [Fun: fun_state_bool,Com: com,Fun_1: fun_state_bool,Com1: com,Com2: com] : while(Fun,Com) != cond(Fun_1,Com1,Com2) ).
tff(fact_1087_com_Osimps_I52_J,axiom,
! [Fun_1: fun_state_bool,Com1: com,Com2: com,Fun: fun_state_bool,Com: com] : cond(Fun_1,Com1,Com2) != while(Fun,Com) ).
tff(fact_1088_expand__Suc,axiom,
! [V: int] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V)))
=> ( hAPP_int_nat(number_number_of_nat,V) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),one_one_nat)) ) ) ).
tff(fact_1089_min__Suc__Suc,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M_1),N_1)) ).
tff(fact_1090_min__diff,axiom,
! [M_1: nat,I_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),I_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),I_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M_1),N_1)),I_1) ).
tff(fact_1091_min__0L,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N_1) = zero_zero_nat ).
tff(fact_1092_min__0R,axiom,
! [N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_1),zero_zero_nat) = zero_zero_nat ).
tff(fact_1093_plus__numeral__code_I9_J,axiom,
! [V: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V)),hAPP_int_int(number_number_of_int,W_1)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),W_1)) ).
tff(fact_1094_times__numeral__code_I5_J,axiom,
! [V: int,W_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V)),hAPP_int_int(number_number_of_int,W_1)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V),W_1)) ).
tff(fact_1095_less__number__of__int__code,axiom,
! [K: int,L: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K)),hAPP_int_int(number_number_of_int,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).
tff(fact_1096_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K)),hAPP_int_int(number_number_of_int,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).
tff(fact_1097_zmult__1,axiom,
! [Z: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z) = Z ).
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_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_1100_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_1101_zle__trans,axiom,
! [K_1: int,I_1: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1)) ) ) ).
tff(fact_1102_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_1103_zle__refl,axiom,
! [W_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),W_1)) ).
tff(fact_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_zadd__left__mono,axiom,
! [K_1: int,I_1: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),I_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),J))) ) ).
tff(fact_1114_zadd__strict__right__mono,axiom,
! [K_1: int,I_1: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_1),K_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J),K_1))) ) ).
tff(fact_1115_zadd__zless__mono,axiom,
! [Z_3: 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_3),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_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),Z))) ) ) ).
tff(fact_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
tff(fact_1123_pos__zmult__eq__1__iff,axiom,
! [N: int,M: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N) = one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
tff(fact_1124_int__one__le__iff__zero__less,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) ) ).
tff(fact_1125_less__bin__lemma,axiom,
! [K: int,L: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),L)),zero_zero_int)) ) ).
tff(fact_1126_zmult__zless__mono2,axiom,
! [K_1: int,I_1: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),J))) ) ) ).
tff(fact_1127_zadd__0__right,axiom,
! [Z: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),zero_zero_int) = Z ).
tff(fact_1128_zadd__0,axiom,
! [Z: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z) = Z ).
tff(fact_1129_le__imp__0__less,axiom,
! [Z: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).
tff(fact_1130_odd__less__0,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)),Z_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) ) ).
tff(fact_1131_odd__nonzero,axiom,
! [Z: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).
tff(fact_1132_zdiv__mono2__neg__lemma,axiom,
! [B_1: int,Q_2: int,R_2: int,B: int,Q_1: int,R_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_2)),R_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),Q_1)),R_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),Q_1)),R_1)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_2)) ) ) ) ) ) ) ).
tff(fact_1133_unique__quotient__lemma__neg,axiom,
! [B_1: int,Q_1: int,R_1: int,Q_2: int,R_2: 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,B_1),Q_1)),R_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_2)),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_1)) ) ) ) ) ).
tff(fact_1134_number__of__is__id,axiom,
! [K_1: int] : hAPP_int_int(number_number_of_int,K_1) = K_1 ).
tff(fact_1135_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
tff(fact_1136_self__quotient__aux2,axiom,
! [R_2: int,Q_2: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),Q_2)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),one_one_int)) ) ) ) ).
tff(fact_1137_self__quotient__aux1,axiom,
! [R_2: int,Q_2: int,A: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))
=> ( ( A = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),Q_2)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_2)) ) ) ) ).
tff(fact_1138_q__pos__lemma,axiom,
! [B: int,Q_1: int,R_1: int] :
( 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,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),Q_1)),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).
tff(fact_1139_q__neg__lemma,axiom,
! [B: int,Q_1: int,R_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),Q_1)),R_1)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).
tff(fact_1140_unique__quotient__lemma,axiom,
! [B_1: int,Q_1: int,R_1: int,Q_2: int,R_2: 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,B_1),Q_1)),R_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_2)),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_2)) ) ) ) ) ).
tff(fact_1141_zdiv__mono2__lemma,axiom,
! [B_1: int,Q_2: int,R_2: int,B: int,Q_1: int,R_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_2)),R_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),Q_1)),R_1) )
=> ( 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,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),Q_1)),R_1)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),B_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_1)) ) ) ) ) ) ) ).
tff(fact_1142_int__less__induct,axiom,
! [Pa: fun_int_bool,I_2: int,K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K))
=> ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),one_one_int)))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),K))
=> ( hBOOL(hAPP_int_bool(Pa,I))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).
tff(fact_1143_int__le__induct,axiom,
! [Pa: fun_int_bool,I_2: int,K: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K))
=> ( hBOOL(hAPP_int_bool(Pa,K))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
=> ( hBOOL(hAPP_int_bool(Pa,I))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).
tff(fact_1144_int__gr__induct,axiom,
! [Pa: fun_int_bool,K: int,I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_2))
=> ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),one_one_int)))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I))
=> ( hBOOL(hAPP_int_bool(Pa,I))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).
tff(fact_1145_int__ge__induct,axiom,
! [Pa: fun_int_bool,K: int,I_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_2))
=> ( hBOOL(hAPP_int_bool(Pa,K))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
=> ( hBOOL(hAPP_int_bool(Pa,I))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).
tff(fact_1146_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_1147_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_1148_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_1149_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_1150_decr__mult__lemma,axiom,
! [K: int,Pa: fun_int_bool,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_2: int] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_2: int] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) ) ) ) ).
tff(fact_1151_incr__mult__lemma,axiom,
! [K: int,Pa: fun_int_bool,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_2: int] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_2: int] :
( hBOOL(hAPP_int_bool(Pa,X_2))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) ) ) ) ).
tff(fact_1152_imp__le__cong,axiom,
! [P_1: bool,Pa: bool,X_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(Pa)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(Pa) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(P_1) ) ) ) ).
tff(fact_1153_conj__le__cong,axiom,
! [P_1: bool,Pa: bool,X_1: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> ( hBOOL(Pa)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(Pa) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(P_1) ) ) ) ).
tff(fact_1154_int__induct,axiom,
! [I_2: int,Pa: fun_int_bool,K: int] :
( hBOOL(hAPP_int_bool(Pa,K))
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
=> ( hBOOL(hAPP_int_bool(Pa,I))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
=> ( ! [I: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
=> ( hBOOL(hAPP_int_bool(Pa,I))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
=> hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).
tff(fact_1155_plusinfinity,axiom,
! [Pa: fun_int_bool,P_1: fun_int_bool,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_2: int,K_2: int] :
( hBOOL(hAPP_int_bool(P_1,X_2))
<=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),D_1)))) )
=> ( ? [Z_2: int] :
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),X_2))
=> ( hBOOL(hAPP_int_bool(Pa,X_2))
<=> hBOOL(hAPP_int_bool(P_1,X_2)) ) )
=> ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
=> ? [X1: int] : hBOOL(hAPP_int_bool(Pa,X1)) ) ) ) ) ).
tff(fact_1156_minusinfinity,axiom,
! [Pa: fun_int_bool,P1: fun_int_bool,D_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_2: int,K_2: int] :
( hBOOL(hAPP_int_bool(P1,X_2))
<=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),D_1)))) )
=> ( ? [Z_2: int] :
! [X_2: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z_2))
=> ( hBOOL(hAPP_int_bool(Pa,X_2))
<=> hBOOL(hAPP_int_bool(P1,X_2)) ) )
=> ( ? [X1: int] : hBOOL(hAPP_int_bool(P1,X1))
=> ? [X1: int] : hBOOL(hAPP_int_bool(Pa,X1)) ) ) ) ) ).
tff(fact_1157_tsub__def,axiom,
! [Y: int,X: int] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = zero_zero_int ) ) ) ).
tff(fact_1158_Nat__Transfer_Otransfer__nat__int__function__closures_I3_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(nat_tsub,X),Y))) ) ) ).
tff(fact_1159_tsub__eq,axiom,
! [Y: int,X: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> ( hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) ) ).
tff(fact_1160_incr__lemma,axiom,
! [Z: int,X: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).
tff(fact_1161_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,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(minus_minus_nat,X),Y)) ).
tff(fact_1162_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_1163_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_1164_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_1165_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
tff(fact_1166_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
tff(fact_1167_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_1168_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_1169_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_1170_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_1171_int__Suc,axiom,
! [M_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_1)) ).
tff(fact_1172_int__Suc0__eq__1,axiom,
hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).
tff(fact_1173_zless__iff__Suc__zadd,axiom,
! [W: int,Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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,hAPP_nat_nat(suc,N_2))) ) ).
tff(fact_1174_zle__int,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_1175_zless__int,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_1176_zmult__int,axiom,
! [M_1: nat,N_1: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) ).
tff(fact_1177_int__mult,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) ).
tff(fact_1178_zadd__int__left,axiom,
! [M_1: nat,N_1: nat,Z: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N_1)),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_1),N_1))),Z) ).
tff(fact_1179_zadd__int,axiom,
! [M_1: nat,N_1: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)) ).
tff(fact_1180_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_1181_abs__int__eq,axiom,
! [M_1: nat] : hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,M_1)) = hAPP_nat_int(semiri1621563631at_int,M_1) ).
tff(fact_1182_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
tff(fact_1183_int__eq__0__conv,axiom,
! [N: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,N) = zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
tff(fact_1184_int__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
tff(fact_1185_int__less__0__conv,axiom,
! [K_1: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_1)),zero_zero_int)) ).
tff(fact_1186_zero__zle__int,axiom,
! [N_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) ).
tff(fact_1187_int__le__0__conv,axiom,
! [N: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N)),zero_zero_int))
<=> ( N = zero_zero_nat ) ) ).
tff(fact_1188_zabs__less__one__iff,axiom,
! [Z_1: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Z_1)),one_one_int))
<=> ( Z_1 = zero_zero_int ) ) ).
tff(fact_1189_abs__zmult__eq__1,axiom,
! [M_1: int,N_1: int] :
( ( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N_1)) = one_one_int )
=> ( hAPP_int_int(abs_abs_int,M_1) = one_one_int ) ) ).
tff(fact_1190_zdiff__int,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) ) ) ).
tff(fact_1191_zero__less__int__conv,axiom,
! [N: nat] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
tff(fact_1192_zmult__zless__mono2__lemma,axiom,
! [K_1: nat,I_1: int,J: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_1)),I_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_1)),J))) ) ) ).
tff(fact_1193_zdiff__int__split,axiom,
! [Pa: fun_int_bool,X_1: nat,Y_1: nat] :
( hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_1))))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
=> hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_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))
=> hBOOL(hAPP_int_bool(Pa,zero_zero_int)) ) ) ) ).
tff(fact_1194_decr__lemma,axiom,
! [X: int,Z: int,D: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).
tff(fact_1195_int__val__lemma,axiom,
! [K: int,F: fun_nat_int,N: nat] :
( ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,N)))
=> ? [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
& ( hAPP_nat_int(F,I) = K ) ) ) ) ) ).
tff(fact_1196_nat0__intermed__int__val,axiom,
! [K: int,F: fun_nat_int,N: nat] :
( ! [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,N)))
=> ? [I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
& ( hAPP_nat_int(F,I) = K ) ) ) ) ) ).
tff(fact_1197_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_1198_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_1199_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_int(semiri1621563631at_int,M) = hAPP_nat_int(semiri1621563631at_int,N) )
<=> ( M = N ) ) ).
%----Helper facts (40)
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_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_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
! [P: bool,Q: int] : hAPP_int_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(cOMBK_bool_nat(P),Q) = P ).
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(cOMBC_int_int_bool(P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_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(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(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(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(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(cOMBS_nat_bool_bool(P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).
tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com,axiom,
! [X: hoare_363350478_state,Y: hoare_363350478_state] :
( ~ hBOOL(hAPP_H1645464494e_bool(hAPP_H705653759e_bool(fequal1026769423_state,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com,axiom,
! [X: hoare_363350478_state,Y: hoare_363350478_state] :
( ( X != Y )
| hBOOL(hAPP_H1645464494e_bool(hAPP_H705653759e_bool(fequal1026769423_state,X),Y)) ) ).
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(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(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(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_044,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____wdcyoqcnvy__,axiom,
! [P: bool,Q: hoare_363350478_state] : hAPP_H1645464494e_bool(cOMBK_574827367_state(P),Q) = P ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel,axiom,
! [P: fun_bool_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(cOMBB_2059354034_state(P),Q),R) = hAPP_bool_bool(P,hAPP_H1645464494e_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,axiom,
! [P: fun_Ho544647106l_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(cOMBS_56025283l_bool(P),Q),R) = hAPP_bool_bool(hAPP_H1173243534l_bool(P,R),hAPP_H1645464494e_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_045,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : hAPP_H1173243534l_bool(hAPP_f1422032445l_bool(cOMBB_877416532_state(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H1645464494e_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,axiom,
! [P: fun_Ho424838195e_bool,Q: hoare_363350478_state,R: hoare_363350478_state] : hAPP_H1645464494e_bool(hAPP_H705653759e_bool(cOMBC_13083556e_bool(P),Q),R) = hAPP_H1645464494e_bool(hAPP_H705653759e_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com___046,axiom,
! [P: fun_Ho177530809l_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(cOMBC_1415895650l_bool(P),Q),R) = hAPP_f568432088l_bool(hAPP_H40373975l_bool(P,R),Q) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(bot_bo39439654e_bool),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_Mirabelle_MGT(c)),bot_bo39439654e_bool))) ).
tff(conj_1,hypothesis,
hBOOL(hAPP_f568432088l_bool(hoare_501514600_state(bot_bo39439654e_bool),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(p,c,q)),bot_bo39439654e_bool))) ).
tff(conj_2,conjecture,
hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(bot_bo39439654e_bool),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hoare_608330478_state(p,c,q)),bot_bo39439654e_bool))) ).
%------------------------------------------------------------------------------