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_thf_l327 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 1312 ( 285 unt; 106 typ; 0 def)
% Number of atoms : 3808 ( 868 equ; 15 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10411 ( 371 ~; 63 |; 127 &;8191 @)
% ( 266 <=>;1383 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 1631 (1631 >; 0 *; 0 +; 0 <<)
% Number of symbols : 108 ( 100 usr; 15 con; 0-4 aty)
% ( 0 !!; 10 ??; 0 @@+; 0 @@-)
% Number of variables : 3228 ( 12 ^;3152 !; 64 ?;3228 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:52:43
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Oloc,type,
loc: $tType ).
thf(ty_ty_tc__Com__Ostate,type,
state: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostate_J,type,
hoare_1708887482_state: $tType ).
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
%----Explicit typings (103)
thf(sy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____nqhfsdfvyv__O,type,
big_se2144116836_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > ( ( hoare_1708887482_state > $o ) > hoare_1708887482_state ) > $o ).
thf(sy_c_Big__Operators_Osemilattice__big_000tc__Int__Oint,type,
big_se913005884ig_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
thf(sy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat,type,
big_se275732192ig_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(sy_c_Com_Ocom_OCond,type,
cond: ( state > $o ) > com > com > com ).
thf(sy_c_Com_Ocom_OLocal,type,
local: loc > ( state > nat ) > com > com ).
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_OSemi,type,
semi: com > com > com ).
thf(sy_c_Com_Ocom_OWhile,type,
while: ( state > $o ) > com > com ).
thf(sy_c_Com_Ocom_Ocom__size,type,
com_size: com > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com,type,
finite185771624_state: ( hoare_1708887482_state > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
finite_card_int: ( int > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
finite_card_nat: ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____nqhfsdfvyv__Ot,type,
finite1816785523_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000tc__Int__Oint,type,
finite1973466193nt_int: ( int > int > int ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Nat__Onat_000tc__Nat__Onat,type,
finite2130160977at_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otrip,type,
finite662762081tate_o: ( hoare_1708887482_state > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Int__Oint_000_062_Itc__Int__Oint_M_Eo_J,type,
finite175163825_int_o: ( int > ( int > $o ) > int > $o ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
finite389864113_nat_o: ( nat > ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__C,type,
finite1625599783_state: ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
finite_finite_int: ( int > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc_,type,
finite850270954_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Int__Oint,type,
finite_fold1Set_int: ( int > int > int ) > ( int > $o ) > int > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Nat__Onat,type,
finite_fold1Set_nat: ( nat > nat > nat ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Co,type,
finite39664218_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_state ).
thf(sy_c_Finite__Set_Ofold1_000tc__Int__Oint,type,
finite_fold1_int: ( int > int > int ) > ( int > $o ) > int ).
thf(sy_c_Finite__Set_Ofold1_000tc__Nat__Onat,type,
finite_fold1_nat: ( nat > nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com,type,
finite309095018_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > hoare_1708887482_state > ( hoare_1708887482_state > $o ) > hoare_1708887482_state ).
thf(sy_c_Finite__Set_Ofold_000tc__Int__Oint_000tc__Int__Oint,type,
finite_fold_int_int: ( int > int > int ) > int > ( int > $o ) > int ).
thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_I,type,
finite2042206440_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > hoare_1708887482_state > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Int__Oint_000tc__Int__Oint,type,
finite772772422nt_int: ( int > int > int ) > int > ( int > $o ) > int > $o ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__Nat__Onat,type,
finite929467206at_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_,type,
finite1615457021_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > ( ( hoare_1708887482_state > $o ) > hoare_1708887482_state ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,type,
finite1626084323ne_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Ot,type,
finite1347568576_state: ( hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state ) > ( ( hoare_1708887482_state > $o ) > hoare_1708887482_state ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,type,
finite1432773856em_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__,type,
minus_2056855718tate_o: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Int__Oint_M_Eo_J,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_Eo,type,
minus_minus_o: $o > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_OMGT,type,
hoare_Mirabelle_MGT: com > hoare_1708887482_state ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Ohoare__derivs_000tc__Com__Ostate,type,
hoare_90032982_state: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Ohoare__valids_000tc__Com__Ostate,type,
hoare_496444244_state: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple_Otriple_000tc__Com__Ostate,type,
hoare_858012674_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1708887482_state ).
thf(sy_c_If_000tc__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
semiri1621563631at_int: nat > int ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Com__Ocom,type,
size_size_com: com > nat ).
thf(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: int > int > int ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__O,type,
bot_bo19817387tate_o: hoare_1708887482_state > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__,type,
ord_le14920683tate_o: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Hoare____Mirabelle____nq,type,
ord_le1728773982te_o_o: ( ( hoare_1708887482_state > $o ) > $o ) > ( ( hoare_1708887482_state > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____nqhfsdfv,type,
ord_le777019615tate_o: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Int__Oint_M_Eo_J,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Predicate_OPowp_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__,type,
powp_H189553652_state: ( hoare_1708887482_state > $o ) > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ost,type,
collec1568722789_state: ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_OCollect_000tc__Int__Oint,type,
collect_int: ( int > $o ) > int > $o ).
thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Osta,type,
insert528405184_state: hoare_1708887482_state > ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o ).
thf(sy_c_Set_Oinsert_000tc__Int__Oint,type,
insert_int: int > ( int > $o ) > int > $o ).
thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
insert_nat: nat > ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__O,type,
the_el864710747_state: ( hoare_1708887482_state > $o ) > hoare_1708887482_state ).
thf(sy_c_Set_Othe__elem_000tc__Int__Oint,type,
the_elem_int: ( int > $o ) > int ).
thf(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
the_elem_nat: ( nat > $o ) > nat ).
thf(sy_c_member_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_Itc__Com__Ostate_J,type,
member451959335_state: hoare_1708887482_state > ( hoare_1708887482_state > $o ) > $o ).
thf(sy_c_member_000tc__Int__Oint,type,
member_int: int > ( int > $o ) > $o ).
thf(sy_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_v_P,type,
p: state > state > $o ).
thf(sy_v_Q,type,
q: state > state > $o ).
thf(sy_v_c,type,
c: com ).
%----Relevant facts (1200)
thf(fact_0_empty,axiom,
! [G_21: hoare_1708887482_state > $o] : ( hoare_90032982_state @ G_21 @ bot_bo19817387tate_o ) ).
thf(fact_1_triple_Oinject,axiom,
! [Fun1_2: state > state > $o,Com_4: com,Fun2_2: state > state > $o,Fun1_1: state > state > $o,Com_3: com,Fun2_1: state > state > $o] :
( ( ( hoare_858012674_state @ Fun1_2 @ Com_4 @ Fun2_2 )
= ( hoare_858012674_state @ Fun1_1 @ Com_3 @ Fun2_1 ) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_4 = Com_3 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
thf(fact_2_hoare__sound,axiom,
! [G_20: hoare_1708887482_state > $o,Ts_7: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ G_20 @ Ts_7 )
=> ( hoare_496444244_state @ G_20 @ Ts_7 ) ) ).
thf(fact_3_cut,axiom,
! [G_19: hoare_1708887482_state > $o,G_18: hoare_1708887482_state > $o,Ts_6: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ G_18 @ Ts_6 )
=> ( ( hoare_90032982_state @ G_19 @ G_18 )
=> ( hoare_90032982_state @ G_19 @ Ts_6 ) ) ) ).
thf(fact_4_hoare__derivs_Oinsert,axiom,
! [Ts_5: hoare_1708887482_state > $o,G_17: hoare_1708887482_state > $o,T_2: hoare_1708887482_state] :
( ( hoare_90032982_state @ G_17 @ ( insert528405184_state @ T_2 @ bot_bo19817387tate_o ) )
=> ( ( hoare_90032982_state @ G_17 @ Ts_5 )
=> ( hoare_90032982_state @ G_17 @ ( insert528405184_state @ T_2 @ Ts_5 ) ) ) ) ).
thf(fact_5_derivs__insertD,axiom,
! [G_16: hoare_1708887482_state > $o,T_1: hoare_1708887482_state,Ts_4: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ G_16 @ ( insert528405184_state @ T_1 @ Ts_4 ) )
=> ( ( hoare_90032982_state @ G_16 @ ( insert528405184_state @ T_1 @ bot_bo19817387tate_o ) )
& ( hoare_90032982_state @ G_16 @ Ts_4 ) ) ) ).
thf(fact_6_conseq2,axiom,
! [Q_13: state > state > $o,G_15: hoare_1708887482_state > $o,P_25: state > state > $o,C_79: com,Q_12: state > state > $o] :
( ( hoare_90032982_state @ G_15 @ ( insert528405184_state @ ( hoare_858012674_state @ P_25 @ C_79 @ Q_12 ) @ bot_bo19817387tate_o ) )
=> ( ! [Z_30: state,S_1: state] :
( ( Q_12 @ Z_30 @ S_1 )
=> ( Q_13 @ Z_30 @ S_1 ) )
=> ( hoare_90032982_state @ G_15 @ ( insert528405184_state @ ( hoare_858012674_state @ P_25 @ C_79 @ Q_13 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_7_conseq1,axiom,
! [P_24: state > state > $o,G_14: hoare_1708887482_state > $o,P_23: state > state > $o,C_78: com,Q_11: state > state > $o] :
( ( hoare_90032982_state @ G_14 @ ( insert528405184_state @ ( hoare_858012674_state @ P_23 @ C_78 @ Q_11 ) @ bot_bo19817387tate_o ) )
=> ( ! [Z_30: state,S_1: state] :
( ( P_24 @ Z_30 @ S_1 )
=> ( P_23 @ Z_30 @ S_1 ) )
=> ( hoare_90032982_state @ G_14 @ ( insert528405184_state @ ( hoare_858012674_state @ P_24 @ C_78 @ Q_11 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_8_insertE,axiom,
! [A_251: int,B_141: int,A_250: int > $o] :
( ( member_int @ A_251 @ ( insert_int @ B_141 @ A_250 ) )
=> ( ( A_251 != B_141 )
=> ( member_int @ A_251 @ A_250 ) ) ) ).
thf(fact_9_insertE,axiom,
! [A_251: nat,B_141: nat,A_250: nat > $o] :
( ( member_nat @ A_251 @ ( insert_nat @ B_141 @ A_250 ) )
=> ( ( A_251 != B_141 )
=> ( member_nat @ A_251 @ A_250 ) ) ) ).
thf(fact_10_insertE,axiom,
! [A_251: hoare_1708887482_state,B_141: hoare_1708887482_state,A_250: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_251 @ ( insert528405184_state @ B_141 @ A_250 ) )
=> ( ( A_251 != B_141 )
=> ( member451959335_state @ A_251 @ A_250 ) ) ) ).
thf(fact_11_insertCI,axiom,
! [B_140: int,A_249: int,B_139: int > $o] :
( ( ~ ( member_int @ A_249 @ B_139 )
=> ( A_249 = B_140 ) )
=> ( member_int @ A_249 @ ( insert_int @ B_140 @ B_139 ) ) ) ).
thf(fact_12_insertCI,axiom,
! [B_140: nat,A_249: nat,B_139: nat > $o] :
( ( ~ ( member_nat @ A_249 @ B_139 )
=> ( A_249 = B_140 ) )
=> ( member_nat @ A_249 @ ( insert_nat @ B_140 @ B_139 ) ) ) ).
thf(fact_13_insertCI,axiom,
! [B_140: hoare_1708887482_state,A_249: hoare_1708887482_state,B_139: hoare_1708887482_state > $o] :
( ( ~ ( member451959335_state @ A_249 @ B_139 )
=> ( A_249 = B_140 ) )
=> ( member451959335_state @ A_249 @ ( insert528405184_state @ B_140 @ B_139 ) ) ) ).
thf(fact_14_conseq12,axiom,
! [Q_10: state > state > $o,P_22: state > state > $o,G_13: hoare_1708887482_state > $o,P_21: state > state > $o,C_77: com,Q_9: state > state > $o] :
( ( hoare_90032982_state @ G_13 @ ( insert528405184_state @ ( hoare_858012674_state @ P_21 @ C_77 @ Q_9 ) @ bot_bo19817387tate_o ) )
=> ( ! [Z_30: state,S_1: state] :
( ( P_22 @ Z_30 @ S_1 )
=> ! [S_2: state] :
( ! [Z_31: state] :
( ( P_21 @ Z_31 @ S_1 )
=> ( Q_9 @ Z_31 @ S_2 ) )
=> ( Q_10 @ Z_30 @ S_2 ) ) )
=> ( hoare_90032982_state @ G_13 @ ( insert528405184_state @ ( hoare_858012674_state @ P_22 @ C_77 @ Q_10 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_15_emptyE,axiom,
! [A_248: int] :
~ ( member_int @ A_248 @ bot_bot_int_o ) ).
thf(fact_16_emptyE,axiom,
! [A_248: nat] :
~ ( member_nat @ A_248 @ bot_bot_nat_o ) ).
thf(fact_17_emptyE,axiom,
! [A_248: hoare_1708887482_state] :
~ ( member451959335_state @ A_248 @ bot_bo19817387tate_o ) ).
thf(fact_18_empty__not__insert,axiom,
! [A_247: int,A_246: int > $o] :
( bot_bot_int_o
!= ( insert_int @ A_247 @ A_246 ) ) ).
thf(fact_19_empty__not__insert,axiom,
! [A_247: nat,A_246: nat > $o] :
( bot_bot_nat_o
!= ( insert_nat @ A_247 @ A_246 ) ) ).
thf(fact_20_empty__not__insert,axiom,
! [A_247: hoare_1708887482_state,A_246: hoare_1708887482_state > $o] :
( bot_bo19817387tate_o
!= ( insert528405184_state @ A_247 @ A_246 ) ) ).
thf(fact_21_insert__not__empty,axiom,
! [A_245: int,A_244: int > $o] :
( ( insert_int @ A_245 @ A_244 )
!= bot_bot_int_o ) ).
thf(fact_22_insert__not__empty,axiom,
! [A_245: nat,A_244: nat > $o] :
( ( insert_nat @ A_245 @ A_244 )
!= bot_bot_nat_o ) ).
thf(fact_23_insert__not__empty,axiom,
! [A_245: hoare_1708887482_state,A_244: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_245 @ A_244 )
!= bot_bo19817387tate_o ) ).
thf(fact_24_singleton__iff,axiom,
! [B_138: int,A_243: int] :
( ( member_int @ B_138 @ ( insert_int @ A_243 @ bot_bot_int_o ) )
<=> ( B_138 = A_243 ) ) ).
thf(fact_25_singleton__iff,axiom,
! [B_138: nat,A_243: nat] :
( ( member_nat @ B_138 @ ( insert_nat @ A_243 @ bot_bot_nat_o ) )
<=> ( B_138 = A_243 ) ) ).
thf(fact_26_singleton__iff,axiom,
! [B_138: hoare_1708887482_state,A_243: hoare_1708887482_state] :
( ( member451959335_state @ B_138 @ ( insert528405184_state @ A_243 @ bot_bo19817387tate_o ) )
<=> ( B_138 = A_243 ) ) ).
thf(fact_27_doubleton__eq__iff,axiom,
! [A_242: int,B_137: int,C_76: int,D_22: int] :
( ( ( insert_int @ A_242 @ ( insert_int @ B_137 @ bot_bot_int_o ) )
= ( insert_int @ C_76 @ ( insert_int @ D_22 @ bot_bot_int_o ) ) )
<=> ( ( ( A_242 = C_76 )
& ( B_137 = D_22 ) )
| ( ( A_242 = D_22 )
& ( B_137 = C_76 ) ) ) ) ).
thf(fact_28_doubleton__eq__iff,axiom,
! [A_242: nat,B_137: nat,C_76: nat,D_22: nat] :
( ( ( insert_nat @ A_242 @ ( insert_nat @ B_137 @ bot_bot_nat_o ) )
= ( insert_nat @ C_76 @ ( insert_nat @ D_22 @ bot_bot_nat_o ) ) )
<=> ( ( ( A_242 = C_76 )
& ( B_137 = D_22 ) )
| ( ( A_242 = D_22 )
& ( B_137 = C_76 ) ) ) ) ).
thf(fact_29_doubleton__eq__iff,axiom,
! [A_242: hoare_1708887482_state,B_137: hoare_1708887482_state,C_76: hoare_1708887482_state,D_22: hoare_1708887482_state] :
( ( ( insert528405184_state @ A_242 @ ( insert528405184_state @ B_137 @ bot_bo19817387tate_o ) )
= ( insert528405184_state @ C_76 @ ( insert528405184_state @ D_22 @ bot_bo19817387tate_o ) ) )
<=> ( ( ( A_242 = C_76 )
& ( B_137 = D_22 ) )
| ( ( A_242 = D_22 )
& ( B_137 = C_76 ) ) ) ) ).
thf(fact_30_equals0D,axiom,
! [A_241: int,A_240: int > $o] :
( ( A_240 = bot_bot_int_o )
=> ~ ( member_int @ A_241 @ A_240 ) ) ).
thf(fact_31_equals0D,axiom,
! [A_241: nat,A_240: nat > $o] :
( ( A_240 = bot_bot_nat_o )
=> ~ ( member_nat @ A_241 @ A_240 ) ) ).
thf(fact_32_equals0D,axiom,
! [A_241: hoare_1708887482_state,A_240: hoare_1708887482_state > $o] :
( ( A_240 = bot_bo19817387tate_o )
=> ~ ( member451959335_state @ A_241 @ A_240 ) ) ).
thf(fact_33_Collect__empty__eq,axiom,
! [P_20: int > $o] :
( ( ( collect_int @ P_20 )
= bot_bot_int_o )
<=> ! [X_1: int] :
~ ( P_20 @ X_1 ) ) ).
thf(fact_34_Collect__empty__eq,axiom,
! [P_20: nat > $o] :
( ( ( collect_nat @ P_20 )
= bot_bot_nat_o )
<=> ! [X_1: nat] :
~ ( P_20 @ X_1 ) ) ).
thf(fact_35_Collect__empty__eq,axiom,
! [P_20: hoare_1708887482_state > $o] :
( ( ( collec1568722789_state @ P_20 )
= bot_bo19817387tate_o )
<=> ! [X_1: hoare_1708887482_state] :
~ ( P_20 @ X_1 ) ) ).
thf(fact_36_empty__iff,axiom,
! [C_75: int] :
~ ( member_int @ C_75 @ bot_bot_int_o ) ).
thf(fact_37_empty__iff,axiom,
! [C_75: nat] :
~ ( member_nat @ C_75 @ bot_bot_nat_o ) ).
thf(fact_38_empty__iff,axiom,
! [C_75: hoare_1708887482_state] :
~ ( member451959335_state @ C_75 @ bot_bo19817387tate_o ) ).
thf(fact_39_empty__Collect__eq,axiom,
! [P_19: int > $o] :
( ( bot_bot_int_o
= ( collect_int @ P_19 ) )
<=> ! [X_1: int] :
~ ( P_19 @ X_1 ) ) ).
thf(fact_40_empty__Collect__eq,axiom,
! [P_19: nat > $o] :
( ( bot_bot_nat_o
= ( collect_nat @ P_19 ) )
<=> ! [X_1: nat] :
~ ( P_19 @ X_1 ) ) ).
thf(fact_41_empty__Collect__eq,axiom,
! [P_19: hoare_1708887482_state > $o] :
( ( bot_bo19817387tate_o
= ( collec1568722789_state @ P_19 ) )
<=> ! [X_1: hoare_1708887482_state] :
~ ( P_19 @ X_1 ) ) ).
thf(fact_42_ex__in__conv,axiom,
! [A_239: int > $o] :
( ? [X_1: int] : ( member_int @ X_1 @ A_239 )
<=> ( A_239 != bot_bot_int_o ) ) ).
thf(fact_43_ex__in__conv,axiom,
! [A_239: nat > $o] :
( ? [X_1: nat] : ( member_nat @ X_1 @ A_239 )
<=> ( A_239 != bot_bot_nat_o ) ) ).
thf(fact_44_ex__in__conv,axiom,
! [A_239: hoare_1708887482_state > $o] :
( ? [X_1: hoare_1708887482_state] : ( member451959335_state @ X_1 @ A_239 )
<=> ( A_239 != bot_bo19817387tate_o ) ) ).
thf(fact_45_all__not__in__conv,axiom,
! [A_238: int > $o] :
( ! [X_1: int] :
~ ( member_int @ X_1 @ A_238 )
<=> ( A_238 = bot_bot_int_o ) ) ).
thf(fact_46_all__not__in__conv,axiom,
! [A_238: nat > $o] :
( ! [X_1: nat] :
~ ( member_nat @ X_1 @ A_238 )
<=> ( A_238 = bot_bot_nat_o ) ) ).
thf(fact_47_all__not__in__conv,axiom,
! [A_238: hoare_1708887482_state > $o] :
( ! [X_1: hoare_1708887482_state] :
~ ( member451959335_state @ X_1 @ A_238 )
<=> ( A_238 = bot_bo19817387tate_o ) ) ).
thf(fact_48_empty__def,axiom,
( bot_bot_int_o
= ( collect_int
@ ^ [X_1: int] : $false ) ) ).
thf(fact_49_empty__def,axiom,
( bot_bot_nat_o
= ( collect_nat
@ ^ [X_1: nat] : $false ) ) ).
thf(fact_50_empty__def,axiom,
( bot_bo19817387tate_o
= ( collec1568722789_state
@ ^ [X_1: hoare_1708887482_state] : $false ) ) ).
thf(fact_51_insert__absorb,axiom,
! [A_237: int,A_236: int > $o] :
( ( member_int @ A_237 @ A_236 )
=> ( ( insert_int @ A_237 @ A_236 )
= A_236 ) ) ).
thf(fact_52_insert__absorb,axiom,
! [A_237: nat,A_236: nat > $o] :
( ( member_nat @ A_237 @ A_236 )
=> ( ( insert_nat @ A_237 @ A_236 )
= A_236 ) ) ).
thf(fact_53_insert__absorb,axiom,
! [A_237: hoare_1708887482_state,A_236: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_237 @ A_236 )
=> ( ( insert528405184_state @ A_237 @ A_236 )
= A_236 ) ) ).
thf(fact_54_insertI2,axiom,
! [B_136: int,A_235: int,B_135: int > $o] :
( ( member_int @ A_235 @ B_135 )
=> ( member_int @ A_235 @ ( insert_int @ B_136 @ B_135 ) ) ) ).
thf(fact_55_insertI2,axiom,
! [B_136: nat,A_235: nat,B_135: nat > $o] :
( ( member_nat @ A_235 @ B_135 )
=> ( member_nat @ A_235 @ ( insert_nat @ B_136 @ B_135 ) ) ) ).
thf(fact_56_insertI2,axiom,
! [B_136: hoare_1708887482_state,A_235: hoare_1708887482_state,B_135: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_235 @ B_135 )
=> ( member451959335_state @ A_235 @ ( insert528405184_state @ B_136 @ B_135 ) ) ) ).
thf(fact_57_insert__ident,axiom,
! [B_134: int > $o,X_109: int,A_234: int > $o] :
( ~ ( member_int @ X_109 @ A_234 )
=> ( ~ ( member_int @ X_109 @ B_134 )
=> ( ( ( insert_int @ X_109 @ A_234 )
= ( insert_int @ X_109 @ B_134 ) )
<=> ( A_234 = B_134 ) ) ) ) ).
thf(fact_58_insert__ident,axiom,
! [B_134: nat > $o,X_109: nat,A_234: nat > $o] :
( ~ ( member_nat @ X_109 @ A_234 )
=> ( ~ ( member_nat @ X_109 @ B_134 )
=> ( ( ( insert_nat @ X_109 @ A_234 )
= ( insert_nat @ X_109 @ B_134 ) )
<=> ( A_234 = B_134 ) ) ) ) ).
thf(fact_59_insert__ident,axiom,
! [B_134: hoare_1708887482_state > $o,X_109: hoare_1708887482_state,A_234: hoare_1708887482_state > $o] :
( ~ ( member451959335_state @ X_109 @ A_234 )
=> ( ~ ( member451959335_state @ X_109 @ B_134 )
=> ( ( ( insert528405184_state @ X_109 @ A_234 )
= ( insert528405184_state @ X_109 @ B_134 ) )
<=> ( A_234 = B_134 ) ) ) ) ).
thf(fact_60_insert__code,axiom,
! [Y_57: int,A_233: int > $o,X_108: int] :
( ( insert_int @ Y_57 @ A_233 @ X_108 )
<=> ( ( Y_57 = X_108 )
| ( A_233 @ X_108 ) ) ) ).
thf(fact_61_insert__code,axiom,
! [Y_57: nat,A_233: nat > $o,X_108: nat] :
( ( insert_nat @ Y_57 @ A_233 @ X_108 )
<=> ( ( Y_57 = X_108 )
| ( A_233 @ X_108 ) ) ) ).
thf(fact_62_insert__code,axiom,
! [Y_57: hoare_1708887482_state,A_233: hoare_1708887482_state > $o,X_108: hoare_1708887482_state] :
( ( insert528405184_state @ Y_57 @ A_233 @ X_108 )
<=> ( ( Y_57 = X_108 )
| ( A_233 @ X_108 ) ) ) ).
thf(fact_63_insert__iff,axiom,
! [A_232: int,B_133: int,A_231: int > $o] :
( ( member_int @ A_232 @ ( insert_int @ B_133 @ A_231 ) )
<=> ( ( A_232 = B_133 )
| ( member_int @ A_232 @ A_231 ) ) ) ).
thf(fact_64_insert__iff,axiom,
! [A_232: nat,B_133: nat,A_231: nat > $o] :
( ( member_nat @ A_232 @ ( insert_nat @ B_133 @ A_231 ) )
<=> ( ( A_232 = B_133 )
| ( member_nat @ A_232 @ A_231 ) ) ) ).
thf(fact_65_insert__iff,axiom,
! [A_232: hoare_1708887482_state,B_133: hoare_1708887482_state,A_231: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_232 @ ( insert528405184_state @ B_133 @ A_231 ) )
<=> ( ( A_232 = B_133 )
| ( member451959335_state @ A_232 @ A_231 ) ) ) ).
thf(fact_66_insert__commute,axiom,
! [X_107: int,Y_56: int,A_230: int > $o] :
( ( insert_int @ X_107 @ ( insert_int @ Y_56 @ A_230 ) )
= ( insert_int @ Y_56 @ ( insert_int @ X_107 @ A_230 ) ) ) ).
thf(fact_67_insert__commute,axiom,
! [X_107: nat,Y_56: nat,A_230: nat > $o] :
( ( insert_nat @ X_107 @ ( insert_nat @ Y_56 @ A_230 ) )
= ( insert_nat @ Y_56 @ ( insert_nat @ X_107 @ A_230 ) ) ) ).
thf(fact_68_insert__commute,axiom,
! [X_107: hoare_1708887482_state,Y_56: hoare_1708887482_state,A_230: hoare_1708887482_state > $o] :
( ( insert528405184_state @ X_107 @ ( insert528405184_state @ Y_56 @ A_230 ) )
= ( insert528405184_state @ Y_56 @ ( insert528405184_state @ X_107 @ A_230 ) ) ) ).
thf(fact_69_insert__absorb2,axiom,
! [X_106: int,A_229: int > $o] :
( ( insert_int @ X_106 @ ( insert_int @ X_106 @ A_229 ) )
= ( insert_int @ X_106 @ A_229 ) ) ).
thf(fact_70_insert__absorb2,axiom,
! [X_106: nat,A_229: nat > $o] :
( ( insert_nat @ X_106 @ ( insert_nat @ X_106 @ A_229 ) )
= ( insert_nat @ X_106 @ A_229 ) ) ).
thf(fact_71_insert__absorb2,axiom,
! [X_106: hoare_1708887482_state,A_229: hoare_1708887482_state > $o] :
( ( insert528405184_state @ X_106 @ ( insert528405184_state @ X_106 @ A_229 ) )
= ( insert528405184_state @ X_106 @ A_229 ) ) ).
thf(fact_72_insert__Collect,axiom,
! [A_228: int,P_18: int > $o] :
( ( insert_int @ A_228 @ ( collect_int @ P_18 ) )
= ( collect_int
@ ^ [U_2: int] : ( (=>) @ ( (~) @ ( U_2 = A_228 ) ) @ ( P_18 @ U_2 ) ) ) ) ).
thf(fact_73_insert__Collect,axiom,
! [A_228: nat,P_18: nat > $o] :
( ( insert_nat @ A_228 @ ( collect_nat @ P_18 ) )
= ( collect_nat
@ ^ [U_2: nat] : ( (=>) @ ( (~) @ ( U_2 = A_228 ) ) @ ( P_18 @ U_2 ) ) ) ) ).
thf(fact_74_insert__Collect,axiom,
! [A_228: hoare_1708887482_state,P_18: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_228 @ ( collec1568722789_state @ P_18 ) )
= ( collec1568722789_state
@ ^ [U_2: hoare_1708887482_state] : ( (=>) @ ( (~) @ ( U_2 = A_228 ) ) @ ( P_18 @ U_2 ) ) ) ) ).
thf(fact_75_insert__compr,axiom,
! [A_227: int,B_132: int > $o] :
( ( insert_int @ A_227 @ B_132 )
= ( collect_int
@ ^ [X_1: int] : ( (|) @ ( X_1 = A_227 ) @ ( member_int @ X_1 @ B_132 ) ) ) ) ).
thf(fact_76_insert__compr,axiom,
! [A_227: nat,B_132: nat > $o] :
( ( insert_nat @ A_227 @ B_132 )
= ( collect_nat
@ ^ [X_1: nat] : ( (|) @ ( X_1 = A_227 ) @ ( member_nat @ X_1 @ B_132 ) ) ) ) ).
thf(fact_77_insert__compr,axiom,
! [A_227: hoare_1708887482_state,B_132: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_227 @ B_132 )
= ( collec1568722789_state
@ ^ [X_1: hoare_1708887482_state] : ( (|) @ ( X_1 = A_227 ) @ ( member451959335_state @ X_1 @ B_132 ) ) ) ) ).
thf(fact_78_insertI1,axiom,
! [A_226: int,B_131: int > $o] : ( member_int @ A_226 @ ( insert_int @ A_226 @ B_131 ) ) ).
thf(fact_79_insertI1,axiom,
! [A_226: nat,B_131: nat > $o] : ( member_nat @ A_226 @ ( insert_nat @ A_226 @ B_131 ) ) ).
thf(fact_80_insertI1,axiom,
! [A_226: hoare_1708887482_state,B_131: hoare_1708887482_state > $o] : ( member451959335_state @ A_226 @ ( insert528405184_state @ A_226 @ B_131 ) ) ).
thf(fact_81_singleton__inject,axiom,
! [A_225: int,B_130: int] :
( ( ( insert_int @ A_225 @ bot_bot_int_o )
= ( insert_int @ B_130 @ bot_bot_int_o ) )
=> ( A_225 = B_130 ) ) ).
thf(fact_82_singleton__inject,axiom,
! [A_225: nat,B_130: nat] :
( ( ( insert_nat @ A_225 @ bot_bot_nat_o )
= ( insert_nat @ B_130 @ bot_bot_nat_o ) )
=> ( A_225 = B_130 ) ) ).
thf(fact_83_singleton__inject,axiom,
! [A_225: hoare_1708887482_state,B_130: hoare_1708887482_state] :
( ( ( insert528405184_state @ A_225 @ bot_bo19817387tate_o )
= ( insert528405184_state @ B_130 @ bot_bo19817387tate_o ) )
=> ( A_225 = B_130 ) ) ).
thf(fact_84_singletonE,axiom,
! [B_129: int,A_224: int] :
( ( member_int @ B_129 @ ( insert_int @ A_224 @ bot_bot_int_o ) )
=> ( B_129 = A_224 ) ) ).
thf(fact_85_singletonE,axiom,
! [B_129: nat,A_224: nat] :
( ( member_nat @ B_129 @ ( insert_nat @ A_224 @ bot_bot_nat_o ) )
=> ( B_129 = A_224 ) ) ).
thf(fact_86_singletonE,axiom,
! [B_129: hoare_1708887482_state,A_224: hoare_1708887482_state] :
( ( member451959335_state @ B_129 @ ( insert528405184_state @ A_224 @ bot_bo19817387tate_o ) )
=> ( B_129 = A_224 ) ) ).
thf(fact_87_the__elem__eq,axiom,
! [X_105: int] :
( ( the_elem_int @ ( insert_int @ X_105 @ bot_bot_int_o ) )
= X_105 ) ).
thf(fact_88_the__elem__eq,axiom,
! [X_105: nat] :
( ( the_elem_nat @ ( insert_nat @ X_105 @ bot_bot_nat_o ) )
= X_105 ) ).
thf(fact_89_the__elem__eq,axiom,
! [X_105: hoare_1708887482_state] :
( ( the_el864710747_state @ ( insert528405184_state @ X_105 @ bot_bo19817387tate_o ) )
= X_105 ) ).
thf(fact_90_bot__apply,axiom,
! [X_104: int] :
( ( bot_bot_int_o @ X_104 )
<=> bot_bot_o ) ).
thf(fact_91_bot__apply,axiom,
! [X_104: nat] :
( ( bot_bot_nat_o @ X_104 )
<=> bot_bot_o ) ).
thf(fact_92_bot__apply,axiom,
! [X_104: hoare_1708887482_state] :
( ( bot_bo19817387tate_o @ X_104 )
<=> bot_bot_o ) ).
thf(fact_93_bot__fun__def,axiom,
! [X_1: int] :
( ( bot_bot_int_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_94_bot__fun__def,axiom,
! [X_1: nat] :
( ( bot_bot_nat_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_95_bot__fun__def,axiom,
! [X_1: hoare_1708887482_state] :
( ( bot_bo19817387tate_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_96_hoare__derivs_OSkip,axiom,
! [G_12: hoare_1708887482_state > $o,P_17: state > state > $o] : ( hoare_90032982_state @ G_12 @ ( insert528405184_state @ ( hoare_858012674_state @ P_17 @ skip @ P_17 ) @ bot_bo19817387tate_o ) ) ).
thf(fact_97_Comp,axiom,
! [D_21: com,R_2: state > state > $o,G_11: hoare_1708887482_state > $o,P_16: state > state > $o,C_74: com,Q_8: state > state > $o] :
( ( hoare_90032982_state @ G_11 @ ( insert528405184_state @ ( hoare_858012674_state @ P_16 @ C_74 @ Q_8 ) @ bot_bo19817387tate_o ) )
=> ( ( hoare_90032982_state @ G_11 @ ( insert528405184_state @ ( hoare_858012674_state @ Q_8 @ D_21 @ R_2 ) @ bot_bo19817387tate_o ) )
=> ( hoare_90032982_state @ G_11 @ ( insert528405184_state @ ( hoare_858012674_state @ P_16 @ ( semi @ C_74 @ D_21 ) @ R_2 ) @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_98_triple_Oexhaust,axiom,
! [Y_55: hoare_1708887482_state] :
~ ! [Fun1: state > state > $o,Com_2: com,Fun2: state > state > $o] :
( Y_55
!= ( hoare_858012674_state @ Fun1 @ Com_2 @ Fun2 ) ) ).
thf(fact_99_Set_Oset__insert,axiom,
! [X_103: int,A_223: int > $o] :
( ( member_int @ X_103 @ A_223 )
=> ~ ! [B_128: int > $o] :
( ( A_223
= ( insert_int @ X_103 @ B_128 ) )
=> ( member_int @ X_103 @ B_128 ) ) ) ).
thf(fact_100_Set_Oset__insert,axiom,
! [X_103: nat,A_223: nat > $o] :
( ( member_nat @ X_103 @ A_223 )
=> ~ ! [B_128: nat > $o] :
( ( A_223
= ( insert_nat @ X_103 @ B_128 ) )
=> ( member_nat @ X_103 @ B_128 ) ) ) ).
thf(fact_101_Set_Oset__insert,axiom,
! [X_103: hoare_1708887482_state,A_223: hoare_1708887482_state > $o] :
( ( member451959335_state @ X_103 @ A_223 )
=> ~ ! [B_128: hoare_1708887482_state > $o] :
( ( A_223
= ( insert528405184_state @ X_103 @ B_128 ) )
=> ( member451959335_state @ X_103 @ B_128 ) ) ) ).
thf(fact_102_mk__disjoint__insert,axiom,
! [A_222: int,A_221: int > $o] :
( ( member_int @ A_222 @ A_221 )
=> ? [B_128: int > $o] :
( ( A_221
= ( insert_int @ A_222 @ B_128 ) )
& ~ ( member_int @ A_222 @ B_128 ) ) ) ).
thf(fact_103_mk__disjoint__insert,axiom,
! [A_222: nat,A_221: nat > $o] :
( ( member_nat @ A_222 @ A_221 )
=> ? [B_128: nat > $o] :
( ( A_221
= ( insert_nat @ A_222 @ B_128 ) )
& ~ ( member_nat @ A_222 @ B_128 ) ) ) ).
thf(fact_104_mk__disjoint__insert,axiom,
! [A_222: hoare_1708887482_state,A_221: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_222 @ A_221 )
=> ? [B_128: hoare_1708887482_state > $o] :
( ( A_221
= ( insert528405184_state @ A_222 @ B_128 ) )
& ~ ( member451959335_state @ A_222 @ B_128 ) ) ) ).
thf(fact_105_equals0I,axiom,
! [A_220: int > $o] :
( ! [Y_7: int] :
~ ( member_int @ Y_7 @ A_220 )
=> ( A_220 = bot_bot_int_o ) ) ).
thf(fact_106_equals0I,axiom,
! [A_220: nat > $o] :
( ! [Y_7: nat] :
~ ( member_nat @ Y_7 @ A_220 )
=> ( A_220 = bot_bot_nat_o ) ) ).
thf(fact_107_equals0I,axiom,
! [A_220: hoare_1708887482_state > $o] :
( ! [Y_7: hoare_1708887482_state] :
~ ( member451959335_state @ Y_7 @ A_220 )
=> ( A_220 = bot_bo19817387tate_o ) ) ).
thf(fact_108_conseq,axiom,
! [Q_6: state > state > $o,G_10: hoare_1708887482_state > $o,C_73: com,P_14: state > state > $o] :
( ! [Z_30: state,S_1: state] :
( ( P_14 @ Z_30 @ S_1 )
=> ? [P_15: state > state > $o,Q_7: state > state > $o] :
( ( hoare_90032982_state @ G_10 @ ( insert528405184_state @ ( hoare_858012674_state @ P_15 @ C_73 @ Q_7 ) @ bot_bo19817387tate_o ) )
& ! [S_2: state] :
( ! [Z_31: state] :
( ( P_15 @ Z_31 @ S_1 )
=> ( Q_7 @ Z_31 @ S_2 ) )
=> ( Q_6 @ Z_30 @ S_2 ) ) ) )
=> ( hoare_90032982_state @ G_10 @ ( insert528405184_state @ ( hoare_858012674_state @ P_14 @ C_73 @ Q_6 ) @ bot_bo19817387tate_o ) ) ) ).
thf(fact_109_nonempty__iff,axiom,
! [A_219: int > $o] :
( ( A_219 != bot_bot_int_o )
<=> ? [X_1: int,B_128: int > $o] :
( ( A_219
= ( insert_int @ X_1 @ B_128 ) )
& ~ ( member_int @ X_1 @ B_128 ) ) ) ).
thf(fact_110_nonempty__iff,axiom,
! [A_219: nat > $o] :
( ( A_219 != bot_bot_nat_o )
<=> ? [X_1: nat,B_128: nat > $o] :
( ( A_219
= ( insert_nat @ X_1 @ B_128 ) )
& ~ ( member_nat @ X_1 @ B_128 ) ) ) ).
thf(fact_111_nonempty__iff,axiom,
! [A_219: hoare_1708887482_state > $o] :
( ( A_219 != bot_bo19817387tate_o )
<=> ? [X_1: hoare_1708887482_state,B_128: hoare_1708887482_state > $o] :
( ( A_219
= ( insert528405184_state @ X_1 @ B_128 ) )
& ~ ( member451959335_state @ X_1 @ B_128 ) ) ) ).
thf(fact_112_bot__empty__eq,axiom,
! [X_1: int] :
( ( bot_bot_int_o @ X_1 )
<=> ( member_int @ X_1 @ bot_bot_int_o ) ) ).
thf(fact_113_bot__empty__eq,axiom,
! [X_1: nat] :
( ( bot_bot_nat_o @ X_1 )
<=> ( member_nat @ X_1 @ bot_bot_nat_o ) ) ).
thf(fact_114_bot__empty__eq,axiom,
! [X_1: hoare_1708887482_state] :
( ( bot_bo19817387tate_o @ X_1 )
<=> ( member451959335_state @ X_1 @ bot_bo19817387tate_o ) ) ).
thf(fact_115_com_Osimps_I13_J,axiom,
! [Com1_1: com,Com2_1: com] :
( ( semi @ Com1_1 @ Com2_1 )
!= skip ) ).
thf(fact_116_com_Osimps_I12_J,axiom,
! [Com1_1: com,Com2_1: com] :
( skip
!= ( semi @ Com1_1 @ Com2_1 ) ) ).
thf(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 ) ) ) ).
thf(fact_118_fold1Set__sing,axiom,
! [F_68: int > int > int,A_218: int,B_127: int] :
( ( finite_fold1Set_int @ F_68 @ ( insert_int @ A_218 @ bot_bot_int_o ) @ B_127 )
<=> ( A_218 = B_127 ) ) ).
thf(fact_119_fold1Set__sing,axiom,
! [F_68: nat > nat > nat,A_218: nat,B_127: nat] :
( ( finite_fold1Set_nat @ F_68 @ ( insert_nat @ A_218 @ bot_bot_nat_o ) @ B_127 )
<=> ( A_218 = B_127 ) ) ).
thf(fact_120_fold1Set__sing,axiom,
! [F_68: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,A_218: hoare_1708887482_state,B_127: hoare_1708887482_state] :
( ( finite850270954_state @ F_68 @ ( insert528405184_state @ A_218 @ bot_bo19817387tate_o ) @ B_127 )
<=> ( A_218 = B_127 ) ) ).
thf(fact_121_folding__one_Osingleton,axiom,
! [X_102: int,F_67: int > int > int,F_66: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_67 @ F_66 )
=> ( ( F_66 @ ( insert_int @ X_102 @ bot_bot_int_o ) )
= X_102 ) ) ).
thf(fact_122_folding__one_Osingleton,axiom,
! [X_102: nat,F_67: nat > nat > nat,F_66: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_67 @ F_66 )
=> ( ( F_66 @ ( insert_nat @ X_102 @ bot_bot_nat_o ) )
= X_102 ) ) ).
thf(fact_123_folding__one_Osingleton,axiom,
! [X_102: hoare_1708887482_state,F_67: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_66: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_67 @ F_66 )
=> ( ( F_66 @ ( insert528405184_state @ X_102 @ bot_bo19817387tate_o ) )
= X_102 ) ) ).
thf(fact_124_fold1__singleton,axiom,
! [F_65: int > int > int,A_217: int] :
( ( finite_fold1_int @ F_65 @ ( insert_int @ A_217 @ bot_bot_int_o ) )
= A_217 ) ).
thf(fact_125_fold1__singleton,axiom,
! [F_65: nat > nat > nat,A_217: nat] :
( ( finite_fold1_nat @ F_65 @ ( insert_nat @ A_217 @ bot_bot_nat_o ) )
= A_217 ) ).
thf(fact_126_fold1__singleton,axiom,
! [F_65: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,A_217: hoare_1708887482_state] :
( ( finite39664218_state @ F_65 @ ( insert528405184_state @ A_217 @ bot_bo19817387tate_o ) )
= A_217 ) ).
thf(fact_127_fold1__singleton__def,axiom,
! [A_216: int,G_9: ( int > $o ) > int,F_64: int > int > int] :
( ( G_9
= ( finite_fold1_int @ F_64 ) )
=> ( ( G_9 @ ( insert_int @ A_216 @ bot_bot_int_o ) )
= A_216 ) ) ).
thf(fact_128_fold1__singleton__def,axiom,
! [A_216: nat,G_9: ( nat > $o ) > nat,F_64: nat > nat > nat] :
( ( G_9
= ( finite_fold1_nat @ F_64 ) )
=> ( ( G_9 @ ( insert_nat @ A_216 @ bot_bot_nat_o ) )
= A_216 ) ) ).
thf(fact_129_fold1__singleton__def,axiom,
! [A_216: hoare_1708887482_state,G_9: ( hoare_1708887482_state > $o ) > hoare_1708887482_state,F_64: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state] :
( ( G_9
= ( finite39664218_state @ F_64 ) )
=> ( ( G_9 @ ( insert528405184_state @ A_216 @ bot_bo19817387tate_o ) )
= A_216 ) ) ).
thf(fact_130_empty__fold1SetE,axiom,
! [F_63: int > int > int,X_101: int] :
~ ( finite_fold1Set_int @ F_63 @ bot_bot_int_o @ X_101 ) ).
thf(fact_131_empty__fold1SetE,axiom,
! [F_63: nat > nat > nat,X_101: nat] :
~ ( finite_fold1Set_nat @ F_63 @ bot_bot_nat_o @ X_101 ) ).
thf(fact_132_empty__fold1SetE,axiom,
! [F_63: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,X_101: hoare_1708887482_state] :
~ ( finite850270954_state @ F_63 @ bot_bo19817387tate_o @ X_101 ) ).
thf(fact_133_fold1Set__nonempty,axiom,
! [F_62: int > int > int,A_215: int > $o,X_100: int] :
( ( finite_fold1Set_int @ F_62 @ A_215 @ X_100 )
=> ( A_215 != bot_bot_int_o ) ) ).
thf(fact_134_fold1Set__nonempty,axiom,
! [F_62: nat > nat > nat,A_215: nat > $o,X_100: nat] :
( ( finite_fold1Set_nat @ F_62 @ A_215 @ X_100 )
=> ( A_215 != bot_bot_nat_o ) ) ).
thf(fact_135_fold1Set__nonempty,axiom,
! [F_62: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,A_215: hoare_1708887482_state > $o,X_100: hoare_1708887482_state] :
( ( finite850270954_state @ F_62 @ A_215 @ X_100 )
=> ( A_215 != bot_bo19817387tate_o ) ) ).
thf(fact_136_subset__singletonD,axiom,
! [A_214: int > $o,X_99: int] :
( ( ord_less_eq_int_o @ A_214 @ ( insert_int @ X_99 @ bot_bot_int_o ) )
=> ( ( A_214 = bot_bot_int_o )
| ( A_214
= ( insert_int @ X_99 @ bot_bot_int_o ) ) ) ) ).
thf(fact_137_subset__singletonD,axiom,
! [A_214: nat > $o,X_99: nat] :
( ( ord_less_eq_nat_o @ A_214 @ ( insert_nat @ X_99 @ bot_bot_nat_o ) )
=> ( ( A_214 = bot_bot_nat_o )
| ( A_214
= ( insert_nat @ X_99 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_138_subset__singletonD,axiom,
! [A_214: hoare_1708887482_state > $o,X_99: hoare_1708887482_state] :
( ( ord_le777019615tate_o @ A_214 @ ( insert528405184_state @ X_99 @ bot_bo19817387tate_o ) )
=> ( ( A_214 = bot_bo19817387tate_o )
| ( A_214
= ( insert528405184_state @ X_99 @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_139_order__refl,axiom,
! [X_98: hoare_1708887482_state > $o] : ( ord_le777019615tate_o @ X_98 @ X_98 ) ).
thf(fact_140_order__refl,axiom,
! [X_98: nat] : ( ord_less_eq_nat @ X_98 @ X_98 ) ).
thf(fact_141_order__refl,axiom,
! [X_98: int] : ( ord_less_eq_int @ X_98 @ X_98 ) ).
thf(fact_142_equalityI,axiom,
! [A_213: hoare_1708887482_state > $o,B_126: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_213 @ B_126 )
=> ( ( ord_le777019615tate_o @ B_126 @ A_213 )
=> ( A_213 = B_126 ) ) ) ).
thf(fact_143_subsetD,axiom,
! [C_72: hoare_1708887482_state,A_212: hoare_1708887482_state > $o,B_125: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_212 @ B_125 )
=> ( ( member451959335_state @ C_72 @ A_212 )
=> ( member451959335_state @ C_72 @ B_125 ) ) ) ).
thf(fact_144_subsetD,axiom,
! [C_72: int,A_212: int > $o,B_125: int > $o] :
( ( ord_less_eq_int_o @ A_212 @ B_125 )
=> ( ( member_int @ C_72 @ A_212 )
=> ( member_int @ C_72 @ B_125 ) ) ) ).
thf(fact_145_subsetD,axiom,
! [C_72: nat,A_212: nat > $o,B_125: nat > $o] :
( ( ord_less_eq_nat_o @ A_212 @ B_125 )
=> ( ( member_nat @ C_72 @ A_212 )
=> ( member_nat @ C_72 @ B_125 ) ) ) ).
thf(fact_146_empty__subsetI,axiom,
! [A_211: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_211 ) ).
thf(fact_147_empty__subsetI,axiom,
! [A_211: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_211 ) ).
thf(fact_148_empty__subsetI,axiom,
! [A_211: hoare_1708887482_state > $o] : ( ord_le777019615tate_o @ bot_bo19817387tate_o @ A_211 ) ).
thf(fact_149_linorder__le__cases,axiom,
! [X_97: nat,Y_54: nat] :
( ~ ( ord_less_eq_nat @ X_97 @ Y_54 )
=> ( ord_less_eq_nat @ Y_54 @ X_97 ) ) ).
thf(fact_150_linorder__le__cases,axiom,
! [X_97: int,Y_54: int] :
( ~ ( ord_less_eq_int @ X_97 @ Y_54 )
=> ( ord_less_eq_int @ Y_54 @ X_97 ) ) ).
thf(fact_151_le__funE,axiom,
! [X_96: hoare_1708887482_state,F_61: hoare_1708887482_state > $o,G_8: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ F_61 @ G_8 )
=> ( ord_less_eq_o @ ( F_61 @ X_96 ) @ ( G_8 @ X_96 ) ) ) ).
thf(fact_152_xt1_I6_J,axiom,
! [Z_29: hoare_1708887482_state > $o,Y_53: hoare_1708887482_state > $o,X_95: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_53 @ X_95 )
=> ( ( ord_le777019615tate_o @ Z_29 @ Y_53 )
=> ( ord_le777019615tate_o @ Z_29 @ X_95 ) ) ) ).
thf(fact_153_xt1_I6_J,axiom,
! [Z_29: nat,Y_53: nat,X_95: nat] :
( ( ord_less_eq_nat @ Y_53 @ X_95 )
=> ( ( ord_less_eq_nat @ Z_29 @ Y_53 )
=> ( ord_less_eq_nat @ Z_29 @ X_95 ) ) ) ).
thf(fact_154_xt1_I6_J,axiom,
! [Z_29: int,Y_53: int,X_95: int] :
( ( ord_less_eq_int @ Y_53 @ X_95 )
=> ( ( ord_less_eq_int @ Z_29 @ Y_53 )
=> ( ord_less_eq_int @ Z_29 @ X_95 ) ) ) ).
thf(fact_155_xt1_I5_J,axiom,
! [Y_52: hoare_1708887482_state > $o,X_94: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_52 @ X_94 )
=> ( ( ord_le777019615tate_o @ X_94 @ Y_52 )
=> ( X_94 = Y_52 ) ) ) ).
thf(fact_156_xt1_I5_J,axiom,
! [Y_52: nat,X_94: nat] :
( ( ord_less_eq_nat @ Y_52 @ X_94 )
=> ( ( ord_less_eq_nat @ X_94 @ Y_52 )
=> ( X_94 = Y_52 ) ) ) ).
thf(fact_157_xt1_I5_J,axiom,
! [Y_52: int,X_94: int] :
( ( ord_less_eq_int @ Y_52 @ X_94 )
=> ( ( ord_less_eq_int @ X_94 @ Y_52 )
=> ( X_94 = Y_52 ) ) ) ).
thf(fact_158_order__trans,axiom,
! [Z_28: hoare_1708887482_state > $o,X_93: hoare_1708887482_state > $o,Y_51: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_93 @ Y_51 )
=> ( ( ord_le777019615tate_o @ Y_51 @ Z_28 )
=> ( ord_le777019615tate_o @ X_93 @ Z_28 ) ) ) ).
thf(fact_159_order__trans,axiom,
! [Z_28: nat,X_93: nat,Y_51: nat] :
( ( ord_less_eq_nat @ X_93 @ Y_51 )
=> ( ( ord_less_eq_nat @ Y_51 @ Z_28 )
=> ( ord_less_eq_nat @ X_93 @ Z_28 ) ) ) ).
thf(fact_160_order__trans,axiom,
! [Z_28: int,X_93: int,Y_51: int] :
( ( ord_less_eq_int @ X_93 @ Y_51 )
=> ( ( ord_less_eq_int @ Y_51 @ Z_28 )
=> ( ord_less_eq_int @ X_93 @ Z_28 ) ) ) ).
thf(fact_161_order__antisym,axiom,
! [X_92: hoare_1708887482_state > $o,Y_50: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_92 @ Y_50 )
=> ( ( ord_le777019615tate_o @ Y_50 @ X_92 )
=> ( X_92 = Y_50 ) ) ) ).
thf(fact_162_order__antisym,axiom,
! [X_92: nat,Y_50: nat] :
( ( ord_less_eq_nat @ X_92 @ Y_50 )
=> ( ( ord_less_eq_nat @ Y_50 @ X_92 )
=> ( X_92 = Y_50 ) ) ) ).
thf(fact_163_order__antisym,axiom,
! [X_92: int,Y_50: int] :
( ( ord_less_eq_int @ X_92 @ Y_50 )
=> ( ( ord_less_eq_int @ Y_50 @ X_92 )
=> ( X_92 = Y_50 ) ) ) ).
thf(fact_164_xt1_I4_J,axiom,
! [C_71: hoare_1708887482_state > $o,B_124: hoare_1708887482_state > $o,A_210: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ B_124 @ A_210 )
=> ( ( B_124 = C_71 )
=> ( ord_le777019615tate_o @ C_71 @ A_210 ) ) ) ).
thf(fact_165_xt1_I4_J,axiom,
! [C_71: nat,B_124: nat,A_210: nat] :
( ( ord_less_eq_nat @ B_124 @ A_210 )
=> ( ( B_124 = C_71 )
=> ( ord_less_eq_nat @ C_71 @ A_210 ) ) ) ).
thf(fact_166_xt1_I4_J,axiom,
! [C_71: int,B_124: int,A_210: int] :
( ( ord_less_eq_int @ B_124 @ A_210 )
=> ( ( B_124 = C_71 )
=> ( ord_less_eq_int @ C_71 @ A_210 ) ) ) ).
thf(fact_167_ord__le__eq__trans,axiom,
! [C_70: hoare_1708887482_state > $o,A_209: hoare_1708887482_state > $o,B_123: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_209 @ B_123 )
=> ( ( B_123 = C_70 )
=> ( ord_le777019615tate_o @ A_209 @ C_70 ) ) ) ).
thf(fact_168_ord__le__eq__trans,axiom,
! [C_70: nat,A_209: nat,B_123: nat] :
( ( ord_less_eq_nat @ A_209 @ B_123 )
=> ( ( B_123 = C_70 )
=> ( ord_less_eq_nat @ A_209 @ C_70 ) ) ) ).
thf(fact_169_ord__le__eq__trans,axiom,
! [C_70: int,A_209: int,B_123: int] :
( ( ord_less_eq_int @ A_209 @ B_123 )
=> ( ( B_123 = C_70 )
=> ( ord_less_eq_int @ A_209 @ C_70 ) ) ) ).
thf(fact_170_xt1_I3_J,axiom,
! [C_69: hoare_1708887482_state > $o,A_208: hoare_1708887482_state > $o,B_122: hoare_1708887482_state > $o] :
( ( A_208 = B_122 )
=> ( ( ord_le777019615tate_o @ C_69 @ B_122 )
=> ( ord_le777019615tate_o @ C_69 @ A_208 ) ) ) ).
thf(fact_171_xt1_I3_J,axiom,
! [C_69: nat,A_208: nat,B_122: nat] :
( ( A_208 = B_122 )
=> ( ( ord_less_eq_nat @ C_69 @ B_122 )
=> ( ord_less_eq_nat @ C_69 @ A_208 ) ) ) ).
thf(fact_172_xt1_I3_J,axiom,
! [C_69: int,A_208: int,B_122: int] :
( ( A_208 = B_122 )
=> ( ( ord_less_eq_int @ C_69 @ B_122 )
=> ( ord_less_eq_int @ C_69 @ A_208 ) ) ) ).
thf(fact_173_ord__eq__le__trans,axiom,
! [C_68: hoare_1708887482_state > $o,A_207: hoare_1708887482_state > $o,B_121: hoare_1708887482_state > $o] :
( ( A_207 = B_121 )
=> ( ( ord_le777019615tate_o @ B_121 @ C_68 )
=> ( ord_le777019615tate_o @ A_207 @ C_68 ) ) ) ).
thf(fact_174_ord__eq__le__trans,axiom,
! [C_68: nat,A_207: nat,B_121: nat] :
( ( A_207 = B_121 )
=> ( ( ord_less_eq_nat @ B_121 @ C_68 )
=> ( ord_less_eq_nat @ A_207 @ C_68 ) ) ) ).
thf(fact_175_ord__eq__le__trans,axiom,
! [C_68: int,A_207: int,B_121: int] :
( ( A_207 = B_121 )
=> ( ( ord_less_eq_int @ B_121 @ C_68 )
=> ( ord_less_eq_int @ A_207 @ C_68 ) ) ) ).
thf(fact_176_predicate1D,axiom,
! [X_91: hoare_1708887482_state,P_13: hoare_1708887482_state > $o,Q_5: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ P_13 @ Q_5 )
=> ( ( P_13 @ X_91 )
=> ( Q_5 @ X_91 ) ) ) ).
thf(fact_177_mem__def,axiom,
! [X_90: hoare_1708887482_state,A_206: hoare_1708887482_state > $o] :
( ( member451959335_state @ X_90 @ A_206 )
<=> ( A_206 @ X_90 ) ) ).
thf(fact_178_mem__def,axiom,
! [X_90: int,A_206: int > $o] :
( ( member_int @ X_90 @ A_206 )
<=> ( A_206 @ X_90 ) ) ).
thf(fact_179_mem__def,axiom,
! [X_90: nat,A_206: nat > $o] :
( ( member_nat @ X_90 @ A_206 )
<=> ( A_206 @ X_90 ) ) ).
thf(fact_180_Collect__def,axiom,
! [P_12: hoare_1708887482_state > $o] :
( ( collec1568722789_state @ P_12 )
= P_12 ) ).
thf(fact_181_order__antisym__conv,axiom,
! [Y_49: hoare_1708887482_state > $o,X_89: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_49 @ X_89 )
=> ( ( ord_le777019615tate_o @ X_89 @ Y_49 )
<=> ( X_89 = Y_49 ) ) ) ).
thf(fact_182_order__antisym__conv,axiom,
! [Y_49: nat,X_89: nat] :
( ( ord_less_eq_nat @ Y_49 @ X_89 )
=> ( ( ord_less_eq_nat @ X_89 @ Y_49 )
<=> ( X_89 = Y_49 ) ) ) ).
thf(fact_183_order__antisym__conv,axiom,
! [Y_49: int,X_89: int] :
( ( ord_less_eq_int @ Y_49 @ X_89 )
=> ( ( ord_less_eq_int @ X_89 @ Y_49 )
<=> ( X_89 = Y_49 ) ) ) ).
thf(fact_184_le__funD,axiom,
! [X_88: hoare_1708887482_state,F_60: hoare_1708887482_state > $o,G_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ F_60 @ G_7 )
=> ( ord_less_eq_o @ ( F_60 @ X_88 ) @ ( G_7 @ X_88 ) ) ) ).
thf(fact_185_order__eq__refl,axiom,
! [X_87: hoare_1708887482_state > $o,Y_48: hoare_1708887482_state > $o] :
( ( X_87 = Y_48 )
=> ( ord_le777019615tate_o @ X_87 @ Y_48 ) ) ).
thf(fact_186_order__eq__refl,axiom,
! [X_87: nat,Y_48: nat] :
( ( X_87 = Y_48 )
=> ( ord_less_eq_nat @ X_87 @ Y_48 ) ) ).
thf(fact_187_order__eq__refl,axiom,
! [X_87: int,Y_48: int] :
( ( X_87 = Y_48 )
=> ( ord_less_eq_int @ X_87 @ Y_48 ) ) ).
thf(fact_188_rev__predicate1D,axiom,
! [Q_4: hoare_1708887482_state > $o,P_11: hoare_1708887482_state > $o,X_86: hoare_1708887482_state] :
( ( P_11 @ X_86 )
=> ( ( ord_le777019615tate_o @ P_11 @ Q_4 )
=> ( Q_4 @ X_86 ) ) ) ).
thf(fact_189_order__eq__iff,axiom,
! [X_85: hoare_1708887482_state > $o,Y_47: hoare_1708887482_state > $o] :
( ( X_85 = Y_47 )
<=> ( ( ord_le777019615tate_o @ X_85 @ Y_47 )
& ( ord_le777019615tate_o @ Y_47 @ X_85 ) ) ) ).
thf(fact_190_order__eq__iff,axiom,
! [X_85: nat,Y_47: nat] :
( ( X_85 = Y_47 )
<=> ( ( ord_less_eq_nat @ X_85 @ Y_47 )
& ( ord_less_eq_nat @ Y_47 @ X_85 ) ) ) ).
thf(fact_191_order__eq__iff,axiom,
! [X_85: int,Y_47: int] :
( ( X_85 = Y_47 )
<=> ( ( ord_less_eq_int @ X_85 @ Y_47 )
& ( ord_less_eq_int @ Y_47 @ X_85 ) ) ) ).
thf(fact_192_linorder__linear,axiom,
! [X_84: nat,Y_46: nat] :
( ( ord_less_eq_nat @ X_84 @ Y_46 )
| ( ord_less_eq_nat @ Y_46 @ X_84 ) ) ).
thf(fact_193_linorder__linear,axiom,
! [X_84: int,Y_46: int] :
( ( ord_less_eq_int @ X_84 @ Y_46 )
| ( ord_less_eq_int @ Y_46 @ X_84 ) ) ).
thf(fact_194_le__fun__def,axiom,
! [F_59: hoare_1708887482_state > $o,G_6: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ F_59 @ G_6 )
<=> ! [X_1: hoare_1708887482_state] : ( ord_less_eq_o @ ( F_59 @ X_1 ) @ ( G_6 @ X_1 ) ) ) ).
thf(fact_195_equalityE,axiom,
! [A_205: hoare_1708887482_state > $o,B_120: hoare_1708887482_state > $o] :
( ( A_205 = B_120 )
=> ~ ( ( ord_le777019615tate_o @ A_205 @ B_120 )
=> ~ ( ord_le777019615tate_o @ B_120 @ A_205 ) ) ) ).
thf(fact_196_subset__trans,axiom,
! [C_67: hoare_1708887482_state > $o,A_204: hoare_1708887482_state > $o,B_119: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_204 @ B_119 )
=> ( ( ord_le777019615tate_o @ B_119 @ C_67 )
=> ( ord_le777019615tate_o @ A_204 @ C_67 ) ) ) ).
thf(fact_197_set__mp,axiom,
! [X_83: hoare_1708887482_state,A_203: hoare_1708887482_state > $o,B_118: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_203 @ B_118 )
=> ( ( member451959335_state @ X_83 @ A_203 )
=> ( member451959335_state @ X_83 @ B_118 ) ) ) ).
thf(fact_198_set__mp,axiom,
! [X_83: int,A_203: int > $o,B_118: int > $o] :
( ( ord_less_eq_int_o @ A_203 @ B_118 )
=> ( ( member_int @ X_83 @ A_203 )
=> ( member_int @ X_83 @ B_118 ) ) ) ).
thf(fact_199_set__mp,axiom,
! [X_83: nat,A_203: nat > $o,B_118: nat > $o] :
( ( ord_less_eq_nat_o @ A_203 @ B_118 )
=> ( ( member_nat @ X_83 @ A_203 )
=> ( member_nat @ X_83 @ B_118 ) ) ) ).
thf(fact_200_set__rev__mp,axiom,
! [B_117: hoare_1708887482_state > $o,X_82: hoare_1708887482_state,A_202: hoare_1708887482_state > $o] :
( ( member451959335_state @ X_82 @ A_202 )
=> ( ( ord_le777019615tate_o @ A_202 @ B_117 )
=> ( member451959335_state @ X_82 @ B_117 ) ) ) ).
thf(fact_201_set__rev__mp,axiom,
! [B_117: int > $o,X_82: int,A_202: int > $o] :
( ( member_int @ X_82 @ A_202 )
=> ( ( ord_less_eq_int_o @ A_202 @ B_117 )
=> ( member_int @ X_82 @ B_117 ) ) ) ).
thf(fact_202_set__rev__mp,axiom,
! [B_117: nat > $o,X_82: nat,A_202: nat > $o] :
( ( member_nat @ X_82 @ A_202 )
=> ( ( ord_less_eq_nat_o @ A_202 @ B_117 )
=> ( member_nat @ X_82 @ B_117 ) ) ) ).
thf(fact_203_in__mono,axiom,
! [X_81: hoare_1708887482_state,A_201: hoare_1708887482_state > $o,B_116: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_201 @ B_116 )
=> ( ( member451959335_state @ X_81 @ A_201 )
=> ( member451959335_state @ X_81 @ B_116 ) ) ) ).
thf(fact_204_in__mono,axiom,
! [X_81: int,A_201: int > $o,B_116: int > $o] :
( ( ord_less_eq_int_o @ A_201 @ B_116 )
=> ( ( member_int @ X_81 @ A_201 )
=> ( member_int @ X_81 @ B_116 ) ) ) ).
thf(fact_205_in__mono,axiom,
! [X_81: nat,A_201: nat > $o,B_116: nat > $o] :
( ( ord_less_eq_nat_o @ A_201 @ B_116 )
=> ( ( member_nat @ X_81 @ A_201 )
=> ( member_nat @ X_81 @ B_116 ) ) ) ).
thf(fact_206_equalityD2,axiom,
! [A_200: hoare_1708887482_state > $o,B_115: hoare_1708887482_state > $o] :
( ( A_200 = B_115 )
=> ( ord_le777019615tate_o @ B_115 @ A_200 ) ) ).
thf(fact_207_equalityD1,axiom,
! [A_199: hoare_1708887482_state > $o,B_114: hoare_1708887482_state > $o] :
( ( A_199 = B_114 )
=> ( ord_le777019615tate_o @ A_199 @ B_114 ) ) ).
thf(fact_208_set__eq__subset,axiom,
! [A_198: hoare_1708887482_state > $o,B_113: hoare_1708887482_state > $o] :
( ( A_198 = B_113 )
<=> ( ( ord_le777019615tate_o @ A_198 @ B_113 )
& ( ord_le777019615tate_o @ B_113 @ A_198 ) ) ) ).
thf(fact_209_subset__refl,axiom,
! [A_197: hoare_1708887482_state > $o] : ( ord_le777019615tate_o @ A_197 @ A_197 ) ).
thf(fact_210_le__bot,axiom,
! [A_196: int > $o] :
( ( ord_less_eq_int_o @ A_196 @ bot_bot_int_o )
=> ( A_196 = bot_bot_int_o ) ) ).
thf(fact_211_le__bot,axiom,
! [A_196: nat > $o] :
( ( ord_less_eq_nat_o @ A_196 @ bot_bot_nat_o )
=> ( A_196 = bot_bot_nat_o ) ) ).
thf(fact_212_le__bot,axiom,
! [A_196: $o] :
( ( ord_less_eq_o @ A_196 @ bot_bot_o )
=> ( A_196
<=> bot_bot_o ) ) ).
thf(fact_213_le__bot,axiom,
! [A_196: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_196 @ bot_bo19817387tate_o )
=> ( A_196 = bot_bo19817387tate_o ) ) ).
thf(fact_214_le__bot,axiom,
! [A_196: nat] :
( ( ord_less_eq_nat @ A_196 @ bot_bot_nat )
=> ( A_196 = bot_bot_nat ) ) ).
thf(fact_215_bot__unique,axiom,
! [A_195: int > $o] :
( ( ord_less_eq_int_o @ A_195 @ bot_bot_int_o )
<=> ( A_195 = bot_bot_int_o ) ) ).
thf(fact_216_bot__unique,axiom,
! [A_195: nat > $o] :
( ( ord_less_eq_nat_o @ A_195 @ bot_bot_nat_o )
<=> ( A_195 = bot_bot_nat_o ) ) ).
thf(fact_217_bot__unique,axiom,
! [A_195: $o] :
( ( ord_less_eq_o @ A_195 @ bot_bot_o )
<=> ( A_195
<=> bot_bot_o ) ) ).
thf(fact_218_bot__unique,axiom,
! [A_195: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_195 @ bot_bo19817387tate_o )
<=> ( A_195 = bot_bo19817387tate_o ) ) ).
thf(fact_219_bot__unique,axiom,
! [A_195: nat] :
( ( ord_less_eq_nat @ A_195 @ bot_bot_nat )
<=> ( A_195 = bot_bot_nat ) ) ).
thf(fact_220_bot__least,axiom,
! [A_194: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_194 ) ).
thf(fact_221_bot__least,axiom,
! [A_194: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_194 ) ).
thf(fact_222_bot__least,axiom,
! [A_194: $o] : ( ord_less_eq_o @ bot_bot_o @ A_194 ) ).
thf(fact_223_bot__least,axiom,
! [A_194: hoare_1708887482_state > $o] : ( ord_le777019615tate_o @ bot_bo19817387tate_o @ A_194 ) ).
thf(fact_224_bot__least,axiom,
! [A_194: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_194 ) ).
thf(fact_225_subset__empty,axiom,
! [A_193: int > $o] :
( ( ord_less_eq_int_o @ A_193 @ bot_bot_int_o )
<=> ( A_193 = bot_bot_int_o ) ) ).
thf(fact_226_subset__empty,axiom,
! [A_193: nat > $o] :
( ( ord_less_eq_nat_o @ A_193 @ bot_bot_nat_o )
<=> ( A_193 = bot_bot_nat_o ) ) ).
thf(fact_227_subset__empty,axiom,
! [A_193: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_193 @ bot_bo19817387tate_o )
<=> ( A_193 = bot_bo19817387tate_o ) ) ).
thf(fact_228_insert__mono,axiom,
! [A_192: int,C_66: int > $o,D_20: int > $o] :
( ( ord_less_eq_int_o @ C_66 @ D_20 )
=> ( ord_less_eq_int_o @ ( insert_int @ A_192 @ C_66 ) @ ( insert_int @ A_192 @ D_20 ) ) ) ).
thf(fact_229_insert__mono,axiom,
! [A_192: nat,C_66: nat > $o,D_20: nat > $o] :
( ( ord_less_eq_nat_o @ C_66 @ D_20 )
=> ( ord_less_eq_nat_o @ ( insert_nat @ A_192 @ C_66 ) @ ( insert_nat @ A_192 @ D_20 ) ) ) ).
thf(fact_230_insert__mono,axiom,
! [A_192: hoare_1708887482_state,C_66: hoare_1708887482_state > $o,D_20: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ C_66 @ D_20 )
=> ( ord_le777019615tate_o @ ( insert528405184_state @ A_192 @ C_66 ) @ ( insert528405184_state @ A_192 @ D_20 ) ) ) ).
thf(fact_231_subset__insertI2,axiom,
! [B_112: int,A_191: int > $o,B_111: int > $o] :
( ( ord_less_eq_int_o @ A_191 @ B_111 )
=> ( ord_less_eq_int_o @ A_191 @ ( insert_int @ B_112 @ B_111 ) ) ) ).
thf(fact_232_subset__insertI2,axiom,
! [B_112: nat,A_191: nat > $o,B_111: nat > $o] :
( ( ord_less_eq_nat_o @ A_191 @ B_111 )
=> ( ord_less_eq_nat_o @ A_191 @ ( insert_nat @ B_112 @ B_111 ) ) ) ).
thf(fact_233_subset__insertI2,axiom,
! [B_112: hoare_1708887482_state,A_191: hoare_1708887482_state > $o,B_111: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_191 @ B_111 )
=> ( ord_le777019615tate_o @ A_191 @ ( insert528405184_state @ B_112 @ B_111 ) ) ) ).
thf(fact_234_subset__insert,axiom,
! [B_110: int > $o,X_80: int,A_190: int > $o] :
( ~ ( member_int @ X_80 @ A_190 )
=> ( ( ord_less_eq_int_o @ A_190 @ ( insert_int @ X_80 @ B_110 ) )
<=> ( ord_less_eq_int_o @ A_190 @ B_110 ) ) ) ).
thf(fact_235_subset__insert,axiom,
! [B_110: nat > $o,X_80: nat,A_190: nat > $o] :
( ~ ( member_nat @ X_80 @ A_190 )
=> ( ( ord_less_eq_nat_o @ A_190 @ ( insert_nat @ X_80 @ B_110 ) )
<=> ( ord_less_eq_nat_o @ A_190 @ B_110 ) ) ) ).
thf(fact_236_subset__insert,axiom,
! [B_110: hoare_1708887482_state > $o,X_80: hoare_1708887482_state,A_190: hoare_1708887482_state > $o] :
( ~ ( member451959335_state @ X_80 @ A_190 )
=> ( ( ord_le777019615tate_o @ A_190 @ ( insert528405184_state @ X_80 @ B_110 ) )
<=> ( ord_le777019615tate_o @ A_190 @ B_110 ) ) ) ).
thf(fact_237_insert__subset,axiom,
! [X_79: int,A_189: int > $o,B_109: int > $o] :
( ( ord_less_eq_int_o @ ( insert_int @ X_79 @ A_189 ) @ B_109 )
<=> ( ( member_int @ X_79 @ B_109 )
& ( ord_less_eq_int_o @ A_189 @ B_109 ) ) ) ).
thf(fact_238_insert__subset,axiom,
! [X_79: nat,A_189: nat > $o,B_109: nat > $o] :
( ( ord_less_eq_nat_o @ ( insert_nat @ X_79 @ A_189 ) @ B_109 )
<=> ( ( member_nat @ X_79 @ B_109 )
& ( ord_less_eq_nat_o @ A_189 @ B_109 ) ) ) ).
thf(fact_239_insert__subset,axiom,
! [X_79: hoare_1708887482_state,A_189: hoare_1708887482_state > $o,B_109: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ ( insert528405184_state @ X_79 @ A_189 ) @ B_109 )
<=> ( ( member451959335_state @ X_79 @ B_109 )
& ( ord_le777019615tate_o @ A_189 @ B_109 ) ) ) ).
thf(fact_240_subset__insertI,axiom,
! [B_108: int > $o,A_188: int] : ( ord_less_eq_int_o @ B_108 @ ( insert_int @ A_188 @ B_108 ) ) ).
thf(fact_241_subset__insertI,axiom,
! [B_108: nat > $o,A_188: nat] : ( ord_less_eq_nat_o @ B_108 @ ( insert_nat @ A_188 @ B_108 ) ) ).
thf(fact_242_subset__insertI,axiom,
! [B_108: hoare_1708887482_state > $o,A_188: hoare_1708887482_state] : ( ord_le777019615tate_o @ B_108 @ ( insert528405184_state @ A_188 @ B_108 ) ) ).
thf(fact_243_thin,axiom,
! [G_5: hoare_1708887482_state > $o,G_4: hoare_1708887482_state > $o,Ts_3: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ G_4 @ Ts_3 )
=> ( ( ord_le777019615tate_o @ G_4 @ G_5 )
=> ( hoare_90032982_state @ G_5 @ Ts_3 ) ) ) ).
thf(fact_244_weaken,axiom,
! [Ts_2: hoare_1708887482_state > $o,G_3: hoare_1708887482_state > $o,Ts_1: hoare_1708887482_state > $o] :
( ( hoare_90032982_state @ G_3 @ Ts_1 )
=> ( ( ord_le777019615tate_o @ Ts_2 @ Ts_1 )
=> ( hoare_90032982_state @ G_3 @ Ts_2 ) ) ) ).
thf(fact_245_asm,axiom,
! [Ts: hoare_1708887482_state > $o,G_2: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Ts @ G_2 )
=> ( hoare_90032982_state @ G_2 @ Ts ) ) ).
thf(fact_246_subsetI,axiom,
! [B_107: hoare_1708887482_state > $o,A_187: hoare_1708887482_state > $o] :
( ! [X_1: hoare_1708887482_state] :
( ( member451959335_state @ X_1 @ A_187 )
=> ( member451959335_state @ X_1 @ B_107 ) )
=> ( ord_le777019615tate_o @ A_187 @ B_107 ) ) ).
thf(fact_247_subsetI,axiom,
! [B_107: int > $o,A_187: int > $o] :
( ! [X_1: int] :
( ( member_int @ X_1 @ A_187 )
=> ( member_int @ X_1 @ B_107 ) )
=> ( ord_less_eq_int_o @ A_187 @ B_107 ) ) ).
thf(fact_248_subsetI,axiom,
! [B_107: nat > $o,A_187: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_187 )
=> ( member_nat @ X_1 @ B_107 ) )
=> ( ord_less_eq_nat_o @ A_187 @ B_107 ) ) ).
thf(fact_249_le__funI,axiom,
! [F_58: hoare_1708887482_state > $o,G_1: hoare_1708887482_state > $o] :
( ! [X_1: hoare_1708887482_state] : ( ord_less_eq_o @ ( F_58 @ X_1 ) @ ( G_1 @ X_1 ) )
=> ( ord_le777019615tate_o @ F_58 @ G_1 ) ) ).
thf(fact_250_predicate1I,axiom,
! [Q_3: hoare_1708887482_state > $o,P_10: hoare_1708887482_state > $o] :
( ! [X_1: hoare_1708887482_state] :
( ( P_10 @ X_1 )
=> ( Q_3 @ X_1 ) )
=> ( ord_le777019615tate_o @ P_10 @ Q_3 ) ) ).
thf(fact_251_Collect__mono,axiom,
! [Q_2: hoare_1708887482_state > $o,P_9: hoare_1708887482_state > $o] :
( ! [X_1: hoare_1708887482_state] :
( ( P_9 @ X_1 )
=> ( Q_2 @ X_1 ) )
=> ( ord_le777019615tate_o @ ( collec1568722789_state @ P_9 ) @ ( collec1568722789_state @ Q_2 ) ) ) ).
thf(fact_252_xt2,axiom,
! [C_65: hoare_1708887482_state > $o,F_57: ( hoare_1708887482_state > $o ) > nat,B_106: hoare_1708887482_state > $o,A_186: nat] :
( ( ord_less_eq_nat @ ( F_57 @ B_106 ) @ A_186 )
=> ( ( ord_le777019615tate_o @ C_65 @ B_106 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_7 @ X_1 )
=> ( ord_less_eq_nat @ ( F_57 @ Y_7 ) @ ( F_57 @ X_1 ) ) )
=> ( ord_less_eq_nat @ ( F_57 @ C_65 ) @ A_186 ) ) ) ) ).
thf(fact_253_xt2,axiom,
! [C_65: hoare_1708887482_state > $o,F_57: ( hoare_1708887482_state > $o ) > int,B_106: hoare_1708887482_state > $o,A_186: int] :
( ( ord_less_eq_int @ ( F_57 @ B_106 ) @ A_186 )
=> ( ( ord_le777019615tate_o @ C_65 @ B_106 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_7 @ X_1 )
=> ( ord_less_eq_int @ ( F_57 @ Y_7 ) @ ( F_57 @ X_1 ) ) )
=> ( ord_less_eq_int @ ( F_57 @ C_65 ) @ A_186 ) ) ) ) ).
thf(fact_254_xt2,axiom,
! [C_65: nat,F_57: nat > hoare_1708887482_state > $o,B_106: nat,A_186: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ ( F_57 @ B_106 ) @ A_186 )
=> ( ( ord_less_eq_nat @ C_65 @ B_106 )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ Y_7 @ X_1 )
=> ( ord_le777019615tate_o @ ( F_57 @ Y_7 ) @ ( F_57 @ X_1 ) ) )
=> ( ord_le777019615tate_o @ ( F_57 @ C_65 ) @ A_186 ) ) ) ) ).
thf(fact_255_xt2,axiom,
! [C_65: int,F_57: int > hoare_1708887482_state > $o,B_106: int,A_186: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ ( F_57 @ B_106 ) @ A_186 )
=> ( ( ord_less_eq_int @ C_65 @ B_106 )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ Y_7 @ X_1 )
=> ( ord_le777019615tate_o @ ( F_57 @ Y_7 ) @ ( F_57 @ X_1 ) ) )
=> ( ord_le777019615tate_o @ ( F_57 @ C_65 ) @ A_186 ) ) ) ) ).
thf(fact_256_xt1_I15_J,axiom,
! [C_64: hoare_1708887482_state > $o,A_185: hoare_1708887482_state > $o,F_56: ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o,B_105: hoare_1708887482_state > $o] :
( ( A_185
= ( F_56 @ B_105 ) )
=> ( ( ord_le777019615tate_o @ C_64 @ B_105 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_7 @ X_1 )
=> ( ord_le777019615tate_o @ ( F_56 @ Y_7 ) @ ( F_56 @ X_1 ) ) )
=> ( ord_le777019615tate_o @ ( F_56 @ C_64 ) @ A_185 ) ) ) ) ).
thf(fact_257_xt1_I15_J,axiom,
! [C_64: nat,A_185: nat,F_56: nat > nat,B_105: nat] :
( ( A_185
= ( F_56 @ B_105 ) )
=> ( ( ord_less_eq_nat @ C_64 @ B_105 )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ Y_7 @ X_1 )
=> ( ord_less_eq_nat @ ( F_56 @ Y_7 ) @ ( F_56 @ X_1 ) ) )
=> ( ord_less_eq_nat @ ( F_56 @ C_64 ) @ A_185 ) ) ) ) ).
thf(fact_258_xt1_I15_J,axiom,
! [C_64: int,A_185: int,F_56: int > int,B_105: int] :
( ( A_185
= ( F_56 @ B_105 ) )
=> ( ( ord_less_eq_int @ C_64 @ B_105 )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ Y_7 @ X_1 )
=> ( ord_less_eq_int @ ( F_56 @ Y_7 ) @ ( F_56 @ X_1 ) ) )
=> ( ord_less_eq_int @ ( F_56 @ C_64 ) @ A_185 ) ) ) ) ).
thf(fact_259_order__subst1,axiom,
! [C_63: hoare_1708887482_state > $o,A_184: nat,F_55: ( hoare_1708887482_state > $o ) > nat,B_104: hoare_1708887482_state > $o] :
( ( ord_less_eq_nat @ A_184 @ ( F_55 @ B_104 ) )
=> ( ( ord_le777019615tate_o @ B_104 @ C_63 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_nat @ ( F_55 @ X_1 ) @ ( F_55 @ Y_7 ) ) )
=> ( ord_less_eq_nat @ A_184 @ ( F_55 @ C_63 ) ) ) ) ) ).
thf(fact_260_order__subst1,axiom,
! [C_63: hoare_1708887482_state > $o,A_184: int,F_55: ( hoare_1708887482_state > $o ) > int,B_104: hoare_1708887482_state > $o] :
( ( ord_less_eq_int @ A_184 @ ( F_55 @ B_104 ) )
=> ( ( ord_le777019615tate_o @ B_104 @ C_63 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_int @ ( F_55 @ X_1 ) @ ( F_55 @ Y_7 ) ) )
=> ( ord_less_eq_int @ A_184 @ ( F_55 @ C_63 ) ) ) ) ) ).
thf(fact_261_order__subst1,axiom,
! [C_63: nat,A_184: hoare_1708887482_state > $o,F_55: nat > hoare_1708887482_state > $o,B_104: nat] :
( ( ord_le777019615tate_o @ A_184 @ ( F_55 @ B_104 ) )
=> ( ( ord_less_eq_nat @ B_104 @ C_63 )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_55 @ X_1 ) @ ( F_55 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ A_184 @ ( F_55 @ C_63 ) ) ) ) ) ).
thf(fact_262_order__subst1,axiom,
! [C_63: int,A_184: hoare_1708887482_state > $o,F_55: int > hoare_1708887482_state > $o,B_104: int] :
( ( ord_le777019615tate_o @ A_184 @ ( F_55 @ B_104 ) )
=> ( ( ord_less_eq_int @ B_104 @ C_63 )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_55 @ X_1 ) @ ( F_55 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ A_184 @ ( F_55 @ C_63 ) ) ) ) ) ).
thf(fact_263_ord__eq__le__subst,axiom,
! [C_62: nat,A_183: hoare_1708887482_state > $o,F_54: nat > hoare_1708887482_state > $o,B_103: nat] :
( ( A_183
= ( F_54 @ B_103 ) )
=> ( ( ord_less_eq_nat @ B_103 @ C_62 )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_54 @ X_1 ) @ ( F_54 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ A_183 @ ( F_54 @ C_62 ) ) ) ) ) ).
thf(fact_264_ord__eq__le__subst,axiom,
! [C_62: int,A_183: hoare_1708887482_state > $o,F_54: int > hoare_1708887482_state > $o,B_103: int] :
( ( A_183
= ( F_54 @ B_103 ) )
=> ( ( ord_less_eq_int @ B_103 @ C_62 )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_54 @ X_1 ) @ ( F_54 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ A_183 @ ( F_54 @ C_62 ) ) ) ) ) ).
thf(fact_265_ord__eq__le__subst,axiom,
! [C_62: hoare_1708887482_state > $o,A_183: nat,F_54: ( hoare_1708887482_state > $o ) > nat,B_103: hoare_1708887482_state > $o] :
( ( A_183
= ( F_54 @ B_103 ) )
=> ( ( ord_le777019615tate_o @ B_103 @ C_62 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_nat @ ( F_54 @ X_1 ) @ ( F_54 @ Y_7 ) ) )
=> ( ord_less_eq_nat @ A_183 @ ( F_54 @ C_62 ) ) ) ) ) ).
thf(fact_266_ord__eq__le__subst,axiom,
! [C_62: hoare_1708887482_state > $o,A_183: int,F_54: ( hoare_1708887482_state > $o ) > int,B_103: hoare_1708887482_state > $o] :
( ( A_183
= ( F_54 @ B_103 ) )
=> ( ( ord_le777019615tate_o @ B_103 @ C_62 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_int @ ( F_54 @ X_1 ) @ ( F_54 @ Y_7 ) ) )
=> ( ord_less_eq_int @ A_183 @ ( F_54 @ C_62 ) ) ) ) ) ).
thf(fact_267_order__subst2,axiom,
! [F_53: nat > hoare_1708887482_state > $o,C_61: hoare_1708887482_state > $o,A_182: nat,B_102: nat] :
( ( ord_less_eq_nat @ A_182 @ B_102 )
=> ( ( ord_le777019615tate_o @ ( F_53 @ B_102 ) @ C_61 )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_53 @ X_1 ) @ ( F_53 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ ( F_53 @ A_182 ) @ C_61 ) ) ) ) ).
thf(fact_268_order__subst2,axiom,
! [F_53: int > hoare_1708887482_state > $o,C_61: hoare_1708887482_state > $o,A_182: int,B_102: int] :
( ( ord_less_eq_int @ A_182 @ B_102 )
=> ( ( ord_le777019615tate_o @ ( F_53 @ B_102 ) @ C_61 )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_53 @ X_1 ) @ ( F_53 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ ( F_53 @ A_182 ) @ C_61 ) ) ) ) ).
thf(fact_269_order__subst2,axiom,
! [F_53: ( hoare_1708887482_state > $o ) > nat,C_61: nat,A_182: hoare_1708887482_state > $o,B_102: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_182 @ B_102 )
=> ( ( ord_less_eq_nat @ ( F_53 @ B_102 ) @ C_61 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_nat @ ( F_53 @ X_1 ) @ ( F_53 @ Y_7 ) ) )
=> ( ord_less_eq_nat @ ( F_53 @ A_182 ) @ C_61 ) ) ) ) ).
thf(fact_270_order__subst2,axiom,
! [F_53: ( hoare_1708887482_state > $o ) > int,C_61: int,A_182: hoare_1708887482_state > $o,B_102: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_182 @ B_102 )
=> ( ( ord_less_eq_int @ ( F_53 @ B_102 ) @ C_61 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_int @ ( F_53 @ X_1 ) @ ( F_53 @ Y_7 ) ) )
=> ( ord_less_eq_int @ ( F_53 @ A_182 ) @ C_61 ) ) ) ) ).
thf(fact_271_ord__le__eq__subst,axiom,
! [F_52: nat > hoare_1708887482_state > $o,C_60: hoare_1708887482_state > $o,A_181: nat,B_101: nat] :
( ( ord_less_eq_nat @ A_181 @ B_101 )
=> ( ( ( F_52 @ B_101 )
= C_60 )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_52 @ X_1 ) @ ( F_52 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ ( F_52 @ A_181 ) @ C_60 ) ) ) ) ).
thf(fact_272_ord__le__eq__subst,axiom,
! [F_52: int > hoare_1708887482_state > $o,C_60: hoare_1708887482_state > $o,A_181: int,B_101: int] :
( ( ord_less_eq_int @ A_181 @ B_101 )
=> ( ( ( F_52 @ B_101 )
= C_60 )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ X_1 @ Y_7 )
=> ( ord_le777019615tate_o @ ( F_52 @ X_1 ) @ ( F_52 @ Y_7 ) ) )
=> ( ord_le777019615tate_o @ ( F_52 @ A_181 ) @ C_60 ) ) ) ) ).
thf(fact_273_ord__le__eq__subst,axiom,
! [F_52: ( hoare_1708887482_state > $o ) > nat,C_60: nat,A_181: hoare_1708887482_state > $o,B_101: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_181 @ B_101 )
=> ( ( ( F_52 @ B_101 )
= C_60 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_nat @ ( F_52 @ X_1 ) @ ( F_52 @ Y_7 ) ) )
=> ( ord_less_eq_nat @ ( F_52 @ A_181 ) @ C_60 ) ) ) ) ).
thf(fact_274_ord__le__eq__subst,axiom,
! [F_52: ( hoare_1708887482_state > $o ) > int,C_60: int,A_181: hoare_1708887482_state > $o,B_101: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_181 @ B_101 )
=> ( ( ( F_52 @ B_101 )
= C_60 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_1 @ Y_7 )
=> ( ord_less_eq_int @ ( F_52 @ X_1 ) @ ( F_52 @ Y_7 ) ) )
=> ( ord_less_eq_int @ ( F_52 @ A_181 ) @ C_60 ) ) ) ) ).
thf(fact_275_xt3,axiom,
! [C_59: hoare_1708887482_state > $o,F_51: nat > hoare_1708887482_state > $o,B_100: nat,A_180: nat] :
( ( ord_less_eq_nat @ B_100 @ A_180 )
=> ( ( ord_le777019615tate_o @ C_59 @ ( F_51 @ B_100 ) )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ Y_7 @ X_1 )
=> ( ord_le777019615tate_o @ ( F_51 @ Y_7 ) @ ( F_51 @ X_1 ) ) )
=> ( ord_le777019615tate_o @ C_59 @ ( F_51 @ A_180 ) ) ) ) ) ).
thf(fact_276_xt3,axiom,
! [C_59: hoare_1708887482_state > $o,F_51: int > hoare_1708887482_state > $o,B_100: int,A_180: int] :
( ( ord_less_eq_int @ B_100 @ A_180 )
=> ( ( ord_le777019615tate_o @ C_59 @ ( F_51 @ B_100 ) )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ Y_7 @ X_1 )
=> ( ord_le777019615tate_o @ ( F_51 @ Y_7 ) @ ( F_51 @ X_1 ) ) )
=> ( ord_le777019615tate_o @ C_59 @ ( F_51 @ A_180 ) ) ) ) ) ).
thf(fact_277_xt3,axiom,
! [C_59: nat,F_51: ( hoare_1708887482_state > $o ) > nat,B_100: hoare_1708887482_state > $o,A_180: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ B_100 @ A_180 )
=> ( ( ord_less_eq_nat @ C_59 @ ( F_51 @ B_100 ) )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_7 @ X_1 )
=> ( ord_less_eq_nat @ ( F_51 @ Y_7 ) @ ( F_51 @ X_1 ) ) )
=> ( ord_less_eq_nat @ C_59 @ ( F_51 @ A_180 ) ) ) ) ) ).
thf(fact_278_xt3,axiom,
! [C_59: int,F_51: ( hoare_1708887482_state > $o ) > int,B_100: hoare_1708887482_state > $o,A_180: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ B_100 @ A_180 )
=> ( ( ord_less_eq_int @ C_59 @ ( F_51 @ B_100 ) )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_7 @ X_1 )
=> ( ord_less_eq_int @ ( F_51 @ Y_7 ) @ ( F_51 @ X_1 ) ) )
=> ( ord_less_eq_int @ C_59 @ ( F_51 @ A_180 ) ) ) ) ) ).
thf(fact_279_xt1_I16_J,axiom,
! [F_50: ( hoare_1708887482_state > $o ) > hoare_1708887482_state > $o,C_58: hoare_1708887482_state > $o,B_99: hoare_1708887482_state > $o,A_179: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ B_99 @ A_179 )
=> ( ( ( F_50 @ B_99 )
= C_58 )
=> ( ! [X_1: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_7 @ X_1 )
=> ( ord_le777019615tate_o @ ( F_50 @ Y_7 ) @ ( F_50 @ X_1 ) ) )
=> ( ord_le777019615tate_o @ C_58 @ ( F_50 @ A_179 ) ) ) ) ) ).
thf(fact_280_xt1_I16_J,axiom,
! [F_50: nat > nat,C_58: nat,B_99: nat,A_179: nat] :
( ( ord_less_eq_nat @ B_99 @ A_179 )
=> ( ( ( F_50 @ B_99 )
= C_58 )
=> ( ! [X_1: nat,Y_7: nat] :
( ( ord_less_eq_nat @ Y_7 @ X_1 )
=> ( ord_less_eq_nat @ ( F_50 @ Y_7 ) @ ( F_50 @ X_1 ) ) )
=> ( ord_less_eq_nat @ C_58 @ ( F_50 @ A_179 ) ) ) ) ) ).
thf(fact_281_xt1_I16_J,axiom,
! [F_50: int > int,C_58: int,B_99: int,A_179: int] :
( ( ord_less_eq_int @ B_99 @ A_179 )
=> ( ( ( F_50 @ B_99 )
= C_58 )
=> ( ! [X_1: int,Y_7: int] :
( ( ord_less_eq_int @ Y_7 @ X_1 )
=> ( ord_less_eq_int @ ( F_50 @ Y_7 ) @ ( F_50 @ X_1 ) ) )
=> ( ord_less_eq_int @ C_58 @ ( F_50 @ A_179 ) ) ) ) ) ).
thf(fact_282_fold1Set_Ointros,axiom,
! [F_49: int > int > int,A_178: int,A_177: int > $o,X_78: int] :
( ( finite772772422nt_int @ F_49 @ A_178 @ A_177 @ X_78 )
=> ( ~ ( member_int @ A_178 @ A_177 )
=> ( finite_fold1Set_int @ F_49 @ ( insert_int @ A_178 @ A_177 ) @ X_78 ) ) ) ).
thf(fact_283_fold1Set_Ointros,axiom,
! [F_49: nat > nat > nat,A_178: nat,A_177: nat > $o,X_78: nat] :
( ( finite929467206at_nat @ F_49 @ A_178 @ A_177 @ X_78 )
=> ( ~ ( member_nat @ A_178 @ A_177 )
=> ( finite_fold1Set_nat @ F_49 @ ( insert_nat @ A_178 @ A_177 ) @ X_78 ) ) ) ).
thf(fact_284_fold1Set_Ointros,axiom,
! [F_49: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,A_178: hoare_1708887482_state,A_177: hoare_1708887482_state > $o,X_78: hoare_1708887482_state] :
( ( finite2042206440_state @ F_49 @ A_178 @ A_177 @ X_78 )
=> ( ~ ( member451959335_state @ A_178 @ A_177 )
=> ( finite850270954_state @ F_49 @ ( insert528405184_state @ A_178 @ A_177 ) @ X_78 ) ) ) ).
thf(fact_285_folding__one_Oinsert,axiom,
! [X_77: int,A_176: int > $o,F_48: int > int > int,F_47: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_48 @ F_47 )
=> ( ( finite_finite_int @ A_176 )
=> ( ~ ( member_int @ X_77 @ A_176 )
=> ( ( A_176 != bot_bot_int_o )
=> ( ( F_47 @ ( insert_int @ X_77 @ A_176 ) )
= ( F_48 @ X_77 @ ( F_47 @ A_176 ) ) ) ) ) ) ) ).
thf(fact_286_folding__one_Oinsert,axiom,
! [X_77: nat,A_176: nat > $o,F_48: nat > nat > nat,F_47: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_48 @ F_47 )
=> ( ( finite_finite_nat @ A_176 )
=> ( ~ ( member_nat @ X_77 @ A_176 )
=> ( ( A_176 != bot_bot_nat_o )
=> ( ( F_47 @ ( insert_nat @ X_77 @ A_176 ) )
= ( F_48 @ X_77 @ ( F_47 @ A_176 ) ) ) ) ) ) ) ).
thf(fact_287_folding__one_Oinsert,axiom,
! [X_77: hoare_1708887482_state,A_176: hoare_1708887482_state > $o,F_48: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_47: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_48 @ F_47 )
=> ( ( finite1625599783_state @ A_176 )
=> ( ~ ( member451959335_state @ X_77 @ A_176 )
=> ( ( A_176 != bot_bo19817387tate_o )
=> ( ( F_47 @ ( insert528405184_state @ X_77 @ A_176 ) )
= ( F_48 @ X_77 @ ( F_47 @ A_176 ) ) ) ) ) ) ) ).
thf(fact_288_Powp__mono,axiom,
! [A_175: hoare_1708887482_state > $o,B_98: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_175 @ B_98 )
=> ( ord_le1728773982te_o_o @ ( powp_H189553652_state @ A_175 ) @ ( powp_H189553652_state @ B_98 ) ) ) ).
thf(fact_289_subset__insert__iff,axiom,
! [A_174: int > $o,X_76: int,B_97: int > $o] :
( ( ord_less_eq_int_o @ A_174 @ ( insert_int @ X_76 @ B_97 ) )
<=> ( ( ( member_int @ X_76 @ A_174 )
=> ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_174 @ ( insert_int @ X_76 @ bot_bot_int_o ) ) @ B_97 ) )
& ( ~ ( member_int @ X_76 @ A_174 )
=> ( ord_less_eq_int_o @ A_174 @ B_97 ) ) ) ) ).
thf(fact_290_subset__insert__iff,axiom,
! [A_174: nat > $o,X_76: nat,B_97: nat > $o] :
( ( ord_less_eq_nat_o @ A_174 @ ( insert_nat @ X_76 @ B_97 ) )
<=> ( ( ( member_nat @ X_76 @ A_174 )
=> ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_174 @ ( insert_nat @ X_76 @ bot_bot_nat_o ) ) @ B_97 ) )
& ( ~ ( member_nat @ X_76 @ A_174 )
=> ( ord_less_eq_nat_o @ A_174 @ B_97 ) ) ) ) ).
thf(fact_291_subset__insert__iff,axiom,
! [A_174: hoare_1708887482_state > $o,X_76: hoare_1708887482_state,B_97: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_174 @ ( insert528405184_state @ X_76 @ B_97 ) )
<=> ( ( ( member451959335_state @ X_76 @ A_174 )
=> ( ord_le777019615tate_o @ ( minus_2056855718tate_o @ A_174 @ ( insert528405184_state @ X_76 @ bot_bo19817387tate_o ) ) @ B_97 ) )
& ( ~ ( member451959335_state @ X_76 @ A_174 )
=> ( ord_le777019615tate_o @ A_174 @ B_97 ) ) ) ) ).
thf(fact_292_diff__single__insert,axiom,
! [A_173: int > $o,X_75: int,B_96: int > $o] :
( ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_173 @ ( insert_int @ X_75 @ bot_bot_int_o ) ) @ B_96 )
=> ( ( member_int @ X_75 @ A_173 )
=> ( ord_less_eq_int_o @ A_173 @ ( insert_int @ X_75 @ B_96 ) ) ) ) ).
thf(fact_293_diff__single__insert,axiom,
! [A_173: nat > $o,X_75: nat,B_96: nat > $o] :
( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_173 @ ( insert_nat @ X_75 @ bot_bot_nat_o ) ) @ B_96 )
=> ( ( member_nat @ X_75 @ A_173 )
=> ( ord_less_eq_nat_o @ A_173 @ ( insert_nat @ X_75 @ B_96 ) ) ) ) ).
thf(fact_294_diff__single__insert,axiom,
! [A_173: hoare_1708887482_state > $o,X_75: hoare_1708887482_state,B_96: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ ( minus_2056855718tate_o @ A_173 @ ( insert528405184_state @ X_75 @ bot_bo19817387tate_o ) ) @ B_96 )
=> ( ( member451959335_state @ X_75 @ A_173 )
=> ( ord_le777019615tate_o @ A_173 @ ( insert528405184_state @ X_75 @ B_96 ) ) ) ) ).
thf(fact_295_folding__one_Oeq__fold,axiom,
! [A_172: int > $o,F_46: int > int > int,F_45: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_46 @ F_45 )
=> ( ( finite_finite_int @ A_172 )
=> ( ( F_45 @ A_172 )
= ( finite_fold1_int @ F_46 @ A_172 ) ) ) ) ).
thf(fact_296_folding__one_Oeq__fold,axiom,
! [A_172: nat > $o,F_46: nat > nat > nat,F_45: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_46 @ F_45 )
=> ( ( finite_finite_nat @ A_172 )
=> ( ( F_45 @ A_172 )
= ( finite_fold1_nat @ F_46 @ A_172 ) ) ) ) ).
thf(fact_297_folding__one_Oeq__fold,axiom,
! [A_172: hoare_1708887482_state > $o,F_46: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_45: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_46 @ F_45 )
=> ( ( finite1625599783_state @ A_172 )
=> ( ( F_45 @ A_172 )
= ( finite39664218_state @ F_46 @ A_172 ) ) ) ) ).
thf(fact_298_folding__one_Oclosed,axiom,
! [A_171: int > $o,F_44: int > int > int,F_43: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_44 @ F_43 )
=> ( ( finite_finite_int @ A_171 )
=> ( ( A_171 != bot_bot_int_o )
=> ( ! [X_1: int,Y_7: int] : ( member_int @ ( F_44 @ X_1 @ Y_7 ) @ ( insert_int @ X_1 @ ( insert_int @ Y_7 @ bot_bot_int_o ) ) )
=> ( member_int @ ( F_43 @ A_171 ) @ A_171 ) ) ) ) ) ).
thf(fact_299_folding__one_Oclosed,axiom,
! [A_171: nat > $o,F_44: nat > nat > nat,F_43: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_44 @ F_43 )
=> ( ( finite_finite_nat @ A_171 )
=> ( ( A_171 != bot_bot_nat_o )
=> ( ! [X_1: nat,Y_7: nat] : ( member_nat @ ( F_44 @ X_1 @ Y_7 ) @ ( insert_nat @ X_1 @ ( insert_nat @ Y_7 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( F_43 @ A_171 ) @ A_171 ) ) ) ) ) ).
thf(fact_300_folding__one_Oclosed,axiom,
! [A_171: hoare_1708887482_state > $o,F_44: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_43: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_44 @ F_43 )
=> ( ( finite1625599783_state @ A_171 )
=> ( ( A_171 != bot_bo19817387tate_o )
=> ( ! [X_1: hoare_1708887482_state,Y_7: hoare_1708887482_state] : ( member451959335_state @ ( F_44 @ X_1 @ Y_7 ) @ ( insert528405184_state @ X_1 @ ( insert528405184_state @ Y_7 @ bot_bo19817387tate_o ) ) )
=> ( member451959335_state @ ( F_43 @ A_171 ) @ A_171 ) ) ) ) ) ).
thf(fact_301_DiffE,axiom,
! [C_57: hoare_1708887482_state,A_170: hoare_1708887482_state > $o,B_95: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_57 @ ( minus_2056855718tate_o @ A_170 @ B_95 ) )
=> ~ ( ( member451959335_state @ C_57 @ A_170 )
=> ( member451959335_state @ C_57 @ B_95 ) ) ) ).
thf(fact_302_DiffE,axiom,
! [C_57: int,A_170: int > $o,B_95: int > $o] :
( ( member_int @ C_57 @ ( minus_minus_int_o @ A_170 @ B_95 ) )
=> ~ ( ( member_int @ C_57 @ A_170 )
=> ( member_int @ C_57 @ B_95 ) ) ) ).
thf(fact_303_DiffE,axiom,
! [C_57: nat,A_170: nat > $o,B_95: nat > $o] :
( ( member_nat @ C_57 @ ( minus_minus_nat_o @ A_170 @ B_95 ) )
=> ~ ( ( member_nat @ C_57 @ A_170 )
=> ( member_nat @ C_57 @ B_95 ) ) ) ).
thf(fact_304_DiffI,axiom,
! [B_94: hoare_1708887482_state > $o,C_56: hoare_1708887482_state,A_169: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_56 @ A_169 )
=> ( ~ ( member451959335_state @ C_56 @ B_94 )
=> ( member451959335_state @ C_56 @ ( minus_2056855718tate_o @ A_169 @ B_94 ) ) ) ) ).
thf(fact_305_DiffI,axiom,
! [B_94: int > $o,C_56: int,A_169: int > $o] :
( ( member_int @ C_56 @ A_169 )
=> ( ~ ( member_int @ C_56 @ B_94 )
=> ( member_int @ C_56 @ ( minus_minus_int_o @ A_169 @ B_94 ) ) ) ) ).
thf(fact_306_DiffI,axiom,
! [B_94: nat > $o,C_56: nat,A_169: nat > $o] :
( ( member_nat @ C_56 @ A_169 )
=> ( ~ ( member_nat @ C_56 @ B_94 )
=> ( member_nat @ C_56 @ ( minus_minus_nat_o @ A_169 @ B_94 ) ) ) ) ).
thf(fact_307_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_int_o ).
thf(fact_308_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_nat_o ).
thf(fact_309_finite_OemptyI,axiom,
finite1625599783_state @ bot_bo19817387tate_o ).
thf(fact_310_finite_OinsertI,axiom,
! [A_168: int,A_167: int > $o] :
( ( finite_finite_int @ A_167 )
=> ( finite_finite_int @ ( insert_int @ A_168 @ A_167 ) ) ) ).
thf(fact_311_finite_OinsertI,axiom,
! [A_168: nat,A_167: nat > $o] :
( ( finite_finite_nat @ A_167 )
=> ( finite_finite_nat @ ( insert_nat @ A_168 @ A_167 ) ) ) ).
thf(fact_312_finite_OinsertI,axiom,
! [A_168: hoare_1708887482_state,A_167: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_167 )
=> ( finite1625599783_state @ ( insert528405184_state @ A_168 @ A_167 ) ) ) ).
thf(fact_313_finite__Diff,axiom,
! [B_93: hoare_1708887482_state > $o,A_166: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_166 )
=> ( finite1625599783_state @ ( minus_2056855718tate_o @ A_166 @ B_93 ) ) ) ).
thf(fact_314_finite__Diff,axiom,
! [B_93: int > $o,A_166: int > $o] :
( ( finite_finite_int @ A_166 )
=> ( finite_finite_int @ ( minus_minus_int_o @ A_166 @ B_93 ) ) ) ).
thf(fact_315_finite__Diff,axiom,
! [B_93: nat > $o,A_166: nat > $o] :
( ( finite_finite_nat @ A_166 )
=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_166 @ B_93 ) ) ) ).
thf(fact_316_finite__Diff__insert,axiom,
! [A_165: int > $o,A_164: int,B_92: int > $o] :
( ( finite_finite_int @ ( minus_minus_int_o @ A_165 @ ( insert_int @ A_164 @ B_92 ) ) )
<=> ( finite_finite_int @ ( minus_minus_int_o @ A_165 @ B_92 ) ) ) ).
thf(fact_317_finite__Diff__insert,axiom,
! [A_165: nat > $o,A_164: nat,B_92: nat > $o] :
( ( finite_finite_nat @ ( minus_minus_nat_o @ A_165 @ ( insert_nat @ A_164 @ B_92 ) ) )
<=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_165 @ B_92 ) ) ) ).
thf(fact_318_finite__Diff__insert,axiom,
! [A_165: hoare_1708887482_state > $o,A_164: hoare_1708887482_state,B_92: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ ( minus_2056855718tate_o @ A_165 @ ( insert528405184_state @ A_164 @ B_92 ) ) )
<=> ( finite1625599783_state @ ( minus_2056855718tate_o @ A_165 @ B_92 ) ) ) ).
thf(fact_319_finite__Diff2,axiom,
! [A_163: hoare_1708887482_state > $o,B_91: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_91 )
=> ( ( finite1625599783_state @ ( minus_2056855718tate_o @ A_163 @ B_91 ) )
<=> ( finite1625599783_state @ A_163 ) ) ) ).
thf(fact_320_finite__Diff2,axiom,
! [A_163: int > $o,B_91: int > $o] :
( ( finite_finite_int @ B_91 )
=> ( ( finite_finite_int @ ( minus_minus_int_o @ A_163 @ B_91 ) )
<=> ( finite_finite_int @ A_163 ) ) ) ).
thf(fact_321_finite__Diff2,axiom,
! [A_163: nat > $o,B_91: nat > $o] :
( ( finite_finite_nat @ B_91 )
=> ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_163 @ B_91 ) )
<=> ( finite_finite_nat @ A_163 ) ) ) ).
thf(fact_322_DiffD2,axiom,
! [C_55: hoare_1708887482_state,A_162: hoare_1708887482_state > $o,B_90: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_55 @ ( minus_2056855718tate_o @ A_162 @ B_90 ) )
=> ~ ( member451959335_state @ C_55 @ B_90 ) ) ).
thf(fact_323_DiffD2,axiom,
! [C_55: int,A_162: int > $o,B_90: int > $o] :
( ( member_int @ C_55 @ ( minus_minus_int_o @ A_162 @ B_90 ) )
=> ~ ( member_int @ C_55 @ B_90 ) ) ).
thf(fact_324_DiffD2,axiom,
! [C_55: nat,A_162: nat > $o,B_90: nat > $o] :
( ( member_nat @ C_55 @ ( minus_minus_nat_o @ A_162 @ B_90 ) )
=> ~ ( member_nat @ C_55 @ B_90 ) ) ).
thf(fact_325_DiffD1,axiom,
! [C_54: hoare_1708887482_state,A_161: hoare_1708887482_state > $o,B_89: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_54 @ ( minus_2056855718tate_o @ A_161 @ B_89 ) )
=> ( member451959335_state @ C_54 @ A_161 ) ) ).
thf(fact_326_DiffD1,axiom,
! [C_54: int,A_161: int > $o,B_89: int > $o] :
( ( member_int @ C_54 @ ( minus_minus_int_o @ A_161 @ B_89 ) )
=> ( member_int @ C_54 @ A_161 ) ) ).
thf(fact_327_DiffD1,axiom,
! [C_54: nat,A_161: nat > $o,B_89: nat > $o] :
( ( member_nat @ C_54 @ ( minus_minus_nat_o @ A_161 @ B_89 ) )
=> ( member_nat @ C_54 @ A_161 ) ) ).
thf(fact_328_Diff__idemp,axiom,
! [A_160: hoare_1708887482_state > $o,B_88: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ ( minus_2056855718tate_o @ A_160 @ B_88 ) @ B_88 )
= ( minus_2056855718tate_o @ A_160 @ B_88 ) ) ).
thf(fact_329_Diff__idemp,axiom,
! [A_160: int > $o,B_88: int > $o] :
( ( minus_minus_int_o @ ( minus_minus_int_o @ A_160 @ B_88 ) @ B_88 )
= ( minus_minus_int_o @ A_160 @ B_88 ) ) ).
thf(fact_330_Diff__idemp,axiom,
! [A_160: nat > $o,B_88: nat > $o] :
( ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_160 @ B_88 ) @ B_88 )
= ( minus_minus_nat_o @ A_160 @ B_88 ) ) ).
thf(fact_331_Diff__iff,axiom,
! [C_53: hoare_1708887482_state,A_159: hoare_1708887482_state > $o,B_87: hoare_1708887482_state > $o] :
( ( member451959335_state @ C_53 @ ( minus_2056855718tate_o @ A_159 @ B_87 ) )
<=> ( ( member451959335_state @ C_53 @ A_159 )
& ~ ( member451959335_state @ C_53 @ B_87 ) ) ) ).
thf(fact_332_Diff__iff,axiom,
! [C_53: int,A_159: int > $o,B_87: int > $o] :
( ( member_int @ C_53 @ ( minus_minus_int_o @ A_159 @ B_87 ) )
<=> ( ( member_int @ C_53 @ A_159 )
& ~ ( member_int @ C_53 @ B_87 ) ) ) ).
thf(fact_333_Diff__iff,axiom,
! [C_53: nat,A_159: nat > $o,B_87: nat > $o] :
( ( member_nat @ C_53 @ ( minus_minus_nat_o @ A_159 @ B_87 ) )
<=> ( ( member_nat @ C_53 @ A_159 )
& ~ ( member_nat @ C_53 @ B_87 ) ) ) ).
thf(fact_334_set__diff__eq,axiom,
! [A_158: hoare_1708887482_state > $o,B_86: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ A_158 @ B_86 )
= ( collec1568722789_state
@ ^ [X_1: hoare_1708887482_state] : ( (&) @ ( member451959335_state @ X_1 @ A_158 ) @ ( (~) @ ( member451959335_state @ X_1 @ B_86 ) ) ) ) ) ).
thf(fact_335_set__diff__eq,axiom,
! [A_158: int > $o,B_86: int > $o] :
( ( minus_minus_int_o @ A_158 @ B_86 )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ A_158 ) @ ( (~) @ ( member_int @ X_1 @ B_86 ) ) ) ) ) ).
thf(fact_336_set__diff__eq,axiom,
! [A_158: nat > $o,B_86: nat > $o] :
( ( minus_minus_nat_o @ A_158 @ B_86 )
= ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ A_158 ) @ ( (~) @ ( member_nat @ X_1 @ B_86 ) ) ) ) ) ).
thf(fact_337_Diff__cancel,axiom,
! [A_157: int > $o] :
( ( minus_minus_int_o @ A_157 @ A_157 )
= bot_bot_int_o ) ).
thf(fact_338_Diff__cancel,axiom,
! [A_157: nat > $o] :
( ( minus_minus_nat_o @ A_157 @ A_157 )
= bot_bot_nat_o ) ).
thf(fact_339_Diff__cancel,axiom,
! [A_157: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ A_157 @ A_157 )
= bot_bo19817387tate_o ) ).
thf(fact_340_Diff__empty,axiom,
! [A_156: int > $o] :
( ( minus_minus_int_o @ A_156 @ bot_bot_int_o )
= A_156 ) ).
thf(fact_341_Diff__empty,axiom,
! [A_156: nat > $o] :
( ( minus_minus_nat_o @ A_156 @ bot_bot_nat_o )
= A_156 ) ).
thf(fact_342_Diff__empty,axiom,
! [A_156: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ A_156 @ bot_bo19817387tate_o )
= A_156 ) ).
thf(fact_343_empty__Diff,axiom,
! [A_155: int > $o] :
( ( minus_minus_int_o @ bot_bot_int_o @ A_155 )
= bot_bot_int_o ) ).
thf(fact_344_empty__Diff,axiom,
! [A_155: nat > $o] :
( ( minus_minus_nat_o @ bot_bot_nat_o @ A_155 )
= bot_bot_nat_o ) ).
thf(fact_345_empty__Diff,axiom,
! [A_155: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ bot_bo19817387tate_o @ A_155 )
= bot_bo19817387tate_o ) ).
thf(fact_346_insert__Diff1,axiom,
! [A_154: int > $o,X_74: int,B_85: int > $o] :
( ( member_int @ X_74 @ B_85 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_74 @ A_154 ) @ B_85 )
= ( minus_minus_int_o @ A_154 @ B_85 ) ) ) ).
thf(fact_347_insert__Diff1,axiom,
! [A_154: nat > $o,X_74: nat,B_85: nat > $o] :
( ( member_nat @ X_74 @ B_85 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_74 @ A_154 ) @ B_85 )
= ( minus_minus_nat_o @ A_154 @ B_85 ) ) ) ).
thf(fact_348_insert__Diff1,axiom,
! [A_154: hoare_1708887482_state > $o,X_74: hoare_1708887482_state,B_85: hoare_1708887482_state > $o] :
( ( member451959335_state @ X_74 @ B_85 )
=> ( ( minus_2056855718tate_o @ ( insert528405184_state @ X_74 @ A_154 ) @ B_85 )
= ( minus_2056855718tate_o @ A_154 @ B_85 ) ) ) ).
thf(fact_349_insert__Diff__if,axiom,
! [A_153: int > $o,X_73: int,B_84: int > $o] :
( ( ( member_int @ X_73 @ B_84 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_73 @ A_153 ) @ B_84 )
= ( minus_minus_int_o @ A_153 @ B_84 ) ) )
& ( ~ ( member_int @ X_73 @ B_84 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_73 @ A_153 ) @ B_84 )
= ( insert_int @ X_73 @ ( minus_minus_int_o @ A_153 @ B_84 ) ) ) ) ) ).
thf(fact_350_insert__Diff__if,axiom,
! [A_153: nat > $o,X_73: nat,B_84: nat > $o] :
( ( ( member_nat @ X_73 @ B_84 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_73 @ A_153 ) @ B_84 )
= ( minus_minus_nat_o @ A_153 @ B_84 ) ) )
& ( ~ ( member_nat @ X_73 @ B_84 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_73 @ A_153 ) @ B_84 )
= ( insert_nat @ X_73 @ ( minus_minus_nat_o @ A_153 @ B_84 ) ) ) ) ) ).
thf(fact_351_insert__Diff__if,axiom,
! [A_153: hoare_1708887482_state > $o,X_73: hoare_1708887482_state,B_84: hoare_1708887482_state > $o] :
( ( ( member451959335_state @ X_73 @ B_84 )
=> ( ( minus_2056855718tate_o @ ( insert528405184_state @ X_73 @ A_153 ) @ B_84 )
= ( minus_2056855718tate_o @ A_153 @ B_84 ) ) )
& ( ~ ( member451959335_state @ X_73 @ B_84 )
=> ( ( minus_2056855718tate_o @ ( insert528405184_state @ X_73 @ A_153 ) @ B_84 )
= ( insert528405184_state @ X_73 @ ( minus_2056855718tate_o @ A_153 @ B_84 ) ) ) ) ) ).
thf(fact_352_double__diff,axiom,
! [C_52: hoare_1708887482_state > $o,A_152: hoare_1708887482_state > $o,B_83: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_152 @ B_83 )
=> ( ( ord_le777019615tate_o @ B_83 @ C_52 )
=> ( ( minus_2056855718tate_o @ B_83 @ ( minus_2056855718tate_o @ C_52 @ A_152 ) )
= A_152 ) ) ) ).
thf(fact_353_double__diff,axiom,
! [C_52: int > $o,A_152: int > $o,B_83: int > $o] :
( ( ord_less_eq_int_o @ A_152 @ B_83 )
=> ( ( ord_less_eq_int_o @ B_83 @ C_52 )
=> ( ( minus_minus_int_o @ B_83 @ ( minus_minus_int_o @ C_52 @ A_152 ) )
= A_152 ) ) ) ).
thf(fact_354_double__diff,axiom,
! [C_52: nat > $o,A_152: nat > $o,B_83: nat > $o] :
( ( ord_less_eq_nat_o @ A_152 @ B_83 )
=> ( ( ord_less_eq_nat_o @ B_83 @ C_52 )
=> ( ( minus_minus_nat_o @ B_83 @ ( minus_minus_nat_o @ C_52 @ A_152 ) )
= A_152 ) ) ) ).
thf(fact_355_Diff__mono,axiom,
! [D_19: hoare_1708887482_state > $o,B_82: hoare_1708887482_state > $o,A_151: hoare_1708887482_state > $o,C_51: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_151 @ C_51 )
=> ( ( ord_le777019615tate_o @ D_19 @ B_82 )
=> ( ord_le777019615tate_o @ ( minus_2056855718tate_o @ A_151 @ B_82 ) @ ( minus_2056855718tate_o @ C_51 @ D_19 ) ) ) ) ).
thf(fact_356_Diff__mono,axiom,
! [D_19: int > $o,B_82: int > $o,A_151: int > $o,C_51: int > $o] :
( ( ord_less_eq_int_o @ A_151 @ C_51 )
=> ( ( ord_less_eq_int_o @ D_19 @ B_82 )
=> ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_151 @ B_82 ) @ ( minus_minus_int_o @ C_51 @ D_19 ) ) ) ) ).
thf(fact_357_Diff__mono,axiom,
! [D_19: nat > $o,B_82: nat > $o,A_151: nat > $o,C_51: nat > $o] :
( ( ord_less_eq_nat_o @ A_151 @ C_51 )
=> ( ( ord_less_eq_nat_o @ D_19 @ B_82 )
=> ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_151 @ B_82 ) @ ( minus_minus_nat_o @ C_51 @ D_19 ) ) ) ) ).
thf(fact_358_Diff__subset,axiom,
! [A_150: hoare_1708887482_state > $o,B_81: hoare_1708887482_state > $o] : ( ord_le777019615tate_o @ ( minus_2056855718tate_o @ A_150 @ B_81 ) @ A_150 ) ).
thf(fact_359_Diff__subset,axiom,
! [A_150: int > $o,B_81: int > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_150 @ B_81 ) @ A_150 ) ).
thf(fact_360_Diff__subset,axiom,
! [A_150: nat > $o,B_81: nat > $o] : ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_150 @ B_81 ) @ A_150 ) ).
thf(fact_361_finite__insert,axiom,
! [A_149: int,A_148: int > $o] :
( ( finite_finite_int @ ( insert_int @ A_149 @ A_148 ) )
<=> ( finite_finite_int @ A_148 ) ) ).
thf(fact_362_finite__insert,axiom,
! [A_149: nat,A_148: nat > $o] :
( ( finite_finite_nat @ ( insert_nat @ A_149 @ A_148 ) )
<=> ( finite_finite_nat @ A_148 ) ) ).
thf(fact_363_finite__insert,axiom,
! [A_149: hoare_1708887482_state,A_148: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ ( insert528405184_state @ A_149 @ A_148 ) )
<=> ( finite1625599783_state @ A_148 ) ) ).
thf(fact_364_finite__subset,axiom,
! [A_147: hoare_1708887482_state > $o,B_80: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_147 @ B_80 )
=> ( ( finite1625599783_state @ B_80 )
=> ( finite1625599783_state @ A_147 ) ) ) ).
thf(fact_365_finite__subset,axiom,
! [A_147: int > $o,B_80: int > $o] :
( ( ord_less_eq_int_o @ A_147 @ B_80 )
=> ( ( finite_finite_int @ B_80 )
=> ( finite_finite_int @ A_147 ) ) ) ).
thf(fact_366_finite__subset,axiom,
! [A_147: nat > $o,B_80: nat > $o] :
( ( ord_less_eq_nat_o @ A_147 @ B_80 )
=> ( ( finite_finite_nat @ B_80 )
=> ( finite_finite_nat @ A_147 ) ) ) ).
thf(fact_367_rev__finite__subset,axiom,
! [A_146: hoare_1708887482_state > $o,B_79: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_79 )
=> ( ( ord_le777019615tate_o @ A_146 @ B_79 )
=> ( finite1625599783_state @ A_146 ) ) ) ).
thf(fact_368_rev__finite__subset,axiom,
! [A_146: int > $o,B_79: int > $o] :
( ( finite_finite_int @ B_79 )
=> ( ( ord_less_eq_int_o @ A_146 @ B_79 )
=> ( finite_finite_int @ A_146 ) ) ) ).
thf(fact_369_rev__finite__subset,axiom,
! [A_146: nat > $o,B_79: nat > $o] :
( ( finite_finite_nat @ B_79 )
=> ( ( ord_less_eq_nat_o @ A_146 @ B_79 )
=> ( finite_finite_nat @ A_146 ) ) ) ).
thf(fact_370_empty__fold__graphE,axiom,
! [F_42: int > int > int,Z_27: int,X_72: int] :
( ( finite772772422nt_int @ F_42 @ Z_27 @ bot_bot_int_o @ X_72 )
=> ( X_72 = Z_27 ) ) ).
thf(fact_371_empty__fold__graphE,axiom,
! [F_42: nat > nat > nat,Z_27: nat,X_72: nat] :
( ( finite929467206at_nat @ F_42 @ Z_27 @ bot_bot_nat_o @ X_72 )
=> ( X_72 = Z_27 ) ) ).
thf(fact_372_empty__fold__graphE,axiom,
! [F_42: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,Z_27: hoare_1708887482_state,X_72: hoare_1708887482_state] :
( ( finite2042206440_state @ F_42 @ Z_27 @ bot_bo19817387tate_o @ X_72 )
=> ( X_72 = Z_27 ) ) ).
thf(fact_373_fold__graph_OemptyI,axiom,
! [F_41: int > int > int,Z_26: int] : ( finite772772422nt_int @ F_41 @ Z_26 @ bot_bot_int_o @ Z_26 ) ).
thf(fact_374_fold__graph_OemptyI,axiom,
! [F_41: nat > nat > nat,Z_26: nat] : ( finite929467206at_nat @ F_41 @ Z_26 @ bot_bot_nat_o @ Z_26 ) ).
thf(fact_375_fold__graph_OemptyI,axiom,
! [F_41: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,Z_26: hoare_1708887482_state] : ( finite2042206440_state @ F_41 @ Z_26 @ bot_bo19817387tate_o @ Z_26 ) ).
thf(fact_376_fold__graph_OinsertI,axiom,
! [F_40: int > int > int,Z_25: int,Y_45: int,X_71: int,A_145: int > $o] :
( ~ ( member_int @ X_71 @ A_145 )
=> ( ( finite772772422nt_int @ F_40 @ Z_25 @ A_145 @ Y_45 )
=> ( finite772772422nt_int @ F_40 @ Z_25 @ ( insert_int @ X_71 @ A_145 ) @ ( F_40 @ X_71 @ Y_45 ) ) ) ) ).
thf(fact_377_fold__graph_OinsertI,axiom,
! [F_40: nat > nat > nat,Z_25: nat,Y_45: nat,X_71: nat,A_145: nat > $o] :
( ~ ( member_nat @ X_71 @ A_145 )
=> ( ( finite929467206at_nat @ F_40 @ Z_25 @ A_145 @ Y_45 )
=> ( finite929467206at_nat @ F_40 @ Z_25 @ ( insert_nat @ X_71 @ A_145 ) @ ( F_40 @ X_71 @ Y_45 ) ) ) ) ).
thf(fact_378_fold__graph_OinsertI,axiom,
! [F_40: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,Z_25: hoare_1708887482_state,Y_45: hoare_1708887482_state,X_71: hoare_1708887482_state,A_145: hoare_1708887482_state > $o] :
( ~ ( member451959335_state @ X_71 @ A_145 )
=> ( ( finite2042206440_state @ F_40 @ Z_25 @ A_145 @ Y_45 )
=> ( finite2042206440_state @ F_40 @ Z_25 @ ( insert528405184_state @ X_71 @ A_145 ) @ ( F_40 @ X_71 @ Y_45 ) ) ) ) ).
thf(fact_379_folding__one_Oremove,axiom,
! [X_70: int,A_144: int > $o,F_39: int > int > int,F_38: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_39 @ F_38 )
=> ( ( finite_finite_int @ A_144 )
=> ( ( member_int @ X_70 @ A_144 )
=> ( ( ( ( minus_minus_int_o @ A_144 @ ( insert_int @ X_70 @ bot_bot_int_o ) )
= bot_bot_int_o )
=> ( ( F_38 @ A_144 )
= X_70 ) )
& ( ( ( minus_minus_int_o @ A_144 @ ( insert_int @ X_70 @ bot_bot_int_o ) )
!= bot_bot_int_o )
=> ( ( F_38 @ A_144 )
= ( F_39 @ X_70 @ ( F_38 @ ( minus_minus_int_o @ A_144 @ ( insert_int @ X_70 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_380_folding__one_Oremove,axiom,
! [X_70: nat,A_144: nat > $o,F_39: nat > nat > nat,F_38: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_39 @ F_38 )
=> ( ( finite_finite_nat @ A_144 )
=> ( ( member_nat @ X_70 @ A_144 )
=> ( ( ( ( minus_minus_nat_o @ A_144 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_38 @ A_144 )
= X_70 ) )
& ( ( ( minus_minus_nat_o @ A_144 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_38 @ A_144 )
= ( F_39 @ X_70 @ ( F_38 @ ( minus_minus_nat_o @ A_144 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_381_folding__one_Oremove,axiom,
! [X_70: hoare_1708887482_state,A_144: hoare_1708887482_state > $o,F_39: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_38: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_39 @ F_38 )
=> ( ( finite1625599783_state @ A_144 )
=> ( ( member451959335_state @ X_70 @ A_144 )
=> ( ( ( ( minus_2056855718tate_o @ A_144 @ ( insert528405184_state @ X_70 @ bot_bo19817387tate_o ) )
= bot_bo19817387tate_o )
=> ( ( F_38 @ A_144 )
= X_70 ) )
& ( ( ( minus_2056855718tate_o @ A_144 @ ( insert528405184_state @ X_70 @ bot_bo19817387tate_o ) )
!= bot_bo19817387tate_o )
=> ( ( F_38 @ A_144 )
= ( F_39 @ X_70 @ ( F_38 @ ( minus_2056855718tate_o @ A_144 @ ( insert528405184_state @ X_70 @ bot_bo19817387tate_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_382_folding__one_Oinsert__remove,axiom,
! [X_69: int,A_143: int > $o,F_37: int > int > int,F_36: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_37 @ F_36 )
=> ( ( finite_finite_int @ A_143 )
=> ( ( ( ( minus_minus_int_o @ A_143 @ ( insert_int @ X_69 @ bot_bot_int_o ) )
= bot_bot_int_o )
=> ( ( F_36 @ ( insert_int @ X_69 @ A_143 ) )
= X_69 ) )
& ( ( ( minus_minus_int_o @ A_143 @ ( insert_int @ X_69 @ bot_bot_int_o ) )
!= bot_bot_int_o )
=> ( ( F_36 @ ( insert_int @ X_69 @ A_143 ) )
= ( F_37 @ X_69 @ ( F_36 @ ( minus_minus_int_o @ A_143 @ ( insert_int @ X_69 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ).
thf(fact_383_folding__one_Oinsert__remove,axiom,
! [X_69: nat,A_143: nat > $o,F_37: nat > nat > nat,F_36: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_37 @ F_36 )
=> ( ( finite_finite_nat @ A_143 )
=> ( ( ( ( minus_minus_nat_o @ A_143 @ ( insert_nat @ X_69 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_36 @ ( insert_nat @ X_69 @ A_143 ) )
= X_69 ) )
& ( ( ( minus_minus_nat_o @ A_143 @ ( insert_nat @ X_69 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_36 @ ( insert_nat @ X_69 @ A_143 ) )
= ( F_37 @ X_69 @ ( F_36 @ ( minus_minus_nat_o @ A_143 @ ( insert_nat @ X_69 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
thf(fact_384_folding__one_Oinsert__remove,axiom,
! [X_69: hoare_1708887482_state,A_143: hoare_1708887482_state > $o,F_37: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_36: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_37 @ F_36 )
=> ( ( finite1625599783_state @ A_143 )
=> ( ( ( ( minus_2056855718tate_o @ A_143 @ ( insert528405184_state @ X_69 @ bot_bo19817387tate_o ) )
= bot_bo19817387tate_o )
=> ( ( F_36 @ ( insert528405184_state @ X_69 @ A_143 ) )
= X_69 ) )
& ( ( ( minus_2056855718tate_o @ A_143 @ ( insert528405184_state @ X_69 @ bot_bo19817387tate_o ) )
!= bot_bo19817387tate_o )
=> ( ( F_36 @ ( insert528405184_state @ X_69 @ A_143 ) )
= ( F_37 @ X_69 @ ( F_36 @ ( minus_2056855718tate_o @ A_143 @ ( insert528405184_state @ X_69 @ bot_bo19817387tate_o ) ) ) ) ) ) ) ) ) ).
thf(fact_385_insert__Diff,axiom,
! [A_142: int,A_141: int > $o] :
( ( member_int @ A_142 @ A_141 )
=> ( ( insert_int @ A_142 @ ( minus_minus_int_o @ A_141 @ ( insert_int @ A_142 @ bot_bot_int_o ) ) )
= A_141 ) ) ).
thf(fact_386_insert__Diff,axiom,
! [A_142: nat,A_141: nat > $o] :
( ( member_nat @ A_142 @ A_141 )
=> ( ( insert_nat @ A_142 @ ( minus_minus_nat_o @ A_141 @ ( insert_nat @ A_142 @ bot_bot_nat_o ) ) )
= A_141 ) ) ).
thf(fact_387_insert__Diff,axiom,
! [A_142: hoare_1708887482_state,A_141: hoare_1708887482_state > $o] :
( ( member451959335_state @ A_142 @ A_141 )
=> ( ( insert528405184_state @ A_142 @ ( minus_2056855718tate_o @ A_141 @ ( insert528405184_state @ A_142 @ bot_bo19817387tate_o ) ) )
= A_141 ) ) ).
thf(fact_388_Diff__insert__absorb,axiom,
! [X_68: int,A_140: int > $o] :
( ~ ( member_int @ X_68 @ A_140 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_68 @ A_140 ) @ ( insert_int @ X_68 @ bot_bot_int_o ) )
= A_140 ) ) ).
thf(fact_389_Diff__insert__absorb,axiom,
! [X_68: nat,A_140: nat > $o] :
( ~ ( member_nat @ X_68 @ A_140 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_140 ) @ ( insert_nat @ X_68 @ bot_bot_nat_o ) )
= A_140 ) ) ).
thf(fact_390_Diff__insert__absorb,axiom,
! [X_68: hoare_1708887482_state,A_140: hoare_1708887482_state > $o] :
( ~ ( member451959335_state @ X_68 @ A_140 )
=> ( ( minus_2056855718tate_o @ ( insert528405184_state @ X_68 @ A_140 ) @ ( insert528405184_state @ X_68 @ bot_bo19817387tate_o ) )
= A_140 ) ) ).
thf(fact_391_insert__Diff__single,axiom,
! [A_139: int,A_138: int > $o] :
( ( insert_int @ A_139 @ ( minus_minus_int_o @ A_138 @ ( insert_int @ A_139 @ bot_bot_int_o ) ) )
= ( insert_int @ A_139 @ A_138 ) ) ).
thf(fact_392_insert__Diff__single,axiom,
! [A_139: nat,A_138: nat > $o] :
( ( insert_nat @ A_139 @ ( minus_minus_nat_o @ A_138 @ ( insert_nat @ A_139 @ bot_bot_nat_o ) ) )
= ( insert_nat @ A_139 @ A_138 ) ) ).
thf(fact_393_insert__Diff__single,axiom,
! [A_139: hoare_1708887482_state,A_138: hoare_1708887482_state > $o] :
( ( insert528405184_state @ A_139 @ ( minus_2056855718tate_o @ A_138 @ ( insert528405184_state @ A_139 @ bot_bo19817387tate_o ) ) )
= ( insert528405184_state @ A_139 @ A_138 ) ) ).
thf(fact_394_Diff__insert2,axiom,
! [A_137: int > $o,A_136: int,B_78: int > $o] :
( ( minus_minus_int_o @ A_137 @ ( insert_int @ A_136 @ B_78 ) )
= ( minus_minus_int_o @ ( minus_minus_int_o @ A_137 @ ( insert_int @ A_136 @ bot_bot_int_o ) ) @ B_78 ) ) ).
thf(fact_395_Diff__insert2,axiom,
! [A_137: nat > $o,A_136: nat,B_78: nat > $o] :
( ( minus_minus_nat_o @ A_137 @ ( insert_nat @ A_136 @ B_78 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_137 @ ( insert_nat @ A_136 @ bot_bot_nat_o ) ) @ B_78 ) ) ).
thf(fact_396_Diff__insert2,axiom,
! [A_137: hoare_1708887482_state > $o,A_136: hoare_1708887482_state,B_78: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ A_137 @ ( insert528405184_state @ A_136 @ B_78 ) )
= ( minus_2056855718tate_o @ ( minus_2056855718tate_o @ A_137 @ ( insert528405184_state @ A_136 @ bot_bo19817387tate_o ) ) @ B_78 ) ) ).
thf(fact_397_Diff__insert,axiom,
! [A_135: int > $o,A_134: int,B_77: int > $o] :
( ( minus_minus_int_o @ A_135 @ ( insert_int @ A_134 @ B_77 ) )
= ( minus_minus_int_o @ ( minus_minus_int_o @ A_135 @ B_77 ) @ ( insert_int @ A_134 @ bot_bot_int_o ) ) ) ).
thf(fact_398_Diff__insert,axiom,
! [A_135: nat > $o,A_134: nat,B_77: nat > $o] :
( ( minus_minus_nat_o @ A_135 @ ( insert_nat @ A_134 @ B_77 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_135 @ B_77 ) @ ( insert_nat @ A_134 @ bot_bot_nat_o ) ) ) ).
thf(fact_399_Diff__insert,axiom,
! [A_135: hoare_1708887482_state > $o,A_134: hoare_1708887482_state,B_77: hoare_1708887482_state > $o] :
( ( minus_2056855718tate_o @ A_135 @ ( insert528405184_state @ A_134 @ B_77 ) )
= ( minus_2056855718tate_o @ ( minus_2056855718tate_o @ A_135 @ B_77 ) @ ( insert528405184_state @ A_134 @ bot_bo19817387tate_o ) ) ) ).
thf(fact_400_finite__empty__induct,axiom,
! [P_8: ( int > $o ) > $o,A_133: int > $o] :
( ( finite_finite_int @ A_133 )
=> ( ( P_8 @ A_133 )
=> ( ! [A_127: int,A_74: int > $o] :
( ( finite_finite_int @ A_74 )
=> ( ( member_int @ A_127 @ A_74 )
=> ( ( P_8 @ A_74 )
=> ( P_8 @ ( minus_minus_int_o @ A_74 @ ( insert_int @ A_127 @ bot_bot_int_o ) ) ) ) ) )
=> ( P_8 @ bot_bot_int_o ) ) ) ) ).
thf(fact_401_finite__empty__induct,axiom,
! [P_8: ( nat > $o ) > $o,A_133: nat > $o] :
( ( finite_finite_nat @ A_133 )
=> ( ( P_8 @ A_133 )
=> ( ! [A_127: nat,A_74: nat > $o] :
( ( finite_finite_nat @ A_74 )
=> ( ( member_nat @ A_127 @ A_74 )
=> ( ( P_8 @ A_74 )
=> ( P_8 @ ( minus_minus_nat_o @ A_74 @ ( insert_nat @ A_127 @ bot_bot_nat_o ) ) ) ) ) )
=> ( P_8 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_402_finite__empty__induct,axiom,
! [P_8: ( hoare_1708887482_state > $o ) > $o,A_133: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_133 )
=> ( ( P_8 @ A_133 )
=> ( ! [A_127: hoare_1708887482_state,A_74: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_74 )
=> ( ( member451959335_state @ A_127 @ A_74 )
=> ( ( P_8 @ A_74 )
=> ( P_8 @ ( minus_2056855718tate_o @ A_74 @ ( insert528405184_state @ A_127 @ bot_bo19817387tate_o ) ) ) ) ) )
=> ( P_8 @ bot_bo19817387tate_o ) ) ) ) ).
thf(fact_403_finite__subset__induct,axiom,
! [P_7: ( int > $o ) > $o,A_132: int > $o,F_35: int > $o] :
( ( finite_finite_int @ F_35 )
=> ( ( ord_less_eq_int_o @ F_35 @ A_132 )
=> ( ( P_7 @ bot_bot_int_o )
=> ( ! [A_127: int,F_24: int > $o] :
( ( finite_finite_int @ F_24 )
=> ( ( member_int @ A_127 @ A_132 )
=> ( ~ ( member_int @ A_127 @ F_24 )
=> ( ( P_7 @ F_24 )
=> ( P_7 @ ( insert_int @ A_127 @ F_24 ) ) ) ) ) )
=> ( P_7 @ F_35 ) ) ) ) ) ).
thf(fact_404_finite__subset__induct,axiom,
! [P_7: ( nat > $o ) > $o,A_132: nat > $o,F_35: nat > $o] :
( ( finite_finite_nat @ F_35 )
=> ( ( ord_less_eq_nat_o @ F_35 @ A_132 )
=> ( ( P_7 @ bot_bot_nat_o )
=> ( ! [A_127: nat,F_24: nat > $o] :
( ( finite_finite_nat @ F_24 )
=> ( ( member_nat @ A_127 @ A_132 )
=> ( ~ ( member_nat @ A_127 @ F_24 )
=> ( ( P_7 @ F_24 )
=> ( P_7 @ ( insert_nat @ A_127 @ F_24 ) ) ) ) ) )
=> ( P_7 @ F_35 ) ) ) ) ) ).
thf(fact_405_finite__subset__induct,axiom,
! [P_7: ( hoare_1708887482_state > $o ) > $o,A_132: hoare_1708887482_state > $o,F_35: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_35 )
=> ( ( ord_le777019615tate_o @ F_35 @ A_132 )
=> ( ( P_7 @ bot_bo19817387tate_o )
=> ( ! [A_127: hoare_1708887482_state,F_24: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_24 )
=> ( ( member451959335_state @ A_127 @ A_132 )
=> ( ~ ( member451959335_state @ A_127 @ F_24 )
=> ( ( P_7 @ F_24 )
=> ( P_7 @ ( insert528405184_state @ A_127 @ F_24 ) ) ) ) ) )
=> ( P_7 @ F_35 ) ) ) ) ) ).
thf(fact_406_insert__fold1SetE,axiom,
! [F_34: int > int > int,A_131: int,X_67: int > $o,X_66: int] :
( ( finite_fold1Set_int @ F_34 @ ( insert_int @ A_131 @ X_67 ) @ X_66 )
=> ~ ! [A_127: int,A_74: int > $o] :
( ( ( insert_int @ A_131 @ X_67 )
= ( insert_int @ A_127 @ A_74 ) )
=> ( ( finite772772422nt_int @ F_34 @ A_127 @ A_74 @ X_66 )
=> ( member_int @ A_127 @ A_74 ) ) ) ) ).
thf(fact_407_insert__fold1SetE,axiom,
! [F_34: nat > nat > nat,A_131: nat,X_67: nat > $o,X_66: nat] :
( ( finite_fold1Set_nat @ F_34 @ ( insert_nat @ A_131 @ X_67 ) @ X_66 )
=> ~ ! [A_127: nat,A_74: nat > $o] :
( ( ( insert_nat @ A_131 @ X_67 )
= ( insert_nat @ A_127 @ A_74 ) )
=> ( ( finite929467206at_nat @ F_34 @ A_127 @ A_74 @ X_66 )
=> ( member_nat @ A_127 @ A_74 ) ) ) ) ).
thf(fact_408_insert__fold1SetE,axiom,
! [F_34: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,A_131: hoare_1708887482_state,X_67: hoare_1708887482_state > $o,X_66: hoare_1708887482_state] :
( ( finite850270954_state @ F_34 @ ( insert528405184_state @ A_131 @ X_67 ) @ X_66 )
=> ~ ! [A_127: hoare_1708887482_state,A_74: hoare_1708887482_state > $o] :
( ( ( insert528405184_state @ A_131 @ X_67 )
= ( insert528405184_state @ A_127 @ A_74 ) )
=> ( ( finite2042206440_state @ F_34 @ A_127 @ A_74 @ X_66 )
=> ( member451959335_state @ A_127 @ A_74 ) ) ) ) ).
thf(fact_409_finite__nonempty__imp__fold1Set,axiom,
! [F_33: int > int > int,A_130: int > $o] :
( ( finite_finite_int @ A_130 )
=> ( ( A_130 != bot_bot_int_o )
=> ( ?? @ int @ ( finite_fold1Set_int @ F_33 @ A_130 ) ) ) ) ).
thf(fact_410_finite__nonempty__imp__fold1Set,axiom,
! [F_33: nat > nat > nat,A_130: nat > $o] :
( ( finite_finite_nat @ A_130 )
=> ( ( A_130 != bot_bot_nat_o )
=> ( ?? @ nat @ ( finite_fold1Set_nat @ F_33 @ A_130 ) ) ) ) ).
thf(fact_411_finite__nonempty__imp__fold1Set,axiom,
! [F_33: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,A_130: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_130 )
=> ( ( A_130 != bot_bo19817387tate_o )
=> ( ?? @ hoare_1708887482_state @ ( finite850270954_state @ F_33 @ A_130 ) ) ) ) ).
thf(fact_412_finite__induct,axiom,
! [P_6: ( int > $o ) > $o,F_32: int > $o] :
( ( finite_finite_int @ F_32 )
=> ( ( P_6 @ bot_bot_int_o )
=> ( ! [X_1: int,F_24: int > $o] :
( ( finite_finite_int @ F_24 )
=> ( ~ ( member_int @ X_1 @ F_24 )
=> ( ( P_6 @ F_24 )
=> ( P_6 @ ( insert_int @ X_1 @ F_24 ) ) ) ) )
=> ( P_6 @ F_32 ) ) ) ) ).
thf(fact_413_finite__induct,axiom,
! [P_6: ( nat > $o ) > $o,F_32: nat > $o] :
( ( finite_finite_nat @ F_32 )
=> ( ( P_6 @ bot_bot_nat_o )
=> ( ! [X_1: nat,F_24: nat > $o] :
( ( finite_finite_nat @ F_24 )
=> ( ~ ( member_nat @ X_1 @ F_24 )
=> ( ( P_6 @ F_24 )
=> ( P_6 @ ( insert_nat @ X_1 @ F_24 ) ) ) ) )
=> ( P_6 @ F_32 ) ) ) ) ).
thf(fact_414_finite__induct,axiom,
! [P_6: ( hoare_1708887482_state > $o ) > $o,F_32: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_32 )
=> ( ( P_6 @ bot_bo19817387tate_o )
=> ( ! [X_1: hoare_1708887482_state,F_24: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_24 )
=> ( ~ ( member451959335_state @ X_1 @ F_24 )
=> ( ( P_6 @ F_24 )
=> ( P_6 @ ( insert528405184_state @ X_1 @ F_24 ) ) ) ) )
=> ( P_6 @ F_32 ) ) ) ) ).
thf(fact_415_finite_Osimps,axiom,
! [A_129: int > $o] :
( ( finite_finite_int @ A_129 )
<=> ( ( A_129 = bot_bot_int_o )
| ? [A_74: int > $o,A_127: int] :
( ( A_129
= ( insert_int @ A_127 @ A_74 ) )
& ( finite_finite_int @ A_74 ) ) ) ) ).
thf(fact_416_finite_Osimps,axiom,
! [A_129: nat > $o] :
( ( finite_finite_nat @ A_129 )
<=> ( ( A_129 = bot_bot_nat_o )
| ? [A_74: nat > $o,A_127: nat] :
( ( A_129
= ( insert_nat @ A_127 @ A_74 ) )
& ( finite_finite_nat @ A_74 ) ) ) ) ).
thf(fact_417_finite_Osimps,axiom,
! [A_129: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_129 )
<=> ( ( A_129 = bot_bo19817387tate_o )
| ? [A_74: hoare_1708887482_state > $o,A_127: hoare_1708887482_state] :
( ( A_129
= ( insert528405184_state @ A_127 @ A_74 ) )
& ( finite1625599783_state @ A_74 ) ) ) ) ).
thf(fact_418_finite__imp__fold__graph,axiom,
! [F_31: int > int > int,Z_24: int,A_128: int > $o] :
( ( finite_finite_int @ A_128 )
=> ( ?? @ int @ ( finite772772422nt_int @ F_31 @ Z_24 @ A_128 ) ) ) ).
thf(fact_419_finite__imp__fold__graph,axiom,
! [F_31: nat > nat > nat,Z_24: nat,A_128: nat > $o] :
( ( finite_finite_nat @ A_128 )
=> ( ?? @ nat @ ( finite929467206at_nat @ F_31 @ Z_24 @ A_128 ) ) ) ).
thf(fact_420_finite__imp__fold__graph,axiom,
! [F_31: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,Z_24: hoare_1708887482_state,A_128: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_128 )
=> ( ?? @ hoare_1708887482_state @ ( finite2042206440_state @ F_31 @ Z_24 @ A_128 ) ) ) ).
thf(fact_421_fold1Set_Osimps,axiom,
! [F_30: int > int > int,A1_1: int > $o,A2_1: int] :
( ( finite_fold1Set_int @ F_30 @ A1_1 @ A2_1 )
<=> ? [A_127: int,A_74: int > $o,X_1: int] :
( ( A1_1
= ( insert_int @ A_127 @ A_74 ) )
& ( A2_1 = X_1 )
& ( finite772772422nt_int @ F_30 @ A_127 @ A_74 @ X_1 )
& ~ ( member_int @ A_127 @ A_74 ) ) ) ).
thf(fact_422_fold1Set_Osimps,axiom,
! [F_30: nat > nat > nat,A1_1: nat > $o,A2_1: nat] :
( ( finite_fold1Set_nat @ F_30 @ A1_1 @ A2_1 )
<=> ? [A_127: nat,A_74: nat > $o,X_1: nat] :
( ( A1_1
= ( insert_nat @ A_127 @ A_74 ) )
& ( A2_1 = X_1 )
& ( finite929467206at_nat @ F_30 @ A_127 @ A_74 @ X_1 )
& ~ ( member_nat @ A_127 @ A_74 ) ) ) ).
thf(fact_423_fold1Set_Osimps,axiom,
! [F_30: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,A1_1: hoare_1708887482_state > $o,A2_1: hoare_1708887482_state] :
( ( finite850270954_state @ F_30 @ A1_1 @ A2_1 )
<=> ? [A_127: hoare_1708887482_state,A_74: hoare_1708887482_state > $o,X_1: hoare_1708887482_state] :
( ( A1_1
= ( insert528405184_state @ A_127 @ A_74 ) )
& ( A2_1 = X_1 )
& ( finite2042206440_state @ F_30 @ A_127 @ A_74 @ X_1 )
& ~ ( member451959335_state @ A_127 @ A_74 ) ) ) ).
thf(fact_424_minus__apply,axiom,
! [A_126: hoare_1708887482_state > $o,B_76: hoare_1708887482_state > $o,X_65: hoare_1708887482_state] :
( ( minus_2056855718tate_o @ A_126 @ B_76 @ X_65 )
<=> ( minus_minus_o @ ( A_126 @ X_65 ) @ ( B_76 @ X_65 ) ) ) ).
thf(fact_425_minus__apply,axiom,
! [A_126: int > $o,B_76: int > $o,X_65: int] :
( ( minus_minus_int_o @ A_126 @ B_76 @ X_65 )
<=> ( minus_minus_o @ ( A_126 @ X_65 ) @ ( B_76 @ X_65 ) ) ) ).
thf(fact_426_minus__apply,axiom,
! [A_126: nat > $o,B_76: nat > $o,X_65: nat] :
( ( minus_minus_nat_o @ A_126 @ B_76 @ X_65 )
<=> ( minus_minus_o @ ( A_126 @ X_65 ) @ ( B_76 @ X_65 ) ) ) ).
thf(fact_427_fun__diff__def,axiom,
! [A_125: hoare_1708887482_state > $o,B_75: hoare_1708887482_state > $o,X_1: hoare_1708887482_state] :
( ( minus_2056855718tate_o @ A_125 @ B_75 @ X_1 )
<=> ( minus_minus_o @ ( A_125 @ X_1 ) @ ( B_75 @ X_1 ) ) ) ).
thf(fact_428_fun__diff__def,axiom,
! [A_125: int > $o,B_75: int > $o,X_1: int] :
( ( minus_minus_int_o @ A_125 @ B_75 @ X_1 )
<=> ( minus_minus_o @ ( A_125 @ X_1 ) @ ( B_75 @ X_1 ) ) ) ).
thf(fact_429_fun__diff__def,axiom,
! [A_125: nat > $o,B_75: nat > $o,X_1: nat] :
( ( minus_minus_nat_o @ A_125 @ B_75 @ X_1 )
<=> ( minus_minus_o @ ( A_125 @ X_1 ) @ ( B_75 @ X_1 ) ) ) ).
thf(fact_430_diff__eq__diff__less__eq,axiom,
! [A_124: int,B_74: int,C_50: int,D_18: int] :
( ( ( minus_minus_int @ A_124 @ B_74 )
= ( minus_minus_int @ C_50 @ D_18 ) )
=> ( ( ord_less_eq_int @ A_124 @ B_74 )
<=> ( ord_less_eq_int @ C_50 @ D_18 ) ) ) ).
thf(fact_431_fold__graph_Osimps,axiom,
! [F_29: int > int > int,Z_23: int,A1: int > $o,A2: int] :
( ( finite772772422nt_int @ F_29 @ Z_23 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bot_int_o )
& ( A2 = Z_23 ) )
| ? [X_1: int,A_74: int > $o,Y_7: int] :
( ( A1
= ( insert_int @ X_1 @ A_74 ) )
& ( A2
= ( F_29 @ X_1 @ Y_7 ) )
& ~ ( member_int @ X_1 @ A_74 )
& ( finite772772422nt_int @ F_29 @ Z_23 @ A_74 @ Y_7 ) ) ) ) ).
thf(fact_432_fold__graph_Osimps,axiom,
! [F_29: nat > nat > nat,Z_23: nat,A1: nat > $o,A2: nat] :
( ( finite929467206at_nat @ F_29 @ Z_23 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bot_nat_o )
& ( A2 = Z_23 ) )
| ? [X_1: nat,A_74: nat > $o,Y_7: nat] :
( ( A1
= ( insert_nat @ X_1 @ A_74 ) )
& ( A2
= ( F_29 @ X_1 @ Y_7 ) )
& ~ ( member_nat @ X_1 @ A_74 )
& ( finite929467206at_nat @ F_29 @ Z_23 @ A_74 @ Y_7 ) ) ) ) ).
thf(fact_433_fold__graph_Osimps,axiom,
! [F_29: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,Z_23: hoare_1708887482_state,A1: hoare_1708887482_state > $o,A2: hoare_1708887482_state] :
( ( finite2042206440_state @ F_29 @ Z_23 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bo19817387tate_o )
& ( A2 = Z_23 ) )
| ? [X_1: hoare_1708887482_state,A_74: hoare_1708887482_state > $o,Y_7: hoare_1708887482_state] :
( ( A1
= ( insert528405184_state @ X_1 @ A_74 ) )
& ( A2
= ( F_29 @ X_1 @ Y_7 ) )
& ~ ( member451959335_state @ X_1 @ A_74 )
& ( finite2042206440_state @ F_29 @ Z_23 @ A_74 @ Y_7 ) ) ) ) ).
thf(fact_434_folding__one__idem_Osubset__idem,axiom,
! [B_73: int > $o,A_123: int > $o,F_28: int > int > int,F_27: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_28 @ F_27 )
=> ( ( finite_finite_int @ A_123 )
=> ( ( B_73 != bot_bot_int_o )
=> ( ( ord_less_eq_int_o @ B_73 @ A_123 )
=> ( ( F_28 @ ( F_27 @ B_73 ) @ ( F_27 @ A_123 ) )
= ( F_27 @ A_123 ) ) ) ) ) ) ).
thf(fact_435_folding__one__idem_Osubset__idem,axiom,
! [B_73: nat > $o,A_123: nat > $o,F_28: nat > nat > nat,F_27: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_28 @ F_27 )
=> ( ( finite_finite_nat @ A_123 )
=> ( ( B_73 != bot_bot_nat_o )
=> ( ( ord_less_eq_nat_o @ B_73 @ A_123 )
=> ( ( F_28 @ ( F_27 @ B_73 ) @ ( F_27 @ A_123 ) )
= ( F_27 @ A_123 ) ) ) ) ) ) ).
thf(fact_436_folding__one__idem_Osubset__idem,axiom,
! [B_73: hoare_1708887482_state > $o,A_123: hoare_1708887482_state > $o,F_28: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_27: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_28 @ F_27 )
=> ( ( finite1625599783_state @ A_123 )
=> ( ( B_73 != bot_bo19817387tate_o )
=> ( ( ord_le777019615tate_o @ B_73 @ A_123 )
=> ( ( F_28 @ ( F_27 @ B_73 ) @ ( F_27 @ A_123 ) )
= ( F_27 @ A_123 ) ) ) ) ) ) ).
thf(fact_437_folding__one__idem_Oinsert__idem,axiom,
! [X_64: int,A_122: int > $o,F_26: int > int > int,F_25: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_26 @ F_25 )
=> ( ( finite_finite_int @ A_122 )
=> ( ( A_122 != bot_bot_int_o )
=> ( ( F_25 @ ( insert_int @ X_64 @ A_122 ) )
= ( F_26 @ X_64 @ ( F_25 @ A_122 ) ) ) ) ) ) ).
thf(fact_438_folding__one__idem_Oinsert__idem,axiom,
! [X_64: nat,A_122: nat > $o,F_26: nat > nat > nat,F_25: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_26 @ F_25 )
=> ( ( finite_finite_nat @ A_122 )
=> ( ( A_122 != bot_bot_nat_o )
=> ( ( F_25 @ ( insert_nat @ X_64 @ A_122 ) )
= ( F_26 @ X_64 @ ( F_25 @ A_122 ) ) ) ) ) ) ).
thf(fact_439_folding__one__idem_Oinsert__idem,axiom,
! [X_64: hoare_1708887482_state,A_122: hoare_1708887482_state > $o,F_26: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_25: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_26 @ F_25 )
=> ( ( finite1625599783_state @ A_122 )
=> ( ( A_122 != bot_bo19817387tate_o )
=> ( ( F_25 @ ( insert528405184_state @ X_64 @ A_122 ) )
= ( F_26 @ X_64 @ ( F_25 @ A_122 ) ) ) ) ) ) ).
thf(fact_440_finite__ne__induct,axiom,
! [P_5: ( int > $o ) > $o,F_23: int > $o] :
( ( finite_finite_int @ F_23 )
=> ( ( F_23 != bot_bot_int_o )
=> ( ! [X_1: int] : ( P_5 @ ( insert_int @ X_1 @ bot_bot_int_o ) )
=> ( ! [X_1: int,F_24: int > $o] :
( ( finite_finite_int @ F_24 )
=> ( ( F_24 != bot_bot_int_o )
=> ( ~ ( member_int @ X_1 @ F_24 )
=> ( ( P_5 @ F_24 )
=> ( P_5 @ ( insert_int @ X_1 @ F_24 ) ) ) ) ) )
=> ( P_5 @ F_23 ) ) ) ) ) ).
thf(fact_441_finite__ne__induct,axiom,
! [P_5: ( nat > $o ) > $o,F_23: nat > $o] :
( ( finite_finite_nat @ F_23 )
=> ( ( F_23 != bot_bot_nat_o )
=> ( ! [X_1: nat] : ( P_5 @ ( insert_nat @ X_1 @ bot_bot_nat_o ) )
=> ( ! [X_1: nat,F_24: nat > $o] :
( ( finite_finite_nat @ F_24 )
=> ( ( F_24 != bot_bot_nat_o )
=> ( ~ ( member_nat @ X_1 @ F_24 )
=> ( ( P_5 @ F_24 )
=> ( P_5 @ ( insert_nat @ X_1 @ F_24 ) ) ) ) ) )
=> ( P_5 @ F_23 ) ) ) ) ) ).
thf(fact_442_finite__ne__induct,axiom,
! [P_5: ( hoare_1708887482_state > $o ) > $o,F_23: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_23 )
=> ( ( F_23 != bot_bo19817387tate_o )
=> ( ! [X_1: hoare_1708887482_state] : ( P_5 @ ( insert528405184_state @ X_1 @ bot_bo19817387tate_o ) )
=> ( ! [X_1: hoare_1708887482_state,F_24: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ F_24 )
=> ( ( F_24 != bot_bo19817387tate_o )
=> ( ~ ( member451959335_state @ X_1 @ F_24 )
=> ( ( P_5 @ F_24 )
=> ( P_5 @ ( insert528405184_state @ X_1 @ F_24 ) ) ) ) ) )
=> ( P_5 @ F_23 ) ) ) ) ) ).
thf(fact_443_folding__one__idem_Oidem,axiom,
! [X_63: hoare_1708887482_state,F_22: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_21: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_22 @ F_21 )
=> ( ( F_22 @ X_63 @ X_63 )
= X_63 ) ) ).
thf(fact_444_folding__one__idem_Oin__idem,axiom,
! [X_62: hoare_1708887482_state,A_121: hoare_1708887482_state > $o,F_20: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_19: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_20 @ F_19 )
=> ( ( finite1625599783_state @ A_121 )
=> ( ( member451959335_state @ X_62 @ A_121 )
=> ( ( F_20 @ X_62 @ ( F_19 @ A_121 ) )
= ( F_19 @ A_121 ) ) ) ) ) ).
thf(fact_445_folding__one__idem_Oin__idem,axiom,
! [X_62: int,A_121: int > $o,F_20: int > int > int,F_19: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_20 @ F_19 )
=> ( ( finite_finite_int @ A_121 )
=> ( ( member_int @ X_62 @ A_121 )
=> ( ( F_20 @ X_62 @ ( F_19 @ A_121 ) )
= ( F_19 @ A_121 ) ) ) ) ) ).
thf(fact_446_folding__one__idem_Oin__idem,axiom,
! [X_62: nat,A_121: nat > $o,F_20: nat > nat > nat,F_19: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_20 @ F_19 )
=> ( ( finite_finite_nat @ A_121 )
=> ( ( member_nat @ X_62 @ A_121 )
=> ( ( F_20 @ X_62 @ ( F_19 @ A_121 ) )
= ( F_19 @ A_121 ) ) ) ) ) ).
thf(fact_447_diff__eq__diff__eq,axiom,
! [A_120: int,B_72: int,C_49: int,D_17: int] :
( ( ( minus_minus_int @ A_120 @ B_72 )
= ( minus_minus_int @ C_49 @ D_17 ) )
=> ( ( A_120 = B_72 )
<=> ( C_49 = D_17 ) ) ) ).
thf(fact_448_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_119: int,Z_22: int,A_118: int > $o,Y_43: int,F_18: int > int > int] :
( ( finite1973466193nt_int @ F_18 )
=> ( ( finite772772422nt_int @ F_18 @ Z_22 @ A_118 @ Y_43 )
=> ( ( member_int @ A_119 @ A_118 )
=> ? [Y_44: int] :
( ( Y_43
= ( F_18 @ A_119 @ Y_44 ) )
& ( finite772772422nt_int @ F_18 @ Z_22 @ ( minus_minus_int_o @ A_118 @ ( insert_int @ A_119 @ bot_bot_int_o ) ) @ Y_44 ) ) ) ) ) ).
thf(fact_449_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_119: nat,Z_22: nat,A_118: nat > $o,Y_43: nat,F_18: nat > nat > nat] :
( ( finite2130160977at_nat @ F_18 )
=> ( ( finite929467206at_nat @ F_18 @ Z_22 @ A_118 @ Y_43 )
=> ( ( member_nat @ A_119 @ A_118 )
=> ? [Y_44: nat] :
( ( Y_43
= ( F_18 @ A_119 @ Y_44 ) )
& ( finite929467206at_nat @ F_18 @ Z_22 @ ( minus_minus_nat_o @ A_118 @ ( insert_nat @ A_119 @ bot_bot_nat_o ) ) @ Y_44 ) ) ) ) ) ).
thf(fact_450_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_119: hoare_1708887482_state,Z_22: hoare_1708887482_state,A_118: hoare_1708887482_state > $o,Y_43: hoare_1708887482_state,F_18: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state] :
( ( finite1816785523_state @ F_18 )
=> ( ( finite2042206440_state @ F_18 @ Z_22 @ A_118 @ Y_43 )
=> ( ( member451959335_state @ A_119 @ A_118 )
=> ? [Y_44: hoare_1708887482_state] :
( ( Y_43
= ( F_18 @ A_119 @ Y_44 ) )
& ( finite2042206440_state @ F_18 @ Z_22 @ ( minus_2056855718tate_o @ A_118 @ ( insert528405184_state @ A_119 @ bot_bo19817387tate_o ) ) @ Y_44 ) ) ) ) ) ).
thf(fact_451_semilattice__big_OF__eq,axiom,
! [A_117: int > $o,F_17: int > int > int,F_16: ( int > $o ) > int] :
( ( big_se913005884ig_int @ F_17 @ F_16 )
=> ( ( finite_finite_int @ A_117 )
=> ( ( F_16 @ A_117 )
= ( finite_fold1_int @ F_17 @ A_117 ) ) ) ) ).
thf(fact_452_semilattice__big_OF__eq,axiom,
! [A_117: nat > $o,F_17: nat > nat > nat,F_16: ( nat > $o ) > nat] :
( ( big_se275732192ig_nat @ F_17 @ F_16 )
=> ( ( finite_finite_nat @ A_117 )
=> ( ( F_16 @ A_117 )
= ( finite_fold1_nat @ F_17 @ A_117 ) ) ) ) ).
thf(fact_453_semilattice__big_OF__eq,axiom,
! [A_117: hoare_1708887482_state > $o,F_17: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_16: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( big_se2144116836_state @ F_17 @ F_16 )
=> ( ( finite1625599783_state @ A_117 )
=> ( ( F_16 @ A_117 )
= ( finite39664218_state @ F_17 @ A_117 ) ) ) ) ).
thf(fact_454_fold__graph__permute__diff,axiom,
! [A_116: nat,B_71: nat,A_115: nat > $o,X_61: nat] :
( ( finite929467206at_nat @ times_times_nat @ B_71 @ A_115 @ X_61 )
=> ( ( member_nat @ A_116 @ A_115 )
=> ( ~ ( member_nat @ B_71 @ A_115 )
=> ( finite929467206at_nat @ times_times_nat @ A_116 @ ( insert_nat @ B_71 @ ( minus_minus_nat_o @ A_115 @ ( insert_nat @ A_116 @ bot_bot_nat_o ) ) ) @ X_61 ) ) ) ) ).
thf(fact_455_fold__graph__permute__diff,axiom,
! [A_116: int,B_71: int,A_115: int > $o,X_61: int] :
( ( finite772772422nt_int @ times_times_int @ B_71 @ A_115 @ X_61 )
=> ( ( member_int @ A_116 @ A_115 )
=> ( ~ ( member_int @ B_71 @ A_115 )
=> ( finite772772422nt_int @ times_times_int @ A_116 @ ( insert_int @ B_71 @ ( minus_minus_int_o @ A_115 @ ( insert_int @ A_116 @ bot_bot_int_o ) ) ) @ X_61 ) ) ) ) ).
thf(fact_456_fold1__insert,axiom,
! [X_60: nat,A_114: nat > $o] :
( ( A_114 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ A_114 )
=> ( ~ ( member_nat @ X_60 @ A_114 )
=> ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ X_60 @ A_114 ) )
= ( times_times_nat @ X_60 @ ( finite_fold1_nat @ times_times_nat @ A_114 ) ) ) ) ) ) ).
thf(fact_457_fold1__insert,axiom,
! [X_60: int,A_114: int > $o] :
( ( A_114 != bot_bot_int_o )
=> ( ( finite_finite_int @ A_114 )
=> ( ~ ( member_int @ X_60 @ A_114 )
=> ( ( finite_fold1_int @ times_times_int @ ( insert_int @ X_60 @ A_114 ) )
= ( times_times_int @ X_60 @ ( finite_fold1_int @ times_times_int @ A_114 ) ) ) ) ) ) ).
thf(fact_458_psubset__insert__iff,axiom,
! [A_113: int > $o,X_59: int,B_70: int > $o] :
( ( ord_less_int_o @ A_113 @ ( insert_int @ X_59 @ B_70 ) )
<=> ( ( ( member_int @ X_59 @ B_70 )
=> ( ord_less_int_o @ A_113 @ B_70 ) )
& ( ~ ( member_int @ X_59 @ B_70 )
=> ( ( ( member_int @ X_59 @ A_113 )
=> ( ord_less_int_o @ ( minus_minus_int_o @ A_113 @ ( insert_int @ X_59 @ bot_bot_int_o ) ) @ B_70 ) )
& ( ~ ( member_int @ X_59 @ A_113 )
=> ( ord_less_eq_int_o @ A_113 @ B_70 ) ) ) ) ) ) ).
thf(fact_459_psubset__insert__iff,axiom,
! [A_113: nat > $o,X_59: nat,B_70: nat > $o] :
( ( ord_less_nat_o @ A_113 @ ( insert_nat @ X_59 @ B_70 ) )
<=> ( ( ( member_nat @ X_59 @ B_70 )
=> ( ord_less_nat_o @ A_113 @ B_70 ) )
& ( ~ ( member_nat @ X_59 @ B_70 )
=> ( ( ( member_nat @ X_59 @ A_113 )
=> ( ord_less_nat_o @ ( minus_minus_nat_o @ A_113 @ ( insert_nat @ X_59 @ bot_bot_nat_o ) ) @ B_70 ) )
& ( ~ ( member_nat @ X_59 @ A_113 )
=> ( ord_less_eq_nat_o @ A_113 @ B_70 ) ) ) ) ) ) ).
thf(fact_460_psubset__insert__iff,axiom,
! [A_113: hoare_1708887482_state > $o,X_59: hoare_1708887482_state,B_70: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_113 @ ( insert528405184_state @ X_59 @ B_70 ) )
<=> ( ( ( member451959335_state @ X_59 @ B_70 )
=> ( ord_le14920683tate_o @ A_113 @ B_70 ) )
& ( ~ ( member451959335_state @ X_59 @ B_70 )
=> ( ( ( member451959335_state @ X_59 @ A_113 )
=> ( ord_le14920683tate_o @ ( minus_2056855718tate_o @ A_113 @ ( insert528405184_state @ X_59 @ bot_bo19817387tate_o ) ) @ B_70 ) )
& ( ~ ( member451959335_state @ X_59 @ A_113 )
=> ( ord_le777019615tate_o @ A_113 @ B_70 ) ) ) ) ) ) ).
thf(fact_461_card__Diff1__le,axiom,
! [X_58: int,A_112: int > $o] :
( ( finite_finite_int @ A_112 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_112 @ ( insert_int @ X_58 @ bot_bot_int_o ) ) ) @ ( finite_card_int @ A_112 ) ) ) ).
thf(fact_462_card__Diff1__le,axiom,
! [X_58: nat,A_112: nat > $o] :
( ( finite_finite_nat @ A_112 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_112 @ ( insert_nat @ X_58 @ bot_bot_nat_o ) ) ) @ ( finite_card_nat @ A_112 ) ) ) ).
thf(fact_463_card__Diff1__le,axiom,
! [X_58: hoare_1708887482_state,A_112: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_112 )
=> ( ord_less_eq_nat @ ( finite185771624_state @ ( minus_2056855718tate_o @ A_112 @ ( insert528405184_state @ X_58 @ bot_bo19817387tate_o ) ) ) @ ( finite185771624_state @ A_112 ) ) ) ).
thf(fact_464_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_111: nat,B_69: nat,C_48: nat] :
( ( times_times_nat @ ( times_times_nat @ A_111 @ B_69 ) @ C_48 )
= ( times_times_nat @ A_111 @ ( times_times_nat @ B_69 @ C_48 ) ) ) ).
thf(fact_465_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_111: int,B_69: int,C_48: int] :
( ( times_times_int @ ( times_times_int @ A_111 @ B_69 ) @ C_48 )
= ( times_times_int @ A_111 @ ( times_times_int @ B_69 @ C_48 ) ) ) ).
thf(fact_466_psubsetD,axiom,
! [C_47: hoare_1708887482_state,A_110: hoare_1708887482_state > $o,B_68: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_110 @ B_68 )
=> ( ( member451959335_state @ C_47 @ A_110 )
=> ( member451959335_state @ C_47 @ B_68 ) ) ) ).
thf(fact_467_psubsetD,axiom,
! [C_47: int,A_110: int > $o,B_68: int > $o] :
( ( ord_less_int_o @ A_110 @ B_68 )
=> ( ( member_int @ C_47 @ A_110 )
=> ( member_int @ C_47 @ B_68 ) ) ) ).
thf(fact_468_psubsetD,axiom,
! [C_47: nat,A_110: nat > $o,B_68: nat > $o] :
( ( ord_less_nat_o @ A_110 @ B_68 )
=> ( ( member_nat @ C_47 @ A_110 )
=> ( member_nat @ C_47 @ B_68 ) ) ) ).
thf(fact_469_psubset__trans,axiom,
! [C_46: hoare_1708887482_state > $o,A_109: hoare_1708887482_state > $o,B_67: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_109 @ B_67 )
=> ( ( ord_le14920683tate_o @ B_67 @ C_46 )
=> ( ord_le14920683tate_o @ A_109 @ C_46 ) ) ) ).
thf(fact_470_comp__fun__commute,axiom,
finite2130160977at_nat @ times_times_nat ).
thf(fact_471_comp__fun__commute,axiom,
finite1973466193nt_int @ times_times_int ).
thf(fact_472_order__less__irrefl,axiom,
! [X_57: hoare_1708887482_state > $o] :
~ ( ord_le14920683tate_o @ X_57 @ X_57 ) ).
thf(fact_473_order__less__irrefl,axiom,
! [X_57: nat] :
~ ( ord_less_nat @ X_57 @ X_57 ) ).
thf(fact_474_order__less__irrefl,axiom,
! [X_57: int] :
~ ( ord_less_int @ X_57 @ X_57 ) ).
thf(fact_475_linorder__neq__iff,axiom,
! [X_56: nat,Y_42: nat] :
( ( X_56 != Y_42 )
<=> ( ( ord_less_nat @ X_56 @ Y_42 )
| ( ord_less_nat @ Y_42 @ X_56 ) ) ) ).
thf(fact_476_linorder__neq__iff,axiom,
! [X_56: int,Y_42: int] :
( ( X_56 != Y_42 )
<=> ( ( ord_less_int @ X_56 @ Y_42 )
| ( ord_less_int @ Y_42 @ X_56 ) ) ) ).
thf(fact_477_not__less__iff__gr__or__eq,axiom,
! [X_55: nat,Y_41: nat] :
( ~ ( ord_less_nat @ X_55 @ Y_41 )
<=> ( ( ord_less_nat @ Y_41 @ X_55 )
| ( X_55 = Y_41 ) ) ) ).
thf(fact_478_not__less__iff__gr__or__eq,axiom,
! [X_55: int,Y_41: int] :
( ~ ( ord_less_int @ X_55 @ Y_41 )
<=> ( ( ord_less_int @ Y_41 @ X_55 )
| ( X_55 = Y_41 ) ) ) ).
thf(fact_479_linorder__less__linear,axiom,
! [X_54: nat,Y_40: nat] :
( ( ord_less_nat @ X_54 @ Y_40 )
| ( X_54 = Y_40 )
| ( ord_less_nat @ Y_40 @ X_54 ) ) ).
thf(fact_480_linorder__less__linear,axiom,
! [X_54: int,Y_40: int] :
( ( ord_less_int @ X_54 @ Y_40 )
| ( X_54 = Y_40 )
| ( ord_less_int @ Y_40 @ X_54 ) ) ).
thf(fact_481_comp__fun__commute_Ofun__left__comm,axiom,
! [X_53: int,Y_39: int,Z_21: int,F_15: int > int > int] :
( ( finite1973466193nt_int @ F_15 )
=> ( ( F_15 @ X_53 @ ( F_15 @ Y_39 @ Z_21 ) )
= ( F_15 @ Y_39 @ ( F_15 @ X_53 @ Z_21 ) ) ) ) ).
thf(fact_482_comp__fun__commute_Ofun__left__comm,axiom,
! [X_53: nat,Y_39: nat,Z_21: nat,F_15: nat > nat > nat] :
( ( finite2130160977at_nat @ F_15 )
=> ( ( F_15 @ X_53 @ ( F_15 @ Y_39 @ Z_21 ) )
= ( F_15 @ Y_39 @ ( F_15 @ X_53 @ Z_21 ) ) ) ) ).
thf(fact_483_linorder__antisym__conv3,axiom,
! [Y_38: nat,X_52: nat] :
( ~ ( ord_less_nat @ Y_38 @ X_52 )
=> ( ~ ( ord_less_nat @ X_52 @ Y_38 )
<=> ( X_52 = Y_38 ) ) ) ).
thf(fact_484_linorder__antisym__conv3,axiom,
! [Y_38: int,X_52: int] :
( ~ ( ord_less_int @ Y_38 @ X_52 )
=> ( ~ ( ord_less_int @ X_52 @ Y_38 )
<=> ( X_52 = Y_38 ) ) ) ).
thf(fact_485_linorder__neqE,axiom,
! [X_51: nat,Y_37: nat] :
( ( X_51 != Y_37 )
=> ( ~ ( ord_less_nat @ X_51 @ Y_37 )
=> ( ord_less_nat @ Y_37 @ X_51 ) ) ) ).
thf(fact_486_linorder__neqE,axiom,
! [X_51: int,Y_37: int] :
( ( X_51 != Y_37 )
=> ( ~ ( ord_less_int @ X_51 @ Y_37 )
=> ( ord_less_int @ Y_37 @ X_51 ) ) ) ).
thf(fact_487_less__imp__neq,axiom,
! [X_50: hoare_1708887482_state > $o,Y_36: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_50 @ Y_36 )
=> ( X_50 != Y_36 ) ) ).
thf(fact_488_less__imp__neq,axiom,
! [X_50: nat,Y_36: nat] :
( ( ord_less_nat @ X_50 @ Y_36 )
=> ( X_50 != Y_36 ) ) ).
thf(fact_489_less__imp__neq,axiom,
! [X_50: int,Y_36: int] :
( ( ord_less_int @ X_50 @ Y_36 )
=> ( X_50 != Y_36 ) ) ).
thf(fact_490_order__less__not__sym,axiom,
! [X_49: hoare_1708887482_state > $o,Y_35: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_49 @ Y_35 )
=> ~ ( ord_le14920683tate_o @ Y_35 @ X_49 ) ) ).
thf(fact_491_order__less__not__sym,axiom,
! [X_49: nat,Y_35: nat] :
( ( ord_less_nat @ X_49 @ Y_35 )
=> ~ ( ord_less_nat @ Y_35 @ X_49 ) ) ).
thf(fact_492_order__less__not__sym,axiom,
! [X_49: int,Y_35: int] :
( ( ord_less_int @ X_49 @ Y_35 )
=> ~ ( ord_less_int @ Y_35 @ X_49 ) ) ).
thf(fact_493_order__less__imp__not__less,axiom,
! [X_48: hoare_1708887482_state > $o,Y_34: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_48 @ Y_34 )
=> ~ ( ord_le14920683tate_o @ Y_34 @ X_48 ) ) ).
thf(fact_494_order__less__imp__not__less,axiom,
! [X_48: nat,Y_34: nat] :
( ( ord_less_nat @ X_48 @ Y_34 )
=> ~ ( ord_less_nat @ Y_34 @ X_48 ) ) ).
thf(fact_495_order__less__imp__not__less,axiom,
! [X_48: int,Y_34: int] :
( ( ord_less_int @ X_48 @ Y_34 )
=> ~ ( ord_less_int @ Y_34 @ X_48 ) ) ).
thf(fact_496_order__less__imp__not__eq,axiom,
! [X_47: hoare_1708887482_state > $o,Y_33: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_47 @ Y_33 )
=> ( X_47 != Y_33 ) ) ).
thf(fact_497_order__less__imp__not__eq,axiom,
! [X_47: nat,Y_33: nat] :
( ( ord_less_nat @ X_47 @ Y_33 )
=> ( X_47 != Y_33 ) ) ).
thf(fact_498_order__less__imp__not__eq,axiom,
! [X_47: int,Y_33: int] :
( ( ord_less_int @ X_47 @ Y_33 )
=> ( X_47 != Y_33 ) ) ).
thf(fact_499_order__less__imp__not__eq2,axiom,
! [X_46: hoare_1708887482_state > $o,Y_32: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_46 @ Y_32 )
=> ( Y_32 != X_46 ) ) ).
thf(fact_500_order__less__imp__not__eq2,axiom,
! [X_46: nat,Y_32: nat] :
( ( ord_less_nat @ X_46 @ Y_32 )
=> ( Y_32 != X_46 ) ) ).
thf(fact_501_order__less__imp__not__eq2,axiom,
! [X_46: int,Y_32: int] :
( ( ord_less_int @ X_46 @ Y_32 )
=> ( Y_32 != X_46 ) ) ).
thf(fact_502_order__less__imp__triv,axiom,
! [P_4: $o,X_45: hoare_1708887482_state > $o,Y_31: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_45 @ Y_31 )
=> ( ( ord_le14920683tate_o @ Y_31 @ X_45 )
=> P_4 ) ) ).
thf(fact_503_order__less__imp__triv,axiom,
! [P_4: $o,X_45: nat,Y_31: nat] :
( ( ord_less_nat @ X_45 @ Y_31 )
=> ( ( ord_less_nat @ Y_31 @ X_45 )
=> P_4 ) ) ).
thf(fact_504_order__less__imp__triv,axiom,
! [P_4: $o,X_45: int,Y_31: int] :
( ( ord_less_int @ X_45 @ Y_31 )
=> ( ( ord_less_int @ Y_31 @ X_45 )
=> P_4 ) ) ).
thf(fact_505_order__less__asym_H,axiom,
! [A_108: hoare_1708887482_state > $o,B_66: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_108 @ B_66 )
=> ~ ( ord_le14920683tate_o @ B_66 @ A_108 ) ) ).
thf(fact_506_order__less__asym_H,axiom,
! [A_108: nat,B_66: nat] :
( ( ord_less_nat @ A_108 @ B_66 )
=> ~ ( ord_less_nat @ B_66 @ A_108 ) ) ).
thf(fact_507_order__less__asym_H,axiom,
! [A_108: int,B_66: int] :
( ( ord_less_int @ A_108 @ B_66 )
=> ~ ( ord_less_int @ B_66 @ A_108 ) ) ).
thf(fact_508_xt1_I9_J,axiom,
! [B_65: hoare_1708887482_state > $o,A_107: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ B_65 @ A_107 )
=> ~ ( ord_le14920683tate_o @ A_107 @ B_65 ) ) ).
thf(fact_509_xt1_I9_J,axiom,
! [B_65: nat,A_107: nat] :
( ( ord_less_nat @ B_65 @ A_107 )
=> ~ ( ord_less_nat @ A_107 @ B_65 ) ) ).
thf(fact_510_xt1_I9_J,axiom,
! [B_65: int,A_107: int] :
( ( ord_less_int @ B_65 @ A_107 )
=> ~ ( ord_less_int @ A_107 @ B_65 ) ) ).
thf(fact_511_ord__eq__less__trans,axiom,
! [C_45: hoare_1708887482_state > $o,A_106: hoare_1708887482_state > $o,B_64: hoare_1708887482_state > $o] :
( ( A_106 = B_64 )
=> ( ( ord_le14920683tate_o @ B_64 @ C_45 )
=> ( ord_le14920683tate_o @ A_106 @ C_45 ) ) ) ).
thf(fact_512_ord__eq__less__trans,axiom,
! [C_45: nat,A_106: nat,B_64: nat] :
( ( A_106 = B_64 )
=> ( ( ord_less_nat @ B_64 @ C_45 )
=> ( ord_less_nat @ A_106 @ C_45 ) ) ) ).
thf(fact_513_ord__eq__less__trans,axiom,
! [C_45: int,A_106: int,B_64: int] :
( ( A_106 = B_64 )
=> ( ( ord_less_int @ B_64 @ C_45 )
=> ( ord_less_int @ A_106 @ C_45 ) ) ) ).
thf(fact_514_xt1_I1_J,axiom,
! [C_44: hoare_1708887482_state > $o,A_105: hoare_1708887482_state > $o,B_63: hoare_1708887482_state > $o] :
( ( A_105 = B_63 )
=> ( ( ord_le14920683tate_o @ C_44 @ B_63 )
=> ( ord_le14920683tate_o @ C_44 @ A_105 ) ) ) ).
thf(fact_515_xt1_I1_J,axiom,
! [C_44: nat,A_105: nat,B_63: nat] :
( ( A_105 = B_63 )
=> ( ( ord_less_nat @ C_44 @ B_63 )
=> ( ord_less_nat @ C_44 @ A_105 ) ) ) ).
thf(fact_516_xt1_I1_J,axiom,
! [C_44: int,A_105: int,B_63: int] :
( ( A_105 = B_63 )
=> ( ( ord_less_int @ C_44 @ B_63 )
=> ( ord_less_int @ C_44 @ A_105 ) ) ) ).
thf(fact_517_ord__less__eq__trans,axiom,
! [C_43: hoare_1708887482_state > $o,A_104: hoare_1708887482_state > $o,B_62: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_104 @ B_62 )
=> ( ( B_62 = C_43 )
=> ( ord_le14920683tate_o @ A_104 @ C_43 ) ) ) ).
thf(fact_518_ord__less__eq__trans,axiom,
! [C_43: nat,A_104: nat,B_62: nat] :
( ( ord_less_nat @ A_104 @ B_62 )
=> ( ( B_62 = C_43 )
=> ( ord_less_nat @ A_104 @ C_43 ) ) ) ).
thf(fact_519_ord__less__eq__trans,axiom,
! [C_43: int,A_104: int,B_62: int] :
( ( ord_less_int @ A_104 @ B_62 )
=> ( ( B_62 = C_43 )
=> ( ord_less_int @ A_104 @ C_43 ) ) ) ).
thf(fact_520_xt1_I2_J,axiom,
! [C_42: hoare_1708887482_state > $o,B_61: hoare_1708887482_state > $o,A_103: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ B_61 @ A_103 )
=> ( ( B_61 = C_42 )
=> ( ord_le14920683tate_o @ C_42 @ A_103 ) ) ) ).
thf(fact_521_xt1_I2_J,axiom,
! [C_42: nat,B_61: nat,A_103: nat] :
( ( ord_less_nat @ B_61 @ A_103 )
=> ( ( B_61 = C_42 )
=> ( ord_less_nat @ C_42 @ A_103 ) ) ) ).
thf(fact_522_xt1_I2_J,axiom,
! [C_42: int,B_61: int,A_103: int] :
( ( ord_less_int @ B_61 @ A_103 )
=> ( ( B_61 = C_42 )
=> ( ord_less_int @ C_42 @ A_103 ) ) ) ).
thf(fact_523_order__less__trans,axiom,
! [Z_20: hoare_1708887482_state > $o,X_44: hoare_1708887482_state > $o,Y_30: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_44 @ Y_30 )
=> ( ( ord_le14920683tate_o @ Y_30 @ Z_20 )
=> ( ord_le14920683tate_o @ X_44 @ Z_20 ) ) ) ).
thf(fact_524_order__less__trans,axiom,
! [Z_20: nat,X_44: nat,Y_30: nat] :
( ( ord_less_nat @ X_44 @ Y_30 )
=> ( ( ord_less_nat @ Y_30 @ Z_20 )
=> ( ord_less_nat @ X_44 @ Z_20 ) ) ) ).
thf(fact_525_order__less__trans,axiom,
! [Z_20: int,X_44: int,Y_30: int] :
( ( ord_less_int @ X_44 @ Y_30 )
=> ( ( ord_less_int @ Y_30 @ Z_20 )
=> ( ord_less_int @ X_44 @ Z_20 ) ) ) ).
thf(fact_526_xt1_I10_J,axiom,
! [Z_19: hoare_1708887482_state > $o,Y_29: hoare_1708887482_state > $o,X_43: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ Y_29 @ X_43 )
=> ( ( ord_le14920683tate_o @ Z_19 @ Y_29 )
=> ( ord_le14920683tate_o @ Z_19 @ X_43 ) ) ) ).
thf(fact_527_xt1_I10_J,axiom,
! [Z_19: nat,Y_29: nat,X_43: nat] :
( ( ord_less_nat @ Y_29 @ X_43 )
=> ( ( ord_less_nat @ Z_19 @ Y_29 )
=> ( ord_less_nat @ Z_19 @ X_43 ) ) ) ).
thf(fact_528_xt1_I10_J,axiom,
! [Z_19: int,Y_29: int,X_43: int] :
( ( ord_less_int @ Y_29 @ X_43 )
=> ( ( ord_less_int @ Z_19 @ Y_29 )
=> ( ord_less_int @ Z_19 @ X_43 ) ) ) ).
thf(fact_529_order__less__asym,axiom,
! [X_42: hoare_1708887482_state > $o,Y_28: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_42 @ Y_28 )
=> ~ ( ord_le14920683tate_o @ Y_28 @ X_42 ) ) ).
thf(fact_530_order__less__asym,axiom,
! [X_42: nat,Y_28: nat] :
( ( ord_less_nat @ X_42 @ Y_28 )
=> ~ ( ord_less_nat @ Y_28 @ X_42 ) ) ).
thf(fact_531_order__less__asym,axiom,
! [X_42: int,Y_28: int] :
( ( ord_less_int @ X_42 @ Y_28 )
=> ~ ( ord_less_int @ Y_28 @ X_42 ) ) ).
thf(fact_532_linorder__cases,axiom,
! [X_41: nat,Y_27: nat] :
( ~ ( ord_less_nat @ X_41 @ Y_27 )
=> ( ( X_41 != Y_27 )
=> ( ord_less_nat @ Y_27 @ X_41 ) ) ) ).
thf(fact_533_linorder__cases,axiom,
! [X_41: int,Y_27: int] :
( ~ ( ord_less_int @ X_41 @ Y_27 )
=> ( ( X_41 != Y_27 )
=> ( ord_less_int @ Y_27 @ X_41 ) ) ) ).
thf(fact_534_less__fun__def,axiom,
! [F_14: hoare_1708887482_state > $o,G: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ F_14 @ G )
<=> ( ( ord_le777019615tate_o @ F_14 @ G )
& ~ ( ord_le777019615tate_o @ G @ F_14 ) ) ) ).
thf(fact_535_linorder__not__less,axiom,
! [X_40: nat,Y_26: nat] :
( ~ ( ord_less_nat @ X_40 @ Y_26 )
<=> ( ord_less_eq_nat @ Y_26 @ X_40 ) ) ).
thf(fact_536_linorder__not__less,axiom,
! [X_40: int,Y_26: int] :
( ~ ( ord_less_int @ X_40 @ Y_26 )
<=> ( ord_less_eq_int @ Y_26 @ X_40 ) ) ).
thf(fact_537_linorder__not__le,axiom,
! [X_39: nat,Y_25: nat] :
( ~ ( ord_less_eq_nat @ X_39 @ Y_25 )
<=> ( ord_less_nat @ Y_25 @ X_39 ) ) ).
thf(fact_538_linorder__not__le,axiom,
! [X_39: int,Y_25: int] :
( ~ ( ord_less_eq_int @ X_39 @ Y_25 )
<=> ( ord_less_int @ Y_25 @ X_39 ) ) ).
thf(fact_539_linorder__le__less__linear,axiom,
! [X_38: nat,Y_24: nat] :
( ( ord_less_eq_nat @ X_38 @ Y_24 )
| ( ord_less_nat @ Y_24 @ X_38 ) ) ).
thf(fact_540_linorder__le__less__linear,axiom,
! [X_38: int,Y_24: int] :
( ( ord_less_eq_int @ X_38 @ Y_24 )
| ( ord_less_int @ Y_24 @ X_38 ) ) ).
thf(fact_541_order__less__le,axiom,
! [X_37: hoare_1708887482_state > $o,Y_23: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_37 @ Y_23 )
<=> ( ( ord_le777019615tate_o @ X_37 @ Y_23 )
& ( X_37 != Y_23 ) ) ) ).
thf(fact_542_order__less__le,axiom,
! [X_37: nat,Y_23: nat] :
( ( ord_less_nat @ X_37 @ Y_23 )
<=> ( ( ord_less_eq_nat @ X_37 @ Y_23 )
& ( X_37 != Y_23 ) ) ) ).
thf(fact_543_order__less__le,axiom,
! [X_37: int,Y_23: int] :
( ( ord_less_int @ X_37 @ Y_23 )
<=> ( ( ord_less_eq_int @ X_37 @ Y_23 )
& ( X_37 != Y_23 ) ) ) ).
thf(fact_544_less__le__not__le,axiom,
! [X_36: hoare_1708887482_state > $o,Y_22: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_36 @ Y_22 )
<=> ( ( ord_le777019615tate_o @ X_36 @ Y_22 )
& ~ ( ord_le777019615tate_o @ Y_22 @ X_36 ) ) ) ).
thf(fact_545_less__le__not__le,axiom,
! [X_36: nat,Y_22: nat] :
( ( ord_less_nat @ X_36 @ Y_22 )
<=> ( ( ord_less_eq_nat @ X_36 @ Y_22 )
& ~ ( ord_less_eq_nat @ Y_22 @ X_36 ) ) ) ).
thf(fact_546_less__le__not__le,axiom,
! [X_36: int,Y_22: int] :
( ( ord_less_int @ X_36 @ Y_22 )
<=> ( ( ord_less_eq_int @ X_36 @ Y_22 )
& ~ ( ord_less_eq_int @ Y_22 @ X_36 ) ) ) ).
thf(fact_547_order__le__less,axiom,
! [X_35: hoare_1708887482_state > $o,Y_21: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_35 @ Y_21 )
<=> ( ( ord_le14920683tate_o @ X_35 @ Y_21 )
| ( X_35 = Y_21 ) ) ) ).
thf(fact_548_order__le__less,axiom,
! [X_35: nat,Y_21: nat] :
( ( ord_less_eq_nat @ X_35 @ Y_21 )
<=> ( ( ord_less_nat @ X_35 @ Y_21 )
| ( X_35 = Y_21 ) ) ) ).
thf(fact_549_order__le__less,axiom,
! [X_35: int,Y_21: int] :
( ( ord_less_eq_int @ X_35 @ Y_21 )
<=> ( ( ord_less_int @ X_35 @ Y_21 )
| ( X_35 = Y_21 ) ) ) ).
thf(fact_550_leI,axiom,
! [X_34: nat,Y_20: nat] :
( ~ ( ord_less_nat @ X_34 @ Y_20 )
=> ( ord_less_eq_nat @ Y_20 @ X_34 ) ) ).
thf(fact_551_leI,axiom,
! [X_34: int,Y_20: int] :
( ~ ( ord_less_int @ X_34 @ Y_20 )
=> ( ord_less_eq_int @ Y_20 @ X_34 ) ) ).
thf(fact_552_not__leE,axiom,
! [Y_19: nat,X_33: nat] :
( ~ ( ord_less_eq_nat @ Y_19 @ X_33 )
=> ( ord_less_nat @ X_33 @ Y_19 ) ) ).
thf(fact_553_not__leE,axiom,
! [Y_19: int,X_33: int] :
( ~ ( ord_less_eq_int @ Y_19 @ X_33 )
=> ( ord_less_int @ X_33 @ Y_19 ) ) ).
thf(fact_554_linorder__antisym__conv1,axiom,
! [X_32: nat,Y_18: nat] :
( ~ ( ord_less_nat @ X_32 @ Y_18 )
=> ( ( ord_less_eq_nat @ X_32 @ Y_18 )
<=> ( X_32 = Y_18 ) ) ) ).
thf(fact_555_linorder__antisym__conv1,axiom,
! [X_32: int,Y_18: int] :
( ~ ( ord_less_int @ X_32 @ Y_18 )
=> ( ( ord_less_eq_int @ X_32 @ Y_18 )
<=> ( X_32 = Y_18 ) ) ) ).
thf(fact_556_order__neq__le__trans,axiom,
! [A_102: hoare_1708887482_state > $o,B_60: hoare_1708887482_state > $o] :
( ( A_102 != B_60 )
=> ( ( ord_le777019615tate_o @ A_102 @ B_60 )
=> ( ord_le14920683tate_o @ A_102 @ B_60 ) ) ) ).
thf(fact_557_order__neq__le__trans,axiom,
! [A_102: nat,B_60: nat] :
( ( A_102 != B_60 )
=> ( ( ord_less_eq_nat @ A_102 @ B_60 )
=> ( ord_less_nat @ A_102 @ B_60 ) ) ) ).
thf(fact_558_order__neq__le__trans,axiom,
! [A_102: int,B_60: int] :
( ( A_102 != B_60 )
=> ( ( ord_less_eq_int @ A_102 @ B_60 )
=> ( ord_less_int @ A_102 @ B_60 ) ) ) ).
thf(fact_559_xt1_I12_J,axiom,
! [A_101: hoare_1708887482_state > $o,B_59: hoare_1708887482_state > $o] :
( ( A_101 != B_59 )
=> ( ( ord_le777019615tate_o @ B_59 @ A_101 )
=> ( ord_le14920683tate_o @ B_59 @ A_101 ) ) ) ).
thf(fact_560_xt1_I12_J,axiom,
! [A_101: nat,B_59: nat] :
( ( A_101 != B_59 )
=> ( ( ord_less_eq_nat @ B_59 @ A_101 )
=> ( ord_less_nat @ B_59 @ A_101 ) ) ) ).
thf(fact_561_xt1_I12_J,axiom,
! [A_101: int,B_59: int] :
( ( A_101 != B_59 )
=> ( ( ord_less_eq_int @ B_59 @ A_101 )
=> ( ord_less_int @ B_59 @ A_101 ) ) ) ).
thf(fact_562_leD,axiom,
! [Y_17: nat,X_31: nat] :
( ( ord_less_eq_nat @ Y_17 @ X_31 )
=> ~ ( ord_less_nat @ X_31 @ Y_17 ) ) ).
thf(fact_563_leD,axiom,
! [Y_17: int,X_31: int] :
( ( ord_less_eq_int @ Y_17 @ X_31 )
=> ~ ( ord_less_int @ X_31 @ Y_17 ) ) ).
thf(fact_564_order__less__imp__le,axiom,
! [X_30: hoare_1708887482_state > $o,Y_16: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_30 @ Y_16 )
=> ( ord_le777019615tate_o @ X_30 @ Y_16 ) ) ).
thf(fact_565_order__less__imp__le,axiom,
! [X_30: nat,Y_16: nat] :
( ( ord_less_nat @ X_30 @ Y_16 )
=> ( ord_less_eq_nat @ X_30 @ Y_16 ) ) ).
thf(fact_566_order__less__imp__le,axiom,
! [X_30: int,Y_16: int] :
( ( ord_less_int @ X_30 @ Y_16 )
=> ( ord_less_eq_int @ X_30 @ Y_16 ) ) ).
thf(fact_567_linorder__antisym__conv2,axiom,
! [X_29: nat,Y_15: nat] :
( ( ord_less_eq_nat @ X_29 @ Y_15 )
=> ( ~ ( ord_less_nat @ X_29 @ Y_15 )
<=> ( X_29 = Y_15 ) ) ) ).
thf(fact_568_linorder__antisym__conv2,axiom,
! [X_29: int,Y_15: int] :
( ( ord_less_eq_int @ X_29 @ Y_15 )
=> ( ~ ( ord_less_int @ X_29 @ Y_15 )
<=> ( X_29 = Y_15 ) ) ) ).
thf(fact_569_order__le__imp__less__or__eq,axiom,
! [X_28: hoare_1708887482_state > $o,Y_14: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_28 @ Y_14 )
=> ( ( ord_le14920683tate_o @ X_28 @ Y_14 )
| ( X_28 = Y_14 ) ) ) ).
thf(fact_570_order__le__imp__less__or__eq,axiom,
! [X_28: nat,Y_14: nat] :
( ( ord_less_eq_nat @ X_28 @ Y_14 )
=> ( ( ord_less_nat @ X_28 @ Y_14 )
| ( X_28 = Y_14 ) ) ) ).
thf(fact_571_order__le__imp__less__or__eq,axiom,
! [X_28: int,Y_14: int] :
( ( ord_less_eq_int @ X_28 @ Y_14 )
=> ( ( ord_less_int @ X_28 @ Y_14 )
| ( X_28 = Y_14 ) ) ) ).
thf(fact_572_order__le__neq__trans,axiom,
! [A_100: hoare_1708887482_state > $o,B_58: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_100 @ B_58 )
=> ( ( A_100 != B_58 )
=> ( ord_le14920683tate_o @ A_100 @ B_58 ) ) ) ).
thf(fact_573_order__le__neq__trans,axiom,
! [A_100: nat,B_58: nat] :
( ( ord_less_eq_nat @ A_100 @ B_58 )
=> ( ( A_100 != B_58 )
=> ( ord_less_nat @ A_100 @ B_58 ) ) ) ).
thf(fact_574_order__le__neq__trans,axiom,
! [A_100: int,B_58: int] :
( ( ord_less_eq_int @ A_100 @ B_58 )
=> ( ( A_100 != B_58 )
=> ( ord_less_int @ A_100 @ B_58 ) ) ) ).
thf(fact_575_xt1_I11_J,axiom,
! [B_57: hoare_1708887482_state > $o,A_99: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ B_57 @ A_99 )
=> ( ( A_99 != B_57 )
=> ( ord_le14920683tate_o @ B_57 @ A_99 ) ) ) ).
thf(fact_576_xt1_I11_J,axiom,
! [B_57: nat,A_99: nat] :
( ( ord_less_eq_nat @ B_57 @ A_99 )
=> ( ( A_99 != B_57 )
=> ( ord_less_nat @ B_57 @ A_99 ) ) ) ).
thf(fact_577_xt1_I11_J,axiom,
! [B_57: int,A_99: int] :
( ( ord_less_eq_int @ B_57 @ A_99 )
=> ( ( A_99 != B_57 )
=> ( ord_less_int @ B_57 @ A_99 ) ) ) ).
thf(fact_578_order__less__le__trans,axiom,
! [Z_18: hoare_1708887482_state > $o,X_27: hoare_1708887482_state > $o,Y_13: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ X_27 @ Y_13 )
=> ( ( ord_le777019615tate_o @ Y_13 @ Z_18 )
=> ( ord_le14920683tate_o @ X_27 @ Z_18 ) ) ) ).
thf(fact_579_order__less__le__trans,axiom,
! [Z_18: nat,X_27: nat,Y_13: nat] :
( ( ord_less_nat @ X_27 @ Y_13 )
=> ( ( ord_less_eq_nat @ Y_13 @ Z_18 )
=> ( ord_less_nat @ X_27 @ Z_18 ) ) ) ).
thf(fact_580_order__less__le__trans,axiom,
! [Z_18: int,X_27: int,Y_13: int] :
( ( ord_less_int @ X_27 @ Y_13 )
=> ( ( ord_less_eq_int @ Y_13 @ Z_18 )
=> ( ord_less_int @ X_27 @ Z_18 ) ) ) ).
thf(fact_581_xt1_I7_J,axiom,
! [Z_17: hoare_1708887482_state > $o,Y_12: hoare_1708887482_state > $o,X_26: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ Y_12 @ X_26 )
=> ( ( ord_le777019615tate_o @ Z_17 @ Y_12 )
=> ( ord_le14920683tate_o @ Z_17 @ X_26 ) ) ) ).
thf(fact_582_xt1_I7_J,axiom,
! [Z_17: nat,Y_12: nat,X_26: nat] :
( ( ord_less_nat @ Y_12 @ X_26 )
=> ( ( ord_less_eq_nat @ Z_17 @ Y_12 )
=> ( ord_less_nat @ Z_17 @ X_26 ) ) ) ).
thf(fact_583_xt1_I7_J,axiom,
! [Z_17: int,Y_12: int,X_26: int] :
( ( ord_less_int @ Y_12 @ X_26 )
=> ( ( ord_less_eq_int @ Z_17 @ Y_12 )
=> ( ord_less_int @ Z_17 @ X_26 ) ) ) ).
thf(fact_584_order__le__less__trans,axiom,
! [Z_16: hoare_1708887482_state > $o,X_25: hoare_1708887482_state > $o,Y_11: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ X_25 @ Y_11 )
=> ( ( ord_le14920683tate_o @ Y_11 @ Z_16 )
=> ( ord_le14920683tate_o @ X_25 @ Z_16 ) ) ) ).
thf(fact_585_order__le__less__trans,axiom,
! [Z_16: nat,X_25: nat,Y_11: nat] :
( ( ord_less_eq_nat @ X_25 @ Y_11 )
=> ( ( ord_less_nat @ Y_11 @ Z_16 )
=> ( ord_less_nat @ X_25 @ Z_16 ) ) ) ).
thf(fact_586_order__le__less__trans,axiom,
! [Z_16: int,X_25: int,Y_11: int] :
( ( ord_less_eq_int @ X_25 @ Y_11 )
=> ( ( ord_less_int @ Y_11 @ Z_16 )
=> ( ord_less_int @ X_25 @ Z_16 ) ) ) ).
thf(fact_587_xt1_I8_J,axiom,
! [Z_15: hoare_1708887482_state > $o,Y_10: hoare_1708887482_state > $o,X_24: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ Y_10 @ X_24 )
=> ( ( ord_le14920683tate_o @ Z_15 @ Y_10 )
=> ( ord_le14920683tate_o @ Z_15 @ X_24 ) ) ) ).
thf(fact_588_xt1_I8_J,axiom,
! [Z_15: nat,Y_10: nat,X_24: nat] :
( ( ord_less_eq_nat @ Y_10 @ X_24 )
=> ( ( ord_less_nat @ Z_15 @ Y_10 )
=> ( ord_less_nat @ Z_15 @ X_24 ) ) ) ).
thf(fact_589_xt1_I8_J,axiom,
! [Z_15: int,Y_10: int,X_24: int] :
( ( ord_less_eq_int @ Y_10 @ X_24 )
=> ( ( ord_less_int @ Z_15 @ Y_10 )
=> ( ord_less_int @ Z_15 @ X_24 ) ) ) ).
thf(fact_590_not__less__bot,axiom,
! [A_98: int > $o] :
~ ( ord_less_int_o @ A_98 @ bot_bot_int_o ) ).
thf(fact_591_not__less__bot,axiom,
! [A_98: nat > $o] :
~ ( ord_less_nat_o @ A_98 @ bot_bot_nat_o ) ).
thf(fact_592_not__less__bot,axiom,
! [A_98: $o] :
~ ( ord_less_o @ A_98 @ bot_bot_o ) ).
thf(fact_593_not__less__bot,axiom,
! [A_98: hoare_1708887482_state > $o] :
~ ( ord_le14920683tate_o @ A_98 @ bot_bo19817387tate_o ) ).
thf(fact_594_not__less__bot,axiom,
! [A_98: nat] :
~ ( ord_less_nat @ A_98 @ bot_bot_nat ) ).
thf(fact_595_bot__less,axiom,
! [A_97: int > $o] :
( ( A_97 != bot_bot_int_o )
<=> ( ord_less_int_o @ bot_bot_int_o @ A_97 ) ) ).
thf(fact_596_bot__less,axiom,
! [A_97: nat > $o] :
( ( A_97 != bot_bot_nat_o )
<=> ( ord_less_nat_o @ bot_bot_nat_o @ A_97 ) ) ).
thf(fact_597_bot__less,axiom,
! [A_97: $o] :
( ~ ( A_97
<=> bot_bot_o )
<=> ( ord_less_o @ bot_bot_o @ A_97 ) ) ).
thf(fact_598_bot__less,axiom,
! [A_97: hoare_1708887482_state > $o] :
( ( A_97 != bot_bo19817387tate_o )
<=> ( ord_le14920683tate_o @ bot_bo19817387tate_o @ A_97 ) ) ).
thf(fact_599_bot__less,axiom,
! [A_97: nat] :
( ( A_97 != bot_bot_nat )
<=> ( ord_less_nat @ bot_bot_nat @ A_97 ) ) ).
thf(fact_600_not__psubset__empty,axiom,
! [A_96: int > $o] :
~ ( ord_less_int_o @ A_96 @ bot_bot_int_o ) ).
thf(fact_601_not__psubset__empty,axiom,
! [A_96: nat > $o] :
~ ( ord_less_nat_o @ A_96 @ bot_bot_nat_o ) ).
thf(fact_602_not__psubset__empty,axiom,
! [A_96: hoare_1708887482_state > $o] :
~ ( ord_le14920683tate_o @ A_96 @ bot_bo19817387tate_o ) ).
thf(fact_603_diff__eq__diff__less,axiom,
! [A_95: int,B_56: int,C_41: int,D_16: int] :
( ( ( minus_minus_int @ A_95 @ B_56 )
= ( minus_minus_int @ C_41 @ D_16 ) )
=> ( ( ord_less_int @ A_95 @ B_56 )
<=> ( ord_less_int @ C_41 @ D_16 ) ) ) ).
thf(fact_604_psubset__eq,axiom,
! [A_94: hoare_1708887482_state > $o,B_55: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_94 @ B_55 )
<=> ( ( ord_le777019615tate_o @ A_94 @ B_55 )
& ( A_94 != B_55 ) ) ) ).
thf(fact_605_subset__iff__psubset__eq,axiom,
! [A_93: hoare_1708887482_state > $o,B_54: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_93 @ B_54 )
<=> ( ( ord_le14920683tate_o @ A_93 @ B_54 )
| ( A_93 = B_54 ) ) ) ).
thf(fact_606_psubset__imp__subset,axiom,
! [A_92: hoare_1708887482_state > $o,B_53: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_92 @ B_53 )
=> ( ord_le777019615tate_o @ A_92 @ B_53 ) ) ).
thf(fact_607_psubset__subset__trans,axiom,
! [C_40: hoare_1708887482_state > $o,A_91: hoare_1708887482_state > $o,B_52: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_91 @ B_52 )
=> ( ( ord_le777019615tate_o @ B_52 @ C_40 )
=> ( ord_le14920683tate_o @ A_91 @ C_40 ) ) ) ).
thf(fact_608_subset__psubset__trans,axiom,
! [C_39: hoare_1708887482_state > $o,A_90: hoare_1708887482_state > $o,B_51: hoare_1708887482_state > $o] :
( ( ord_le777019615tate_o @ A_90 @ B_51 )
=> ( ( ord_le14920683tate_o @ B_51 @ C_39 )
=> ( ord_le14920683tate_o @ A_90 @ C_39 ) ) ) ).
thf(fact_609_card__insert__le,axiom,
! [X_23: int,A_89: int > $o] :
( ( finite_finite_int @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A_89 ) @ ( finite_card_int @ ( insert_int @ X_23 @ A_89 ) ) ) ) ).
thf(fact_610_card__insert__le,axiom,
! [X_23: nat,A_89: nat > $o] :
( ( finite_finite_nat @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A_89 ) @ ( finite_card_nat @ ( insert_nat @ X_23 @ A_89 ) ) ) ) ).
thf(fact_611_card__insert__le,axiom,
! [X_23: hoare_1708887482_state,A_89: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_89 )
=> ( ord_less_eq_nat @ ( finite185771624_state @ A_89 ) @ ( finite185771624_state @ ( insert528405184_state @ X_23 @ A_89 ) ) ) ) ).
thf(fact_612_card__mono,axiom,
! [A_88: hoare_1708887482_state > $o,B_50: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_50 )
=> ( ( ord_le777019615tate_o @ A_88 @ B_50 )
=> ( ord_less_eq_nat @ ( finite185771624_state @ A_88 ) @ ( finite185771624_state @ B_50 ) ) ) ) ).
thf(fact_613_card__mono,axiom,
! [A_88: int > $o,B_50: int > $o] :
( ( finite_finite_int @ B_50 )
=> ( ( ord_less_eq_int_o @ A_88 @ B_50 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A_88 ) @ ( finite_card_int @ B_50 ) ) ) ) ).
thf(fact_614_card__mono,axiom,
! [A_88: nat > $o,B_50: nat > $o] :
( ( finite_finite_nat @ B_50 )
=> ( ( ord_less_eq_nat_o @ A_88 @ B_50 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A_88 ) @ ( finite_card_nat @ B_50 ) ) ) ) ).
thf(fact_615_card__seteq,axiom,
! [A_87: hoare_1708887482_state > $o,B_49: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_49 )
=> ( ( ord_le777019615tate_o @ A_87 @ B_49 )
=> ( ( ord_less_eq_nat @ ( finite185771624_state @ B_49 ) @ ( finite185771624_state @ A_87 ) )
=> ( A_87 = B_49 ) ) ) ) ).
thf(fact_616_card__seteq,axiom,
! [A_87: int > $o,B_49: int > $o] :
( ( finite_finite_int @ B_49 )
=> ( ( ord_less_eq_int_o @ A_87 @ B_49 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B_49 ) @ ( finite_card_int @ A_87 ) )
=> ( A_87 = B_49 ) ) ) ) ).
thf(fact_617_card__seteq,axiom,
! [A_87: nat > $o,B_49: nat > $o] :
( ( finite_finite_nat @ B_49 )
=> ( ( ord_less_eq_nat_o @ A_87 @ B_49 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B_49 ) @ ( finite_card_nat @ A_87 ) )
=> ( A_87 = B_49 ) ) ) ) ).
thf(fact_618_diff__card__le__card__Diff,axiom,
! [A_86: hoare_1708887482_state > $o,B_48: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_48 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite185771624_state @ A_86 ) @ ( finite185771624_state @ B_48 ) ) @ ( finite185771624_state @ ( minus_2056855718tate_o @ A_86 @ B_48 ) ) ) ) ).
thf(fact_619_diff__card__le__card__Diff,axiom,
! [A_86: int > $o,B_48: int > $o] :
( ( finite_finite_int @ B_48 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A_86 ) @ ( finite_card_int @ B_48 ) ) @ ( finite_card_int @ ( minus_minus_int_o @ A_86 @ B_48 ) ) ) ) ).
thf(fact_620_diff__card__le__card__Diff,axiom,
! [A_86: nat > $o,B_48: nat > $o] :
( ( finite_finite_nat @ B_48 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A_86 ) @ ( finite_card_nat @ B_48 ) ) @ ( finite_card_nat @ ( minus_minus_nat_o @ A_86 @ B_48 ) ) ) ) ).
thf(fact_621_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_9: int,Z_14: int,A_85: int > $o,X_22: int,F_13: int > int > int] :
( ( finite1973466193nt_int @ F_13 )
=> ( ( finite772772422nt_int @ F_13 @ Z_14 @ A_85 @ X_22 )
=> ( ( finite772772422nt_int @ F_13 @ Z_14 @ A_85 @ Y_9 )
=> ( Y_9 = X_22 ) ) ) ) ).
thf(fact_622_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_9: nat,Z_14: nat,A_85: nat > $o,X_22: nat,F_13: nat > nat > nat] :
( ( finite2130160977at_nat @ F_13 )
=> ( ( finite929467206at_nat @ F_13 @ Z_14 @ A_85 @ X_22 )
=> ( ( finite929467206at_nat @ F_13 @ Z_14 @ A_85 @ Y_9 )
=> ( Y_9 = X_22 ) ) ) ) ).
thf(fact_623_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_9: hoare_1708887482_state,Z_14: hoare_1708887482_state,A_85: hoare_1708887482_state > $o,X_22: hoare_1708887482_state,F_13: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state] :
( ( finite1816785523_state @ F_13 )
=> ( ( finite2042206440_state @ F_13 @ Z_14 @ A_85 @ X_22 )
=> ( ( finite2042206440_state @ F_13 @ Z_14 @ A_85 @ Y_9 )
=> ( Y_9 = X_22 ) ) ) ) ).
thf(fact_624_fold__graph__insert__swap,axiom,
! [Z_13: int,B_47: int,A_84: int > $o,Y_8: int] :
( ( finite772772422nt_int @ times_times_int @ B_47 @ A_84 @ Y_8 )
=> ( ~ ( member_int @ B_47 @ A_84 )
=> ( finite772772422nt_int @ times_times_int @ Z_13 @ ( insert_int @ B_47 @ A_84 ) @ ( times_times_int @ Z_13 @ Y_8 ) ) ) ) ).
thf(fact_625_fold__graph__insert__swap,axiom,
! [Z_13: nat,B_47: nat,A_84: nat > $o,Y_8: nat] :
( ( finite929467206at_nat @ times_times_nat @ B_47 @ A_84 @ Y_8 )
=> ( ~ ( member_nat @ B_47 @ A_84 )
=> ( finite929467206at_nat @ times_times_nat @ Z_13 @ ( insert_nat @ B_47 @ A_84 ) @ ( times_times_nat @ Z_13 @ Y_8 ) ) ) ) ).
thf(fact_626_card__Diff__subset,axiom,
! [A_83: hoare_1708887482_state > $o,B_46: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_46 )
=> ( ( ord_le777019615tate_o @ B_46 @ A_83 )
=> ( ( finite185771624_state @ ( minus_2056855718tate_o @ A_83 @ B_46 ) )
= ( minus_minus_nat @ ( finite185771624_state @ A_83 ) @ ( finite185771624_state @ B_46 ) ) ) ) ) ).
thf(fact_627_card__Diff__subset,axiom,
! [A_83: int > $o,B_46: int > $o] :
( ( finite_finite_int @ B_46 )
=> ( ( ord_less_eq_int_o @ B_46 @ A_83 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_83 @ B_46 ) )
= ( minus_minus_nat @ ( finite_card_int @ A_83 ) @ ( finite_card_int @ B_46 ) ) ) ) ) ).
thf(fact_628_card__Diff__subset,axiom,
! [A_83: nat > $o,B_46: nat > $o] :
( ( finite_finite_nat @ B_46 )
=> ( ( ord_less_eq_nat_o @ B_46 @ A_83 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_83 @ B_46 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A_83 ) @ ( finite_card_nat @ B_46 ) ) ) ) ) ).
thf(fact_629_fold1__in,axiom,
! [A_82: int > $o] :
( ( finite_finite_int @ A_82 )
=> ( ( A_82 != bot_bot_int_o )
=> ( ! [X_1: int,Y_7: int] : ( member_int @ ( times_times_int @ X_1 @ Y_7 ) @ ( insert_int @ X_1 @ ( insert_int @ Y_7 @ bot_bot_int_o ) ) )
=> ( member_int @ ( finite_fold1_int @ times_times_int @ A_82 ) @ A_82 ) ) ) ) ).
thf(fact_630_fold1__in,axiom,
! [A_82: nat > $o] :
( ( finite_finite_nat @ A_82 )
=> ( ( A_82 != bot_bot_nat_o )
=> ( ! [X_1: nat,Y_7: nat] : ( member_nat @ ( times_times_nat @ X_1 @ Y_7 ) @ ( insert_nat @ X_1 @ ( insert_nat @ Y_7 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( finite_fold1_nat @ times_times_nat @ A_82 ) @ A_82 ) ) ) ) ).
thf(fact_631_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_12: int,X_21: int,A_81: int > $o,V_2: int,F_12: int > int > int] :
( ( finite1973466193nt_int @ F_12 )
=> ( ( finite772772422nt_int @ F_12 @ Z_12 @ ( insert_int @ X_21 @ A_81 ) @ V_2 )
=> ( ~ ( member_int @ X_21 @ A_81 )
=> ~ ! [Y_7: int] :
( ( V_2
= ( F_12 @ X_21 @ Y_7 ) )
=> ~ ( finite772772422nt_int @ F_12 @ Z_12 @ A_81 @ Y_7 ) ) ) ) ) ).
thf(fact_632_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_12: nat,X_21: nat,A_81: nat > $o,V_2: nat,F_12: nat > nat > nat] :
( ( finite2130160977at_nat @ F_12 )
=> ( ( finite929467206at_nat @ F_12 @ Z_12 @ ( insert_nat @ X_21 @ A_81 ) @ V_2 )
=> ( ~ ( member_nat @ X_21 @ A_81 )
=> ~ ! [Y_7: nat] :
( ( V_2
= ( F_12 @ X_21 @ Y_7 ) )
=> ~ ( finite929467206at_nat @ F_12 @ Z_12 @ A_81 @ Y_7 ) ) ) ) ) ).
thf(fact_633_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_12: hoare_1708887482_state,X_21: hoare_1708887482_state,A_81: hoare_1708887482_state > $o,V_2: hoare_1708887482_state,F_12: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state] :
( ( finite1816785523_state @ F_12 )
=> ( ( finite2042206440_state @ F_12 @ Z_12 @ ( insert528405184_state @ X_21 @ A_81 ) @ V_2 )
=> ( ~ ( member451959335_state @ X_21 @ A_81 )
=> ~ ! [Y_7: hoare_1708887482_state] :
( ( V_2
= ( F_12 @ X_21 @ Y_7 ) )
=> ~ ( finite2042206440_state @ F_12 @ Z_12 @ A_81 @ Y_7 ) ) ) ) ) ).
thf(fact_634_psubset__imp__ex__mem,axiom,
! [A_80: hoare_1708887482_state > $o,B_45: hoare_1708887482_state > $o] :
( ( ord_le14920683tate_o @ A_80 @ B_45 )
=> ? [B_42: hoare_1708887482_state] : ( member451959335_state @ B_42 @ ( minus_2056855718tate_o @ B_45 @ A_80 ) ) ) ).
thf(fact_635_psubset__imp__ex__mem,axiom,
! [A_80: int > $o,B_45: int > $o] :
( ( ord_less_int_o @ A_80 @ B_45 )
=> ? [B_42: int] : ( member_int @ B_42 @ ( minus_minus_int_o @ B_45 @ A_80 ) ) ) ).
thf(fact_636_psubset__imp__ex__mem,axiom,
! [A_80: nat > $o,B_45: nat > $o] :
( ( ord_less_nat_o @ A_80 @ B_45 )
=> ? [B_42: nat] : ( member_nat @ B_42 @ ( minus_minus_nat_o @ B_45 @ A_80 ) ) ) ).
thf(fact_637_psubset__card__mono,axiom,
! [A_79: hoare_1708887482_state > $o,B_44: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_44 )
=> ( ( ord_le14920683tate_o @ A_79 @ B_44 )
=> ( ord_less_nat @ ( finite185771624_state @ A_79 ) @ ( finite185771624_state @ B_44 ) ) ) ) ).
thf(fact_638_psubset__card__mono,axiom,
! [A_79: int > $o,B_44: int > $o] :
( ( finite_finite_int @ B_44 )
=> ( ( ord_less_int_o @ A_79 @ B_44 )
=> ( ord_less_nat @ ( finite_card_int @ A_79 ) @ ( finite_card_int @ B_44 ) ) ) ) ).
thf(fact_639_psubset__card__mono,axiom,
! [A_79: nat > $o,B_44: nat > $o] :
( ( finite_finite_nat @ B_44 )
=> ( ( ord_less_nat_o @ A_79 @ B_44 )
=> ( ord_less_nat @ ( finite_card_nat @ A_79 ) @ ( finite_card_nat @ B_44 ) ) ) ) ).
thf(fact_640_card__psubset,axiom,
! [A_78: hoare_1708887482_state > $o,B_43: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ B_43 )
=> ( ( ord_le777019615tate_o @ A_78 @ B_43 )
=> ( ( ord_less_nat @ ( finite185771624_state @ A_78 ) @ ( finite185771624_state @ B_43 ) )
=> ( ord_le14920683tate_o @ A_78 @ B_43 ) ) ) ) ).
thf(fact_641_card__psubset,axiom,
! [A_78: int > $o,B_43: int > $o] :
( ( finite_finite_int @ B_43 )
=> ( ( ord_less_eq_int_o @ A_78 @ B_43 )
=> ( ( ord_less_nat @ ( finite_card_int @ A_78 ) @ ( finite_card_int @ B_43 ) )
=> ( ord_less_int_o @ A_78 @ B_43 ) ) ) ) ).
thf(fact_642_card__psubset,axiom,
! [A_78: nat > $o,B_43: nat > $o] :
( ( finite_finite_nat @ B_43 )
=> ( ( ord_less_eq_nat_o @ A_78 @ B_43 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A_78 ) @ ( finite_card_nat @ B_43 ) )
=> ( ord_less_nat_o @ A_78 @ B_43 ) ) ) ) ).
thf(fact_643_card__Diff1__less,axiom,
! [X_20: int,A_77: int > $o] :
( ( finite_finite_int @ A_77 )
=> ( ( member_int @ X_20 @ A_77 )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_77 @ ( insert_int @ X_20 @ bot_bot_int_o ) ) ) @ ( finite_card_int @ A_77 ) ) ) ) ).
thf(fact_644_card__Diff1__less,axiom,
! [X_20: nat,A_77: nat > $o] :
( ( finite_finite_nat @ A_77 )
=> ( ( member_nat @ X_20 @ A_77 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_77 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) ) ) @ ( finite_card_nat @ A_77 ) ) ) ) ).
thf(fact_645_card__Diff1__less,axiom,
! [X_20: hoare_1708887482_state,A_77: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_77 )
=> ( ( member451959335_state @ X_20 @ A_77 )
=> ( ord_less_nat @ ( finite185771624_state @ ( minus_2056855718tate_o @ A_77 @ ( insert528405184_state @ X_20 @ bot_bo19817387tate_o ) ) ) @ ( finite185771624_state @ A_77 ) ) ) ) ).
thf(fact_646_card__Diff2__less,axiom,
! [Y_6: int,X_19: int,A_76: int > $o] :
( ( finite_finite_int @ A_76 )
=> ( ( member_int @ X_19 @ A_76 )
=> ( ( member_int @ Y_6 @ A_76 )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_int_o @ ( minus_minus_int_o @ A_76 @ ( insert_int @ X_19 @ bot_bot_int_o ) ) @ ( insert_int @ Y_6 @ bot_bot_int_o ) ) ) @ ( finite_card_int @ A_76 ) ) ) ) ) ).
thf(fact_647_card__Diff2__less,axiom,
! [Y_6: nat,X_19: nat,A_76: nat > $o] :
( ( finite_finite_nat @ A_76 )
=> ( ( member_nat @ X_19 @ A_76 )
=> ( ( member_nat @ Y_6 @ A_76 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_76 @ ( insert_nat @ X_19 @ bot_bot_nat_o ) ) @ ( insert_nat @ Y_6 @ bot_bot_nat_o ) ) ) @ ( finite_card_nat @ A_76 ) ) ) ) ) ).
thf(fact_648_card__Diff2__less,axiom,
! [Y_6: hoare_1708887482_state,X_19: hoare_1708887482_state,A_76: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_76 )
=> ( ( member451959335_state @ X_19 @ A_76 )
=> ( ( member451959335_state @ Y_6 @ A_76 )
=> ( ord_less_nat @ ( finite185771624_state @ ( minus_2056855718tate_o @ ( minus_2056855718tate_o @ A_76 @ ( insert528405184_state @ X_19 @ bot_bo19817387tate_o ) ) @ ( insert528405184_state @ Y_6 @ bot_bo19817387tate_o ) ) ) @ ( finite185771624_state @ A_76 ) ) ) ) ) ).
thf(fact_649_finite__linorder__max__induct,axiom,
! [P_3: ( int > $o ) > $o,A_75: int > $o] :
( ( finite_finite_int @ A_75 )
=> ( ( P_3 @ bot_bot_int_o )
=> ( ! [B_42: int,A_74: int > $o] :
( ( finite_finite_int @ A_74 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ A_74 )
=> ( ord_less_int @ X_1 @ B_42 ) )
=> ( ( P_3 @ A_74 )
=> ( P_3 @ ( insert_int @ B_42 @ A_74 ) ) ) ) )
=> ( P_3 @ A_75 ) ) ) ) ).
thf(fact_650_finite__linorder__max__induct,axiom,
! [P_3: ( nat > $o ) > $o,A_75: nat > $o] :
( ( finite_finite_nat @ A_75 )
=> ( ( P_3 @ bot_bot_nat_o )
=> ( ! [B_42: nat,A_74: nat > $o] :
( ( finite_finite_nat @ A_74 )
=> ( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_74 )
=> ( ord_less_nat @ X_1 @ B_42 ) )
=> ( ( P_3 @ A_74 )
=> ( P_3 @ ( insert_nat @ B_42 @ A_74 ) ) ) ) )
=> ( P_3 @ A_75 ) ) ) ) ).
thf(fact_651_diff__less__mono,axiom,
! [C_38: nat,A: nat,B_1: nat] :
( ( ord_less_nat @ A @ B_1 )
=> ( ( ord_less_eq_nat @ C_38 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C_38 ) @ ( minus_minus_nat @ B_1 @ C_38 ) ) ) ) ).
thf(fact_652_less__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_653_finite__linorder__min__induct,axiom,
! [P_2: ( int > $o ) > $o,A_73: int > $o] :
( ( finite_finite_int @ A_73 )
=> ( ( P_2 @ bot_bot_int_o )
=> ( ! [B_42: int,A_74: int > $o] :
( ( finite_finite_int @ A_74 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ A_74 )
=> ( ord_less_int @ B_42 @ X_1 ) )
=> ( ( P_2 @ A_74 )
=> ( P_2 @ ( insert_int @ B_42 @ A_74 ) ) ) ) )
=> ( P_2 @ A_73 ) ) ) ) ).
thf(fact_654_finite__linorder__min__induct,axiom,
! [P_2: ( nat > $o ) > $o,A_73: nat > $o] :
( ( finite_finite_nat @ A_73 )
=> ( ( P_2 @ bot_bot_nat_o )
=> ( ! [B_42: nat,A_74: nat > $o] :
( ( finite_finite_nat @ A_74 )
=> ( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_74 )
=> ( ord_less_nat @ B_42 @ X_1 ) )
=> ( ( P_2 @ A_74 )
=> ( P_2 @ ( insert_nat @ B_42 @ A_74 ) ) ) ) )
=> ( P_2 @ A_73 ) ) ) ) ).
thf(fact_655_le__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_656_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_657_eq__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
<=> ( M = N ) ) ) ) ).
thf(fact_658_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_659_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_660_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_661_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_662_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_663_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_664_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( P @ N @ M ) )
=> ( ( ( M = N )
=> ( P @ N @ M ) )
=> ( ( ( ord_less_nat @ N @ M )
=> ( P @ N @ M ) )
=> ( P @ N @ M ) ) ) ) ).
thf(fact_665_mult__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_666_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_667_le__trans,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I_1 @ K ) ) ) ).
thf(fact_668_mult__le__mono2,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ).
thf(fact_669_mult__le__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
thf(fact_670_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_671_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_672_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_673_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_674_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_675_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_676_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_677_diff__commute,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ K ) @ J ) ) ).
thf(fact_678_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_679_le__eq__less__or__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_680_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_681_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_682_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_683_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_684_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
thf(fact_685_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_686_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_687_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
thf(fact_688_diff__diff__cancel,axiom,
! [I_1: nat,N: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I_1 ) )
= I_1 ) ) ).
thf(fact_689_less__mono__imp__le__mono,axiom,
! [I_1: nat,J: nat,F: nat > nat] :
( ! [I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J_1 ) ) )
=> ( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( F @ I_1 ) @ ( F @ J ) ) ) ) ).
thf(fact_690_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_691_card__Diff__singleton__if,axiom,
! [X_18: int,A_72: int > $o] :
( ( finite_finite_int @ A_72 )
=> ( ( ( member_int @ X_18 @ A_72 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_72 @ ( insert_int @ X_18 @ bot_bot_int_o ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A_72 ) @ one_one_nat ) ) )
& ( ~ ( member_int @ X_18 @ A_72 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_72 @ ( insert_int @ X_18 @ bot_bot_int_o ) ) )
= ( finite_card_int @ A_72 ) ) ) ) ) ).
thf(fact_692_card__Diff__singleton__if,axiom,
! [X_18: nat,A_72: nat > $o] :
( ( finite_finite_nat @ A_72 )
=> ( ( ( member_nat @ X_18 @ A_72 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_72 @ ( insert_nat @ X_18 @ bot_bot_nat_o ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A_72 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X_18 @ A_72 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_72 @ ( insert_nat @ X_18 @ bot_bot_nat_o ) ) )
= ( finite_card_nat @ A_72 ) ) ) ) ) ).
thf(fact_693_card__Diff__singleton__if,axiom,
! [X_18: hoare_1708887482_state,A_72: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_72 )
=> ( ( ( member451959335_state @ X_18 @ A_72 )
=> ( ( finite185771624_state @ ( minus_2056855718tate_o @ A_72 @ ( insert528405184_state @ X_18 @ bot_bo19817387tate_o ) ) )
= ( minus_minus_nat @ ( finite185771624_state @ A_72 ) @ one_one_nat ) ) )
& ( ~ ( member451959335_state @ X_18 @ A_72 )
=> ( ( finite185771624_state @ ( minus_2056855718tate_o @ A_72 @ ( insert528405184_state @ X_18 @ bot_bo19817387tate_o ) ) )
= ( finite185771624_state @ A_72 ) ) ) ) ) ).
thf(fact_694_card__Diff__singleton,axiom,
! [X_17: int,A_71: int > $o] :
( ( finite_finite_int @ A_71 )
=> ( ( member_int @ X_17 @ A_71 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_71 @ ( insert_int @ X_17 @ bot_bot_int_o ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A_71 ) @ one_one_nat ) ) ) ) ).
thf(fact_695_card__Diff__singleton,axiom,
! [X_17: nat,A_71: nat > $o] :
( ( finite_finite_nat @ A_71 )
=> ( ( member_nat @ X_17 @ A_71 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_71 @ ( insert_nat @ X_17 @ bot_bot_nat_o ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A_71 ) @ one_one_nat ) ) ) ) ).
thf(fact_696_card__Diff__singleton,axiom,
! [X_17: hoare_1708887482_state,A_71: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_71 )
=> ( ( member451959335_state @ X_17 @ A_71 )
=> ( ( finite185771624_state @ ( minus_2056855718tate_o @ A_71 @ ( insert528405184_state @ X_17 @ bot_bo19817387tate_o ) ) )
= ( minus_minus_nat @ ( finite185771624_state @ A_71 ) @ one_one_nat ) ) ) ) ).
thf(fact_697_card__Diff__insert,axiom,
! [B_41: int > $o,A_70: int,A_69: int > $o] :
( ( finite_finite_int @ A_69 )
=> ( ( member_int @ A_70 @ A_69 )
=> ( ~ ( member_int @ A_70 @ B_41 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_69 @ ( insert_int @ A_70 @ B_41 ) ) )
= ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_69 @ B_41 ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_698_card__Diff__insert,axiom,
! [B_41: nat > $o,A_70: nat,A_69: nat > $o] :
( ( finite_finite_nat @ A_69 )
=> ( ( member_nat @ A_70 @ A_69 )
=> ( ~ ( member_nat @ A_70 @ B_41 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_69 @ ( insert_nat @ A_70 @ B_41 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_69 @ B_41 ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_699_card__Diff__insert,axiom,
! [B_41: hoare_1708887482_state > $o,A_70: hoare_1708887482_state,A_69: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_69 )
=> ( ( member451959335_state @ A_70 @ A_69 )
=> ( ~ ( member451959335_state @ A_70 @ B_41 )
=> ( ( finite185771624_state @ ( minus_2056855718tate_o @ A_69 @ ( insert528405184_state @ A_70 @ B_41 ) ) )
= ( minus_minus_nat @ ( finite185771624_state @ ( minus_2056855718tate_o @ A_69 @ B_41 ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_700_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_11: int,X_16: int,A_68: int > $o,F_11: int > int > int] :
( ( finite1973466193nt_int @ F_11 )
=> ( ( finite_finite_int @ A_68 )
=> ( ( finite_fold_int_int @ F_11 @ Z_11 @ ( insert_int @ X_16 @ A_68 ) )
= ( F_11 @ X_16 @ ( finite_fold_int_int @ F_11 @ Z_11 @ ( minus_minus_int_o @ A_68 @ ( insert_int @ X_16 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_701_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_11: nat,X_16: nat,A_68: nat > $o,F_11: nat > nat > nat] :
( ( finite2130160977at_nat @ F_11 )
=> ( ( finite_finite_nat @ A_68 )
=> ( ( finite_fold_nat_nat @ F_11 @ Z_11 @ ( insert_nat @ X_16 @ A_68 ) )
= ( F_11 @ X_16 @ ( finite_fold_nat_nat @ F_11 @ Z_11 @ ( minus_minus_nat_o @ A_68 @ ( insert_nat @ X_16 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_702_nat__mult__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K )
= ( times_times_nat @ M @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_703_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_704_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_705_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_706_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_707_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_708_one__reorient,axiom,
! [X_15: int] :
( ( one_one_int = X_15 )
<=> ( X_15 = one_one_int ) ) ).
thf(fact_709_one__reorient,axiom,
! [X_15: nat] :
( ( one_one_nat = X_15 )
<=> ( X_15 = one_one_nat ) ) ).
thf(fact_710_mult__1__left,axiom,
! [A_67: int] :
( ( times_times_int @ one_one_int @ A_67 )
= A_67 ) ).
thf(fact_711_mult__1__left,axiom,
! [A_67: nat] :
( ( times_times_nat @ one_one_nat @ A_67 )
= A_67 ) ).
thf(fact_712_mult__1,axiom,
! [A_66: int] :
( ( times_times_int @ one_one_int @ A_66 )
= A_66 ) ).
thf(fact_713_mult__1,axiom,
! [A_66: nat] :
( ( times_times_nat @ one_one_nat @ A_66 )
= A_66 ) ).
thf(fact_714_mult__1__right,axiom,
! [A_65: int] :
( ( times_times_int @ A_65 @ one_one_int )
= A_65 ) ).
thf(fact_715_mult__1__right,axiom,
! [A_65: nat] :
( ( times_times_nat @ A_65 @ one_one_nat )
= A_65 ) ).
thf(fact_716_mult_Ocomm__neutral,axiom,
! [A_64: int] :
( ( times_times_int @ A_64 @ one_one_int )
= A_64 ) ).
thf(fact_717_mult_Ocomm__neutral,axiom,
! [A_64: nat] :
( ( times_times_nat @ A_64 @ one_one_nat )
= A_64 ) ).
thf(fact_718_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_14: int,Z_10: int,A_63: int > $o,F_10: int > int > int] :
( ( finite1973466193nt_int @ F_10 )
=> ( ( finite_finite_int @ A_63 )
=> ( ( F_10 @ X_14 @ ( finite_fold_int_int @ F_10 @ Z_10 @ A_63 ) )
= ( finite_fold_int_int @ F_10 @ ( F_10 @ X_14 @ Z_10 ) @ A_63 ) ) ) ) ).
thf(fact_719_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_14: nat,Z_10: nat,A_63: nat > $o,F_10: nat > nat > nat] :
( ( finite2130160977at_nat @ F_10 )
=> ( ( finite_finite_nat @ A_63 )
=> ( ( F_10 @ X_14 @ ( finite_fold_nat_nat @ F_10 @ Z_10 @ A_63 ) )
= ( finite_fold_nat_nat @ F_10 @ ( F_10 @ X_14 @ Z_10 ) @ A_63 ) ) ) ) ).
thf(fact_720_comp__fun__commute_Ofold__equality,axiom,
! [Z_9: int,A_62: int > $o,Y_5: int,F_9: int > int > int] :
( ( finite1973466193nt_int @ F_9 )
=> ( ( finite772772422nt_int @ F_9 @ Z_9 @ A_62 @ Y_5 )
=> ( ( finite_fold_int_int @ F_9 @ Z_9 @ A_62 )
= Y_5 ) ) ) ).
thf(fact_721_comp__fun__commute_Ofold__equality,axiom,
! [Z_9: nat,A_62: nat > $o,Y_5: nat,F_9: nat > nat > nat] :
( ( finite2130160977at_nat @ F_9 )
=> ( ( finite929467206at_nat @ F_9 @ Z_9 @ A_62 @ Y_5 )
=> ( ( finite_fold_nat_nat @ F_9 @ Z_9 @ A_62 )
= Y_5 ) ) ) ).
thf(fact_722_comp__fun__commute_Ofold__equality,axiom,
! [Z_9: hoare_1708887482_state,A_62: hoare_1708887482_state > $o,Y_5: hoare_1708887482_state,F_9: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state] :
( ( finite1816785523_state @ F_9 )
=> ( ( finite2042206440_state @ F_9 @ Z_9 @ A_62 @ Y_5 )
=> ( ( finite309095018_state @ F_9 @ Z_9 @ A_62 )
= Y_5 ) ) ) ).
thf(fact_723_comp__fun__commute_Ofold__insert2,axiom,
! [Z_8: int,X_13: int,A_61: int > $o,F_8: int > int > int] :
( ( finite1973466193nt_int @ F_8 )
=> ( ( finite_finite_int @ A_61 )
=> ( ~ ( member_int @ X_13 @ A_61 )
=> ( ( finite_fold_int_int @ F_8 @ Z_8 @ ( insert_int @ X_13 @ A_61 ) )
= ( finite_fold_int_int @ F_8 @ ( F_8 @ X_13 @ Z_8 ) @ A_61 ) ) ) ) ) ).
thf(fact_724_comp__fun__commute_Ofold__insert2,axiom,
! [Z_8: nat,X_13: nat,A_61: nat > $o,F_8: nat > nat > nat] :
( ( finite2130160977at_nat @ F_8 )
=> ( ( finite_finite_nat @ A_61 )
=> ( ~ ( member_nat @ X_13 @ A_61 )
=> ( ( finite_fold_nat_nat @ F_8 @ Z_8 @ ( insert_nat @ X_13 @ A_61 ) )
= ( finite_fold_nat_nat @ F_8 @ ( F_8 @ X_13 @ Z_8 ) @ A_61 ) ) ) ) ) ).
thf(fact_725_comp__fun__commute_Ofold__insert,axiom,
! [Z_7: int,X_12: int,A_60: int > $o,F_7: int > int > int] :
( ( finite1973466193nt_int @ F_7 )
=> ( ( finite_finite_int @ A_60 )
=> ( ~ ( member_int @ X_12 @ A_60 )
=> ( ( finite_fold_int_int @ F_7 @ Z_7 @ ( insert_int @ X_12 @ A_60 ) )
= ( F_7 @ X_12 @ ( finite_fold_int_int @ F_7 @ Z_7 @ A_60 ) ) ) ) ) ) ).
thf(fact_726_comp__fun__commute_Ofold__insert,axiom,
! [Z_7: nat,X_12: nat,A_60: nat > $o,F_7: nat > nat > nat] :
( ( finite2130160977at_nat @ F_7 )
=> ( ( finite_finite_nat @ A_60 )
=> ( ~ ( member_nat @ X_12 @ A_60 )
=> ( ( finite_fold_nat_nat @ F_7 @ Z_7 @ ( insert_nat @ X_12 @ A_60 ) )
= ( F_7 @ X_12 @ ( finite_fold_nat_nat @ F_7 @ Z_7 @ A_60 ) ) ) ) ) ) ).
thf(fact_727_folding__one_Oeq__fold_H,axiom,
! [X_11: int,A_59: int > $o,F_6: int > int > int,F_5: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_6 @ F_5 )
=> ( ( finite_finite_int @ A_59 )
=> ( ~ ( member_int @ X_11 @ A_59 )
=> ( ( F_5 @ ( insert_int @ X_11 @ A_59 ) )
= ( finite_fold_int_int @ F_6 @ X_11 @ A_59 ) ) ) ) ) ).
thf(fact_728_folding__one_Oeq__fold_H,axiom,
! [X_11: nat,A_59: nat > $o,F_6: nat > nat > nat,F_5: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_6 @ F_5 )
=> ( ( finite_finite_nat @ A_59 )
=> ( ~ ( member_nat @ X_11 @ A_59 )
=> ( ( F_5 @ ( insert_nat @ X_11 @ A_59 ) )
= ( finite_fold_nat_nat @ F_6 @ X_11 @ A_59 ) ) ) ) ) ).
thf(fact_729_folding__one_Oeq__fold_H,axiom,
! [X_11: hoare_1708887482_state,A_59: hoare_1708887482_state > $o,F_6: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_5: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1615457021_state @ F_6 @ F_5 )
=> ( ( finite1625599783_state @ A_59 )
=> ( ~ ( member451959335_state @ X_11 @ A_59 )
=> ( ( F_5 @ ( insert528405184_state @ X_11 @ A_59 ) )
= ( finite309095018_state @ F_6 @ X_11 @ A_59 ) ) ) ) ) ).
thf(fact_730_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_58: int,A_57: int > $o,F_4: int > int > int,F_3: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_4 @ F_3 )
=> ( ( finite_finite_int @ A_57 )
=> ( ( F_3 @ ( insert_int @ A_58 @ A_57 ) )
= ( finite_fold_int_int @ F_4 @ A_58 @ A_57 ) ) ) ) ).
thf(fact_731_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_58: nat,A_57: nat > $o,F_4: nat > nat > nat,F_3: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_4 @ F_3 )
=> ( ( finite_finite_nat @ A_57 )
=> ( ( F_3 @ ( insert_nat @ A_58 @ A_57 ) )
= ( finite_fold_nat_nat @ F_4 @ A_58 @ A_57 ) ) ) ) ).
thf(fact_732_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_58: hoare_1708887482_state,A_57: hoare_1708887482_state > $o,F_4: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state,F_3: ( hoare_1708887482_state > $o ) > hoare_1708887482_state] :
( ( finite1347568576_state @ F_4 @ F_3 )
=> ( ( finite1625599783_state @ A_57 )
=> ( ( F_3 @ ( insert528405184_state @ A_58 @ A_57 ) )
= ( finite309095018_state @ F_4 @ A_58 @ A_57 ) ) ) ) ).
thf(fact_733_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_6: int,A_56: int > $o,F_2: int > int > int] :
( ( finite1973466193nt_int @ F_2 )
=> ( ( finite_finite_int @ A_56 )
=> ( finite772772422nt_int @ F_2 @ Z_6 @ A_56 @ ( finite_fold_int_int @ F_2 @ Z_6 @ A_56 ) ) ) ) ).
thf(fact_734_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_6: nat,A_56: nat > $o,F_2: nat > nat > nat] :
( ( finite2130160977at_nat @ F_2 )
=> ( ( finite_finite_nat @ A_56 )
=> ( finite929467206at_nat @ F_2 @ Z_6 @ A_56 @ ( finite_fold_nat_nat @ F_2 @ Z_6 @ A_56 ) ) ) ) ).
thf(fact_735_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_6: hoare_1708887482_state,A_56: hoare_1708887482_state > $o,F_2: hoare_1708887482_state > hoare_1708887482_state > hoare_1708887482_state] :
( ( finite1816785523_state @ F_2 )
=> ( ( finite1625599783_state @ A_56 )
=> ( finite2042206440_state @ F_2 @ Z_6 @ A_56 @ ( finite309095018_state @ F_2 @ Z_6 @ A_56 ) ) ) ) ).
thf(fact_736_fold1__eq__fold,axiom,
! [A_55: int,A_54: int > $o] :
( ( finite_finite_int @ A_54 )
=> ( ~ ( member_int @ A_55 @ A_54 )
=> ( ( finite_fold1_int @ times_times_int @ ( insert_int @ A_55 @ A_54 ) )
= ( finite_fold_int_int @ times_times_int @ A_55 @ A_54 ) ) ) ) ).
thf(fact_737_fold1__eq__fold,axiom,
! [A_55: nat,A_54: nat > $o] :
( ( finite_finite_nat @ A_54 )
=> ( ~ ( member_nat @ A_55 @ A_54 )
=> ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ A_55 @ A_54 ) )
= ( finite_fold_nat_nat @ times_times_nat @ A_55 @ A_54 ) ) ) ) ).
thf(fact_738_comp__fun__commute_Ofold__rec,axiom,
! [Z_5: int,X_10: int,A_53: int > $o,F_1: int > int > int] :
( ( finite1973466193nt_int @ F_1 )
=> ( ( finite_finite_int @ A_53 )
=> ( ( member_int @ X_10 @ A_53 )
=> ( ( finite_fold_int_int @ F_1 @ Z_5 @ A_53 )
= ( F_1 @ X_10 @ ( finite_fold_int_int @ F_1 @ Z_5 @ ( minus_minus_int_o @ A_53 @ ( insert_int @ X_10 @ bot_bot_int_o ) ) ) ) ) ) ) ) ).
thf(fact_739_comp__fun__commute_Ofold__rec,axiom,
! [Z_5: nat,X_10: nat,A_53: nat > $o,F_1: nat > nat > nat] :
( ( finite2130160977at_nat @ F_1 )
=> ( ( finite_finite_nat @ A_53 )
=> ( ( member_nat @ X_10 @ A_53 )
=> ( ( finite_fold_nat_nat @ F_1 @ Z_5 @ A_53 )
= ( F_1 @ X_10 @ ( finite_fold_nat_nat @ F_1 @ Z_5 @ ( minus_minus_nat_o @ A_53 @ ( insert_nat @ X_10 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ).
thf(fact_740_less__1__mult,axiom,
! [N_2: int,M_7: int] :
( ( ord_less_int @ one_one_int @ M_7 )
=> ( ( ord_less_int @ one_one_int @ N_2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M_7 @ N_2 ) ) ) ) ).
thf(fact_741_less__1__mult,axiom,
! [N_2: nat,M_7: nat] :
( ( ord_less_nat @ one_one_nat @ M_7 )
=> ( ( ord_less_nat @ one_one_nat @ N_2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M_7 @ N_2 ) ) ) ) ).
thf(fact_742_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_52: int] :
( ( times_times_int @ one_one_int @ A_52 )
= A_52 ) ).
thf(fact_743_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_52: nat] :
( ( times_times_nat @ one_one_nat @ A_52 )
= A_52 ) ).
thf(fact_744_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_51: int] :
( ( times_times_int @ A_51 @ one_one_int )
= A_51 ) ).
thf(fact_745_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_51: nat] :
( ( times_times_nat @ A_51 @ one_one_nat )
= A_51 ) ).
thf(fact_746_card_Oremove,axiom,
! [X_9: int,A_50: int > $o] :
( ( finite_finite_int @ A_50 )
=> ( ( member_int @ X_9 @ A_50 )
=> ( ( finite_card_int @ A_50 )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_50 @ ( insert_int @ X_9 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_747_card_Oremove,axiom,
! [X_9: nat,A_50: nat > $o] :
( ( finite_finite_nat @ A_50 )
=> ( ( member_nat @ X_9 @ A_50 )
=> ( ( finite_card_nat @ A_50 )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_50 @ ( insert_nat @ X_9 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_748_card_Oremove,axiom,
! [X_9: hoare_1708887482_state,A_50: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_50 )
=> ( ( member451959335_state @ X_9 @ A_50 )
=> ( ( finite185771624_state @ A_50 )
= ( plus_plus_nat @ one_one_nat @ ( finite185771624_state @ ( minus_2056855718tate_o @ A_50 @ ( insert528405184_state @ X_9 @ bot_bo19817387tate_o ) ) ) ) ) ) ) ).
thf(fact_749_card_Oinsert__remove,axiom,
! [X_8: int,A_49: int > $o] :
( ( finite_finite_int @ A_49 )
=> ( ( finite_card_int @ ( insert_int @ X_8 @ A_49 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_49 @ ( insert_int @ X_8 @ bot_bot_int_o ) ) ) ) ) ) ).
thf(fact_750_card_Oinsert__remove,axiom,
! [X_8: nat,A_49: nat > $o] :
( ( finite_finite_nat @ A_49 )
=> ( ( finite_card_nat @ ( insert_nat @ X_8 @ A_49 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_49 @ ( insert_nat @ X_8 @ bot_bot_nat_o ) ) ) ) ) ) ).
thf(fact_751_card_Oinsert__remove,axiom,
! [X_8: hoare_1708887482_state,A_49: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_49 )
=> ( ( finite185771624_state @ ( insert528405184_state @ X_8 @ A_49 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite185771624_state @ ( minus_2056855718tate_o @ A_49 @ ( insert528405184_state @ X_8 @ bot_bo19817387tate_o ) ) ) ) ) ) ).
thf(fact_752_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_7: int,Y_4: int,Z_4: int] :
( ( times_times_int @ X_7 @ ( plus_plus_int @ Y_4 @ Z_4 ) )
= ( plus_plus_int @ ( times_times_int @ X_7 @ Y_4 ) @ ( times_times_int @ X_7 @ Z_4 ) ) ) ).
thf(fact_753_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_7: nat,Y_4: nat,Z_4: nat] :
( ( times_times_nat @ X_7 @ ( plus_plus_nat @ Y_4 @ Z_4 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_7 @ Y_4 ) @ ( times_times_nat @ X_7 @ Z_4 ) ) ) ).
thf(fact_754_crossproduct__noteq,axiom,
! [C_37: int,D_15: int,A_48: int,B_40: int] :
( ( ( A_48 != B_40 )
& ( C_37 != D_15 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_48 @ C_37 ) @ ( times_times_int @ B_40 @ D_15 ) )
!= ( plus_plus_int @ ( times_times_int @ A_48 @ D_15 ) @ ( times_times_int @ B_40 @ C_37 ) ) ) ) ).
thf(fact_755_crossproduct__noteq,axiom,
! [C_37: nat,D_15: nat,A_48: nat,B_40: nat] :
( ( ( A_48 != B_40 )
& ( C_37 != D_15 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_48 @ C_37 ) @ ( times_times_nat @ B_40 @ D_15 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_48 @ D_15 ) @ ( times_times_nat @ B_40 @ C_37 ) ) ) ) ).
thf(fact_756_comm__semiring__class_Odistrib,axiom,
! [A_47: int,B_39: int,C_36: int] :
( ( times_times_int @ ( plus_plus_int @ A_47 @ B_39 ) @ C_36 )
= ( plus_plus_int @ ( times_times_int @ A_47 @ C_36 ) @ ( times_times_int @ B_39 @ C_36 ) ) ) ).
thf(fact_757_comm__semiring__class_Odistrib,axiom,
! [A_47: nat,B_39: nat,C_36: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_47 @ B_39 ) @ C_36 )
= ( plus_plus_nat @ ( times_times_nat @ A_47 @ C_36 ) @ ( times_times_nat @ B_39 @ C_36 ) ) ) ).
thf(fact_758_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_46: int,B_38: int,C_35: int] :
( ( times_times_int @ ( plus_plus_int @ A_46 @ B_38 ) @ C_35 )
= ( plus_plus_int @ ( times_times_int @ A_46 @ C_35 ) @ ( times_times_int @ B_38 @ C_35 ) ) ) ).
thf(fact_759_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_46: nat,B_38: nat,C_35: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_46 @ B_38 ) @ C_35 )
= ( plus_plus_nat @ ( times_times_nat @ A_46 @ C_35 ) @ ( times_times_nat @ B_38 @ C_35 ) ) ) ).
thf(fact_760_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_45: int,M_6: int,B_37: int] :
( ( plus_plus_int @ ( times_times_int @ A_45 @ M_6 ) @ ( times_times_int @ B_37 @ M_6 ) )
= ( times_times_int @ ( plus_plus_int @ A_45 @ B_37 ) @ M_6 ) ) ).
thf(fact_761_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_45: nat,M_6: nat,B_37: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_45 @ M_6 ) @ ( times_times_nat @ B_37 @ M_6 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_45 @ B_37 ) @ M_6 ) ) ).
thf(fact_762_combine__common__factor,axiom,
! [A_44: int,E_6: int,B_36: int,C_34: int] :
( ( plus_plus_int @ ( times_times_int @ A_44 @ E_6 ) @ ( plus_plus_int @ ( times_times_int @ B_36 @ E_6 ) @ C_34 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_44 @ B_36 ) @ E_6 ) @ C_34 ) ) ).
thf(fact_763_combine__common__factor,axiom,
! [A_44: nat,E_6: nat,B_36: nat,C_34: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_44 @ E_6 ) @ ( plus_plus_nat @ ( times_times_nat @ B_36 @ E_6 ) @ C_34 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_44 @ B_36 ) @ E_6 ) @ C_34 ) ) ).
thf(fact_764_crossproduct__eq,axiom,
! [W_2: int,Y_3: int,X_6: int,Z_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_2 @ Y_3 ) @ ( times_times_int @ X_6 @ Z_3 ) )
= ( plus_plus_int @ ( times_times_int @ W_2 @ Z_3 ) @ ( times_times_int @ X_6 @ Y_3 ) ) )
<=> ( ( W_2 = X_6 )
| ( Y_3 = Z_3 ) ) ) ).
thf(fact_765_crossproduct__eq,axiom,
! [W_2: nat,Y_3: nat,X_6: nat,Z_3: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_2 @ Y_3 ) @ ( times_times_nat @ X_6 @ Z_3 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_2 @ Z_3 ) @ ( times_times_nat @ X_6 @ Y_3 ) ) )
<=> ( ( W_2 = X_6 )
| ( Y_3 = Z_3 ) ) ) ).
thf(fact_766_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_43: int,C_33: int] :
( ( plus_plus_int @ A_43 @ C_33 )
= ( plus_plus_int @ C_33 @ A_43 ) ) ).
thf(fact_767_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_43: nat,C_33: nat] :
( ( plus_plus_nat @ A_43 @ C_33 )
= ( plus_plus_nat @ C_33 @ A_43 ) ) ).
thf(fact_768_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_42: int,C_32: int,D_14: int] :
( ( plus_plus_int @ A_42 @ ( plus_plus_int @ C_32 @ D_14 ) )
= ( plus_plus_int @ C_32 @ ( plus_plus_int @ A_42 @ D_14 ) ) ) ).
thf(fact_769_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_42: nat,C_32: nat,D_14: nat] :
( ( plus_plus_nat @ A_42 @ ( plus_plus_nat @ C_32 @ D_14 ) )
= ( plus_plus_nat @ C_32 @ ( plus_plus_nat @ A_42 @ D_14 ) ) ) ).
thf(fact_770_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_41: int,C_31: int,D_13: int] :
( ( plus_plus_int @ A_41 @ ( plus_plus_int @ C_31 @ D_13 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_41 @ C_31 ) @ D_13 ) ) ).
thf(fact_771_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_41: nat,C_31: nat,D_13: nat] :
( ( plus_plus_nat @ A_41 @ ( plus_plus_nat @ C_31 @ D_13 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_41 @ C_31 ) @ D_13 ) ) ).
thf(fact_772_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_40: int,B_35: int,C_30: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_40 @ B_35 ) @ C_30 )
= ( plus_plus_int @ A_40 @ ( plus_plus_int @ B_35 @ C_30 ) ) ) ).
thf(fact_773_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_40: nat,B_35: nat,C_30: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_40 @ B_35 ) @ C_30 )
= ( plus_plus_nat @ A_40 @ ( plus_plus_nat @ B_35 @ C_30 ) ) ) ).
thf(fact_774_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_39: int,B_34: int,C_29: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_39 @ B_34 ) @ C_29 )
= ( plus_plus_int @ ( plus_plus_int @ A_39 @ C_29 ) @ B_34 ) ) ).
thf(fact_775_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_39: nat,B_34: nat,C_29: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_39 @ B_34 ) @ C_29 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_39 @ C_29 ) @ B_34 ) ) ).
thf(fact_776_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_38: int,B_33: int,C_28: int,D_12: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_38 @ B_33 ) @ ( plus_plus_int @ C_28 @ D_12 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_38 @ C_28 ) @ ( plus_plus_int @ B_33 @ D_12 ) ) ) ).
thf(fact_777_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_38: nat,B_33: nat,C_28: nat,D_12: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_38 @ B_33 ) @ ( plus_plus_nat @ C_28 @ D_12 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_38 @ C_28 ) @ ( plus_plus_nat @ B_33 @ D_12 ) ) ) ).
thf(fact_778_add__right__imp__eq,axiom,
! [B_32: int,A_37: int,C_27: int] :
( ( ( plus_plus_int @ B_32 @ A_37 )
= ( plus_plus_int @ C_27 @ A_37 ) )
=> ( B_32 = C_27 ) ) ).
thf(fact_779_add__right__imp__eq,axiom,
! [B_32: nat,A_37: nat,C_27: nat] :
( ( ( plus_plus_nat @ B_32 @ A_37 )
= ( plus_plus_nat @ C_27 @ A_37 ) )
=> ( B_32 = C_27 ) ) ).
thf(fact_780_add__imp__eq,axiom,
! [A_36: int,B_31: int,C_26: int] :
( ( ( plus_plus_int @ A_36 @ B_31 )
= ( plus_plus_int @ A_36 @ C_26 ) )
=> ( B_31 = C_26 ) ) ).
thf(fact_781_add__imp__eq,axiom,
! [A_36: nat,B_31: nat,C_26: nat] :
( ( ( plus_plus_nat @ A_36 @ B_31 )
= ( plus_plus_nat @ A_36 @ C_26 ) )
=> ( B_31 = C_26 ) ) ).
thf(fact_782_add__left__imp__eq,axiom,
! [A_35: int,B_30: int,C_25: int] :
( ( ( plus_plus_int @ A_35 @ B_30 )
= ( plus_plus_int @ A_35 @ C_25 ) )
=> ( B_30 = C_25 ) ) ).
thf(fact_783_add__left__imp__eq,axiom,
! [A_35: nat,B_30: nat,C_25: nat] :
( ( ( plus_plus_nat @ A_35 @ B_30 )
= ( plus_plus_nat @ A_35 @ C_25 ) )
=> ( B_30 = C_25 ) ) ).
thf(fact_784_add__right__cancel,axiom,
! [B_29: int,A_34: int,C_24: int] :
( ( ( plus_plus_int @ B_29 @ A_34 )
= ( plus_plus_int @ C_24 @ A_34 ) )
<=> ( B_29 = C_24 ) ) ).
thf(fact_785_add__right__cancel,axiom,
! [B_29: nat,A_34: nat,C_24: nat] :
( ( ( plus_plus_nat @ B_29 @ A_34 )
= ( plus_plus_nat @ C_24 @ A_34 ) )
<=> ( B_29 = C_24 ) ) ).
thf(fact_786_add__left__cancel,axiom,
! [A_33: int,B_28: int,C_23: int] :
( ( ( plus_plus_int @ A_33 @ B_28 )
= ( plus_plus_int @ A_33 @ C_23 ) )
<=> ( B_28 = C_23 ) ) ).
thf(fact_787_add__left__cancel,axiom,
! [A_33: nat,B_28: nat,C_23: nat] :
( ( ( plus_plus_nat @ A_33 @ B_28 )
= ( plus_plus_nat @ A_33 @ C_23 ) )
<=> ( B_28 = C_23 ) ) ).
thf(fact_788_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_32: int,B_27: int,C_22: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_32 @ B_27 ) @ C_22 )
= ( plus_plus_int @ A_32 @ ( plus_plus_int @ B_27 @ C_22 ) ) ) ).
thf(fact_789_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_32: nat,B_27: nat,C_22: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_32 @ B_27 ) @ C_22 )
= ( plus_plus_nat @ A_32 @ ( plus_plus_nat @ B_27 @ C_22 ) ) ) ).
thf(fact_790_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N @ K ) )
<=> ( M = N ) ) ).
thf(fact_791_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N ) )
<=> ( M = N ) ) ).
thf(fact_792_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K ) ) ) ).
thf(fact_793_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z ) ) ) ).
thf(fact_794_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_795_add__le__cancel__right,axiom,
! [A_31: int,C_21: int,B_26: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_31 @ C_21 ) @ ( plus_plus_int @ B_26 @ C_21 ) )
<=> ( ord_less_eq_int @ A_31 @ B_26 ) ) ).
thf(fact_796_add__le__cancel__right,axiom,
! [A_31: nat,C_21: nat,B_26: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_31 @ C_21 ) @ ( plus_plus_nat @ B_26 @ C_21 ) )
<=> ( ord_less_eq_nat @ A_31 @ B_26 ) ) ).
thf(fact_797_add__le__cancel__left,axiom,
! [C_20: int,A_30: int,B_25: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_20 @ A_30 ) @ ( plus_plus_int @ C_20 @ B_25 ) )
<=> ( ord_less_eq_int @ A_30 @ B_25 ) ) ).
thf(fact_798_add__le__cancel__left,axiom,
! [C_20: nat,A_30: nat,B_25: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_20 @ A_30 ) @ ( plus_plus_nat @ C_20 @ B_25 ) )
<=> ( ord_less_eq_nat @ A_30 @ B_25 ) ) ).
thf(fact_799_add__right__mono,axiom,
! [C_19: int,A_29: int,B_24: int] :
( ( ord_less_eq_int @ A_29 @ B_24 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_29 @ C_19 ) @ ( plus_plus_int @ B_24 @ C_19 ) ) ) ).
thf(fact_800_add__right__mono,axiom,
! [C_19: nat,A_29: nat,B_24: nat] :
( ( ord_less_eq_nat @ A_29 @ B_24 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_29 @ C_19 ) @ ( plus_plus_nat @ B_24 @ C_19 ) ) ) ).
thf(fact_801_add__left__mono,axiom,
! [C_18: int,A_28: int,B_23: int] :
( ( ord_less_eq_int @ A_28 @ B_23 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_18 @ A_28 ) @ ( plus_plus_int @ C_18 @ B_23 ) ) ) ).
thf(fact_802_add__left__mono,axiom,
! [C_18: nat,A_28: nat,B_23: nat] :
( ( ord_less_eq_nat @ A_28 @ B_23 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_18 @ A_28 ) @ ( plus_plus_nat @ C_18 @ B_23 ) ) ) ).
thf(fact_803_add__mono,axiom,
! [C_17: int,D_11: int,A_27: int,B_22: int] :
( ( ord_less_eq_int @ A_27 @ B_22 )
=> ( ( ord_less_eq_int @ C_17 @ D_11 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_27 @ C_17 ) @ ( plus_plus_int @ B_22 @ D_11 ) ) ) ) ).
thf(fact_804_add__mono,axiom,
! [C_17: nat,D_11: nat,A_27: nat,B_22: nat] :
( ( ord_less_eq_nat @ A_27 @ B_22 )
=> ( ( ord_less_eq_nat @ C_17 @ D_11 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_27 @ C_17 ) @ ( plus_plus_nat @ B_22 @ D_11 ) ) ) ) ).
thf(fact_805_add__le__imp__le__right,axiom,
! [A_26: int,C_16: int,B_21: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_26 @ C_16 ) @ ( plus_plus_int @ B_21 @ C_16 ) )
=> ( ord_less_eq_int @ A_26 @ B_21 ) ) ).
thf(fact_806_add__le__imp__le__right,axiom,
! [A_26: nat,C_16: nat,B_21: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_26 @ C_16 ) @ ( plus_plus_nat @ B_21 @ C_16 ) )
=> ( ord_less_eq_nat @ A_26 @ B_21 ) ) ).
thf(fact_807_add__le__imp__le__left,axiom,
! [C_15: int,A_25: int,B_20: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_15 @ A_25 ) @ ( plus_plus_int @ C_15 @ B_20 ) )
=> ( ord_less_eq_int @ A_25 @ B_20 ) ) ).
thf(fact_808_add__le__imp__le__left,axiom,
! [C_15: nat,A_25: nat,B_20: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_15 @ A_25 ) @ ( plus_plus_nat @ C_15 @ B_20 ) )
=> ( ord_less_eq_nat @ A_25 @ B_20 ) ) ).
thf(fact_809_add__less__imp__less__left,axiom,
! [C_14: int,A_24: int,B_19: int] :
( ( ord_less_int @ ( plus_plus_int @ C_14 @ A_24 ) @ ( plus_plus_int @ C_14 @ B_19 ) )
=> ( ord_less_int @ A_24 @ B_19 ) ) ).
thf(fact_810_add__less__imp__less__left,axiom,
! [C_14: nat,A_24: nat,B_19: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_14 @ A_24 ) @ ( plus_plus_nat @ C_14 @ B_19 ) )
=> ( ord_less_nat @ A_24 @ B_19 ) ) ).
thf(fact_811_add__less__imp__less__right,axiom,
! [A_23: int,C_13: int,B_18: int] :
( ( ord_less_int @ ( plus_plus_int @ A_23 @ C_13 ) @ ( plus_plus_int @ B_18 @ C_13 ) )
=> ( ord_less_int @ A_23 @ B_18 ) ) ).
thf(fact_812_add__less__imp__less__right,axiom,
! [A_23: nat,C_13: nat,B_18: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_23 @ C_13 ) @ ( plus_plus_nat @ B_18 @ C_13 ) )
=> ( ord_less_nat @ A_23 @ B_18 ) ) ).
thf(fact_813_add__strict__mono,axiom,
! [C_12: int,D_10: int,A_22: int,B_17: int] :
( ( ord_less_int @ A_22 @ B_17 )
=> ( ( ord_less_int @ C_12 @ D_10 )
=> ( ord_less_int @ ( plus_plus_int @ A_22 @ C_12 ) @ ( plus_plus_int @ B_17 @ D_10 ) ) ) ) ).
thf(fact_814_add__strict__mono,axiom,
! [C_12: nat,D_10: nat,A_22: nat,B_17: nat] :
( ( ord_less_nat @ A_22 @ B_17 )
=> ( ( ord_less_nat @ C_12 @ D_10 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_22 @ C_12 ) @ ( plus_plus_nat @ B_17 @ D_10 ) ) ) ) ).
thf(fact_815_add__strict__left__mono,axiom,
! [C_11: int,A_21: int,B_16: int] :
( ( ord_less_int @ A_21 @ B_16 )
=> ( ord_less_int @ ( plus_plus_int @ C_11 @ A_21 ) @ ( plus_plus_int @ C_11 @ B_16 ) ) ) ).
thf(fact_816_add__strict__left__mono,axiom,
! [C_11: nat,A_21: nat,B_16: nat] :
( ( ord_less_nat @ A_21 @ B_16 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_11 @ A_21 ) @ ( plus_plus_nat @ C_11 @ B_16 ) ) ) ).
thf(fact_817_add__strict__right__mono,axiom,
! [C_10: int,A_20: int,B_15: int] :
( ( ord_less_int @ A_20 @ B_15 )
=> ( ord_less_int @ ( plus_plus_int @ A_20 @ C_10 ) @ ( plus_plus_int @ B_15 @ C_10 ) ) ) ).
thf(fact_818_add__strict__right__mono,axiom,
! [C_10: nat,A_20: nat,B_15: nat] :
( ( ord_less_nat @ A_20 @ B_15 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_20 @ C_10 ) @ ( plus_plus_nat @ B_15 @ C_10 ) ) ) ).
thf(fact_819_add__less__cancel__left,axiom,
! [C_9: int,A_19: int,B_14: int] :
( ( ord_less_int @ ( plus_plus_int @ C_9 @ A_19 ) @ ( plus_plus_int @ C_9 @ B_14 ) )
<=> ( ord_less_int @ A_19 @ B_14 ) ) ).
thf(fact_820_add__less__cancel__left,axiom,
! [C_9: nat,A_19: nat,B_14: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_9 @ A_19 ) @ ( plus_plus_nat @ C_9 @ B_14 ) )
<=> ( ord_less_nat @ A_19 @ B_14 ) ) ).
thf(fact_821_add__less__cancel__right,axiom,
! [A_18: int,C_8: int,B_13: int] :
( ( ord_less_int @ ( plus_plus_int @ A_18 @ C_8 ) @ ( plus_plus_int @ B_13 @ C_8 ) )
<=> ( ord_less_int @ A_18 @ B_13 ) ) ).
thf(fact_822_add__less__cancel__right,axiom,
! [A_18: nat,C_8: nat,B_13: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_18 @ C_8 ) @ ( plus_plus_nat @ B_13 @ C_8 ) )
<=> ( ord_less_nat @ A_18 @ B_13 ) ) ).
thf(fact_823_add__diff__cancel,axiom,
! [A_17: int,B_12: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_17 @ B_12 ) @ B_12 )
= A_17 ) ).
thf(fact_824_diff__add__cancel,axiom,
! [A_16: int,B_11: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_16 @ B_11 ) @ B_11 )
= A_16 ) ).
thf(fact_825_add__lessD1,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
=> ( ord_less_nat @ I_1 @ K ) ) ).
thf(fact_826_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_827_add__less__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_828_add__less__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_829_trans__less__add2,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_830_trans__less__add1,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_831_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_832_not__add__less2,axiom,
! [J: nat,I_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I_1 ) @ I_1 ) ).
thf(fact_833_not__add__less1,axiom,
! [I_1: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ I_1 ) ).
thf(fact_834_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_835_termination__basic__simps_I1_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_836_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_837_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_838_le__iff__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ? [K_1: nat] :
( N
= ( plus_plus_nat @ M @ K_1 ) ) ) ).
thf(fact_839_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_840_trans__le__add1,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_841_trans__le__add2,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_842_add__le__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_843_add__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_844_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
thf(fact_845_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_846_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_847_termination__basic__simps_I3_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_848_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_849_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_850_diff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_851_diff__diff__left,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
= ( minus_minus_nat @ I_1 @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_852_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_853_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_854_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_855_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_856_less__add__one,axiom,
! [A_15: int] : ( ord_less_int @ A_15 @ ( plus_plus_int @ A_15 @ one_one_int ) ) ).
thf(fact_857_less__add__one,axiom,
! [A_15: nat] : ( ord_less_nat @ A_15 @ ( plus_plus_nat @ A_15 @ one_one_nat ) ) ).
thf(fact_858_eq__add__iff2,axiom,
! [A_14: int,E_5: int,C_7: int,B_10: int,D_9: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_14 @ E_5 ) @ C_7 )
= ( plus_plus_int @ ( times_times_int @ B_10 @ E_5 ) @ D_9 ) )
<=> ( C_7
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_10 @ A_14 ) @ E_5 ) @ D_9 ) ) ) ).
thf(fact_859_eq__add__iff1,axiom,
! [A_13: int,E_4: int,C_6: int,B_9: int,D_8: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_13 @ E_4 ) @ C_6 )
= ( plus_plus_int @ ( times_times_int @ B_9 @ E_4 ) @ D_8 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_13 @ B_9 ) @ E_4 ) @ C_6 )
= D_8 ) ) ).
thf(fact_860_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_12: int,M_5: int] :
( ( plus_plus_int @ ( times_times_int @ A_12 @ M_5 ) @ M_5 )
= ( times_times_int @ ( plus_plus_int @ A_12 @ one_one_int ) @ M_5 ) ) ).
thf(fact_861_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_12: nat,M_5: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_12 @ M_5 ) @ M_5 )
= ( times_times_nat @ ( plus_plus_nat @ A_12 @ one_one_nat ) @ M_5 ) ) ).
thf(fact_862_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4: int,A_11: int] :
( ( plus_plus_int @ M_4 @ ( times_times_int @ A_11 @ M_4 ) )
= ( times_times_int @ ( plus_plus_int @ A_11 @ one_one_int ) @ M_4 ) ) ).
thf(fact_863_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_4: nat,A_11: nat] :
( ( plus_plus_nat @ M_4 @ ( times_times_nat @ A_11 @ M_4 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_11 @ one_one_nat ) @ M_4 ) ) ).
thf(fact_864_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_3: int] :
( ( plus_plus_int @ M_3 @ M_3 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_3 ) ) ).
thf(fact_865_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_3: nat] :
( ( plus_plus_nat @ M_3 @ M_3 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_3 ) ) ).
thf(fact_866_comp__fun__idem__insert,axiom,
finite175163825_int_o @ insert_int ).
thf(fact_867_comp__fun__idem__insert,axiom,
finite389864113_nat_o @ insert_nat ).
thf(fact_868_comp__fun__idem__insert,axiom,
finite662762081tate_o @ insert528405184_state ).
thf(fact_869_add__less__le__mono,axiom,
! [C_5: int,D_7: int,A_10: int,B_8: int] :
( ( ord_less_int @ A_10 @ B_8 )
=> ( ( ord_less_eq_int @ C_5 @ D_7 )
=> ( ord_less_int @ ( plus_plus_int @ A_10 @ C_5 ) @ ( plus_plus_int @ B_8 @ D_7 ) ) ) ) ).
thf(fact_870_add__less__le__mono,axiom,
! [C_5: nat,D_7: nat,A_10: nat,B_8: nat] :
( ( ord_less_nat @ A_10 @ B_8 )
=> ( ( ord_less_eq_nat @ C_5 @ D_7 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_10 @ C_5 ) @ ( plus_plus_nat @ B_8 @ D_7 ) ) ) ) ).
thf(fact_871_add__le__less__mono,axiom,
! [C_4: int,D_6: int,A_9: int,B_7: int] :
( ( ord_less_eq_int @ A_9 @ B_7 )
=> ( ( ord_less_int @ C_4 @ D_6 )
=> ( ord_less_int @ ( plus_plus_int @ A_9 @ C_4 ) @ ( plus_plus_int @ B_7 @ D_6 ) ) ) ) ).
thf(fact_872_add__le__less__mono,axiom,
! [C_4: nat,D_6: nat,A_9: nat,B_7: nat] :
( ( ord_less_eq_nat @ A_9 @ B_7 )
=> ( ( ord_less_nat @ C_4 @ D_6 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_9 @ C_4 ) @ ( plus_plus_nat @ B_7 @ D_6 ) ) ) ) ).
thf(fact_873_le__add__iff2,axiom,
! [A_8: int,E_3: int,C_3: int,B_6: int,D_5: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_8 @ E_3 ) @ C_3 ) @ ( plus_plus_int @ ( times_times_int @ B_6 @ E_3 ) @ D_5 ) )
<=> ( ord_less_eq_int @ C_3 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_6 @ A_8 ) @ E_3 ) @ D_5 ) ) ) ).
thf(fact_874_le__add__iff1,axiom,
! [A_7: int,E_2: int,C_2: int,B_5: int,D_4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_7 @ E_2 ) @ C_2 ) @ ( plus_plus_int @ ( times_times_int @ B_5 @ E_2 ) @ D_4 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_7 @ B_5 ) @ E_2 ) @ C_2 ) @ D_4 ) ) ).
thf(fact_875_less__add__iff2,axiom,
! [A_6: int,E_1: int,C_1: int,B_4: int,D_3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_6 @ E_1 ) @ C_1 ) @ ( plus_plus_int @ ( times_times_int @ B_4 @ E_1 ) @ D_3 ) )
<=> ( ord_less_int @ C_1 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_4 @ A_6 ) @ E_1 ) @ D_3 ) ) ) ).
thf(fact_876_less__add__iff1,axiom,
! [A_5: int,E: int,C: int,B_3: int,D_2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_5 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B_3 @ E ) @ D_2 ) )
<=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_5 @ B_3 ) @ E ) @ C ) @ D_2 ) ) ).
thf(fact_877_less__diff__conv,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( ord_less_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ).
thf(fact_878_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_879_diff__add__assoc2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 ) ) ) ).
thf(fact_880_add__diff__assoc2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K ) ) ) ).
thf(fact_881_diff__add__assoc,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
= ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
thf(fact_882_le__imp__diff__is__add,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ( minus_minus_nat @ J @ I_1 )
= K )
<=> ( J
= ( plus_plus_nat @ K @ I_1 ) ) ) ) ).
thf(fact_883_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N ) @ N )
= M ) ) ).
thf(fact_884_le__diff__conv2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
thf(fact_885_add__diff__assoc,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K ) ) ) ).
thf(fact_886_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_887_le__add__diff,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) ) ).
thf(fact_888_le__diff__conv,axiom,
! [J: nat,K: nat,I_1: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I_1 @ K ) ) ) ).
thf(fact_889_diff__diff__right,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
thf(fact_890_linorder__neqE__linordered__idom,axiom,
! [X_5: int,Y_2: int] :
( ( X_5 != Y_2 )
=> ( ~ ( ord_less_int @ X_5 @ Y_2 )
=> ( ord_less_int @ Y_2 @ X_5 ) ) ) ).
thf(fact_891_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_4: int,B_2: int] :
( ( times_times_int @ A_4 @ B_2 )
= ( times_times_int @ B_2 @ A_4 ) ) ).
thf(fact_892_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_4: nat,B_2: nat] :
( ( times_times_nat @ A_4 @ B_2 )
= ( times_times_nat @ B_2 @ A_4 ) ) ).
thf(fact_893_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ Lx_6 @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ Rx_6 @ ( times_times_int @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_894_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: nat,Rx_6: nat,Ry_4: nat] :
( ( times_times_nat @ Lx_6 @ ( times_times_nat @ Rx_6 @ Ry_4 ) )
= ( times_times_nat @ Rx_6 @ ( times_times_nat @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_895_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ Lx_5 @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ ( times_times_int @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_896_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: nat,Rx_5: nat,Ry_3: nat] :
( ( times_times_nat @ Lx_5 @ ( times_times_nat @ Rx_5 @ Ry_3 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_897_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_898_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: nat,Ly_4: nat,Rx_4: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_nat @ Lx_4 @ ( times_times_nat @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_899_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_900_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_3: nat,Rx_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_nat @ ( times_times_nat @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_901_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly_2 ) @ ( times_times_int @ Rx_2 @ Ry_2 ) )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_902_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: nat,Ly_2: nat,Rx_2: nat,Ry_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_2 @ Ly_2 ) @ ( times_times_nat @ Rx_2 @ Ry_2 ) )
= ( times_times_nat @ Lx_2 @ ( times_times_nat @ Ly_2 @ ( times_times_nat @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_903_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ Rx_1 @ ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_904_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_1 @ Ly_1 ) @ ( times_times_nat @ Rx_1 @ Ry_1 ) )
= ( times_times_nat @ Rx_1 @ ( times_times_nat @ ( times_times_nat @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_905_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] :
( ( times_times_int @ ( times_times_int @ Lx @ Ly ) @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ ( times_times_int @ Lx @ Rx ) @ ( times_times_int @ Ly @ Ry ) ) ) ).
thf(fact_906_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx @ Ly ) @ ( times_times_nat @ Rx @ Ry ) )
= ( times_times_nat @ ( times_times_nat @ Lx @ Rx ) @ ( times_times_nat @ Ly @ Ry ) ) ) ).
thf(fact_907_card_Oinsert,axiom,
! [X_4: int,A_3: int > $o] :
( ( finite_finite_int @ A_3 )
=> ( ~ ( member_int @ X_4 @ A_3 )
=> ( ( finite_card_int @ ( insert_int @ X_4 @ A_3 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ A_3 ) ) ) ) ) ).
thf(fact_908_card_Oinsert,axiom,
! [X_4: nat,A_3: nat > $o] :
( ( finite_finite_nat @ A_3 )
=> ( ~ ( member_nat @ X_4 @ A_3 )
=> ( ( finite_card_nat @ ( insert_nat @ X_4 @ A_3 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ A_3 ) ) ) ) ) ).
thf(fact_909_card_Oinsert,axiom,
! [X_4: hoare_1708887482_state,A_3: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_3 )
=> ( ~ ( member451959335_state @ X_4 @ A_3 )
=> ( ( finite185771624_state @ ( insert528405184_state @ X_4 @ A_3 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite185771624_state @ A_3 ) ) ) ) ) ).
thf(fact_910_nat__less__add__iff1,axiom,
! [U_1: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U_1 ) @ M ) @ N ) ) ) ).
thf(fact_911_nat__less__add__iff2,axiom,
! [U_1: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
<=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U_1 ) @ N ) ) ) ) ).
thf(fact_912_nat__le__add__iff1,axiom,
! [U_1: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U_1 ) @ M ) @ N ) ) ) ).
thf(fact_913_nat__diff__add__eq1,axiom,
! [U_1: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U_1 ) @ M ) @ N ) ) ) ).
thf(fact_914_nat__eq__add__iff1,axiom,
! [U_1: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U_1 ) @ M )
= N ) ) ) ).
thf(fact_915_left__add__mult__distrib,axiom,
! [I_1: nat,U_1: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I_1 @ J ) @ U_1 ) @ K ) ) ).
thf(fact_916_nat__eq__add__iff2,axiom,
! [U_1: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U_1 ) @ N ) ) ) ) ).
thf(fact_917_nat__diff__add__eq2,axiom,
! [U_1: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U_1 ) @ N ) ) ) ) ).
thf(fact_918_nat__le__add__iff2,axiom,
! [U_1: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U_1 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U_1 ) @ N ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U_1 ) @ N ) ) ) ) ).
thf(fact_919_mono__nat__linear__lb,axiom,
! [M: nat,K: nat,F: nat > nat] :
( ! [M_1: nat,N_1: nat] :
( ( ord_less_nat @ M_1 @ N_1 )
=> ( ord_less_nat @ ( F @ M_1 ) @ ( F @ N_1 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_920_convex__bound__lt,axiom,
! [V_1: int,U: int,Y_1: int,X_3: int,A_2: int] :
( ( ord_less_int @ X_3 @ A_2 )
=> ( ( ord_less_int @ Y_1 @ A_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_1 )
=> ( ( ( plus_plus_int @ U @ V_1 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X_3 ) @ ( times_times_int @ V_1 @ Y_1 ) ) @ A_2 ) ) ) ) ) ) ).
thf(fact_921_card__Suc__Diff1,axiom,
! [X_2: int,A_1: int > $o] :
( ( finite_finite_int @ A_1 )
=> ( ( member_int @ X_2 @ A_1 )
=> ( ( suc @ ( finite_card_int @ ( minus_minus_int_o @ A_1 @ ( insert_int @ X_2 @ bot_bot_int_o ) ) ) )
= ( finite_card_int @ A_1 ) ) ) ) ).
thf(fact_922_card__Suc__Diff1,axiom,
! [X_2: nat,A_1: nat > $o] :
( ( finite_finite_nat @ A_1 )
=> ( ( member_nat @ X_2 @ A_1 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_nat_o @ A_1 @ ( insert_nat @ X_2 @ bot_bot_nat_o ) ) ) )
= ( finite_card_nat @ A_1 ) ) ) ) ).
thf(fact_923_card__Suc__Diff1,axiom,
! [X_2: hoare_1708887482_state,A_1: hoare_1708887482_state > $o] :
( ( finite1625599783_state @ A_1 )
=> ( ( member451959335_state @ X_2 @ A_1 )
=> ( ( suc @ ( finite185771624_state @ ( minus_2056855718tate_o @ A_1 @ ( insert528405184_state @ X_2 @ bot_bo19817387tate_o ) ) ) )
= ( finite185771624_state @ A_1 ) ) ) ) ).
thf(fact_924_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_925_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_926_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_927_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_928_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_929_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_930_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_931_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
thf(fact_932_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
<=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
thf(fact_933_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_934_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_935_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_936_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
thf(fact_937_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_938_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
thf(fact_939_less__trans__Suc,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I_1 ) @ K ) ) ) ).
thf(fact_940_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
thf(fact_941_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_942_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_943_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_944_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_945_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_946_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_947_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
thf(fact_948_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_949_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_950_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
thf(fact_951_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_952_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_953_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_954_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
thf(fact_955_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( M = N ) ) ).
thf(fact_956_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_957_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_958_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_959_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_960_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_961_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_962_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_963_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_964_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_965_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_966_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_967_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_968_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_969_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_970_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
thf(fact_971_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_972_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_973_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_974_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_975_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_976_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_977_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_978_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_979_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_980_nat_Oinject,axiom,
! [Nat_3: nat,Nat_1: nat] :
( ( ( suc @ Nat_3 )
= ( suc @ Nat_1 ) )
<=> ( Nat_3 = Nat_1 ) ) ).
thf(fact_981_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_982_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] :
( ( suc @ Nat_2 )
!= zero_zero_nat ) ).
thf(fact_983_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_984_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_985_nat_Osimps_I2_J,axiom,
! [Nat_1: nat] :
( zero_zero_nat
!= ( suc @ Nat_1 ) ) ).
thf(fact_986_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_987_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
<=> ? [M_1: nat] :
( N
= ( suc @ M_1 ) ) ) ).
thf(fact_988_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_989_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( M = zero_zero_nat )
| ? [J_1: nat] :
( ( M
= ( suc @ J_1 ) )
& ( ord_less_nat @ J_1 @ N ) ) ) ) ).
thf(fact_990_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
thf(fact_991_one__less__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
thf(fact_992_n__less__n__mult__m,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
thf(fact_993_n__less__m__mult__n,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
thf(fact_994_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
thf(fact_995_diff__Suc__less,axiom,
! [I_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I_1 ) ) @ N ) ) ).
thf(fact_996_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_997_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
thf(fact_998_less__add__Suc1,axiom,
! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ I_1 @ M ) ) ) ).
thf(fact_999_less__add__Suc2,axiom,
! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ M @ I_1 ) ) ) ).
thf(fact_1000_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_1001_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1002_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_1003_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_1004_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_1005_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1006_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1007_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_1008_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
thf(fact_1009_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1010_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_1011_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_1012_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_1013_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1014_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
thf(fact_1015_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_1016_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1017_diff__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
thf(fact_1018_mult__less__mono2,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
thf(fact_1019_mult__less__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
thf(fact_1020_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1021_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1022_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_1023_nat__mult__less__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1024_nat__mult__eq__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( M = N ) ) ) ).
thf(fact_1025_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_1026_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_1027_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1028_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
thf(fact_1029_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J ) ) ) ) ).
thf(fact_1030_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ M )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
thf(fact_1031_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B_1: nat] :
( ( P @ ( minus_minus_nat @ A @ B_1 ) )
<=> ( ( ( ord_less_nat @ A @ B_1 )
=> ( P @ zero_zero_nat ) )
& ! [D_1: nat] :
( ( A
= ( plus_plus_nat @ B_1 @ D_1 ) )
=> ( P @ D_1 ) ) ) ) ).
thf(fact_1032_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B_1: nat] :
( ( P @ ( minus_minus_nat @ A @ B_1 ) )
<=> ~ ( ( ( ord_less_nat @ A @ B_1 )
& ~ ( P @ zero_zero_nat ) )
| ? [D_1: nat] :
( ( A
= ( plus_plus_nat @ B_1 @ D_1 ) )
& ~ ( P @ D_1 ) ) ) ) ).
thf(fact_1033_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1034_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1035_nat__mult__le__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1036_ex__least__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K_1 )
=> ~ ( P @ I ) )
& ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
thf(fact_1037_mult__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
thf(fact_1038_add__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N )
= N ) )
& ( ( M != zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N )
= ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
thf(fact_1039_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
thf(fact_1040_Suc__eq__plus1__left,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ one_one_nat @ N ) ) ).
thf(fact_1041_Suc__eq__plus1,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ N @ one_one_nat ) ) ).
thf(fact_1042_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
thf(fact_1043_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
thf(fact_1044_inc__induct,axiom,
! [P: nat > $o,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( P @ J )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1045_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K_1 )
=> ~ ( P @ I ) )
& ( P @ K_1 ) ) ) ) ).
thf(fact_1046_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_1047_less__imp__add__positive,axiom,
! [I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ? [K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ( plus_plus_nat @ I_1 @ K_1 )
= J ) ) ) ).
thf(fact_1048_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M_1: nat] :
( N
= ( suc @ M_1 ) ) ) ).
thf(fact_1049_zero__induct__lemma,axiom,
! [I_1: nat,P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ ( minus_minus_nat @ K @ I_1 ) ) ) ) ).
thf(fact_1050_Suc__le__D,axiom,
! [N: nat,M_2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M_2 )
=> ? [M_1: nat] :
( M_2
= ( suc @ M_1 ) ) ) ).
thf(fact_1051_lessE,axiom,
! [I_1: nat,K: nat] :
( ( ord_less_nat @ I_1 @ K )
=> ( ( K
!= ( suc @ I_1 ) )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ) ).
thf(fact_1052_Suc__lessE,axiom,
! [I_1: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I_1 ) @ K )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ).
thf(fact_1053_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M_1: nat] :
( N
= ( suc @ M_1 ) ) ) ).
thf(fact_1054_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
thf(fact_1055_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_1056_nat__induct,axiom,
! [N: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N_1: nat] :
( ( P @ N_1 )
=> ( P @ ( suc @ N_1 ) ) )
=> ( P @ N ) ) ) ).
thf(fact_1057_com_Osize_I4_J,axiom,
! [Com1: com,Com2: com] :
( ( com_size @ ( semi @ Com1 @ Com2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( com_size @ Com1 ) @ ( com_size @ Com2 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1058_com_Osize_I12_J,axiom,
! [Com1: com,Com2: com] :
( ( size_size_com @ ( semi @ Com1 @ Com2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ Com1 ) @ ( size_size_com @ Com2 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1059_com_Osize_I9_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
thf(fact_1060_com_Osize_I1_J,axiom,
( ( com_size @ skip )
= zero_zero_nat ) ).
thf(fact_1061_com_Osize_I5_J,axiom,
! [Fun_1: state > $o,Com1: com,Com2: com] :
( ( com_size @ ( cond @ Fun_1 @ Com1 @ Com2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( com_size @ Com1 ) @ ( com_size @ Com2 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1062_com_Osimps_I45_J,axiom,
! [Fun: state > $o,Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
( ( cond @ Fun @ Com1_1 @ Com2_1 )
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_1063_com_Osimps_I44_J,axiom,
! [Com1: com,Com2: com,Fun: state > $o,Com1_1: com,Com2_1: com] :
( ( semi @ Com1 @ Com2 )
!= ( cond @ Fun @ Com1_1 @ Com2_1 ) ) ).
thf(fact_1064_com_Osimps_I15_J,axiom,
! [Fun: state > $o,Com1_1: com,Com2_1: com] :
( ( cond @ Fun @ Com1_1 @ Com2_1 )
!= skip ) ).
thf(fact_1065_com_Osimps_I14_J,axiom,
! [Fun: state > $o,Com1_1: com,Com2_1: com] :
( skip
!= ( cond @ Fun @ Com1_1 @ Com2_1 ) ) ).
thf(fact_1066_com_Osimps_I4_J,axiom,
! [Fun_1: state > $o,Com1: com,Com2: com,Fun: state > $o,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 ) ) ) ).
thf(fact_1067_com_Osize_I13_J,axiom,
! [Fun_1: state > $o,Com1: com,Com2: com] :
( ( size_size_com @ ( cond @ Fun_1 @ Com1 @ Com2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ Com1 ) @ ( size_size_com @ Com2 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1068_com_Osize_I11_J,axiom,
! [Loc: loc,Fun_1: state > nat,Com_1: com] :
( ( size_size_com @ ( local @ Loc @ Fun_1 @ Com_1 ) )
= ( plus_plus_nat @ ( size_size_com @ Com_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1069_com_Osize_I3_J,axiom,
! [Loc: loc,Fun_1: state > nat,Com_1: com] :
( ( com_size @ ( local @ Loc @ Fun_1 @ Com_1 ) )
= ( plus_plus_nat @ ( com_size @ Com_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1070_com_Osimps_I36_J,axiom,
! [Loc: loc,Fun_1: state > nat,Com_1: com,Fun: state > $o,Com1_1: com,Com2_1: com] :
( ( local @ Loc @ Fun_1 @ Com_1 )
!= ( cond @ Fun @ Com1_1 @ Com2_1 ) ) ).
thf(fact_1071_com_Osimps_I37_J,axiom,
! [Fun: state > $o,Com1_1: com,Com2_1: com,Loc: loc,Fun_1: state > nat,Com_1: com] :
( ( cond @ Fun @ Com1_1 @ Com2_1 )
!= ( local @ Loc @ Fun_1 @ Com_1 ) ) ).
thf(fact_1072_com_Osimps_I2_J,axiom,
! [Loc: loc,Fun_1: state > nat,Com_1: com,Loc_1: loc,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 ) ) ) ).
thf(fact_1073_com_Osimps_I10_J,axiom,
! [Loc_1: loc,Fun: state > nat,Com: com] :
( skip
!= ( local @ Loc_1 @ Fun @ Com ) ) ).
thf(fact_1074_com_Osimps_I11_J,axiom,
! [Loc_1: loc,Fun: state > nat,Com: com] :
( ( local @ Loc_1 @ Fun @ Com )
!= skip ) ).
thf(fact_1075_com_Osimps_I35_J,axiom,
! [Com1_1: com,Com2_1: com,Loc: loc,Fun_1: state > nat,Com_1: com] :
( ( semi @ Com1_1 @ Com2_1 )
!= ( local @ Loc @ Fun_1 @ Com_1 ) ) ).
thf(fact_1076_com_Osimps_I34_J,axiom,
! [Loc: loc,Fun_1: state > nat,Com_1: com,Com1_1: com,Com2_1: com] :
( ( local @ Loc @ Fun_1 @ Com_1 )
!= ( semi @ Com1_1 @ Com2_1 ) ) ).
thf(fact_1077_com_Osize_I6_J,axiom,
! [Fun_1: state > $o,Com_1: com] :
( ( com_size @ ( while @ Fun_1 @ Com_1 ) )
= ( plus_plus_nat @ ( com_size @ Com_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1078_com_Osize_I14_J,axiom,
! [Fun_1: state > $o,Com_1: com] :
( ( size_size_com @ ( while @ Fun_1 @ Com_1 ) )
= ( plus_plus_nat @ ( size_size_com @ Com_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_1079_com_Osimps_I39_J,axiom,
! [Fun: state > $o,Com: com,Loc: loc,Fun_1: state > nat,Com_1: com] :
( ( while @ Fun @ Com )
!= ( local @ Loc @ Fun_1 @ Com_1 ) ) ).
thf(fact_1080_com_Osimps_I38_J,axiom,
! [Loc: loc,Fun_1: state > nat,Com_1: com,Fun: state > $o,Com: com] :
( ( local @ Loc @ Fun_1 @ Com_1 )
!= ( while @ Fun @ Com ) ) ).
thf(fact_1081_com_Osimps_I47_J,axiom,
! [Fun: state > $o,Com: com,Com1: com,Com2: com] :
( ( while @ Fun @ Com )
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_1082_com_Osimps_I46_J,axiom,
! [Com1: com,Com2: com,Fun: state > $o,Com: com] :
( ( semi @ Com1 @ Com2 )
!= ( while @ Fun @ Com ) ) ).
thf(fact_1083_com_Osimps_I17_J,axiom,
! [Fun: state > $o,Com: com] :
( ( while @ Fun @ Com )
!= skip ) ).
thf(fact_1084_com_Osimps_I16_J,axiom,
! [Fun: state > $o,Com: com] :
( skip
!= ( while @ Fun @ Com ) ) ).
thf(fact_1085_com_Osimps_I5_J,axiom,
! [Fun_1: state > $o,Com_1: com,Fun: state > $o,Com: com] :
( ( ( while @ Fun_1 @ Com_1 )
= ( while @ Fun @ Com ) )
<=> ( ( Fun_1 = Fun )
& ( Com_1 = Com ) ) ) ).
thf(fact_1086_com_Osimps_I53_J,axiom,
! [Fun: state > $o,Com: com,Fun_1: state > $o,Com1: com,Com2: com] :
( ( while @ Fun @ Com )
!= ( cond @ Fun_1 @ Com1 @ Com2 ) ) ).
thf(fact_1087_com_Osimps_I52_J,axiom,
! [Fun_1: state > $o,Com1: com,Com2: com,Fun: state > $o,Com: com] :
( ( cond @ Fun_1 @ Com1 @ Com2 )
!= ( while @ Fun @ Com ) ) ).
thf(fact_1088_expand__Suc,axiom,
! [V: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
=> ( ( number_number_of_nat @ V )
= ( suc @ ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat ) ) ) ) ).
thf(fact_1089_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
thf(fact_1090_min__diff,axiom,
! [M: nat,I_1: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I_1 ) @ ( minus_minus_nat @ N @ I_1 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I_1 ) ) ).
thf(fact_1091_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_1092_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_1093_plus__numeral__code_I9_J,axiom,
! [V: int,W: int] :
( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) ) ).
thf(fact_1094_times__numeral__code_I5_J,axiom,
! [V: int,W: int] :
( ( times_times_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( times_times_int @ V @ W ) ) ) ).
thf(fact_1095_less__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_1096_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_1097_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_1098_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_1099_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_1100_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_1101_zle__trans,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_eq_int @ I_1 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I_1 @ K ) ) ) ).
thf(fact_1102_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_1103_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_1104_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_1105_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_1106_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
= ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
thf(fact_1107_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_1108_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_1109_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_1110_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ Z1 @ Z2 ) @ Z3 )
= ( plus_plus_int @ Z1 @ ( plus_plus_int @ Z2 @ Z3 ) ) ) ).
thf(fact_1111_zadd__left__commute,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) )
= ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z ) ) ) ).
thf(fact_1112_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_1113_zadd__left__mono,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_eq_int @ I_1 @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I_1 ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_1114_zadd__strict__right__mono,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ord_less_int @ ( plus_plus_int @ I_1 @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
thf(fact_1115_zadd__zless__mono,axiom,
! [Z_2: int,Z: int,W_1: int,W: int] :
( ( ord_less_int @ W_1 @ W )
=> ( ( ord_less_eq_int @ Z_2 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_1 @ Z_2 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_1116_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_1117_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_1118_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
thf(fact_1119_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
thf(fact_1120_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_1121_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_1122_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_1123_pos__zmult__eq__1__iff,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
thf(fact_1124_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
<=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
thf(fact_1125_less__bin__lemma,axiom,
! [K: int,L: int] :
( ( ord_less_int @ K @ L )
<=> ( ord_less_int @ ( minus_minus_int @ K @ L ) @ zero_zero_int ) ) ).
thf(fact_1126_zmult__zless__mono2,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I_1 ) @ ( times_times_int @ K @ J ) ) ) ) ).
thf(fact_1127_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_1128_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_1129_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
thf(fact_1130_odd__less__0,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
<=> ( ord_less_int @ Z @ zero_zero_int ) ) ).
thf(fact_1131_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_1132_zdiv__mono2__neg__lemma,axiom,
! [B_1: int,Q_1: int,R_1: int,B: int,Q: int,R: int] :
( ( ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) @ zero_zero_int )
=> ( ( ord_less_int @ R_1 @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ B_1 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
thf(fact_1133_unique__quotient__lemma__neg,axiom,
! [B_1: int,Q: int,R: int,Q_1: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( ord_less_int @ B_1 @ R_1 )
=> ( ( ord_less_int @ B_1 @ R )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
thf(fact_1134_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_1135_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_1136_self__quotient__aux2,axiom,
! [R_1: int,Q_1: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q_1 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ord_less_eq_int @ Q_1 @ one_one_int ) ) ) ) ).
thf(fact_1137_self__quotient__aux1,axiom,
! [R_1: int,Q_1: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q_1 ) ) )
=> ( ( ord_less_int @ R_1 @ A )
=> ( ord_less_eq_int @ one_one_int @ Q_1 ) ) ) ) ).
thf(fact_1138_q__pos__lemma,axiom,
! [B: int,Q: int,R: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_int @ R @ B )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ Q ) ) ) ) ).
thf(fact_1139_q__neg__lemma,axiom,
! [B: int,Q: int,R: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ Q @ zero_zero_int ) ) ) ) ).
thf(fact_1140_unique__quotient__lemma,axiom,
! [B_1: int,Q: int,R: int,Q_1: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B_1 )
=> ( ( ord_less_int @ R_1 @ B_1 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
thf(fact_1141_zdiv__mono2__lemma,axiom,
! [B_1: int,Q_1: int,R_1: int,B: int,Q: int,R: int] :
( ( ( plus_plus_int @ ( times_times_int @ B_1 @ Q_1 ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R ) )
=> ( ( ord_less_int @ R @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ B_1 )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
thf(fact_1142_int__less__induct,axiom,
! [P: int > $o,I_1: int,K: int] :
( ( ord_less_int @ I_1 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1143_int__le__induct,axiom,
! [P: int > $o,I_1: int,K: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1144_int__gr__induct,axiom,
! [P: int > $o,K: int,I_1: int] :
( ( ord_less_int @ K @ I_1 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1145_int__ge__induct,axiom,
! [P: int > $o,K: int,I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1146_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_1147_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_1148_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_1149_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_1150_decr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1151_incr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1152_imp__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P_1 ) ) ) ).
thf(fact_1153_conj__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
& P_1 ) ) ) ).
thf(fact_1154_int__induct,axiom,
! [I_1: int,P: int > $o,K: int] :
( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1155_plusinfinity,axiom,
! [P: int > $o,P_1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K_1: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ? [Z_1: int] :
! [X_1: int] :
( ( ord_less_int @ Z_1 @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( ?? @ int @ P_1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_1156_minusinfinity,axiom,
! [P: int > $o,P1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K_1: int] :
( ( P1 @ X_1 )
<=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ? [Z_1: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z_1 )
=> ( ( P @ X_1 )
<=> ( P1 @ X_1 ) ) )
=> ( ( ?? @ int @ P1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_1157_tsub__def,axiom,
! [Y: int,X: int] :
( ( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) )
& ( ~ ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= zero_zero_int ) ) ) ).
thf(fact_1158_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( nat_tsub @ X @ Y ) ) ) ) ).
thf(fact_1159_tsub__eq,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_1160_incr__lemma,axiom,
! [Z: int,X: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
thf(fact_1161_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X: nat,Y: nat] :
( ( nat_tsub @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) ) ).
thf(fact_1162_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z ) ) ).
thf(fact_1163_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( P @ X_1 ) )
<=> ? [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_1164_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P @ X_1 ) )
<=> ! [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_1165_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_1166_transfer__int__nat__numerals_I1_J,axiom,
( zero_zero_int
= ( semiri1621563631at_int @ zero_zero_nat ) ) ).
thf(fact_1167_transfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_1168_transfer__int__nat__relations_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_nat @ X @ Y ) ) ).
thf(fact_1169_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( times_times_nat @ X @ Y ) ) ) ).
thf(fact_1170_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ X @ Y ) ) ) ).
thf(fact_1171_int__Suc,axiom,
! [M: nat] :
( ( semiri1621563631at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_1172_int__Suc0__eq__1,axiom,
( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
thf(fact_1173_zless__iff__Suc__zadd,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
<=> ? [N_1: nat] :
( Z
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_1174_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1175_zless__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1176_zmult__int,axiom,
! [M: nat,N: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_1177_int__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_1178_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
thf(fact_1179_zadd__int,axiom,
! [M: nat,N: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_1180_zle__iff__zadd,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ Z )
<=> ? [N_1: nat] :
( Z
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_1181_abs__int__eq,axiom,
! [M: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
= ( semiri1621563631at_int @ M ) ) ).
thf(fact_1182_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_1183_int__eq__0__conv,axiom,
! [N: nat] :
( ( ( semiri1621563631at_int @ N )
= zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_1184_int__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_1185_int__less__0__conv,axiom,
! [K: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
thf(fact_1186_zero__zle__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_1187_int__le__0__conv,axiom,
! [N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_1188_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
<=> ( Z = zero_zero_int ) ) ).
thf(fact_1189_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
thf(fact_1190_zdiff__int,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_1191_zero__less__int__conv,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_1192_zmult__zless__mono2__lemma,axiom,
! [K: nat,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ I_1 ) @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ J ) ) ) ) ).
thf(fact_1193_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) )
<=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
thf(fact_1194_decr__lemma,axiom,
! [X: int,Z: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
thf(fact_1195_int__val__lemma,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
thf(fact_1196_nat0__intermed__int__val,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
thf(fact_1197_transfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_1198_int__if__cong,axiom,
! [X: nat,Y: nat,P: $o] :
( ( P
=> ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) )
& ( ~ P
=> ( ( semiri1621563631at_int @ Y )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) ) ) ).
thf(fact_1199_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N ) )
<=> ( M = N ) ) ).
%----Helper facts (3)
thf(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
%----Conjectures (3)
thf(conj_0,hypothesis,
hoare_90032982_state @ bot_bo19817387tate_o @ ( insert528405184_state @ ( hoare_Mirabelle_MGT @ c ) @ bot_bo19817387tate_o ) ).
thf(conj_1,hypothesis,
hoare_496444244_state @ bot_bo19817387tate_o @ ( insert528405184_state @ ( hoare_858012674_state @ p @ c @ q ) @ bot_bo19817387tate_o ) ).
thf(conj_2,conjecture,
hoare_90032982_state @ bot_bo19817387tate_o @ ( insert528405184_state @ ( hoare_858012674_state @ p @ c @ q ) @ bot_bo19817387tate_o ) ).
%------------------------------------------------------------------------------