TPTP Problem File: SWW470^3.p
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%------------------------------------------------------------------------------
% File : SWW470^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 200, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_1000_thf_l200 [Bla11]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0, 1.00 v7.2.0, 0.75 v7.1.0, 0.00 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.80 v5.3.0
% Syntax : Number of formulae : 1366 ( 301 unt; 158 typ; 0 def)
% Number of atoms : 4545 (1056 equ; 93 cnn)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 11387 ( 329 ~; 52 |; 193 &;9241 @)
% ( 203 <=>;1363 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 2520 (2520 >; 0 *; 0 +; 0 <<)
% Number of symbols : 159 ( 151 usr; 15 con; 0-4 aty)
% ( 0 !!; 6 ??; 0 @@+; 0 @@-)
% Number of variables : 3579 ( 205 ^;3300 !; 74 ?;3579 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:35:33
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
thf(ty_ty_t__a,type,
x_a: $tType ).
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Ostate,type,
state: $tType ).
thf(ty_ty_tc__Com__Ovname,type,
vname: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
hoare_2091234717iple_a: $tType ).
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
%----Explicit typings (154)
thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Hoare____Mirabelle_,type,
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thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Int__Oint_000tc__Na,type,
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thf(sy_c_Big__Operators_Olinorder__class_OMax_000tc__Int__Oint,type,
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thf(sy_c_Big__Operators_Olinorder__class_OMin_000tc__Int__Oint,type,
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thf(sy_c_Big__Operators_Olinorder__class_OMin_000tc__Nat__Onat,type,
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thf(sy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____nqhfsdfvyv__O,type,
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thf(sy_c_Big__Operators_Osemilattice__big_000tc__Int__Oint,type,
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thf(sy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat,type,
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thf(sy_c_Com_Ocom_OAss,type,
ass: vname > ( state > nat ) > com ).
thf(sy_c_Com_Ocom_OSKIP,type,
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thf(sy_c_Com_Ocom_OSemi,type,
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thf(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
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thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Int__Oint_000_062_Itc__Int__Oint_M_E,type,
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thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otrip,type,
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thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__,type,
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thf(sy_c_Finite__Set_Ofold1_000tc__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J__007,type,
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thf(sy_c_Finite__Set_Ofold_000tc__Int__Oint_000_062_Itc__Int__Oint_M_Eo_J,type,
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thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_I,type,
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thf(sy_c_Finite__Set_Ofold__graph_000tc__Int__Oint_000_062_Itc__Int__Oint_M_Eo_J,type,
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thf(sy_c_Finite__Set_Ofold__graph_000tc__Int__Oint_000tc__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_,type,
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thf(sy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____nqhfsdfvyv__Ot,type,
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thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__,type,
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thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Int__Oint_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_000_Eo,type,
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thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
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thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_000tc__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
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thf(sy_c_HOL_OThe_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_M_E,type,
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thf(sy_c_HOL_OThe_000_062_Itc__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Hoare__Mirabelle__nqhfsdfvyv_Otriple_Otriple_000t__a,type,
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thf(sy_c_Natural_Oupdate,type,
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thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
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thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Int__Oint_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____nqhfsdfv,type,
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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_Omax_000tc__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_000tc__Int__Oint,type,
ord_min_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otrip,type,
partia443170835iple_a: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a ).
thf(sy_c_Partial__Function_Oflat__lub_000tc__Int__Oint,type,
partial_flat_lub_int: int > ( int > $o ) > int ).
thf(sy_c_Partial__Function_Oflat__lub_000tc__Nat__Onat,type,
partial_flat_lub_nat: nat > ( nat > $o ) > nat ).
thf(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Int__Oint,type,
ord_at875362053st_int: int > int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Nat__Onat,type,
ord_at238088361st_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OgreaterThanAtMost_000tc__Nat__Onat,type,
ord_gr375877188st_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OlessThan_000tc__Int__Oint,type,
ord_lessThan_int: int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OlessThan_000tc__Nat__Onat,type,
ord_lessThan_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_000_062_Itc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
collec1008234059le_a_o: ( ( hoare_2091234717iple_a > $o ) > $o ) > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_Set_OCollect_000_062_Itc__Int__Oint_M_Eo_J,type,
collect_int_o: ( ( int > $o ) > $o ) > ( int > $o ) > $o ).
thf(sy_c_Set_OCollect_000_062_Itc__Nat__Onat_M_Eo_J,type,
collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
collec992574898iple_a: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $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_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000tc__,type,
image_1661191109iple_a: ( hoare_2091234717iple_a > hoare_2091234717iple_a ) > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000tc___009,type,
image_263112078_a_int: ( hoare_2091234717iple_a > int ) > ( hoare_2091234717iple_a > $o ) > int > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_000tc___010,type,
image_1773322034_a_nat: ( hoare_2091234717iple_a > nat ) > ( hoare_2091234717iple_a > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otripl,type,
image_338319932iple_a: ( int > hoare_2091234717iple_a ) > ( int > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Int__Oint,type,
image_int_int: ( int > int ) > ( int > $o ) > int > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Nat__Onat,type,
image_int_nat: ( int > nat ) > ( int > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otripl,type,
image_359186840iple_a: ( nat > hoare_2091234717iple_a ) > ( nat > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
image_nat_int: ( nat > int ) > ( nat > $o ) > int > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
insert1597628439iple_a: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o ).
thf(sy_c_Set_Oinsert_000tc__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_It__a_J,type,
the_el13400124iple_a: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a ).
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_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
fequal1604381340iple_a: hoare_2091234717iple_a > hoare_2091234717iple_a > $o ).
thf(sy_c_fequal_000tc__Int__Oint,type,
fequal_int: int > int > $o ).
thf(sy_c_fequal_000tc__Nat__Onat,type,
fequal_nat: nat > nat > $o ).
thf(sy_c_member_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J,type,
member290856304iple_a: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > $o ).
thf(sy_c_member_000tc__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_G,type,
g: hoare_2091234717iple_a > $o ).
thf(sy_v_P,type,
p: x_a > state > $o ).
thf(sy_v_b,type,
b: state > $o ).
thf(sy_v_c,type,
c: com ).
%----Relevant facts (1198)
thf(fact_0_empty,axiom,
! [G_37: hoare_2091234717iple_a > $o] : ( hoare_1467856363rivs_a @ G_37 @ bot_bo1791335050le_a_o ) ).
thf(fact_1_triple_Oinject,axiom,
! [Fun1_2: x_a > state > $o,Com_2: com,Fun2_2: x_a > state > $o,Fun1_1: x_a > state > $o,Com_1: com,Fun2_1: x_a > state > $o] :
( ( ( hoare_657976383iple_a @ Fun1_2 @ Com_2 @ Fun2_2 )
= ( hoare_657976383iple_a @ Fun1_1 @ Com_1 @ Fun2_1 ) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_2 = Com_1 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
thf(fact_2_cut,axiom,
! [G_36: hoare_2091234717iple_a > $o,G_35: hoare_2091234717iple_a > $o,Ts_4: hoare_2091234717iple_a > $o] :
( ( hoare_1467856363rivs_a @ G_35 @ Ts_4 )
=> ( ( hoare_1467856363rivs_a @ G_36 @ G_35 )
=> ( hoare_1467856363rivs_a @ G_36 @ Ts_4 ) ) ) ).
thf(fact_3_hoare__derivs_Oinsert,axiom,
! [Ts_3: hoare_2091234717iple_a > $o,G_34: hoare_2091234717iple_a > $o,T_2: hoare_2091234717iple_a] :
( ( hoare_1467856363rivs_a @ G_34 @ ( insert1597628439iple_a @ T_2 @ bot_bo1791335050le_a_o ) )
=> ( ( hoare_1467856363rivs_a @ G_34 @ Ts_3 )
=> ( hoare_1467856363rivs_a @ G_34 @ ( insert1597628439iple_a @ T_2 @ Ts_3 ) ) ) ) ).
thf(fact_4_constant,axiom,
! [G_33: hoare_2091234717iple_a > $o,P_33: x_a > state > $o,C_60: com,Q_16: x_a > state > $o,C_59: $o] :
( ( C_59
=> ( hoare_1467856363rivs_a @ G_33 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_33 @ C_60 @ Q_16 ) @ bot_bo1791335050le_a_o ) ) )
=> ( hoare_1467856363rivs_a @ G_33
@ ( insert1597628439iple_a
@ ( hoare_657976383iple_a
@ ^ [Z: x_a,S: state] : ( (&) @ ( P_33 @ Z @ S ) @ C_59 )
@ C_60
@ Q_16 )
@ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_5_escape,axiom,
! [G_32: hoare_2091234717iple_a > $o,C_58: com,Q_15: x_a > state > $o,P_32: x_a > state > $o] :
( ! [Z: x_a,S: state] :
( ( P_32 @ Z @ S )
=> ( hoare_1467856363rivs_a @ G_32
@ ( insert1597628439iple_a
@ ( hoare_657976383iple_a
@ ^ [Za: x_a,S_5: state] : S_5 = S
@ C_58
@ ^ [Z_38: x_a] : ( Q_15 @ Z ) )
@ bot_bo1791335050le_a_o ) ) )
=> ( hoare_1467856363rivs_a @ G_32 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_32 @ C_58 @ Q_15 ) @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_6_conseq2,axiom,
! [Q_14: x_a > state > $o,G_31: hoare_2091234717iple_a > $o,P_31: x_a > state > $o,C_57: com,Q_13: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_31 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_31 @ C_57 @ Q_13 ) @ bot_bo1791335050le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( Q_13 @ Z @ S )
=> ( Q_14 @ Z @ S ) )
=> ( hoare_1467856363rivs_a @ G_31 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_31 @ C_57 @ Q_14 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_7_conseq1,axiom,
! [P_30: x_a > state > $o,G_30: hoare_2091234717iple_a > $o,P_29: x_a > state > $o,C_56: com,Q_12: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_30 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_29 @ C_56 @ Q_12 ) @ bot_bo1791335050le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( P_30 @ Z @ S )
=> ( P_29 @ Z @ S ) )
=> ( hoare_1467856363rivs_a @ G_30 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_30 @ C_56 @ Q_12 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_8_conseq12,axiom,
! [Q_11: x_a > state > $o,P_28: x_a > state > $o,G_29: hoare_2091234717iple_a > $o,P_27: x_a > state > $o,C_55: com,Q_10: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_29 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_27 @ C_55 @ Q_10 ) @ bot_bo1791335050le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( P_28 @ Z @ S )
=> ! [S_5: state] :
( ! [Z_38: x_a] :
( ( P_27 @ Z_38 @ S )
=> ( Q_10 @ Z_38 @ S_5 ) )
=> ( Q_11 @ Z @ S_5 ) ) )
=> ( hoare_1467856363rivs_a @ G_29 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_28 @ C_55 @ Q_11 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_9_insertE,axiom,
! [A_255: nat,B_120: nat,A_254: nat > $o] :
( ( member_nat @ A_255 @ ( insert_nat @ B_120 @ A_254 ) )
=> ( ( A_255 != B_120 )
=> ( member_nat @ A_255 @ A_254 ) ) ) ).
thf(fact_10_insertE,axiom,
! [A_255: hoare_2091234717iple_a,B_120: hoare_2091234717iple_a,A_254: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_255 @ ( insert1597628439iple_a @ B_120 @ A_254 ) )
=> ( ( A_255 != B_120 )
=> ( member290856304iple_a @ A_255 @ A_254 ) ) ) ).
thf(fact_11_insertE,axiom,
! [A_255: int,B_120: int,A_254: int > $o] :
( ( member_int @ A_255 @ ( insert_int @ B_120 @ A_254 ) )
=> ( ( A_255 != B_120 )
=> ( member_int @ A_255 @ A_254 ) ) ) ).
thf(fact_12_insertCI,axiom,
! [B_119: nat,A_253: nat,B_118: nat > $o] :
( ( ~ ( member_nat @ A_253 @ B_118 )
=> ( A_253 = B_119 ) )
=> ( member_nat @ A_253 @ ( insert_nat @ B_119 @ B_118 ) ) ) ).
thf(fact_13_insertCI,axiom,
! [B_119: hoare_2091234717iple_a,A_253: hoare_2091234717iple_a,B_118: hoare_2091234717iple_a > $o] :
( ( ~ ( member290856304iple_a @ A_253 @ B_118 )
=> ( A_253 = B_119 ) )
=> ( member290856304iple_a @ A_253 @ ( insert1597628439iple_a @ B_119 @ B_118 ) ) ) ).
thf(fact_14_insertCI,axiom,
! [B_119: int,A_253: int,B_118: int > $o] :
( ( ~ ( member_int @ A_253 @ B_118 )
=> ( A_253 = B_119 ) )
=> ( member_int @ A_253 @ ( insert_int @ B_119 @ B_118 ) ) ) ).
thf(fact_15_emptyE,axiom,
! [A_252: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ A_252 @ bot_bo1791335050le_a_o ) ).
thf(fact_16_emptyE,axiom,
! [A_252: nat] :
~ ( member_nat @ A_252 @ bot_bot_nat_o ) ).
thf(fact_17_emptyE,axiom,
! [A_252: int] :
~ ( member_int @ A_252 @ bot_bot_int_o ) ).
thf(fact_18_singleton__conv2,axiom,
! [A_251: nat] :
( ( collect_nat @ ( fequal_nat @ A_251 ) )
= ( insert_nat @ A_251 @ bot_bot_nat_o ) ) ).
thf(fact_19_singleton__conv2,axiom,
! [A_251: hoare_2091234717iple_a] :
( ( collec992574898iple_a @ ( fequal1604381340iple_a @ A_251 ) )
= ( insert1597628439iple_a @ A_251 @ bot_bo1791335050le_a_o ) ) ).
thf(fact_20_singleton__conv2,axiom,
! [A_251: int] :
( ( collect_int @ ( fequal_int @ A_251 ) )
= ( insert_int @ A_251 @ bot_bot_int_o ) ) ).
thf(fact_21_singleton__conv,axiom,
! [A_250: nat] :
( ( collect_nat
@ ^ [X_1: nat] : X_1 = A_250 )
= ( insert_nat @ A_250 @ bot_bot_nat_o ) ) ).
thf(fact_22_singleton__conv,axiom,
! [A_250: hoare_2091234717iple_a] :
( ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : X_1 = A_250 )
= ( insert1597628439iple_a @ A_250 @ bot_bo1791335050le_a_o ) ) ).
thf(fact_23_singleton__conv,axiom,
! [A_250: int] :
( ( collect_int
@ ^ [X_1: int] : X_1 = A_250 )
= ( insert_int @ A_250 @ bot_bot_int_o ) ) ).
thf(fact_24_Collect__conv__if2,axiom,
! [P_26: nat > $o,A_249: nat] :
( ( ( P_26 @ A_249 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( A_249 = X_1 ) @ ( P_26 @ X_1 ) ) )
= ( insert_nat @ A_249 @ bot_bot_nat_o ) ) )
& ( ~ ( P_26 @ A_249 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( A_249 = X_1 ) @ ( P_26 @ X_1 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_25_Collect__conv__if2,axiom,
! [P_26: hoare_2091234717iple_a > $o,A_249: hoare_2091234717iple_a] :
( ( ( P_26 @ A_249 )
=> ( ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( A_249 = X_1 ) @ ( P_26 @ X_1 ) ) )
= ( insert1597628439iple_a @ A_249 @ bot_bo1791335050le_a_o ) ) )
& ( ~ ( P_26 @ A_249 )
=> ( ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( A_249 = X_1 ) @ ( P_26 @ X_1 ) ) )
= bot_bo1791335050le_a_o ) ) ) ).
thf(fact_26_Collect__conv__if2,axiom,
! [P_26: int > $o,A_249: int] :
( ( ( P_26 @ A_249 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( A_249 = X_1 ) @ ( P_26 @ X_1 ) ) )
= ( insert_int @ A_249 @ bot_bot_int_o ) ) )
& ( ~ ( P_26 @ A_249 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( A_249 = X_1 ) @ ( P_26 @ X_1 ) ) )
= bot_bot_int_o ) ) ) ).
thf(fact_27_Collect__conv__if,axiom,
! [P_25: nat > $o,A_248: nat] :
( ( ( P_25 @ A_248 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( X_1 = A_248 ) @ ( P_25 @ X_1 ) ) )
= ( insert_nat @ A_248 @ bot_bot_nat_o ) ) )
& ( ~ ( P_25 @ A_248 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( X_1 = A_248 ) @ ( P_25 @ X_1 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_28_Collect__conv__if,axiom,
! [P_25: hoare_2091234717iple_a > $o,A_248: hoare_2091234717iple_a] :
( ( ( P_25 @ A_248 )
=> ( ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( X_1 = A_248 ) @ ( P_25 @ X_1 ) ) )
= ( insert1597628439iple_a @ A_248 @ bot_bo1791335050le_a_o ) ) )
& ( ~ ( P_25 @ A_248 )
=> ( ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( X_1 = A_248 ) @ ( P_25 @ X_1 ) ) )
= bot_bo1791335050le_a_o ) ) ) ).
thf(fact_29_Collect__conv__if,axiom,
! [P_25: int > $o,A_248: int] :
( ( ( P_25 @ A_248 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( X_1 = A_248 ) @ ( P_25 @ X_1 ) ) )
= ( insert_int @ A_248 @ bot_bot_int_o ) ) )
& ( ~ ( P_25 @ A_248 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( X_1 = A_248 ) @ ( P_25 @ X_1 ) ) )
= bot_bot_int_o ) ) ) ).
thf(fact_30_equals0D,axiom,
! [A_247: hoare_2091234717iple_a,A_246: hoare_2091234717iple_a > $o] :
( ( A_246 = bot_bo1791335050le_a_o )
=> ~ ( member290856304iple_a @ A_247 @ A_246 ) ) ).
thf(fact_31_equals0D,axiom,
! [A_247: nat,A_246: nat > $o] :
( ( A_246 = bot_bot_nat_o )
=> ~ ( member_nat @ A_247 @ A_246 ) ) ).
thf(fact_32_equals0D,axiom,
! [A_247: int,A_246: int > $o] :
( ( A_246 = bot_bot_int_o )
=> ~ ( member_int @ A_247 @ A_246 ) ) ).
thf(fact_33_Collect__empty__eq,axiom,
! [P_24: hoare_2091234717iple_a > $o] :
( ( ( collec992574898iple_a @ P_24 )
= bot_bo1791335050le_a_o )
<=> ! [X_1: hoare_2091234717iple_a] :
~ ( P_24 @ X_1 ) ) ).
thf(fact_34_Collect__empty__eq,axiom,
! [P_24: int > $o] :
( ( ( collect_int @ P_24 )
= bot_bot_int_o )
<=> ! [X_1: int] :
~ ( P_24 @ X_1 ) ) ).
thf(fact_35_Collect__empty__eq,axiom,
! [P_24: nat > $o] :
( ( ( collect_nat @ P_24 )
= bot_bot_nat_o )
<=> ! [X_1: nat] :
~ ( P_24 @ X_1 ) ) ).
thf(fact_36_empty__iff,axiom,
! [C_54: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ C_54 @ bot_bo1791335050le_a_o ) ).
thf(fact_37_empty__iff,axiom,
! [C_54: nat] :
~ ( member_nat @ C_54 @ bot_bot_nat_o ) ).
thf(fact_38_empty__iff,axiom,
! [C_54: int] :
~ ( member_int @ C_54 @ bot_bot_int_o ) ).
thf(fact_39_empty__Collect__eq,axiom,
! [P_23: hoare_2091234717iple_a > $o] :
( ( bot_bo1791335050le_a_o
= ( collec992574898iple_a @ P_23 ) )
<=> ! [X_1: hoare_2091234717iple_a] :
~ ( P_23 @ X_1 ) ) ).
thf(fact_40_empty__Collect__eq,axiom,
! [P_23: int > $o] :
( ( bot_bot_int_o
= ( collect_int @ P_23 ) )
<=> ! [X_1: int] :
~ ( P_23 @ X_1 ) ) ).
thf(fact_41_empty__Collect__eq,axiom,
! [P_23: nat > $o] :
( ( bot_bot_nat_o
= ( collect_nat @ P_23 ) )
<=> ! [X_1: nat] :
~ ( P_23 @ X_1 ) ) ).
thf(fact_42_ex__in__conv,axiom,
! [A_245: hoare_2091234717iple_a > $o] :
( ? [X_1: hoare_2091234717iple_a] : ( member290856304iple_a @ X_1 @ A_245 )
<=> ( A_245 != bot_bo1791335050le_a_o ) ) ).
thf(fact_43_ex__in__conv,axiom,
! [A_245: nat > $o] :
( ? [X_1: nat] : ( member_nat @ X_1 @ A_245 )
<=> ( A_245 != bot_bot_nat_o ) ) ).
thf(fact_44_ex__in__conv,axiom,
! [A_245: int > $o] :
( ? [X_1: int] : ( member_int @ X_1 @ A_245 )
<=> ( A_245 != bot_bot_int_o ) ) ).
thf(fact_45_all__not__in__conv,axiom,
! [A_244: hoare_2091234717iple_a > $o] :
( ! [X_1: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ X_1 @ A_244 )
<=> ( A_244 = bot_bo1791335050le_a_o ) ) ).
thf(fact_46_all__not__in__conv,axiom,
! [A_244: nat > $o] :
( ! [X_1: nat] :
~ ( member_nat @ X_1 @ A_244 )
<=> ( A_244 = bot_bot_nat_o ) ) ).
thf(fact_47_all__not__in__conv,axiom,
! [A_244: int > $o] :
( ! [X_1: int] :
~ ( member_int @ X_1 @ A_244 )
<=> ( A_244 = bot_bot_int_o ) ) ).
thf(fact_48_empty__def,axiom,
( bot_bo1791335050le_a_o
= ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : $false ) ) ).
thf(fact_49_empty__def,axiom,
( bot_bot_int_o
= ( collect_int
@ ^ [X_1: int] : $false ) ) ).
thf(fact_50_empty__def,axiom,
( bot_bot_nat_o
= ( collect_nat
@ ^ [X_1: nat] : $false ) ) ).
thf(fact_51_insert__absorb,axiom,
! [A_243: nat,A_242: nat > $o] :
( ( member_nat @ A_243 @ A_242 )
=> ( ( insert_nat @ A_243 @ A_242 )
= A_242 ) ) ).
thf(fact_52_insert__absorb,axiom,
! [A_243: hoare_2091234717iple_a,A_242: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_243 @ A_242 )
=> ( ( insert1597628439iple_a @ A_243 @ A_242 )
= A_242 ) ) ).
thf(fact_53_insert__absorb,axiom,
! [A_243: int,A_242: int > $o] :
( ( member_int @ A_243 @ A_242 )
=> ( ( insert_int @ A_243 @ A_242 )
= A_242 ) ) ).
thf(fact_54_insertI2,axiom,
! [B_117: nat,A_241: nat,B_116: nat > $o] :
( ( member_nat @ A_241 @ B_116 )
=> ( member_nat @ A_241 @ ( insert_nat @ B_117 @ B_116 ) ) ) ).
thf(fact_55_insertI2,axiom,
! [B_117: hoare_2091234717iple_a,A_241: hoare_2091234717iple_a,B_116: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_241 @ B_116 )
=> ( member290856304iple_a @ A_241 @ ( insert1597628439iple_a @ B_117 @ B_116 ) ) ) ).
thf(fact_56_insertI2,axiom,
! [B_117: int,A_241: int,B_116: int > $o] :
( ( member_int @ A_241 @ B_116 )
=> ( member_int @ A_241 @ ( insert_int @ B_117 @ B_116 ) ) ) ).
thf(fact_57_insert__ident,axiom,
! [B_115: nat > $o,X_102: nat,A_240: nat > $o] :
( ~ ( member_nat @ X_102 @ A_240 )
=> ( ~ ( member_nat @ X_102 @ B_115 )
=> ( ( ( insert_nat @ X_102 @ A_240 )
= ( insert_nat @ X_102 @ B_115 ) )
<=> ( A_240 = B_115 ) ) ) ) ).
thf(fact_58_insert__ident,axiom,
! [B_115: hoare_2091234717iple_a > $o,X_102: hoare_2091234717iple_a,A_240: hoare_2091234717iple_a > $o] :
( ~ ( member290856304iple_a @ X_102 @ A_240 )
=> ( ~ ( member290856304iple_a @ X_102 @ B_115 )
=> ( ( ( insert1597628439iple_a @ X_102 @ A_240 )
= ( insert1597628439iple_a @ X_102 @ B_115 ) )
<=> ( A_240 = B_115 ) ) ) ) ).
thf(fact_59_insert__ident,axiom,
! [B_115: int > $o,X_102: int,A_240: int > $o] :
( ~ ( member_int @ X_102 @ A_240 )
=> ( ~ ( member_int @ X_102 @ B_115 )
=> ( ( ( insert_int @ X_102 @ A_240 )
= ( insert_int @ X_102 @ B_115 ) )
<=> ( A_240 = B_115 ) ) ) ) ).
thf(fact_60_insert__code,axiom,
! [Y_40: nat,A_239: nat > $o,X_101: nat] :
( ( insert_nat @ Y_40 @ A_239 @ X_101 )
<=> ( ( Y_40 = X_101 )
| ( A_239 @ X_101 ) ) ) ).
thf(fact_61_insert__code,axiom,
! [Y_40: hoare_2091234717iple_a,A_239: hoare_2091234717iple_a > $o,X_101: hoare_2091234717iple_a] :
( ( insert1597628439iple_a @ Y_40 @ A_239 @ X_101 )
<=> ( ( Y_40 = X_101 )
| ( A_239 @ X_101 ) ) ) ).
thf(fact_62_insert__code,axiom,
! [Y_40: int,A_239: int > $o,X_101: int] :
( ( insert_int @ Y_40 @ A_239 @ X_101 )
<=> ( ( Y_40 = X_101 )
| ( A_239 @ X_101 ) ) ) ).
thf(fact_63_insert__iff,axiom,
! [A_238: nat,B_114: nat,A_237: nat > $o] :
( ( member_nat @ A_238 @ ( insert_nat @ B_114 @ A_237 ) )
<=> ( ( A_238 = B_114 )
| ( member_nat @ A_238 @ A_237 ) ) ) ).
thf(fact_64_insert__iff,axiom,
! [A_238: hoare_2091234717iple_a,B_114: hoare_2091234717iple_a,A_237: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_238 @ ( insert1597628439iple_a @ B_114 @ A_237 ) )
<=> ( ( A_238 = B_114 )
| ( member290856304iple_a @ A_238 @ A_237 ) ) ) ).
thf(fact_65_insert__iff,axiom,
! [A_238: int,B_114: int,A_237: int > $o] :
( ( member_int @ A_238 @ ( insert_int @ B_114 @ A_237 ) )
<=> ( ( A_238 = B_114 )
| ( member_int @ A_238 @ A_237 ) ) ) ).
thf(fact_66_insert__commute,axiom,
! [X_100: nat,Y_39: nat,A_236: nat > $o] :
( ( insert_nat @ X_100 @ ( insert_nat @ Y_39 @ A_236 ) )
= ( insert_nat @ Y_39 @ ( insert_nat @ X_100 @ A_236 ) ) ) ).
thf(fact_67_insert__commute,axiom,
! [X_100: hoare_2091234717iple_a,Y_39: hoare_2091234717iple_a,A_236: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ X_100 @ ( insert1597628439iple_a @ Y_39 @ A_236 ) )
= ( insert1597628439iple_a @ Y_39 @ ( insert1597628439iple_a @ X_100 @ A_236 ) ) ) ).
thf(fact_68_insert__commute,axiom,
! [X_100: int,Y_39: int,A_236: int > $o] :
( ( insert_int @ X_100 @ ( insert_int @ Y_39 @ A_236 ) )
= ( insert_int @ Y_39 @ ( insert_int @ X_100 @ A_236 ) ) ) ).
thf(fact_69_insert__absorb2,axiom,
! [X_99: nat,A_235: nat > $o] :
( ( insert_nat @ X_99 @ ( insert_nat @ X_99 @ A_235 ) )
= ( insert_nat @ X_99 @ A_235 ) ) ).
thf(fact_70_insert__absorb2,axiom,
! [X_99: hoare_2091234717iple_a,A_235: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ X_99 @ ( insert1597628439iple_a @ X_99 @ A_235 ) )
= ( insert1597628439iple_a @ X_99 @ A_235 ) ) ).
thf(fact_71_insert__absorb2,axiom,
! [X_99: int,A_235: int > $o] :
( ( insert_int @ X_99 @ ( insert_int @ X_99 @ A_235 ) )
= ( insert_int @ X_99 @ A_235 ) ) ).
thf(fact_72_insert__Collect,axiom,
! [A_234: nat,P_22: nat > $o] :
( ( insert_nat @ A_234 @ ( collect_nat @ P_22 ) )
= ( collect_nat
@ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_234 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
thf(fact_73_insert__Collect,axiom,
! [A_234: hoare_2091234717iple_a,P_22: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_234 @ ( collec992574898iple_a @ P_22 ) )
= ( collec992574898iple_a
@ ^ [U_1: hoare_2091234717iple_a] : ( (=>) @ ( (~) @ ( U_1 = A_234 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
thf(fact_74_insert__Collect,axiom,
! [A_234: int,P_22: int > $o] :
( ( insert_int @ A_234 @ ( collect_int @ P_22 ) )
= ( collect_int
@ ^ [U_1: int] : ( (=>) @ ( (~) @ ( U_1 = A_234 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
thf(fact_75_insert__compr,axiom,
! [A_233: nat,B_113: nat > $o] :
( ( insert_nat @ A_233 @ B_113 )
= ( collect_nat
@ ^ [X_1: nat] : ( (|) @ ( X_1 = A_233 ) @ ( member_nat @ X_1 @ B_113 ) ) ) ) ).
thf(fact_76_insert__compr,axiom,
! [A_233: hoare_2091234717iple_a,B_113: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_233 @ B_113 )
= ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (|) @ ( X_1 = A_233 ) @ ( member290856304iple_a @ X_1 @ B_113 ) ) ) ) ).
thf(fact_77_insert__compr,axiom,
! [A_233: int,B_113: int > $o] :
( ( insert_int @ A_233 @ B_113 )
= ( collect_int
@ ^ [X_1: int] : ( (|) @ ( X_1 = A_233 ) @ ( member_int @ X_1 @ B_113 ) ) ) ) ).
thf(fact_78_insertI1,axiom,
! [A_232: nat,B_112: nat > $o] : ( member_nat @ A_232 @ ( insert_nat @ A_232 @ B_112 ) ) ).
thf(fact_79_insertI1,axiom,
! [A_232: hoare_2091234717iple_a,B_112: hoare_2091234717iple_a > $o] : ( member290856304iple_a @ A_232 @ ( insert1597628439iple_a @ A_232 @ B_112 ) ) ).
thf(fact_80_insertI1,axiom,
! [A_232: int,B_112: int > $o] : ( member_int @ A_232 @ ( insert_int @ A_232 @ B_112 ) ) ).
thf(fact_81_insert__compr__raw,axiom,
! [X_1: nat,Xa: nat > $o] :
( ( insert_nat @ X_1 @ Xa )
= ( collect_nat
@ ^ [Y_28: nat] : ( (|) @ ( Y_28 = X_1 ) @ ( member_nat @ Y_28 @ Xa ) ) ) ) ).
thf(fact_82_insert__compr__raw,axiom,
! [X_1: hoare_2091234717iple_a,Xa: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ X_1 @ Xa )
= ( collec992574898iple_a
@ ^ [Y_28: hoare_2091234717iple_a] : ( (|) @ ( Y_28 = X_1 ) @ ( member290856304iple_a @ Y_28 @ Xa ) ) ) ) ).
thf(fact_83_insert__compr__raw,axiom,
! [X_1: int,Xa: int > $o] :
( ( insert_int @ X_1 @ Xa )
= ( collect_int
@ ^ [Y_28: int] : ( (|) @ ( Y_28 = X_1 ) @ ( member_int @ Y_28 @ Xa ) ) ) ) ).
thf(fact_84_singleton__inject,axiom,
! [A_231: nat,B_111: nat] :
( ( ( insert_nat @ A_231 @ bot_bot_nat_o )
= ( insert_nat @ B_111 @ bot_bot_nat_o ) )
=> ( A_231 = B_111 ) ) ).
thf(fact_85_singleton__inject,axiom,
! [A_231: hoare_2091234717iple_a,B_111: hoare_2091234717iple_a] :
( ( ( insert1597628439iple_a @ A_231 @ bot_bo1791335050le_a_o )
= ( insert1597628439iple_a @ B_111 @ bot_bo1791335050le_a_o ) )
=> ( A_231 = B_111 ) ) ).
thf(fact_86_singleton__inject,axiom,
! [A_231: int,B_111: int] :
( ( ( insert_int @ A_231 @ bot_bot_int_o )
= ( insert_int @ B_111 @ bot_bot_int_o ) )
=> ( A_231 = B_111 ) ) ).
thf(fact_87_singletonE,axiom,
! [B_110: nat,A_230: nat] :
( ( member_nat @ B_110 @ ( insert_nat @ A_230 @ bot_bot_nat_o ) )
=> ( B_110 = A_230 ) ) ).
thf(fact_88_singletonE,axiom,
! [B_110: hoare_2091234717iple_a,A_230: hoare_2091234717iple_a] :
( ( member290856304iple_a @ B_110 @ ( insert1597628439iple_a @ A_230 @ bot_bo1791335050le_a_o ) )
=> ( B_110 = A_230 ) ) ).
thf(fact_89_singletonE,axiom,
! [B_110: int,A_230: int] :
( ( member_int @ B_110 @ ( insert_int @ A_230 @ bot_bot_int_o ) )
=> ( B_110 = A_230 ) ) ).
thf(fact_90_doubleton__eq__iff,axiom,
! [A_229: nat,B_109: nat,C_53: nat,D_15: nat] :
( ( ( insert_nat @ A_229 @ ( insert_nat @ B_109 @ bot_bot_nat_o ) )
= ( insert_nat @ C_53 @ ( insert_nat @ D_15 @ bot_bot_nat_o ) ) )
<=> ( ( ( A_229 = C_53 )
& ( B_109 = D_15 ) )
| ( ( A_229 = D_15 )
& ( B_109 = C_53 ) ) ) ) ).
thf(fact_91_doubleton__eq__iff,axiom,
! [A_229: hoare_2091234717iple_a,B_109: hoare_2091234717iple_a,C_53: hoare_2091234717iple_a,D_15: hoare_2091234717iple_a] :
( ( ( insert1597628439iple_a @ A_229 @ ( insert1597628439iple_a @ B_109 @ bot_bo1791335050le_a_o ) )
= ( insert1597628439iple_a @ C_53 @ ( insert1597628439iple_a @ D_15 @ bot_bo1791335050le_a_o ) ) )
<=> ( ( ( A_229 = C_53 )
& ( B_109 = D_15 ) )
| ( ( A_229 = D_15 )
& ( B_109 = C_53 ) ) ) ) ).
thf(fact_92_doubleton__eq__iff,axiom,
! [A_229: int,B_109: int,C_53: int,D_15: int] :
( ( ( insert_int @ A_229 @ ( insert_int @ B_109 @ bot_bot_int_o ) )
= ( insert_int @ C_53 @ ( insert_int @ D_15 @ bot_bot_int_o ) ) )
<=> ( ( ( A_229 = C_53 )
& ( B_109 = D_15 ) )
| ( ( A_229 = D_15 )
& ( B_109 = C_53 ) ) ) ) ).
thf(fact_93_singleton__iff,axiom,
! [B_108: nat,A_228: nat] :
( ( member_nat @ B_108 @ ( insert_nat @ A_228 @ bot_bot_nat_o ) )
<=> ( B_108 = A_228 ) ) ).
thf(fact_94_singleton__iff,axiom,
! [B_108: hoare_2091234717iple_a,A_228: hoare_2091234717iple_a] :
( ( member290856304iple_a @ B_108 @ ( insert1597628439iple_a @ A_228 @ bot_bo1791335050le_a_o ) )
<=> ( B_108 = A_228 ) ) ).
thf(fact_95_singleton__iff,axiom,
! [B_108: int,A_228: int] :
( ( member_int @ B_108 @ ( insert_int @ A_228 @ bot_bot_int_o ) )
<=> ( B_108 = A_228 ) ) ).
thf(fact_96_insert__not__empty,axiom,
! [A_227: nat,A_226: nat > $o] :
( ( insert_nat @ A_227 @ A_226 )
!= bot_bot_nat_o ) ).
thf(fact_97_insert__not__empty,axiom,
! [A_227: hoare_2091234717iple_a,A_226: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_227 @ A_226 )
!= bot_bo1791335050le_a_o ) ).
thf(fact_98_insert__not__empty,axiom,
! [A_227: int,A_226: int > $o] :
( ( insert_int @ A_227 @ A_226 )
!= bot_bot_int_o ) ).
thf(fact_99_empty__not__insert,axiom,
! [A_225: nat,A_224: nat > $o] :
( bot_bot_nat_o
!= ( insert_nat @ A_225 @ A_224 ) ) ).
thf(fact_100_empty__not__insert,axiom,
! [A_225: hoare_2091234717iple_a,A_224: hoare_2091234717iple_a > $o] :
( bot_bo1791335050le_a_o
!= ( insert1597628439iple_a @ A_225 @ A_224 ) ) ).
thf(fact_101_empty__not__insert,axiom,
! [A_225: int,A_224: int > $o] :
( bot_bot_int_o
!= ( insert_int @ A_225 @ A_224 ) ) ).
thf(fact_102_the__elem__eq,axiom,
! [X_98: nat] :
( ( the_elem_nat @ ( insert_nat @ X_98 @ bot_bot_nat_o ) )
= X_98 ) ).
thf(fact_103_the__elem__eq,axiom,
! [X_98: hoare_2091234717iple_a] :
( ( the_el13400124iple_a @ ( insert1597628439iple_a @ X_98 @ bot_bo1791335050le_a_o ) )
= X_98 ) ).
thf(fact_104_the__elem__eq,axiom,
! [X_98: int] :
( ( the_elem_int @ ( insert_int @ X_98 @ bot_bot_int_o ) )
= X_98 ) ).
thf(fact_105_bot__apply,axiom,
! [X_97: hoare_2091234717iple_a] :
( ( bot_bo1791335050le_a_o @ X_97 )
<=> bot_bot_o ) ).
thf(fact_106_bot__apply,axiom,
! [X_97: int] :
( ( bot_bot_int_o @ X_97 )
<=> bot_bot_o ) ).
thf(fact_107_bot__apply,axiom,
! [X_97: nat] :
( ( bot_bot_nat_o @ X_97 )
<=> bot_bot_o ) ).
thf(fact_108_bot__fun__def,axiom,
! [X_1: hoare_2091234717iple_a] :
( ( bot_bo1791335050le_a_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_109_bot__fun__def,axiom,
! [X_1: int] :
( ( bot_bot_int_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_110_bot__fun__def,axiom,
! [X_1: nat] :
( ( bot_bot_nat_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_111_hoare__derivs_OSkip,axiom,
! [G_28: hoare_2091234717iple_a > $o,P_21: x_a > state > $o] : ( hoare_1467856363rivs_a @ G_28 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_21 @ skip @ P_21 ) @ bot_bo1791335050le_a_o ) ) ).
thf(fact_112_Comp,axiom,
! [D_14: com,R_4: x_a > state > $o,G_27: hoare_2091234717iple_a > $o,P_20: x_a > state > $o,C_52: com,Q_9: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_27 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_20 @ C_52 @ Q_9 ) @ bot_bo1791335050le_a_o ) )
=> ( ( hoare_1467856363rivs_a @ G_27 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ Q_9 @ D_14 @ R_4 ) @ bot_bo1791335050le_a_o ) )
=> ( hoare_1467856363rivs_a @ G_27 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_20 @ ( semi @ C_52 @ D_14 ) @ R_4 ) @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_113_triple_Oexhaust,axiom,
! [Y_38: hoare_2091234717iple_a] :
~ ! [Fun1: x_a > state > $o,Com: com,Fun2: x_a > state > $o] :
( Y_38
!= ( hoare_657976383iple_a @ Fun1 @ Com @ Fun2 ) ) ).
thf(fact_114_Set_Oset__insert,axiom,
! [X_96: nat,A_223: nat > $o] :
( ( member_nat @ X_96 @ A_223 )
=> ~ ! [B_55: nat > $o] :
( ( A_223
= ( insert_nat @ X_96 @ B_55 ) )
=> ( member_nat @ X_96 @ B_55 ) ) ) ).
thf(fact_115_Set_Oset__insert,axiom,
! [X_96: hoare_2091234717iple_a,A_223: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_96 @ A_223 )
=> ~ ! [B_55: hoare_2091234717iple_a > $o] :
( ( A_223
= ( insert1597628439iple_a @ X_96 @ B_55 ) )
=> ( member290856304iple_a @ X_96 @ B_55 ) ) ) ).
thf(fact_116_Set_Oset__insert,axiom,
! [X_96: int,A_223: int > $o] :
( ( member_int @ X_96 @ A_223 )
=> ~ ! [B_55: int > $o] :
( ( A_223
= ( insert_int @ X_96 @ B_55 ) )
=> ( member_int @ X_96 @ B_55 ) ) ) ).
thf(fact_117_mk__disjoint__insert,axiom,
! [A_222: nat,A_221: nat > $o] :
( ( member_nat @ A_222 @ A_221 )
=> ? [B_55: nat > $o] :
( ( A_221
= ( insert_nat @ A_222 @ B_55 ) )
& ~ ( member_nat @ A_222 @ B_55 ) ) ) ).
thf(fact_118_mk__disjoint__insert,axiom,
! [A_222: hoare_2091234717iple_a,A_221: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_222 @ A_221 )
=> ? [B_55: hoare_2091234717iple_a > $o] :
( ( A_221
= ( insert1597628439iple_a @ A_222 @ B_55 ) )
& ~ ( member290856304iple_a @ A_222 @ B_55 ) ) ) ).
thf(fact_119_mk__disjoint__insert,axiom,
! [A_222: int,A_221: int > $o] :
( ( member_int @ A_222 @ A_221 )
=> ? [B_55: int > $o] :
( ( A_221
= ( insert_int @ A_222 @ B_55 ) )
& ~ ( member_int @ A_222 @ B_55 ) ) ) ).
thf(fact_120_equals0I,axiom,
! [A_220: hoare_2091234717iple_a > $o] :
( ! [Y_28: hoare_2091234717iple_a] :
~ ( member290856304iple_a @ Y_28 @ A_220 )
=> ( A_220 = bot_bo1791335050le_a_o ) ) ).
thf(fact_121_equals0I,axiom,
! [A_220: nat > $o] :
( ! [Y_28: nat] :
~ ( member_nat @ Y_28 @ A_220 )
=> ( A_220 = bot_bot_nat_o ) ) ).
thf(fact_122_equals0I,axiom,
! [A_220: int > $o] :
( ! [Y_28: int] :
~ ( member_int @ Y_28 @ A_220 )
=> ( A_220 = bot_bot_int_o ) ) ).
thf(fact_123_conseq,axiom,
! [Q_7: x_a > state > $o,G_26: hoare_2091234717iple_a > $o,C_51: com,P_18: x_a > state > $o] :
( ! [Z: x_a,S: state] :
( ( P_18 @ Z @ S )
=> ? [P_19: x_a > state > $o,Q_8: x_a > state > $o] :
( ( hoare_1467856363rivs_a @ G_26 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_19 @ C_51 @ Q_8 ) @ bot_bo1791335050le_a_o ) )
& ! [S_5: state] :
( ! [Z_38: x_a] :
( ( P_19 @ Z_38 @ S )
=> ( Q_8 @ Z_38 @ S_5 ) )
=> ( Q_7 @ Z @ S_5 ) ) ) )
=> ( hoare_1467856363rivs_a @ G_26 @ ( insert1597628439iple_a @ ( hoare_657976383iple_a @ P_18 @ C_51 @ Q_7 ) @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_124_nonempty__iff,axiom,
! [A_219: nat > $o] :
( ( A_219 != bot_bot_nat_o )
<=> ? [X_1: nat,B_55: nat > $o] :
( ( A_219
= ( insert_nat @ X_1 @ B_55 ) )
& ~ ( member_nat @ X_1 @ B_55 ) ) ) ).
thf(fact_125_nonempty__iff,axiom,
! [A_219: hoare_2091234717iple_a > $o] :
( ( A_219 != bot_bo1791335050le_a_o )
<=> ? [X_1: hoare_2091234717iple_a,B_55: hoare_2091234717iple_a > $o] :
( ( A_219
= ( insert1597628439iple_a @ X_1 @ B_55 ) )
& ~ ( member290856304iple_a @ X_1 @ B_55 ) ) ) ).
thf(fact_126_nonempty__iff,axiom,
! [A_219: int > $o] :
( ( A_219 != bot_bot_int_o )
<=> ? [X_1: int,B_55: int > $o] :
( ( A_219
= ( insert_int @ X_1 @ B_55 ) )
& ~ ( member_int @ X_1 @ B_55 ) ) ) ).
thf(fact_127_com_Osimps_I13_J,axiom,
! [Com1: com,Com2: com] :
( ( semi @ Com1 @ Com2 )
!= skip ) ).
thf(fact_128_com_Osimps_I12_J,axiom,
! [Com1: com,Com2: com] :
( skip
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_129_the__elem__def,axiom,
! [X_95: nat > $o] :
( ( the_elem_nat @ X_95 )
= ( the_nat
@ ^ [X_1: nat] :
( X_95
= ( insert_nat @ X_1 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_130_the__elem__def,axiom,
! [X_95: hoare_2091234717iple_a > $o] :
( ( the_el13400124iple_a @ X_95 )
= ( the_Ho1471183438iple_a
@ ^ [X_1: hoare_2091234717iple_a] :
( X_95
= ( insert1597628439iple_a @ X_1 @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_131_the__elem__def,axiom,
! [X_95: int > $o] :
( ( the_elem_int @ X_95 )
= ( the_int
@ ^ [X_1: int] :
( X_95
= ( insert_int @ X_1 @ bot_bot_int_o ) ) ) ) ).
thf(fact_132_com_Osimps_I3_J,axiom,
! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
( ( ( semi @ Com1_1 @ Com2_1 )
= ( semi @ Com1 @ Com2 ) )
<=> ( ( Com1_1 = Com1 )
& ( Com2_1 = Com2 ) ) ) ).
thf(fact_133_fold1Set__sing,axiom,
! [F_105: nat > nat > nat,A_218: nat,B_107: nat] :
( ( finite_fold1Set_nat @ F_105 @ ( insert_nat @ A_218 @ bot_bot_nat_o ) @ B_107 )
<=> ( A_218 = B_107 ) ) ).
thf(fact_134_fold1Set__sing,axiom,
! [F_105: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A_218: hoare_2091234717iple_a,B_107: hoare_2091234717iple_a] :
( ( finite114877549iple_a @ F_105 @ ( insert1597628439iple_a @ A_218 @ bot_bo1791335050le_a_o ) @ B_107 )
<=> ( A_218 = B_107 ) ) ).
thf(fact_135_fold1Set__sing,axiom,
! [F_105: int > int > int,A_218: int,B_107: int] :
( ( finite_fold1Set_int @ F_105 @ ( insert_int @ A_218 @ bot_bot_int_o ) @ B_107 )
<=> ( A_218 = B_107 ) ) ).
thf(fact_136_folding__one_Osingleton,axiom,
! [X_94: nat,F_104: nat > nat > nat,F_103: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_104 @ F_103 )
=> ( ( F_103 @ ( insert_nat @ X_94 @ bot_bot_nat_o ) )
= X_94 ) ) ).
thf(fact_137_folding__one_Osingleton,axiom,
! [X_94: hoare_2091234717iple_a,F_104: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_103: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_104 @ F_103 )
=> ( ( F_103 @ ( insert1597628439iple_a @ X_94 @ bot_bo1791335050le_a_o ) )
= X_94 ) ) ).
thf(fact_138_folding__one_Osingleton,axiom,
! [X_94: int,F_104: int > int > int,F_103: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_104 @ F_103 )
=> ( ( F_103 @ ( insert_int @ X_94 @ bot_bot_int_o ) )
= X_94 ) ) ).
thf(fact_139_bot__empty__eq,axiom,
! [X_1: hoare_2091234717iple_a] :
( ( bot_bo1791335050le_a_o @ X_1 )
<=> ( member290856304iple_a @ X_1 @ bot_bo1791335050le_a_o ) ) ).
thf(fact_140_bot__empty__eq,axiom,
! [X_1: nat] :
( ( bot_bot_nat_o @ X_1 )
<=> ( member_nat @ X_1 @ bot_bot_nat_o ) ) ).
thf(fact_141_bot__empty__eq,axiom,
! [X_1: int] :
( ( bot_bot_int_o @ X_1 )
<=> ( member_int @ X_1 @ bot_bot_int_o ) ) ).
thf(fact_142_Ass,axiom,
! [G_25: hoare_2091234717iple_a > $o,P_17: x_a > state > $o,X_93: vname,A_217: state > nat] :
( hoare_1467856363rivs_a @ G_25
@ ( insert1597628439iple_a
@ ( hoare_657976383iple_a
@ ^ [Z: x_a,S: state] : ( P_17 @ Z @ ( update @ S @ X_93 @ ( A_217 @ S ) ) )
@ ( ass @ X_93 @ A_217 )
@ P_17 )
@ bot_bo1791335050le_a_o ) ) ).
thf(fact_143_fold1__singleton,axiom,
! [F_102: nat > nat > nat,A_216: nat] :
( ( finite_fold1_nat @ F_102 @ ( insert_nat @ A_216 @ bot_bot_nat_o ) )
= A_216 ) ).
thf(fact_144_fold1__singleton,axiom,
! [F_102: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A_216: hoare_2091234717iple_a] :
( ( finite2106937597iple_a @ F_102 @ ( insert1597628439iple_a @ A_216 @ bot_bo1791335050le_a_o ) )
= A_216 ) ).
thf(fact_145_fold1__singleton,axiom,
! [F_102: int > int > int,A_216: int] :
( ( finite_fold1_int @ F_102 @ ( insert_int @ A_216 @ bot_bot_int_o ) )
= A_216 ) ).
thf(fact_146_fold1__singleton__def,axiom,
! [A_215: nat,G_24: ( nat > $o ) > nat,F_101: nat > nat > nat] :
( ( G_24
= ( finite_fold1_nat @ F_101 ) )
=> ( ( G_24 @ ( insert_nat @ A_215 @ bot_bot_nat_o ) )
= A_215 ) ) ).
thf(fact_147_fold1__singleton__def,axiom,
! [A_215: hoare_2091234717iple_a,G_24: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a,F_101: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( G_24
= ( finite2106937597iple_a @ F_101 ) )
=> ( ( G_24 @ ( insert1597628439iple_a @ A_215 @ bot_bo1791335050le_a_o ) )
= A_215 ) ) ).
thf(fact_148_fold1__singleton__def,axiom,
! [A_215: int,G_24: ( int > $o ) > int,F_101: int > int > int] :
( ( G_24
= ( finite_fold1_int @ F_101 ) )
=> ( ( G_24 @ ( insert_int @ A_215 @ bot_bot_int_o ) )
= A_215 ) ) ).
thf(fact_149_image__constant__conv,axiom,
! [C_50: nat,A_214: hoare_2091234717iple_a > $o] :
( ( ( A_214 = bot_bo1791335050le_a_o )
=> ( ( image_1773322034_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : C_50
@ A_214 )
= bot_bot_nat_o ) )
& ( ( A_214 != bot_bo1791335050le_a_o )
=> ( ( image_1773322034_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : C_50
@ A_214 )
= ( insert_nat @ C_50 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_150_image__constant__conv,axiom,
! [C_50: nat,A_214: int > $o] :
( ( ( A_214 = bot_bot_int_o )
=> ( ( image_int_nat
@ ^ [X_1: int] : C_50
@ A_214 )
= bot_bot_nat_o ) )
& ( ( A_214 != bot_bot_int_o )
=> ( ( image_int_nat
@ ^ [X_1: int] : C_50
@ A_214 )
= ( insert_nat @ C_50 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_151_image__constant__conv,axiom,
! [C_50: hoare_2091234717iple_a,A_214: hoare_2091234717iple_a > $o] :
( ( ( A_214 = bot_bo1791335050le_a_o )
=> ( ( image_1661191109iple_a
@ ^ [X_1: hoare_2091234717iple_a] : C_50
@ A_214 )
= bot_bo1791335050le_a_o ) )
& ( ( A_214 != bot_bo1791335050le_a_o )
=> ( ( image_1661191109iple_a
@ ^ [X_1: hoare_2091234717iple_a] : C_50
@ A_214 )
= ( insert1597628439iple_a @ C_50 @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_152_image__constant__conv,axiom,
! [C_50: int,A_214: int > $o] :
( ( ( A_214 = bot_bot_int_o )
=> ( ( image_int_int
@ ^ [X_1: int] : C_50
@ A_214 )
= bot_bot_int_o ) )
& ( ( A_214 != bot_bot_int_o )
=> ( ( image_int_int
@ ^ [X_1: int] : C_50
@ A_214 )
= ( insert_int @ C_50 @ bot_bot_int_o ) ) ) ) ).
thf(fact_153_image__constant__conv,axiom,
! [C_50: nat,A_214: nat > $o] :
( ( ( A_214 = bot_bot_nat_o )
=> ( ( image_nat_nat
@ ^ [X_1: nat] : C_50
@ A_214 )
= bot_bot_nat_o ) )
& ( ( A_214 != bot_bot_nat_o )
=> ( ( image_nat_nat
@ ^ [X_1: nat] : C_50
@ A_214 )
= ( insert_nat @ C_50 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_154_image__constant__conv,axiom,
! [C_50: int,A_214: nat > $o] :
( ( ( A_214 = bot_bot_nat_o )
=> ( ( image_nat_int
@ ^ [X_1: nat] : C_50
@ A_214 )
= bot_bot_int_o ) )
& ( ( A_214 != bot_bot_nat_o )
=> ( ( image_nat_int
@ ^ [X_1: nat] : C_50
@ A_214 )
= ( insert_int @ C_50 @ bot_bot_int_o ) ) ) ) ).
thf(fact_155_image__eqI,axiom,
! [A_213: hoare_2091234717iple_a > $o,B_106: hoare_2091234717iple_a,F_100: hoare_2091234717iple_a > hoare_2091234717iple_a,X_92: hoare_2091234717iple_a] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member290856304iple_a @ X_92 @ A_213 )
=> ( member290856304iple_a @ B_106 @ ( image_1661191109iple_a @ F_100 @ A_213 ) ) ) ) ).
thf(fact_156_image__eqI,axiom,
! [A_213: nat > $o,B_106: hoare_2091234717iple_a,F_100: nat > hoare_2091234717iple_a,X_92: nat] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member_nat @ X_92 @ A_213 )
=> ( member290856304iple_a @ B_106 @ ( image_359186840iple_a @ F_100 @ A_213 ) ) ) ) ).
thf(fact_157_image__eqI,axiom,
! [A_213: int > $o,B_106: hoare_2091234717iple_a,F_100: int > hoare_2091234717iple_a,X_92: int] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member_int @ X_92 @ A_213 )
=> ( member290856304iple_a @ B_106 @ ( image_338319932iple_a @ F_100 @ A_213 ) ) ) ) ).
thf(fact_158_image__eqI,axiom,
! [A_213: hoare_2091234717iple_a > $o,B_106: nat,F_100: hoare_2091234717iple_a > nat,X_92: hoare_2091234717iple_a] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member290856304iple_a @ X_92 @ A_213 )
=> ( member_nat @ B_106 @ ( image_1773322034_a_nat @ F_100 @ A_213 ) ) ) ) ).
thf(fact_159_image__eqI,axiom,
! [A_213: int > $o,B_106: int,F_100: int > int,X_92: int] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member_int @ X_92 @ A_213 )
=> ( member_int @ B_106 @ ( image_int_int @ F_100 @ A_213 ) ) ) ) ).
thf(fact_160_image__eqI,axiom,
! [A_213: hoare_2091234717iple_a > $o,B_106: int,F_100: hoare_2091234717iple_a > int,X_92: hoare_2091234717iple_a] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member290856304iple_a @ X_92 @ A_213 )
=> ( member_int @ B_106 @ ( image_263112078_a_int @ F_100 @ A_213 ) ) ) ) ).
thf(fact_161_image__eqI,axiom,
! [A_213: nat > $o,B_106: nat,F_100: nat > nat,X_92: nat] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member_nat @ X_92 @ A_213 )
=> ( member_nat @ B_106 @ ( image_nat_nat @ F_100 @ A_213 ) ) ) ) ).
thf(fact_162_image__eqI,axiom,
! [A_213: nat > $o,B_106: int,F_100: nat > int,X_92: nat] :
( ( B_106
= ( F_100 @ X_92 ) )
=> ( ( member_nat @ X_92 @ A_213 )
=> ( member_int @ B_106 @ ( image_nat_int @ F_100 @ A_213 ) ) ) ) ).
thf(fact_163_fold1__def,axiom,
! [F_99: int > int > int,A_212: int > $o] :
( ( finite_fold1_int @ F_99 @ A_212 )
= ( the_int @ ( finite_fold1Set_int @ F_99 @ A_212 ) ) ) ).
thf(fact_164_fold1__def,axiom,
! [F_99: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A_212: hoare_2091234717iple_a > $o] :
( ( finite2106937597iple_a @ F_99 @ A_212 )
= ( the_Ho1471183438iple_a @ ( finite114877549iple_a @ F_99 @ A_212 ) ) ) ).
thf(fact_165_fold1__def,axiom,
! [F_99: nat > nat > nat,A_212: nat > $o] :
( ( finite_fold1_nat @ F_99 @ A_212 )
= ( the_nat @ ( finite_fold1Set_nat @ F_99 @ A_212 ) ) ) ).
thf(fact_166_com_Osimps_I1_J,axiom,
! [Vname_1: vname,Fun_1: state > nat,Vname: vname,Fun: state > nat] :
( ( ( ass @ Vname_1 @ Fun_1 )
= ( ass @ Vname @ Fun ) )
<=> ( ( Vname_1 = Vname )
& ( Fun_1 = Fun ) ) ) ).
thf(fact_167_rev__image__eqI,axiom,
! [B_105: hoare_2091234717iple_a,F_98: hoare_2091234717iple_a > hoare_2091234717iple_a,X_91: hoare_2091234717iple_a,A_211: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member290856304iple_a @ B_105 @ ( image_1661191109iple_a @ F_98 @ A_211 ) ) ) ) ).
thf(fact_168_rev__image__eqI,axiom,
! [B_105: hoare_2091234717iple_a,F_98: nat > hoare_2091234717iple_a,X_91: nat,A_211: nat > $o] :
( ( member_nat @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member290856304iple_a @ B_105 @ ( image_359186840iple_a @ F_98 @ A_211 ) ) ) ) ).
thf(fact_169_rev__image__eqI,axiom,
! [B_105: hoare_2091234717iple_a,F_98: int > hoare_2091234717iple_a,X_91: int,A_211: int > $o] :
( ( member_int @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member290856304iple_a @ B_105 @ ( image_338319932iple_a @ F_98 @ A_211 ) ) ) ) ).
thf(fact_170_rev__image__eqI,axiom,
! [B_105: nat,F_98: hoare_2091234717iple_a > nat,X_91: hoare_2091234717iple_a,A_211: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member_nat @ B_105 @ ( image_1773322034_a_nat @ F_98 @ A_211 ) ) ) ) ).
thf(fact_171_rev__image__eqI,axiom,
! [B_105: int,F_98: int > int,X_91: int,A_211: int > $o] :
( ( member_int @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member_int @ B_105 @ ( image_int_int @ F_98 @ A_211 ) ) ) ) ).
thf(fact_172_rev__image__eqI,axiom,
! [B_105: int,F_98: hoare_2091234717iple_a > int,X_91: hoare_2091234717iple_a,A_211: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member_int @ B_105 @ ( image_263112078_a_int @ F_98 @ A_211 ) ) ) ) ).
thf(fact_173_rev__image__eqI,axiom,
! [B_105: nat,F_98: nat > nat,X_91: nat,A_211: nat > $o] :
( ( member_nat @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member_nat @ B_105 @ ( image_nat_nat @ F_98 @ A_211 ) ) ) ) ).
thf(fact_174_rev__image__eqI,axiom,
! [B_105: int,F_98: nat > int,X_91: nat,A_211: nat > $o] :
( ( member_nat @ X_91 @ A_211 )
=> ( ( B_105
= ( F_98 @ X_91 ) )
=> ( member_int @ B_105 @ ( image_nat_int @ F_98 @ A_211 ) ) ) ) ).
thf(fact_175_imageI,axiom,
! [F_97: hoare_2091234717iple_a > hoare_2091234717iple_a,X_90: hoare_2091234717iple_a,A_210: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_90 @ A_210 )
=> ( member290856304iple_a @ ( F_97 @ X_90 ) @ ( image_1661191109iple_a @ F_97 @ A_210 ) ) ) ).
thf(fact_176_imageI,axiom,
! [F_97: nat > hoare_2091234717iple_a,X_90: nat,A_210: nat > $o] :
( ( member_nat @ X_90 @ A_210 )
=> ( member290856304iple_a @ ( F_97 @ X_90 ) @ ( image_359186840iple_a @ F_97 @ A_210 ) ) ) ).
thf(fact_177_imageI,axiom,
! [F_97: int > hoare_2091234717iple_a,X_90: int,A_210: int > $o] :
( ( member_int @ X_90 @ A_210 )
=> ( member290856304iple_a @ ( F_97 @ X_90 ) @ ( image_338319932iple_a @ F_97 @ A_210 ) ) ) ).
thf(fact_178_imageI,axiom,
! [F_97: hoare_2091234717iple_a > nat,X_90: hoare_2091234717iple_a,A_210: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_90 @ A_210 )
=> ( member_nat @ ( F_97 @ X_90 ) @ ( image_1773322034_a_nat @ F_97 @ A_210 ) ) ) ).
thf(fact_179_imageI,axiom,
! [F_97: int > int,X_90: int,A_210: int > $o] :
( ( member_int @ X_90 @ A_210 )
=> ( member_int @ ( F_97 @ X_90 ) @ ( image_int_int @ F_97 @ A_210 ) ) ) ).
thf(fact_180_imageI,axiom,
! [F_97: hoare_2091234717iple_a > int,X_90: hoare_2091234717iple_a,A_210: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_90 @ A_210 )
=> ( member_int @ ( F_97 @ X_90 ) @ ( image_263112078_a_int @ F_97 @ A_210 ) ) ) ).
thf(fact_181_imageI,axiom,
! [F_97: nat > nat,X_90: nat,A_210: nat > $o] :
( ( member_nat @ X_90 @ A_210 )
=> ( member_nat @ ( F_97 @ X_90 ) @ ( image_nat_nat @ F_97 @ A_210 ) ) ) ).
thf(fact_182_imageI,axiom,
! [F_97: nat > int,X_90: nat,A_210: nat > $o] :
( ( member_nat @ X_90 @ A_210 )
=> ( member_int @ ( F_97 @ X_90 ) @ ( image_nat_int @ F_97 @ A_210 ) ) ) ).
thf(fact_183_image__iff,axiom,
! [Z_37: hoare_2091234717iple_a,F_96: hoare_2091234717iple_a > hoare_2091234717iple_a,A_209: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ Z_37 @ ( image_1661191109iple_a @ F_96 @ A_209 ) )
<=> ? [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_209 )
& ( Z_37
= ( F_96 @ X_1 ) ) ) ) ).
thf(fact_184_image__iff,axiom,
! [Z_37: int,F_96: int > int,A_209: int > $o] :
( ( member_int @ Z_37 @ ( image_int_int @ F_96 @ A_209 ) )
<=> ? [X_1: int] :
( ( member_int @ X_1 @ A_209 )
& ( Z_37
= ( F_96 @ X_1 ) ) ) ) ).
thf(fact_185_image__iff,axiom,
! [Z_37: nat,F_96: nat > nat,A_209: nat > $o] :
( ( member_nat @ Z_37 @ ( image_nat_nat @ F_96 @ A_209 ) )
<=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_209 )
& ( Z_37
= ( F_96 @ X_1 ) ) ) ) ).
thf(fact_186_image__iff,axiom,
! [Z_37: int,F_96: nat > int,A_209: nat > $o] :
( ( member_int @ Z_37 @ ( image_nat_int @ F_96 @ A_209 ) )
<=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_209 )
& ( Z_37
= ( F_96 @ X_1 ) ) ) ) ).
thf(fact_187_image__image,axiom,
! [F_95: nat > int,G_23: int > nat,A_208: int > $o] :
( ( image_nat_int @ F_95 @ ( image_int_nat @ G_23 @ A_208 ) )
= ( image_int_int
@ ^ [X_1: int] : ( F_95 @ ( G_23 @ X_1 ) )
@ A_208 ) ) ).
thf(fact_188_image__image,axiom,
! [F_95: int > int,G_23: nat > int,A_208: nat > $o] :
( ( image_int_int @ F_95 @ ( image_nat_int @ G_23 @ A_208 ) )
= ( image_nat_int
@ ^ [X_1: nat] : ( F_95 @ ( G_23 @ X_1 ) )
@ A_208 ) ) ).
thf(fact_189_image__ident,axiom,
! [Y_37: int > $o] :
( ( image_int_int
@ ^ [X_1: int] : X_1
@ Y_37 )
= Y_37 ) ).
thf(fact_190_image__ident,axiom,
! [Y_37: hoare_2091234717iple_a > $o] :
( ( image_1661191109iple_a
@ ^ [X_1: hoare_2091234717iple_a] : X_1
@ Y_37 )
= Y_37 ) ).
thf(fact_191_image__ident,axiom,
! [Y_37: nat > $o] :
( ( image_nat_nat
@ ^ [X_1: nat] : X_1
@ Y_37 )
= Y_37 ) ).
thf(fact_192_com_Osimps_I24_J,axiom,
! [Vname_1: vname,Fun_1: state > nat,Com1: com,Com2: com] :
( ( ass @ Vname_1 @ Fun_1 )
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_193_com_Osimps_I25_J,axiom,
! [Com1: com,Com2: com,Vname_1: vname,Fun_1: state > nat] :
( ( semi @ Com1 @ Com2 )
!= ( ass @ Vname_1 @ Fun_1 ) ) ).
thf(fact_194_com_Osimps_I8_J,axiom,
! [Vname: vname,Fun: state > nat] :
( skip
!= ( ass @ Vname @ Fun ) ) ).
thf(fact_195_mem__def,axiom,
! [X_89: hoare_2091234717iple_a,A_207: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_89 @ A_207 )
<=> ( A_207 @ X_89 ) ) ).
thf(fact_196_mem__def,axiom,
! [X_89: nat,A_207: nat > $o] :
( ( member_nat @ X_89 @ A_207 )
<=> ( A_207 @ X_89 ) ) ).
thf(fact_197_mem__def,axiom,
! [X_89: int,A_207: int > $o] :
( ( member_int @ X_89 @ A_207 )
<=> ( A_207 @ X_89 ) ) ).
thf(fact_198_Collect__def,axiom,
! [P_16: int > $o] :
( ( collect_int @ P_16 )
= P_16 ) ).
thf(fact_199_Collect__def,axiom,
! [P_16: hoare_2091234717iple_a > $o] :
( ( collec992574898iple_a @ P_16 )
= P_16 ) ).
thf(fact_200_Collect__def,axiom,
! [P_16: nat > $o] :
( ( collect_nat @ P_16 )
= P_16 ) ).
thf(fact_201_com_Osimps_I9_J,axiom,
! [Vname: vname,Fun: state > nat] :
( ( ass @ Vname @ Fun )
!= skip ) ).
thf(fact_202_image__is__empty,axiom,
! [F_94: hoare_2091234717iple_a > hoare_2091234717iple_a,A_206: hoare_2091234717iple_a > $o] :
( ( ( image_1661191109iple_a @ F_94 @ A_206 )
= bot_bo1791335050le_a_o )
<=> ( A_206 = bot_bo1791335050le_a_o ) ) ).
thf(fact_203_image__is__empty,axiom,
! [F_94: int > int,A_206: int > $o] :
( ( ( image_int_int @ F_94 @ A_206 )
= bot_bot_int_o )
<=> ( A_206 = bot_bot_int_o ) ) ).
thf(fact_204_image__is__empty,axiom,
! [F_94: nat > nat,A_206: nat > $o] :
( ( ( image_nat_nat @ F_94 @ A_206 )
= bot_bot_nat_o )
<=> ( A_206 = bot_bot_nat_o ) ) ).
thf(fact_205_image__is__empty,axiom,
! [F_94: nat > int,A_206: nat > $o] :
( ( ( image_nat_int @ F_94 @ A_206 )
= bot_bot_int_o )
<=> ( A_206 = bot_bot_nat_o ) ) ).
thf(fact_206_image__empty,axiom,
! [F_93: hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( image_1661191109iple_a @ F_93 @ bot_bo1791335050le_a_o )
= bot_bo1791335050le_a_o ) ).
thf(fact_207_image__empty,axiom,
! [F_93: int > int] :
( ( image_int_int @ F_93 @ bot_bot_int_o )
= bot_bot_int_o ) ).
thf(fact_208_image__empty,axiom,
! [F_93: nat > nat] :
( ( image_nat_nat @ F_93 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
thf(fact_209_image__empty,axiom,
! [F_93: nat > int] :
( ( image_nat_int @ F_93 @ bot_bot_nat_o )
= bot_bot_int_o ) ).
thf(fact_210_empty__is__image,axiom,
! [F_92: hoare_2091234717iple_a > hoare_2091234717iple_a,A_205: hoare_2091234717iple_a > $o] :
( ( bot_bo1791335050le_a_o
= ( image_1661191109iple_a @ F_92 @ A_205 ) )
<=> ( A_205 = bot_bo1791335050le_a_o ) ) ).
thf(fact_211_empty__is__image,axiom,
! [F_92: int > int,A_205: int > $o] :
( ( bot_bot_int_o
= ( image_int_int @ F_92 @ A_205 ) )
<=> ( A_205 = bot_bot_int_o ) ) ).
thf(fact_212_empty__is__image,axiom,
! [F_92: nat > nat,A_205: nat > $o] :
( ( bot_bot_nat_o
= ( image_nat_nat @ F_92 @ A_205 ) )
<=> ( A_205 = bot_bot_nat_o ) ) ).
thf(fact_213_empty__is__image,axiom,
! [F_92: nat > int,A_205: nat > $o] :
( ( bot_bot_int_o
= ( image_nat_int @ F_92 @ A_205 ) )
<=> ( A_205 = bot_bot_nat_o ) ) ).
thf(fact_214_insert__image,axiom,
! [F_91: int > nat,X_88: int,A_204: int > $o] :
( ( member_int @ X_88 @ A_204 )
=> ( ( insert_nat @ ( F_91 @ X_88 ) @ ( image_int_nat @ F_91 @ A_204 ) )
= ( image_int_nat @ F_91 @ A_204 ) ) ) ).
thf(fact_215_insert__image,axiom,
! [F_91: hoare_2091234717iple_a > hoare_2091234717iple_a,X_88: hoare_2091234717iple_a,A_204: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_88 @ A_204 )
=> ( ( insert1597628439iple_a @ ( F_91 @ X_88 ) @ ( image_1661191109iple_a @ F_91 @ A_204 ) )
= ( image_1661191109iple_a @ F_91 @ A_204 ) ) ) ).
thf(fact_216_insert__image,axiom,
! [F_91: int > int,X_88: int,A_204: int > $o] :
( ( member_int @ X_88 @ A_204 )
=> ( ( insert_int @ ( F_91 @ X_88 ) @ ( image_int_int @ F_91 @ A_204 ) )
= ( image_int_int @ F_91 @ A_204 ) ) ) ).
thf(fact_217_insert__image,axiom,
! [F_91: hoare_2091234717iple_a > int,X_88: hoare_2091234717iple_a,A_204: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_88 @ A_204 )
=> ( ( insert_int @ ( F_91 @ X_88 ) @ ( image_263112078_a_int @ F_91 @ A_204 ) )
= ( image_263112078_a_int @ F_91 @ A_204 ) ) ) ).
thf(fact_218_insert__image,axiom,
! [F_91: nat > nat,X_88: nat,A_204: nat > $o] :
( ( member_nat @ X_88 @ A_204 )
=> ( ( insert_nat @ ( F_91 @ X_88 ) @ ( image_nat_nat @ F_91 @ A_204 ) )
= ( image_nat_nat @ F_91 @ A_204 ) ) ) ).
thf(fact_219_insert__image,axiom,
! [F_91: nat > int,X_88: nat,A_204: nat > $o] :
( ( member_nat @ X_88 @ A_204 )
=> ( ( insert_int @ ( F_91 @ X_88 ) @ ( image_nat_int @ F_91 @ A_204 ) )
= ( image_nat_int @ F_91 @ A_204 ) ) ) ).
thf(fact_220_image__insert,axiom,
! [F_90: hoare_2091234717iple_a > nat,A_203: hoare_2091234717iple_a,B_104: hoare_2091234717iple_a > $o] :
( ( image_1773322034_a_nat @ F_90 @ ( insert1597628439iple_a @ A_203 @ B_104 ) )
= ( insert_nat @ ( F_90 @ A_203 ) @ ( image_1773322034_a_nat @ F_90 @ B_104 ) ) ) ).
thf(fact_221_image__insert,axiom,
! [F_90: int > nat,A_203: int,B_104: int > $o] :
( ( image_int_nat @ F_90 @ ( insert_int @ A_203 @ B_104 ) )
= ( insert_nat @ ( F_90 @ A_203 ) @ ( image_int_nat @ F_90 @ B_104 ) ) ) ).
thf(fact_222_image__insert,axiom,
! [F_90: hoare_2091234717iple_a > hoare_2091234717iple_a,A_203: hoare_2091234717iple_a,B_104: hoare_2091234717iple_a > $o] :
( ( image_1661191109iple_a @ F_90 @ ( insert1597628439iple_a @ A_203 @ B_104 ) )
= ( insert1597628439iple_a @ ( F_90 @ A_203 ) @ ( image_1661191109iple_a @ F_90 @ B_104 ) ) ) ).
thf(fact_223_image__insert,axiom,
! [F_90: nat > hoare_2091234717iple_a,A_203: nat,B_104: nat > $o] :
( ( image_359186840iple_a @ F_90 @ ( insert_nat @ A_203 @ B_104 ) )
= ( insert1597628439iple_a @ ( F_90 @ A_203 ) @ ( image_359186840iple_a @ F_90 @ B_104 ) ) ) ).
thf(fact_224_image__insert,axiom,
! [F_90: int > int,A_203: int,B_104: int > $o] :
( ( image_int_int @ F_90 @ ( insert_int @ A_203 @ B_104 ) )
= ( insert_int @ ( F_90 @ A_203 ) @ ( image_int_int @ F_90 @ B_104 ) ) ) ).
thf(fact_225_image__insert,axiom,
! [F_90: nat > nat,A_203: nat,B_104: nat > $o] :
( ( image_nat_nat @ F_90 @ ( insert_nat @ A_203 @ B_104 ) )
= ( insert_nat @ ( F_90 @ A_203 ) @ ( image_nat_nat @ F_90 @ B_104 ) ) ) ).
thf(fact_226_image__insert,axiom,
! [F_90: nat > int,A_203: nat,B_104: nat > $o] :
( ( image_nat_int @ F_90 @ ( insert_nat @ A_203 @ B_104 ) )
= ( insert_int @ ( F_90 @ A_203 ) @ ( image_nat_int @ F_90 @ B_104 ) ) ) ).
thf(fact_227_fold1Set__nonempty,axiom,
! [F_89: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A_202: hoare_2091234717iple_a > $o,X_87: hoare_2091234717iple_a] :
( ( finite114877549iple_a @ F_89 @ A_202 @ X_87 )
=> ( A_202 != bot_bo1791335050le_a_o ) ) ).
thf(fact_228_fold1Set__nonempty,axiom,
! [F_89: int > int > int,A_202: int > $o,X_87: int] :
( ( finite_fold1Set_int @ F_89 @ A_202 @ X_87 )
=> ( A_202 != bot_bot_int_o ) ) ).
thf(fact_229_fold1Set__nonempty,axiom,
! [F_89: nat > nat > nat,A_202: nat > $o,X_87: nat] :
( ( finite_fold1Set_nat @ F_89 @ A_202 @ X_87 )
=> ( A_202 != bot_bot_nat_o ) ) ).
thf(fact_230_empty__fold1SetE,axiom,
! [F_88: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,X_86: hoare_2091234717iple_a] :
~ ( finite114877549iple_a @ F_88 @ bot_bo1791335050le_a_o @ X_86 ) ).
thf(fact_231_empty__fold1SetE,axiom,
! [F_88: int > int > int,X_86: int] :
~ ( finite_fold1Set_int @ F_88 @ bot_bot_int_o @ X_86 ) ).
thf(fact_232_empty__fold1SetE,axiom,
! [F_88: nat > nat > nat,X_86: nat] :
~ ( finite_fold1Set_nat @ F_88 @ bot_bot_nat_o @ X_86 ) ).
thf(fact_233_image__constant,axiom,
! [C_49: nat,X_85: hoare_2091234717iple_a,A_201: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_85 @ A_201 )
=> ( ( image_1773322034_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : C_49
@ A_201 )
= ( insert_nat @ C_49 @ bot_bot_nat_o ) ) ) ).
thf(fact_234_image__constant,axiom,
! [C_49: nat,X_85: int,A_201: int > $o] :
( ( member_int @ X_85 @ A_201 )
=> ( ( image_int_nat
@ ^ [X_1: int] : C_49
@ A_201 )
= ( insert_nat @ C_49 @ bot_bot_nat_o ) ) ) ).
thf(fact_235_image__constant,axiom,
! [C_49: hoare_2091234717iple_a,X_85: hoare_2091234717iple_a,A_201: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_85 @ A_201 )
=> ( ( image_1661191109iple_a
@ ^ [X_1: hoare_2091234717iple_a] : C_49
@ A_201 )
= ( insert1597628439iple_a @ C_49 @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_236_image__constant,axiom,
! [C_49: int,X_85: int,A_201: int > $o] :
( ( member_int @ X_85 @ A_201 )
=> ( ( image_int_int
@ ^ [X_1: int] : C_49
@ A_201 )
= ( insert_int @ C_49 @ bot_bot_int_o ) ) ) ).
thf(fact_237_image__constant,axiom,
! [C_49: int,X_85: hoare_2091234717iple_a,A_201: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_85 @ A_201 )
=> ( ( image_263112078_a_int
@ ^ [X_1: hoare_2091234717iple_a] : C_49
@ A_201 )
= ( insert_int @ C_49 @ bot_bot_int_o ) ) ) ).
thf(fact_238_image__constant,axiom,
! [C_49: nat,X_85: nat,A_201: nat > $o] :
( ( member_nat @ X_85 @ A_201 )
=> ( ( image_nat_nat
@ ^ [X_1: nat] : C_49
@ A_201 )
= ( insert_nat @ C_49 @ bot_bot_nat_o ) ) ) ).
thf(fact_239_image__constant,axiom,
! [C_49: int,X_85: nat,A_201: nat > $o] :
( ( member_nat @ X_85 @ A_201 )
=> ( ( image_nat_int
@ ^ [X_1: nat] : C_49
@ A_201 )
= ( insert_int @ C_49 @ bot_bot_int_o ) ) ) ).
thf(fact_240_imageE,axiom,
! [B_103: hoare_2091234717iple_a,F_87: hoare_2091234717iple_a > hoare_2091234717iple_a,A_200: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ B_103 @ ( image_1661191109iple_a @ F_87 @ A_200 ) )
=> ~ ! [X_1: hoare_2091234717iple_a] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member290856304iple_a @ X_1 @ A_200 ) ) ) ).
thf(fact_241_imageE,axiom,
! [B_103: nat,F_87: hoare_2091234717iple_a > nat,A_200: hoare_2091234717iple_a > $o] :
( ( member_nat @ B_103 @ ( image_1773322034_a_nat @ F_87 @ A_200 ) )
=> ~ ! [X_1: hoare_2091234717iple_a] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member290856304iple_a @ X_1 @ A_200 ) ) ) ).
thf(fact_242_imageE,axiom,
! [B_103: int,F_87: hoare_2091234717iple_a > int,A_200: hoare_2091234717iple_a > $o] :
( ( member_int @ B_103 @ ( image_263112078_a_int @ F_87 @ A_200 ) )
=> ~ ! [X_1: hoare_2091234717iple_a] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member290856304iple_a @ X_1 @ A_200 ) ) ) ).
thf(fact_243_imageE,axiom,
! [B_103: hoare_2091234717iple_a,F_87: nat > hoare_2091234717iple_a,A_200: nat > $o] :
( ( member290856304iple_a @ B_103 @ ( image_359186840iple_a @ F_87 @ A_200 ) )
=> ~ ! [X_1: nat] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member_nat @ X_1 @ A_200 ) ) ) ).
thf(fact_244_imageE,axiom,
! [B_103: int,F_87: int > int,A_200: int > $o] :
( ( member_int @ B_103 @ ( image_int_int @ F_87 @ A_200 ) )
=> ~ ! [X_1: int] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member_int @ X_1 @ A_200 ) ) ) ).
thf(fact_245_imageE,axiom,
! [B_103: hoare_2091234717iple_a,F_87: int > hoare_2091234717iple_a,A_200: int > $o] :
( ( member290856304iple_a @ B_103 @ ( image_338319932iple_a @ F_87 @ A_200 ) )
=> ~ ! [X_1: int] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member_int @ X_1 @ A_200 ) ) ) ).
thf(fact_246_imageE,axiom,
! [B_103: nat,F_87: nat > nat,A_200: nat > $o] :
( ( member_nat @ B_103 @ ( image_nat_nat @ F_87 @ A_200 ) )
=> ~ ! [X_1: nat] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member_nat @ X_1 @ A_200 ) ) ) ).
thf(fact_247_imageE,axiom,
! [B_103: int,F_87: nat > int,A_200: nat > $o] :
( ( member_int @ B_103 @ ( image_nat_int @ F_87 @ A_200 ) )
=> ~ ! [X_1: nat] :
( ( B_103
= ( F_87 @ X_1 ) )
=> ~ ( member_nat @ X_1 @ A_200 ) ) ) ).
thf(fact_248_the__sym__eq__trivial,axiom,
! [X_84: int] :
( ( the_int @ ( fequal_int @ X_84 ) )
= X_84 ) ).
thf(fact_249_the__sym__eq__trivial,axiom,
! [X_84: hoare_2091234717iple_a] :
( ( the_Ho1471183438iple_a @ ( fequal1604381340iple_a @ X_84 ) )
= X_84 ) ).
thf(fact_250_the__sym__eq__trivial,axiom,
! [X_84: nat] :
( ( the_nat @ ( fequal_nat @ X_84 ) )
= X_84 ) ).
thf(fact_251_the__eq__trivial,axiom,
! [A_199: int] :
( ( the_int
@ ^ [X_1: int] : X_1 = A_199 )
= A_199 ) ).
thf(fact_252_the__eq__trivial,axiom,
! [A_199: hoare_2091234717iple_a] :
( ( the_Ho1471183438iple_a
@ ^ [X_1: hoare_2091234717iple_a] : X_1 = A_199 )
= A_199 ) ).
thf(fact_253_the__eq__trivial,axiom,
! [A_199: nat] :
( ( the_nat
@ ^ [X_1: nat] : X_1 = A_199 )
= A_199 ) ).
thf(fact_254_If__def,axiom,
! [X_83: int,Y_36: int,P_15: $o] :
( ( P_15
=> ( X_83
= ( the_int
@ ^ [Z_36: int] : ( (&) @ ( (=>) @ P_15 @ ( Z_36 = X_83 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_36 = Y_36 ) ) ) ) ) )
& ( ~ P_15
=> ( Y_36
= ( the_int
@ ^ [Z_36: int] : ( (&) @ ( (=>) @ P_15 @ ( Z_36 = X_83 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_36 = Y_36 ) ) ) ) ) ) ) ).
thf(fact_255_If__def,axiom,
! [X_83: hoare_2091234717iple_a,Y_36: hoare_2091234717iple_a,P_15: $o] :
( ( P_15
=> ( X_83
= ( the_Ho1471183438iple_a
@ ^ [Z_36: hoare_2091234717iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_36 = X_83 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_36 = Y_36 ) ) ) ) ) )
& ( ~ P_15
=> ( Y_36
= ( the_Ho1471183438iple_a
@ ^ [Z_36: hoare_2091234717iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_36 = X_83 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_36 = Y_36 ) ) ) ) ) ) ) ).
thf(fact_256_If__def,axiom,
! [X_83: nat,Y_36: nat,P_15: $o] :
( ( P_15
=> ( X_83
= ( the_nat
@ ^ [Z_36: nat] : ( (&) @ ( (=>) @ P_15 @ ( Z_36 = X_83 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_36 = Y_36 ) ) ) ) ) )
& ( ~ P_15
=> ( Y_36
= ( the_nat
@ ^ [Z_36: nat] : ( (&) @ ( (=>) @ P_15 @ ( Z_36 = X_83 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_36 = Y_36 ) ) ) ) ) ) ) ).
thf(fact_257_image__cong,axiom,
! [F_86: hoare_2091234717iple_a > hoare_2091234717iple_a,G_22: hoare_2091234717iple_a > hoare_2091234717iple_a,M_5: hoare_2091234717iple_a > $o,N_6: hoare_2091234717iple_a > $o] :
( ( M_5 = N_6 )
=> ( ! [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ N_6 )
=> ( ( F_86 @ X_1 )
= ( G_22 @ X_1 ) ) )
=> ( ( image_1661191109iple_a @ F_86 @ M_5 )
= ( image_1661191109iple_a @ G_22 @ N_6 ) ) ) ) ).
thf(fact_258_image__cong,axiom,
! [F_86: int > int,G_22: int > int,M_5: int > $o,N_6: int > $o] :
( ( M_5 = N_6 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ N_6 )
=> ( ( F_86 @ X_1 )
= ( G_22 @ X_1 ) ) )
=> ( ( image_int_int @ F_86 @ M_5 )
= ( image_int_int @ G_22 @ N_6 ) ) ) ) ).
thf(fact_259_image__cong,axiom,
! [F_86: nat > nat,G_22: nat > nat,M_5: nat > $o,N_6: nat > $o] :
( ( M_5 = N_6 )
=> ( ! [X_1: nat] :
( ( member_nat @ X_1 @ N_6 )
=> ( ( F_86 @ X_1 )
= ( G_22 @ X_1 ) ) )
=> ( ( image_nat_nat @ F_86 @ M_5 )
= ( image_nat_nat @ G_22 @ N_6 ) ) ) ) ).
thf(fact_260_image__cong,axiom,
! [F_86: nat > int,G_22: nat > int,M_5: nat > $o,N_6: nat > $o] :
( ( M_5 = N_6 )
=> ( ! [X_1: nat] :
( ( member_nat @ X_1 @ N_6 )
=> ( ( F_86 @ X_1 )
= ( G_22 @ X_1 ) ) )
=> ( ( image_nat_int @ F_86 @ M_5 )
= ( image_nat_int @ G_22 @ N_6 ) ) ) ) ).
thf(fact_261_fold1Set_Ointros,axiom,
! [F_85: nat > nat > nat,A_198: nat,A_197: nat > $o,X_82: nat] :
( ( finite929467206at_nat @ F_85 @ A_198 @ A_197 @ X_82 )
=> ( ~ ( member_nat @ A_198 @ A_197 )
=> ( finite_fold1Set_nat @ F_85 @ ( insert_nat @ A_198 @ A_197 ) @ X_82 ) ) ) ).
thf(fact_262_fold1Set_Ointros,axiom,
! [F_85: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A_198: hoare_2091234717iple_a,A_197: hoare_2091234717iple_a > $o,X_82: hoare_2091234717iple_a] :
( ( finite1218641926iple_a @ F_85 @ A_198 @ A_197 @ X_82 )
=> ( ~ ( member290856304iple_a @ A_198 @ A_197 )
=> ( finite114877549iple_a @ F_85 @ ( insert1597628439iple_a @ A_198 @ A_197 ) @ X_82 ) ) ) ).
thf(fact_263_fold1Set_Ointros,axiom,
! [F_85: int > int > int,A_198: int,A_197: int > $o,X_82: int] :
( ( finite772772422nt_int @ F_85 @ A_198 @ A_197 @ X_82 )
=> ( ~ ( member_int @ A_198 @ A_197 )
=> ( finite_fold1Set_int @ F_85 @ ( insert_int @ A_198 @ A_197 ) @ X_82 ) ) ) ).
thf(fact_264_the__equality,axiom,
! [P_14: int > $o,A_196: int] :
( ( P_14 @ A_196 )
=> ( ! [X_1: int] :
( ( P_14 @ X_1 )
=> ( X_1 = A_196 ) )
=> ( ( the_int @ P_14 )
= A_196 ) ) ) ).
thf(fact_265_the__equality,axiom,
! [P_14: hoare_2091234717iple_a > $o,A_196: hoare_2091234717iple_a] :
( ( P_14 @ A_196 )
=> ( ! [X_1: hoare_2091234717iple_a] :
( ( P_14 @ X_1 )
=> ( X_1 = A_196 ) )
=> ( ( the_Ho1471183438iple_a @ P_14 )
= A_196 ) ) ) ).
thf(fact_266_the__equality,axiom,
! [P_14: nat > $o,A_196: nat] :
( ( P_14 @ A_196 )
=> ( ! [X_1: nat] :
( ( P_14 @ X_1 )
=> ( X_1 = A_196 ) )
=> ( ( the_nat @ P_14 )
= A_196 ) ) ) ).
thf(fact_267_folding__one_Oinsert,axiom,
! [X_81: nat,A_195: nat > $o,F_84: nat > nat > nat,F_83: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_84 @ F_83 )
=> ( ( finite_finite_nat @ A_195 )
=> ( ~ ( member_nat @ X_81 @ A_195 )
=> ( ( A_195 != bot_bot_nat_o )
=> ( ( F_83 @ ( insert_nat @ X_81 @ A_195 ) )
= ( F_84 @ X_81 @ ( F_83 @ A_195 ) ) ) ) ) ) ) ).
thf(fact_268_folding__one_Oinsert,axiom,
! [X_81: hoare_2091234717iple_a,A_195: hoare_2091234717iple_a > $o,F_84: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_83: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_84 @ F_83 )
=> ( ( finite232261744iple_a @ A_195 )
=> ( ~ ( member290856304iple_a @ X_81 @ A_195 )
=> ( ( A_195 != bot_bo1791335050le_a_o )
=> ( ( F_83 @ ( insert1597628439iple_a @ X_81 @ A_195 ) )
= ( F_84 @ X_81 @ ( F_83 @ A_195 ) ) ) ) ) ) ) ).
thf(fact_269_folding__one_Oinsert,axiom,
! [X_81: int,A_195: int > $o,F_84: int > int > int,F_83: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_84 @ F_83 )
=> ( ( finite_finite_int @ A_195 )
=> ( ~ ( member_int @ X_81 @ A_195 )
=> ( ( A_195 != bot_bot_int_o )
=> ( ( F_83 @ ( insert_int @ X_81 @ A_195 ) )
= ( F_84 @ X_81 @ ( F_83 @ A_195 ) ) ) ) ) ) ) ).
thf(fact_270_folding__one_Oeq__fold,axiom,
! [A_194: hoare_2091234717iple_a > $o,F_82: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_81: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_82 @ F_81 )
=> ( ( finite232261744iple_a @ A_194 )
=> ( ( F_81 @ A_194 )
= ( finite2106937597iple_a @ F_82 @ A_194 ) ) ) ) ).
thf(fact_271_folding__one_Oeq__fold,axiom,
! [A_194: nat > $o,F_82: nat > nat > nat,F_81: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_82 @ F_81 )
=> ( ( finite_finite_nat @ A_194 )
=> ( ( F_81 @ A_194 )
= ( finite_fold1_nat @ F_82 @ A_194 ) ) ) ) ).
thf(fact_272_folding__one_Oeq__fold,axiom,
! [A_194: int > $o,F_82: int > int > int,F_81: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_82 @ F_81 )
=> ( ( finite_finite_int @ A_194 )
=> ( ( F_81 @ A_194 )
= ( finite_fold1_int @ F_82 @ A_194 ) ) ) ) ).
thf(fact_273_folding__one_Oclosed,axiom,
! [A_193: nat > $o,F_80: nat > nat > nat,F_79: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_80 @ F_79 )
=> ( ( finite_finite_nat @ A_193 )
=> ( ( A_193 != bot_bot_nat_o )
=> ( ! [X_1: nat,Y_28: nat] : ( member_nat @ ( F_80 @ X_1 @ Y_28 ) @ ( insert_nat @ X_1 @ ( insert_nat @ Y_28 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( F_79 @ A_193 ) @ A_193 ) ) ) ) ) ).
thf(fact_274_folding__one_Oclosed,axiom,
! [A_193: hoare_2091234717iple_a > $o,F_80: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_79: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_80 @ F_79 )
=> ( ( finite232261744iple_a @ A_193 )
=> ( ( A_193 != bot_bo1791335050le_a_o )
=> ( ! [X_1: hoare_2091234717iple_a,Y_28: hoare_2091234717iple_a] : ( member290856304iple_a @ ( F_80 @ X_1 @ Y_28 ) @ ( insert1597628439iple_a @ X_1 @ ( insert1597628439iple_a @ Y_28 @ bot_bo1791335050le_a_o ) ) )
=> ( member290856304iple_a @ ( F_79 @ A_193 ) @ A_193 ) ) ) ) ) ).
thf(fact_275_folding__one_Oclosed,axiom,
! [A_193: int > $o,F_80: int > int > int,F_79: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_80 @ F_79 )
=> ( ( finite_finite_int @ A_193 )
=> ( ( A_193 != bot_bot_int_o )
=> ( ! [X_1: int,Y_28: int] : ( member_int @ ( F_80 @ X_1 @ Y_28 ) @ ( insert_int @ X_1 @ ( insert_int @ Y_28 @ bot_bot_int_o ) ) )
=> ( member_int @ ( F_79 @ A_193 ) @ A_193 ) ) ) ) ) ).
thf(fact_276_finite__Collect__conjI,axiom,
! [Q_6: hoare_2091234717iple_a > $o,P_13: hoare_2091234717iple_a > $o] :
( ( ( finite232261744iple_a @ ( collec992574898iple_a @ P_13 ) )
| ( finite232261744iple_a @ ( collec992574898iple_a @ Q_6 ) ) )
=> ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_6 @ X_1 ) ) ) ) ) ).
thf(fact_277_finite__Collect__conjI,axiom,
! [Q_6: int > $o,P_13: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P_13 ) )
| ( finite_finite_int @ ( collect_int @ Q_6 ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_6 @ X_1 ) ) ) ) ) ).
thf(fact_278_finite__Collect__conjI,axiom,
! [Q_6: nat > $o,P_13: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P_13 ) )
| ( finite_finite_nat @ ( collect_nat @ Q_6 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_6 @ X_1 ) ) ) ) ) ).
thf(fact_279_finite_OemptyI,axiom,
finite232261744iple_a @ bot_bo1791335050le_a_o ).
thf(fact_280_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_nat_o ).
thf(fact_281_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_int_o ).
thf(fact_282_finite_OinsertI,axiom,
! [A_192: nat,A_191: nat > $o] :
( ( finite_finite_nat @ A_191 )
=> ( finite_finite_nat @ ( insert_nat @ A_192 @ A_191 ) ) ) ).
thf(fact_283_finite_OinsertI,axiom,
! [A_192: hoare_2091234717iple_a,A_191: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_191 )
=> ( finite232261744iple_a @ ( insert1597628439iple_a @ A_192 @ A_191 ) ) ) ).
thf(fact_284_finite_OinsertI,axiom,
! [A_192: int,A_191: int > $o] :
( ( finite_finite_int @ A_191 )
=> ( finite_finite_int @ ( insert_int @ A_192 @ A_191 ) ) ) ).
thf(fact_285_finite__imageI,axiom,
! [H_5: hoare_2091234717iple_a > hoare_2091234717iple_a,F_78: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_78 )
=> ( finite232261744iple_a @ ( image_1661191109iple_a @ H_5 @ F_78 ) ) ) ).
thf(fact_286_finite__imageI,axiom,
! [H_5: nat > hoare_2091234717iple_a,F_78: nat > $o] :
( ( finite_finite_nat @ F_78 )
=> ( finite232261744iple_a @ ( image_359186840iple_a @ H_5 @ F_78 ) ) ) ).
thf(fact_287_finite__imageI,axiom,
! [H_5: int > hoare_2091234717iple_a,F_78: int > $o] :
( ( finite_finite_int @ F_78 )
=> ( finite232261744iple_a @ ( image_338319932iple_a @ H_5 @ F_78 ) ) ) ).
thf(fact_288_finite__imageI,axiom,
! [H_5: hoare_2091234717iple_a > nat,F_78: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_78 )
=> ( finite_finite_nat @ ( image_1773322034_a_nat @ H_5 @ F_78 ) ) ) ).
thf(fact_289_finite__imageI,axiom,
! [H_5: int > int,F_78: int > $o] :
( ( finite_finite_int @ F_78 )
=> ( finite_finite_int @ ( image_int_int @ H_5 @ F_78 ) ) ) ).
thf(fact_290_finite__imageI,axiom,
! [H_5: hoare_2091234717iple_a > int,F_78: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_78 )
=> ( finite_finite_int @ ( image_263112078_a_int @ H_5 @ F_78 ) ) ) ).
thf(fact_291_finite__imageI,axiom,
! [H_5: nat > nat,F_78: nat > $o] :
( ( finite_finite_nat @ F_78 )
=> ( finite_finite_nat @ ( image_nat_nat @ H_5 @ F_78 ) ) ) ).
thf(fact_292_finite__imageI,axiom,
! [H_5: nat > int,F_78: nat > $o] :
( ( finite_finite_nat @ F_78 )
=> ( finite_finite_int @ ( image_nat_int @ H_5 @ F_78 ) ) ) ).
thf(fact_293_finite__Collect__disjI,axiom,
! [P_12: hoare_2091234717iple_a > $o,Q_5: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_5 @ X_1 ) ) ) )
<=> ( ( finite232261744iple_a @ ( collec992574898iple_a @ P_12 ) )
& ( finite232261744iple_a @ ( collec992574898iple_a @ Q_5 ) ) ) ) ).
thf(fact_294_finite__Collect__disjI,axiom,
! [P_12: int > $o,Q_5: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_5 @ X_1 ) ) ) )
<=> ( ( finite_finite_int @ ( collect_int @ P_12 ) )
& ( finite_finite_int @ ( collect_int @ Q_5 ) ) ) ) ).
thf(fact_295_finite__Collect__disjI,axiom,
! [P_12: nat > $o,Q_5: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X_1: nat] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_5 @ X_1 ) ) ) )
<=> ( ( finite_finite_nat @ ( collect_nat @ P_12 ) )
& ( finite_finite_nat @ ( collect_nat @ Q_5 ) ) ) ) ).
thf(fact_296_finite__insert,axiom,
! [A_190: nat,A_189: nat > $o] :
( ( finite_finite_nat @ ( insert_nat @ A_190 @ A_189 ) )
<=> ( finite_finite_nat @ A_189 ) ) ).
thf(fact_297_finite__insert,axiom,
! [A_190: hoare_2091234717iple_a,A_189: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ ( insert1597628439iple_a @ A_190 @ A_189 ) )
<=> ( finite232261744iple_a @ A_189 ) ) ).
thf(fact_298_finite__insert,axiom,
! [A_190: int,A_189: int > $o] :
( ( finite_finite_int @ ( insert_int @ A_190 @ A_189 ) )
<=> ( finite_finite_int @ A_189 ) ) ).
thf(fact_299_fold__graph_OemptyI,axiom,
! [F_77: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,Z_35: hoare_2091234717iple_a] : ( finite1218641926iple_a @ F_77 @ Z_35 @ bot_bo1791335050le_a_o @ Z_35 ) ).
thf(fact_300_fold__graph_OemptyI,axiom,
! [F_77: int > int > int,Z_35: int] : ( finite772772422nt_int @ F_77 @ Z_35 @ bot_bot_int_o @ Z_35 ) ).
thf(fact_301_fold__graph_OemptyI,axiom,
! [F_77: nat > nat > nat,Z_35: nat] : ( finite929467206at_nat @ F_77 @ Z_35 @ bot_bot_nat_o @ Z_35 ) ).
thf(fact_302_empty__fold__graphE,axiom,
! [F_76: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,Z_34: hoare_2091234717iple_a,X_80: hoare_2091234717iple_a] :
( ( finite1218641926iple_a @ F_76 @ Z_34 @ bot_bo1791335050le_a_o @ X_80 )
=> ( X_80 = Z_34 ) ) ).
thf(fact_303_empty__fold__graphE,axiom,
! [F_76: int > int > int,Z_34: int,X_80: int] :
( ( finite772772422nt_int @ F_76 @ Z_34 @ bot_bot_int_o @ X_80 )
=> ( X_80 = Z_34 ) ) ).
thf(fact_304_empty__fold__graphE,axiom,
! [F_76: nat > nat > nat,Z_34: nat,X_80: nat] :
( ( finite929467206at_nat @ F_76 @ Z_34 @ bot_bot_nat_o @ X_80 )
=> ( X_80 = Z_34 ) ) ).
thf(fact_305_fold__graph_OinsertI,axiom,
! [F_75: nat > nat > nat,Z_33: nat,Y_35: nat,X_79: nat,A_188: nat > $o] :
( ~ ( member_nat @ X_79 @ A_188 )
=> ( ( finite929467206at_nat @ F_75 @ Z_33 @ A_188 @ Y_35 )
=> ( finite929467206at_nat @ F_75 @ Z_33 @ ( insert_nat @ X_79 @ A_188 ) @ ( F_75 @ X_79 @ Y_35 ) ) ) ) ).
thf(fact_306_fold__graph_OinsertI,axiom,
! [F_75: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,Z_33: hoare_2091234717iple_a,Y_35: hoare_2091234717iple_a,X_79: hoare_2091234717iple_a,A_188: hoare_2091234717iple_a > $o] :
( ~ ( member290856304iple_a @ X_79 @ A_188 )
=> ( ( finite1218641926iple_a @ F_75 @ Z_33 @ A_188 @ Y_35 )
=> ( finite1218641926iple_a @ F_75 @ Z_33 @ ( insert1597628439iple_a @ X_79 @ A_188 ) @ ( F_75 @ X_79 @ Y_35 ) ) ) ) ).
thf(fact_307_fold__graph_OinsertI,axiom,
! [F_75: int > int > int,Z_33: int,Y_35: int,X_79: int,A_188: int > $o] :
( ~ ( member_int @ X_79 @ A_188 )
=> ( ( finite772772422nt_int @ F_75 @ Z_33 @ A_188 @ Y_35 )
=> ( finite772772422nt_int @ F_75 @ Z_33 @ ( insert_int @ X_79 @ A_188 ) @ ( F_75 @ X_79 @ Y_35 ) ) ) ) ).
thf(fact_308_insert__fold1SetE,axiom,
! [F_74: nat > nat > nat,A_187: nat,X_78: nat > $o,X_77: nat] :
( ( finite_fold1Set_nat @ F_74 @ ( insert_nat @ A_187 @ X_78 ) @ X_77 )
=> ~ ! [A_19: nat,A_134: nat > $o] :
( ( ( insert_nat @ A_187 @ X_78 )
= ( insert_nat @ A_19 @ A_134 ) )
=> ( ( finite929467206at_nat @ F_74 @ A_19 @ A_134 @ X_77 )
=> ( member_nat @ A_19 @ A_134 ) ) ) ) ).
thf(fact_309_insert__fold1SetE,axiom,
! [F_74: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A_187: hoare_2091234717iple_a,X_78: hoare_2091234717iple_a > $o,X_77: hoare_2091234717iple_a] :
( ( finite114877549iple_a @ F_74 @ ( insert1597628439iple_a @ A_187 @ X_78 ) @ X_77 )
=> ~ ! [A_19: hoare_2091234717iple_a,A_134: hoare_2091234717iple_a > $o] :
( ( ( insert1597628439iple_a @ A_187 @ X_78 )
= ( insert1597628439iple_a @ A_19 @ A_134 ) )
=> ( ( finite1218641926iple_a @ F_74 @ A_19 @ A_134 @ X_77 )
=> ( member290856304iple_a @ A_19 @ A_134 ) ) ) ) ).
thf(fact_310_insert__fold1SetE,axiom,
! [F_74: int > int > int,A_187: int,X_78: int > $o,X_77: int] :
( ( finite_fold1Set_int @ F_74 @ ( insert_int @ A_187 @ X_78 ) @ X_77 )
=> ~ ! [A_19: int,A_134: int > $o] :
( ( ( insert_int @ A_187 @ X_78 )
= ( insert_int @ A_19 @ A_134 ) )
=> ( ( finite772772422nt_int @ F_74 @ A_19 @ A_134 @ X_77 )
=> ( member_int @ A_19 @ A_134 ) ) ) ) ).
thf(fact_311_finite__nonempty__imp__fold1Set,axiom,
! [F_73: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A_186: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_186 )
=> ( ( A_186 != bot_bo1791335050le_a_o )
=> ( ?? @ hoare_2091234717iple_a @ ( finite114877549iple_a @ F_73 @ A_186 ) ) ) ) ).
thf(fact_312_finite__nonempty__imp__fold1Set,axiom,
! [F_73: nat > nat > nat,A_186: nat > $o] :
( ( finite_finite_nat @ A_186 )
=> ( ( A_186 != bot_bot_nat_o )
=> ( ?? @ nat @ ( finite_fold1Set_nat @ F_73 @ A_186 ) ) ) ) ).
thf(fact_313_finite__nonempty__imp__fold1Set,axiom,
! [F_73: int > int > int,A_186: int > $o] :
( ( finite_finite_int @ A_186 )
=> ( ( A_186 != bot_bot_int_o )
=> ( ?? @ int @ ( finite_fold1Set_int @ F_73 @ A_186 ) ) ) ) ).
thf(fact_314_finite_Osimps,axiom,
! [A_185: nat > $o] :
( ( finite_finite_nat @ A_185 )
<=> ( ( A_185 = bot_bot_nat_o )
| ? [A_134: nat > $o,A_19: nat] :
( ( A_185
= ( insert_nat @ A_19 @ A_134 ) )
& ( finite_finite_nat @ A_134 ) ) ) ) ).
thf(fact_315_finite_Osimps,axiom,
! [A_185: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_185 )
<=> ( ( A_185 = bot_bo1791335050le_a_o )
| ? [A_134: hoare_2091234717iple_a > $o,A_19: hoare_2091234717iple_a] :
( ( A_185
= ( insert1597628439iple_a @ A_19 @ A_134 ) )
& ( finite232261744iple_a @ A_134 ) ) ) ) ).
thf(fact_316_finite_Osimps,axiom,
! [A_185: int > $o] :
( ( finite_finite_int @ A_185 )
<=> ( ( A_185 = bot_bot_int_o )
| ? [A_134: int > $o,A_19: int] :
( ( A_185
= ( insert_int @ A_19 @ A_134 ) )
& ( finite_finite_int @ A_134 ) ) ) ) ).
thf(fact_317_finite__induct,axiom,
! [P_11: ( nat > $o ) > $o,F_72: nat > $o] :
( ( finite_finite_nat @ F_72 )
=> ( ( P_11 @ bot_bot_nat_o )
=> ( ! [X_1: nat,F_6: nat > $o] :
( ( finite_finite_nat @ F_6 )
=> ( ~ ( member_nat @ X_1 @ F_6 )
=> ( ( P_11 @ F_6 )
=> ( P_11 @ ( insert_nat @ X_1 @ F_6 ) ) ) ) )
=> ( P_11 @ F_72 ) ) ) ) ).
thf(fact_318_finite__induct,axiom,
! [P_11: ( hoare_2091234717iple_a > $o ) > $o,F_72: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_72 )
=> ( ( P_11 @ bot_bo1791335050le_a_o )
=> ( ! [X_1: hoare_2091234717iple_a,F_6: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_6 )
=> ( ~ ( member290856304iple_a @ X_1 @ F_6 )
=> ( ( P_11 @ F_6 )
=> ( P_11 @ ( insert1597628439iple_a @ X_1 @ F_6 ) ) ) ) )
=> ( P_11 @ F_72 ) ) ) ) ).
thf(fact_319_finite__induct,axiom,
! [P_11: ( int > $o ) > $o,F_72: int > $o] :
( ( finite_finite_int @ F_72 )
=> ( ( P_11 @ bot_bot_int_o )
=> ( ! [X_1: int,F_6: int > $o] :
( ( finite_finite_int @ F_6 )
=> ( ~ ( member_int @ X_1 @ F_6 )
=> ( ( P_11 @ F_6 )
=> ( P_11 @ ( insert_int @ X_1 @ F_6 ) ) ) ) )
=> ( P_11 @ F_72 ) ) ) ) ).
thf(fact_320_pigeonhole__infinite,axiom,
! [F_71: hoare_2091234717iple_a > hoare_2091234717iple_a,A_184: hoare_2091234717iple_a > $o] :
( ~ ( finite232261744iple_a @ A_184 )
=> ( ( finite232261744iple_a @ ( image_1661191109iple_a @ F_71 @ A_184 ) )
=> ? [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_184 )
& ~ ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [A_19: hoare_2091234717iple_a] :
( (&) @ ( member290856304iple_a @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_321_pigeonhole__infinite,axiom,
! [F_71: hoare_2091234717iple_a > nat,A_184: hoare_2091234717iple_a > $o] :
( ~ ( finite232261744iple_a @ A_184 )
=> ( ( finite_finite_nat @ ( image_1773322034_a_nat @ F_71 @ A_184 ) )
=> ? [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_184 )
& ~ ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [A_19: hoare_2091234717iple_a] :
( (&) @ ( member290856304iple_a @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_322_pigeonhole__infinite,axiom,
! [F_71: int > nat,A_184: int > $o] :
( ~ ( finite_finite_int @ A_184 )
=> ( ( finite_finite_nat @ ( image_int_nat @ F_71 @ A_184 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_184 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_19: int] :
( (&) @ ( member_int @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_323_pigeonhole__infinite,axiom,
! [F_71: hoare_2091234717iple_a > int,A_184: hoare_2091234717iple_a > $o] :
( ~ ( finite232261744iple_a @ A_184 )
=> ( ( finite_finite_int @ ( image_263112078_a_int @ F_71 @ A_184 ) )
=> ? [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_184 )
& ~ ( finite232261744iple_a
@ ( collec992574898iple_a
@ ^ [A_19: hoare_2091234717iple_a] :
( (&) @ ( member290856304iple_a @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_324_pigeonhole__infinite,axiom,
! [F_71: nat > hoare_2091234717iple_a,A_184: nat > $o] :
( ~ ( finite_finite_nat @ A_184 )
=> ( ( finite232261744iple_a @ ( image_359186840iple_a @ F_71 @ A_184 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_184 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_19: nat] :
( (&) @ ( member_nat @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_325_pigeonhole__infinite,axiom,
! [F_71: int > int,A_184: int > $o] :
( ~ ( finite_finite_int @ A_184 )
=> ( ( finite_finite_int @ ( image_int_int @ F_71 @ A_184 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_184 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_19: int] :
( (&) @ ( member_int @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_326_pigeonhole__infinite,axiom,
! [F_71: int > hoare_2091234717iple_a,A_184: int > $o] :
( ~ ( finite_finite_int @ A_184 )
=> ( ( finite232261744iple_a @ ( image_338319932iple_a @ F_71 @ A_184 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_184 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_19: int] :
( (&) @ ( member_int @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_327_pigeonhole__infinite,axiom,
! [F_71: nat > nat,A_184: nat > $o] :
( ~ ( finite_finite_nat @ A_184 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F_71 @ A_184 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_184 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_19: nat] :
( (&) @ ( member_nat @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_328_pigeonhole__infinite,axiom,
! [F_71: nat > int,A_184: nat > $o] :
( ~ ( finite_finite_nat @ A_184 )
=> ( ( finite_finite_int @ ( image_nat_int @ F_71 @ A_184 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_184 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_19: nat] :
( (&) @ ( member_nat @ A_19 @ A_184 )
@ ( ( F_71 @ A_19 )
= ( F_71 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_329_finite__imp__fold__graph,axiom,
! [F_70: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,Z_32: hoare_2091234717iple_a,A_183: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_183 )
=> ( ?? @ hoare_2091234717iple_a @ ( finite1218641926iple_a @ F_70 @ Z_32 @ A_183 ) ) ) ).
thf(fact_330_finite__imp__fold__graph,axiom,
! [F_70: nat > nat > nat,Z_32: nat,A_183: nat > $o] :
( ( finite_finite_nat @ A_183 )
=> ( ?? @ nat @ ( finite929467206at_nat @ F_70 @ Z_32 @ A_183 ) ) ) ).
thf(fact_331_finite__imp__fold__graph,axiom,
! [F_70: int > int > int,Z_32: int,A_183: int > $o] :
( ( finite_finite_int @ A_183 )
=> ( ?? @ int @ ( finite772772422nt_int @ F_70 @ Z_32 @ A_183 ) ) ) ).
thf(fact_332_fold1Set_Osimps,axiom,
! [F_69: nat > nat > nat,A1_1: nat > $o,A2_1: nat] :
( ( finite_fold1Set_nat @ F_69 @ A1_1 @ A2_1 )
<=> ? [A_19: nat,A_134: nat > $o,X_1: nat] :
( ( A1_1
= ( insert_nat @ A_19 @ A_134 ) )
& ( A2_1 = X_1 )
& ( finite929467206at_nat @ F_69 @ A_19 @ A_134 @ X_1 )
& ~ ( member_nat @ A_19 @ A_134 ) ) ) ).
thf(fact_333_fold1Set_Osimps,axiom,
! [F_69: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,A1_1: hoare_2091234717iple_a > $o,A2_1: hoare_2091234717iple_a] :
( ( finite114877549iple_a @ F_69 @ A1_1 @ A2_1 )
<=> ? [A_19: hoare_2091234717iple_a,A_134: hoare_2091234717iple_a > $o,X_1: hoare_2091234717iple_a] :
( ( A1_1
= ( insert1597628439iple_a @ A_19 @ A_134 ) )
& ( A2_1 = X_1 )
& ( finite1218641926iple_a @ F_69 @ A_19 @ A_134 @ X_1 )
& ~ ( member290856304iple_a @ A_19 @ A_134 ) ) ) ).
thf(fact_334_fold1Set_Osimps,axiom,
! [F_69: int > int > int,A1_1: int > $o,A2_1: int] :
( ( finite_fold1Set_int @ F_69 @ A1_1 @ A2_1 )
<=> ? [A_19: int,A_134: int > $o,X_1: int] :
( ( A1_1
= ( insert_int @ A_19 @ A_134 ) )
& ( A2_1 = X_1 )
& ( finite772772422nt_int @ F_69 @ A_19 @ A_134 @ X_1 )
& ~ ( member_int @ A_19 @ A_134 ) ) ) ).
thf(fact_335_theI,axiom,
! [P_10: int > $o,A_182: int] :
( ( P_10 @ A_182 )
=> ( ! [X_1: int] :
( ( P_10 @ X_1 )
=> ( X_1 = A_182 ) )
=> ( P_10 @ ( the_int @ P_10 ) ) ) ) ).
thf(fact_336_theI,axiom,
! [P_10: hoare_2091234717iple_a > $o,A_182: hoare_2091234717iple_a] :
( ( P_10 @ A_182 )
=> ( ! [X_1: hoare_2091234717iple_a] :
( ( P_10 @ X_1 )
=> ( X_1 = A_182 ) )
=> ( P_10 @ ( the_Ho1471183438iple_a @ P_10 ) ) ) ) ).
thf(fact_337_theI,axiom,
! [P_10: nat > $o,A_182: nat] :
( ( P_10 @ A_182 )
=> ( ! [X_1: nat] :
( ( P_10 @ X_1 )
=> ( X_1 = A_182 ) )
=> ( P_10 @ ( the_nat @ P_10 ) ) ) ) ).
thf(fact_338_the1__equality,axiom,
! [A_181: int,P_9: int > $o] :
( ? [X_1: int] :
( ( P_9 @ X_1 )
& ! [Y_28: int] :
( ( P_9 @ Y_28 )
=> ( Y_28 = X_1 ) ) )
=> ( ( P_9 @ A_181 )
=> ( ( the_int @ P_9 )
= A_181 ) ) ) ).
thf(fact_339_the1__equality,axiom,
! [A_181: hoare_2091234717iple_a,P_9: hoare_2091234717iple_a > $o] :
( ? [X_1: hoare_2091234717iple_a] :
( ( P_9 @ X_1 )
& ! [Y_28: hoare_2091234717iple_a] :
( ( P_9 @ Y_28 )
=> ( Y_28 = X_1 ) ) )
=> ( ( P_9 @ A_181 )
=> ( ( the_Ho1471183438iple_a @ P_9 )
= A_181 ) ) ) ).
thf(fact_340_the1__equality,axiom,
! [A_181: nat,P_9: nat > $o] :
( ? [X_1: nat] :
( ( P_9 @ X_1 )
& ! [Y_28: nat] :
( ( P_9 @ Y_28 )
=> ( Y_28 = X_1 ) ) )
=> ( ( P_9 @ A_181 )
=> ( ( the_nat @ P_9 )
= A_181 ) ) ) ).
thf(fact_341_theI_H,axiom,
! [P_8: int > $o] :
( ? [X_1: int] :
( ( P_8 @ X_1 )
& ! [Y_28: int] :
( ( P_8 @ Y_28 )
=> ( Y_28 = X_1 ) ) )
=> ( P_8 @ ( the_int @ P_8 ) ) ) ).
thf(fact_342_theI_H,axiom,
! [P_8: hoare_2091234717iple_a > $o] :
( ? [X_1: hoare_2091234717iple_a] :
( ( P_8 @ X_1 )
& ! [Y_28: hoare_2091234717iple_a] :
( ( P_8 @ Y_28 )
=> ( Y_28 = X_1 ) ) )
=> ( P_8 @ ( the_Ho1471183438iple_a @ P_8 ) ) ) ).
thf(fact_343_theI_H,axiom,
! [P_8: nat > $o] :
( ? [X_1: nat] :
( ( P_8 @ X_1 )
& ! [Y_28: nat] :
( ( P_8 @ Y_28 )
=> ( Y_28 = X_1 ) ) )
=> ( P_8 @ ( the_nat @ P_8 ) ) ) ).
thf(fact_344_fold__graph_Osimps,axiom,
! [F_68: nat > nat > nat,Z_31: nat,A1: nat > $o,A2: nat] :
( ( finite929467206at_nat @ F_68 @ Z_31 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bot_nat_o )
& ( A2 = Z_31 ) )
| ? [X_1: nat,A_134: nat > $o,Y_28: nat] :
( ( A1
= ( insert_nat @ X_1 @ A_134 ) )
& ( A2
= ( F_68 @ X_1 @ Y_28 ) )
& ~ ( member_nat @ X_1 @ A_134 )
& ( finite929467206at_nat @ F_68 @ Z_31 @ A_134 @ Y_28 ) ) ) ) ).
thf(fact_345_fold__graph_Osimps,axiom,
! [F_68: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,Z_31: hoare_2091234717iple_a,A1: hoare_2091234717iple_a > $o,A2: hoare_2091234717iple_a] :
( ( finite1218641926iple_a @ F_68 @ Z_31 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bo1791335050le_a_o )
& ( A2 = Z_31 ) )
| ? [X_1: hoare_2091234717iple_a,A_134: hoare_2091234717iple_a > $o,Y_28: hoare_2091234717iple_a] :
( ( A1
= ( insert1597628439iple_a @ X_1 @ A_134 ) )
& ( A2
= ( F_68 @ X_1 @ Y_28 ) )
& ~ ( member290856304iple_a @ X_1 @ A_134 )
& ( finite1218641926iple_a @ F_68 @ Z_31 @ A_134 @ Y_28 ) ) ) ) ).
thf(fact_346_fold__graph_Osimps,axiom,
! [F_68: int > int > int,Z_31: int,A1: int > $o,A2: int] :
( ( finite772772422nt_int @ F_68 @ Z_31 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bot_int_o )
& ( A2 = Z_31 ) )
| ? [X_1: int,A_134: int > $o,Y_28: int] :
( ( A1
= ( insert_int @ X_1 @ A_134 ) )
& ( A2
= ( F_68 @ X_1 @ Y_28 ) )
& ~ ( member_int @ X_1 @ A_134 )
& ( finite772772422nt_int @ F_68 @ Z_31 @ A_134 @ Y_28 ) ) ) ) ).
thf(fact_347_folding__one__idem_Oinsert__idem,axiom,
! [X_76: nat,A_180: nat > $o,F_67: nat > nat > nat,F_66: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_67 @ F_66 )
=> ( ( finite_finite_nat @ A_180 )
=> ( ( A_180 != bot_bot_nat_o )
=> ( ( F_66 @ ( insert_nat @ X_76 @ A_180 ) )
= ( F_67 @ X_76 @ ( F_66 @ A_180 ) ) ) ) ) ) ).
thf(fact_348_folding__one__idem_Oinsert__idem,axiom,
! [X_76: hoare_2091234717iple_a,A_180: hoare_2091234717iple_a > $o,F_67: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_66: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_67 @ F_66 )
=> ( ( finite232261744iple_a @ A_180 )
=> ( ( A_180 != bot_bo1791335050le_a_o )
=> ( ( F_66 @ ( insert1597628439iple_a @ X_76 @ A_180 ) )
= ( F_67 @ X_76 @ ( F_66 @ A_180 ) ) ) ) ) ) ).
thf(fact_349_folding__one__idem_Oinsert__idem,axiom,
! [X_76: int,A_180: int > $o,F_67: int > int > int,F_66: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_67 @ F_66 )
=> ( ( finite_finite_int @ A_180 )
=> ( ( A_180 != bot_bot_int_o )
=> ( ( F_66 @ ( insert_int @ X_76 @ A_180 ) )
= ( F_67 @ X_76 @ ( F_66 @ A_180 ) ) ) ) ) ) ).
thf(fact_350_finite__ne__induct,axiom,
! [P_7: ( nat > $o ) > $o,F_65: nat > $o] :
( ( finite_finite_nat @ F_65 )
=> ( ( F_65 != bot_bot_nat_o )
=> ( ! [X_1: nat] : ( P_7 @ ( insert_nat @ X_1 @ bot_bot_nat_o ) )
=> ( ! [X_1: nat,F_6: nat > $o] :
( ( finite_finite_nat @ F_6 )
=> ( ( F_6 != bot_bot_nat_o )
=> ( ~ ( member_nat @ X_1 @ F_6 )
=> ( ( P_7 @ F_6 )
=> ( P_7 @ ( insert_nat @ X_1 @ F_6 ) ) ) ) ) )
=> ( P_7 @ F_65 ) ) ) ) ) ).
thf(fact_351_finite__ne__induct,axiom,
! [P_7: ( hoare_2091234717iple_a > $o ) > $o,F_65: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_65 )
=> ( ( F_65 != bot_bo1791335050le_a_o )
=> ( ! [X_1: hoare_2091234717iple_a] : ( P_7 @ ( insert1597628439iple_a @ X_1 @ bot_bo1791335050le_a_o ) )
=> ( ! [X_1: hoare_2091234717iple_a,F_6: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_6 )
=> ( ( F_6 != bot_bo1791335050le_a_o )
=> ( ~ ( member290856304iple_a @ X_1 @ F_6 )
=> ( ( P_7 @ F_6 )
=> ( P_7 @ ( insert1597628439iple_a @ X_1 @ F_6 ) ) ) ) ) )
=> ( P_7 @ F_65 ) ) ) ) ) ).
thf(fact_352_finite__ne__induct,axiom,
! [P_7: ( int > $o ) > $o,F_65: int > $o] :
( ( finite_finite_int @ F_65 )
=> ( ( F_65 != bot_bot_int_o )
=> ( ! [X_1: int] : ( P_7 @ ( insert_int @ X_1 @ bot_bot_int_o ) )
=> ( ! [X_1: int,F_6: int > $o] :
( ( finite_finite_int @ F_6 )
=> ( ( F_6 != bot_bot_int_o )
=> ( ~ ( member_int @ X_1 @ F_6 )
=> ( ( P_7 @ F_6 )
=> ( P_7 @ ( insert_int @ X_1 @ F_6 ) ) ) ) ) )
=> ( P_7 @ F_65 ) ) ) ) ) ).
thf(fact_353_semilattice__big_OF__eq,axiom,
! [A_179: hoare_2091234717iple_a > $o,F_64: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_63: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( big_se96866163iple_a @ F_64 @ F_63 )
=> ( ( finite232261744iple_a @ A_179 )
=> ( ( F_63 @ A_179 )
= ( finite2106937597iple_a @ F_64 @ A_179 ) ) ) ) ).
thf(fact_354_semilattice__big_OF__eq,axiom,
! [A_179: nat > $o,F_64: nat > nat > nat,F_63: ( nat > $o ) > nat] :
( ( big_se275732192ig_nat @ F_64 @ F_63 )
=> ( ( finite_finite_nat @ A_179 )
=> ( ( F_63 @ A_179 )
= ( finite_fold1_nat @ F_64 @ A_179 ) ) ) ) ).
thf(fact_355_semilattice__big_OF__eq,axiom,
! [A_179: int > $o,F_64: int > int > int,F_63: ( int > $o ) > int] :
( ( big_se913005884ig_int @ F_64 @ F_63 )
=> ( ( finite_finite_int @ A_179 )
=> ( ( F_63 @ A_179 )
= ( finite_fold1_int @ F_64 @ A_179 ) ) ) ) ).
thf(fact_356_folding__one_Oremove,axiom,
! [X_75: nat,A_178: nat > $o,F_62: nat > nat > nat,F_61: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_62 @ F_61 )
=> ( ( finite_finite_nat @ A_178 )
=> ( ( member_nat @ X_75 @ A_178 )
=> ( ( ( ( minus_minus_nat_o @ A_178 @ ( insert_nat @ X_75 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_61 @ A_178 )
= X_75 ) )
& ( ( ( minus_minus_nat_o @ A_178 @ ( insert_nat @ X_75 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_61 @ A_178 )
= ( F_62 @ X_75 @ ( F_61 @ ( minus_minus_nat_o @ A_178 @ ( insert_nat @ X_75 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_357_folding__one_Oremove,axiom,
! [X_75: hoare_2091234717iple_a,A_178: hoare_2091234717iple_a > $o,F_62: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_61: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_62 @ F_61 )
=> ( ( finite232261744iple_a @ A_178 )
=> ( ( member290856304iple_a @ X_75 @ A_178 )
=> ( ( ( ( minus_836160335le_a_o @ A_178 @ ( insert1597628439iple_a @ X_75 @ bot_bo1791335050le_a_o ) )
= bot_bo1791335050le_a_o )
=> ( ( F_61 @ A_178 )
= X_75 ) )
& ( ( ( minus_836160335le_a_o @ A_178 @ ( insert1597628439iple_a @ X_75 @ bot_bo1791335050le_a_o ) )
!= bot_bo1791335050le_a_o )
=> ( ( F_61 @ A_178 )
= ( F_62 @ X_75 @ ( F_61 @ ( minus_836160335le_a_o @ A_178 @ ( insert1597628439iple_a @ X_75 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_358_folding__one_Oremove,axiom,
! [X_75: int,A_178: int > $o,F_62: int > int > int,F_61: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_62 @ F_61 )
=> ( ( finite_finite_int @ A_178 )
=> ( ( member_int @ X_75 @ A_178 )
=> ( ( ( ( minus_minus_int_o @ A_178 @ ( insert_int @ X_75 @ bot_bot_int_o ) )
= bot_bot_int_o )
=> ( ( F_61 @ A_178 )
= X_75 ) )
& ( ( ( minus_minus_int_o @ A_178 @ ( insert_int @ X_75 @ bot_bot_int_o ) )
!= bot_bot_int_o )
=> ( ( F_61 @ A_178 )
= ( F_62 @ X_75 @ ( F_61 @ ( minus_minus_int_o @ A_178 @ ( insert_int @ X_75 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_359_folding__one_Oinsert__remove,axiom,
! [X_74: nat,A_177: nat > $o,F_60: nat > nat > nat,F_59: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_60 @ F_59 )
=> ( ( finite_finite_nat @ A_177 )
=> ( ( ( ( minus_minus_nat_o @ A_177 @ ( insert_nat @ X_74 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_59 @ ( insert_nat @ X_74 @ A_177 ) )
= X_74 ) )
& ( ( ( minus_minus_nat_o @ A_177 @ ( insert_nat @ X_74 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_59 @ ( insert_nat @ X_74 @ A_177 ) )
= ( F_60 @ X_74 @ ( F_59 @ ( minus_minus_nat_o @ A_177 @ ( insert_nat @ X_74 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
thf(fact_360_folding__one_Oinsert__remove,axiom,
! [X_74: hoare_2091234717iple_a,A_177: hoare_2091234717iple_a > $o,F_60: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_59: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_60 @ F_59 )
=> ( ( finite232261744iple_a @ A_177 )
=> ( ( ( ( minus_836160335le_a_o @ A_177 @ ( insert1597628439iple_a @ X_74 @ bot_bo1791335050le_a_o ) )
= bot_bo1791335050le_a_o )
=> ( ( F_59 @ ( insert1597628439iple_a @ X_74 @ A_177 ) )
= X_74 ) )
& ( ( ( minus_836160335le_a_o @ A_177 @ ( insert1597628439iple_a @ X_74 @ bot_bo1791335050le_a_o ) )
!= bot_bo1791335050le_a_o )
=> ( ( F_59 @ ( insert1597628439iple_a @ X_74 @ A_177 ) )
= ( F_60 @ X_74 @ ( F_59 @ ( minus_836160335le_a_o @ A_177 @ ( insert1597628439iple_a @ X_74 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ) ) ).
thf(fact_361_folding__one_Oinsert__remove,axiom,
! [X_74: int,A_177: int > $o,F_60: int > int > int,F_59: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_60 @ F_59 )
=> ( ( finite_finite_int @ A_177 )
=> ( ( ( ( minus_minus_int_o @ A_177 @ ( insert_int @ X_74 @ bot_bot_int_o ) )
= bot_bot_int_o )
=> ( ( F_59 @ ( insert_int @ X_74 @ A_177 ) )
= X_74 ) )
& ( ( ( minus_minus_int_o @ A_177 @ ( insert_int @ X_74 @ bot_bot_int_o ) )
!= bot_bot_int_o )
=> ( ( F_59 @ ( insert_int @ X_74 @ A_177 ) )
= ( F_60 @ X_74 @ ( F_59 @ ( minus_minus_int_o @ A_177 @ ( insert_int @ X_74 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ).
thf(fact_362_fold1__insert,axiom,
! [X_73: nat,A_176: nat > $o] :
( ( A_176 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ A_176 )
=> ( ~ ( member_nat @ X_73 @ A_176 )
=> ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ X_73 @ A_176 ) )
= ( times_times_nat @ X_73 @ ( finite_fold1_nat @ times_times_nat @ A_176 ) ) ) ) ) ) ).
thf(fact_363_fold1__insert,axiom,
! [X_73: int,A_176: int > $o] :
( ( A_176 != bot_bot_int_o )
=> ( ( finite_finite_int @ A_176 )
=> ( ~ ( member_int @ X_73 @ A_176 )
=> ( ( finite_fold1_int @ times_times_int @ ( insert_int @ X_73 @ A_176 ) )
= ( times_times_int @ X_73 @ ( finite_fold1_int @ times_times_int @ A_176 ) ) ) ) ) ) ).
thf(fact_364_DiffI,axiom,
! [B_102: hoare_2091234717iple_a > $o,C_48: hoare_2091234717iple_a,A_175: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_48 @ A_175 )
=> ( ~ ( member290856304iple_a @ C_48 @ B_102 )
=> ( member290856304iple_a @ C_48 @ ( minus_836160335le_a_o @ A_175 @ B_102 ) ) ) ) ).
thf(fact_365_DiffI,axiom,
! [B_102: nat > $o,C_48: nat,A_175: nat > $o] :
( ( member_nat @ C_48 @ A_175 )
=> ( ~ ( member_nat @ C_48 @ B_102 )
=> ( member_nat @ C_48 @ ( minus_minus_nat_o @ A_175 @ B_102 ) ) ) ) ).
thf(fact_366_DiffI,axiom,
! [B_102: int > $o,C_48: int,A_175: int > $o] :
( ( member_int @ C_48 @ A_175 )
=> ( ~ ( member_int @ C_48 @ B_102 )
=> ( member_int @ C_48 @ ( minus_minus_int_o @ A_175 @ B_102 ) ) ) ) ).
thf(fact_367_DiffE,axiom,
! [C_47: hoare_2091234717iple_a,A_174: hoare_2091234717iple_a > $o,B_101: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_47 @ ( minus_836160335le_a_o @ A_174 @ B_101 ) )
=> ~ ( ( member290856304iple_a @ C_47 @ A_174 )
=> ( member290856304iple_a @ C_47 @ B_101 ) ) ) ).
thf(fact_368_DiffE,axiom,
! [C_47: nat,A_174: nat > $o,B_101: nat > $o] :
( ( member_nat @ C_47 @ ( minus_minus_nat_o @ A_174 @ B_101 ) )
=> ~ ( ( member_nat @ C_47 @ A_174 )
=> ( member_nat @ C_47 @ B_101 ) ) ) ).
thf(fact_369_DiffE,axiom,
! [C_47: int,A_174: int > $o,B_101: int > $o] :
( ( member_int @ C_47 @ ( minus_minus_int_o @ A_174 @ B_101 ) )
=> ~ ( ( member_int @ C_47 @ A_174 )
=> ( member_int @ C_47 @ B_101 ) ) ) ).
thf(fact_370_finite__Diff,axiom,
! [B_100: hoare_2091234717iple_a > $o,A_173: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_173 )
=> ( finite232261744iple_a @ ( minus_836160335le_a_o @ A_173 @ B_100 ) ) ) ).
thf(fact_371_finite__Diff,axiom,
! [B_100: nat > $o,A_173: nat > $o] :
( ( finite_finite_nat @ A_173 )
=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_173 @ B_100 ) ) ) ).
thf(fact_372_finite__Diff,axiom,
! [B_100: int > $o,A_173: int > $o] :
( ( finite_finite_int @ A_173 )
=> ( finite_finite_int @ ( minus_minus_int_o @ A_173 @ B_100 ) ) ) ).
thf(fact_373_folding__one__idem_Oidem,axiom,
! [X_72: int,F_58: int > int > int,F_57: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_58 @ F_57 )
=> ( ( F_58 @ X_72 @ X_72 )
= X_72 ) ) ).
thf(fact_374_folding__one__idem_Oidem,axiom,
! [X_72: hoare_2091234717iple_a,F_58: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_57: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_58 @ F_57 )
=> ( ( F_58 @ X_72 @ X_72 )
= X_72 ) ) ).
thf(fact_375_folding__one__idem_Oidem,axiom,
! [X_72: nat,F_58: nat > nat > nat,F_57: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_58 @ F_57 )
=> ( ( F_58 @ X_72 @ X_72 )
= X_72 ) ) ).
thf(fact_376_set__diff__eq,axiom,
! [A_172: hoare_2091234717iple_a > $o,B_99: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_172 @ B_99 )
= ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( member290856304iple_a @ X_1 @ A_172 ) @ ( (~) @ ( member290856304iple_a @ X_1 @ B_99 ) ) ) ) ) ).
thf(fact_377_set__diff__eq,axiom,
! [A_172: nat > $o,B_99: nat > $o] :
( ( minus_minus_nat_o @ A_172 @ B_99 )
= ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ A_172 ) @ ( (~) @ ( member_nat @ X_1 @ B_99 ) ) ) ) ) ).
thf(fact_378_set__diff__eq,axiom,
! [A_172: int > $o,B_99: int > $o] :
( ( minus_minus_int_o @ A_172 @ B_99 )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ A_172 ) @ ( (~) @ ( member_int @ X_1 @ B_99 ) ) ) ) ) ).
thf(fact_379_Diff__iff,axiom,
! [C_46: hoare_2091234717iple_a,A_171: hoare_2091234717iple_a > $o,B_98: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_46 @ ( minus_836160335le_a_o @ A_171 @ B_98 ) )
<=> ( ( member290856304iple_a @ C_46 @ A_171 )
& ~ ( member290856304iple_a @ C_46 @ B_98 ) ) ) ).
thf(fact_380_Diff__iff,axiom,
! [C_46: nat,A_171: nat > $o,B_98: nat > $o] :
( ( member_nat @ C_46 @ ( minus_minus_nat_o @ A_171 @ B_98 ) )
<=> ( ( member_nat @ C_46 @ A_171 )
& ~ ( member_nat @ C_46 @ B_98 ) ) ) ).
thf(fact_381_Diff__iff,axiom,
! [C_46: int,A_171: int > $o,B_98: int > $o] :
( ( member_int @ C_46 @ ( minus_minus_int_o @ A_171 @ B_98 ) )
<=> ( ( member_int @ C_46 @ A_171 )
& ~ ( member_int @ C_46 @ B_98 ) ) ) ).
thf(fact_382_Diff__idemp,axiom,
! [A_170: int > $o,B_97: int > $o] :
( ( minus_minus_int_o @ ( minus_minus_int_o @ A_170 @ B_97 ) @ B_97 )
= ( minus_minus_int_o @ A_170 @ B_97 ) ) ).
thf(fact_383_Diff__idemp,axiom,
! [A_170: hoare_2091234717iple_a > $o,B_97: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ ( minus_836160335le_a_o @ A_170 @ B_97 ) @ B_97 )
= ( minus_836160335le_a_o @ A_170 @ B_97 ) ) ).
thf(fact_384_Diff__idemp,axiom,
! [A_170: nat > $o,B_97: nat > $o] :
( ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_170 @ B_97 ) @ B_97 )
= ( minus_minus_nat_o @ A_170 @ B_97 ) ) ).
thf(fact_385_DiffD1,axiom,
! [C_45: hoare_2091234717iple_a,A_169: hoare_2091234717iple_a > $o,B_96: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_45 @ ( minus_836160335le_a_o @ A_169 @ B_96 ) )
=> ( member290856304iple_a @ C_45 @ A_169 ) ) ).
thf(fact_386_DiffD1,axiom,
! [C_45: nat,A_169: nat > $o,B_96: nat > $o] :
( ( member_nat @ C_45 @ ( minus_minus_nat_o @ A_169 @ B_96 ) )
=> ( member_nat @ C_45 @ A_169 ) ) ).
thf(fact_387_DiffD1,axiom,
! [C_45: int,A_169: int > $o,B_96: int > $o] :
( ( member_int @ C_45 @ ( minus_minus_int_o @ A_169 @ B_96 ) )
=> ( member_int @ C_45 @ A_169 ) ) ).
thf(fact_388_DiffD2,axiom,
! [C_44: hoare_2091234717iple_a,A_168: hoare_2091234717iple_a > $o,B_95: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ C_44 @ ( minus_836160335le_a_o @ A_168 @ B_95 ) )
=> ~ ( member290856304iple_a @ C_44 @ B_95 ) ) ).
thf(fact_389_DiffD2,axiom,
! [C_44: nat,A_168: nat > $o,B_95: nat > $o] :
( ( member_nat @ C_44 @ ( minus_minus_nat_o @ A_168 @ B_95 ) )
=> ~ ( member_nat @ C_44 @ B_95 ) ) ).
thf(fact_390_DiffD2,axiom,
! [C_44: int,A_168: int > $o,B_95: int > $o] :
( ( member_int @ C_44 @ ( minus_minus_int_o @ A_168 @ B_95 ) )
=> ~ ( member_int @ C_44 @ B_95 ) ) ).
thf(fact_391_empty__Diff,axiom,
! [A_167: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ bot_bo1791335050le_a_o @ A_167 )
= bot_bo1791335050le_a_o ) ).
thf(fact_392_empty__Diff,axiom,
! [A_167: int > $o] :
( ( minus_minus_int_o @ bot_bot_int_o @ A_167 )
= bot_bot_int_o ) ).
thf(fact_393_empty__Diff,axiom,
! [A_167: nat > $o] :
( ( minus_minus_nat_o @ bot_bot_nat_o @ A_167 )
= bot_bot_nat_o ) ).
thf(fact_394_Diff__empty,axiom,
! [A_166: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_166 @ bot_bo1791335050le_a_o )
= A_166 ) ).
thf(fact_395_Diff__empty,axiom,
! [A_166: int > $o] :
( ( minus_minus_int_o @ A_166 @ bot_bot_int_o )
= A_166 ) ).
thf(fact_396_Diff__empty,axiom,
! [A_166: nat > $o] :
( ( minus_minus_nat_o @ A_166 @ bot_bot_nat_o )
= A_166 ) ).
thf(fact_397_Diff__cancel,axiom,
! [A_165: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_165 @ A_165 )
= bot_bo1791335050le_a_o ) ).
thf(fact_398_Diff__cancel,axiom,
! [A_165: int > $o] :
( ( minus_minus_int_o @ A_165 @ A_165 )
= bot_bot_int_o ) ).
thf(fact_399_Diff__cancel,axiom,
! [A_165: nat > $o] :
( ( minus_minus_nat_o @ A_165 @ A_165 )
= bot_bot_nat_o ) ).
thf(fact_400_finite__Diff2,axiom,
! [A_164: hoare_2091234717iple_a > $o,B_94: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ B_94 )
=> ( ( finite232261744iple_a @ ( minus_836160335le_a_o @ A_164 @ B_94 ) )
<=> ( finite232261744iple_a @ A_164 ) ) ) ).
thf(fact_401_finite__Diff2,axiom,
! [A_164: nat > $o,B_94: nat > $o] :
( ( finite_finite_nat @ B_94 )
=> ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_164 @ B_94 ) )
<=> ( finite_finite_nat @ A_164 ) ) ) ).
thf(fact_402_finite__Diff2,axiom,
! [A_164: int > $o,B_94: int > $o] :
( ( finite_finite_int @ B_94 )
=> ( ( finite_finite_int @ ( minus_minus_int_o @ A_164 @ B_94 ) )
<=> ( finite_finite_int @ A_164 ) ) ) ).
thf(fact_403_insert__Diff__if,axiom,
! [A_163: nat > $o,X_71: nat,B_93: nat > $o] :
( ( ( member_nat @ X_71 @ B_93 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_71 @ A_163 ) @ B_93 )
= ( minus_minus_nat_o @ A_163 @ B_93 ) ) )
& ( ~ ( member_nat @ X_71 @ B_93 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_71 @ A_163 ) @ B_93 )
= ( insert_nat @ X_71 @ ( minus_minus_nat_o @ A_163 @ B_93 ) ) ) ) ) ).
thf(fact_404_insert__Diff__if,axiom,
! [A_163: hoare_2091234717iple_a > $o,X_71: hoare_2091234717iple_a,B_93: hoare_2091234717iple_a > $o] :
( ( ( member290856304iple_a @ X_71 @ B_93 )
=> ( ( minus_836160335le_a_o @ ( insert1597628439iple_a @ X_71 @ A_163 ) @ B_93 )
= ( minus_836160335le_a_o @ A_163 @ B_93 ) ) )
& ( ~ ( member290856304iple_a @ X_71 @ B_93 )
=> ( ( minus_836160335le_a_o @ ( insert1597628439iple_a @ X_71 @ A_163 ) @ B_93 )
= ( insert1597628439iple_a @ X_71 @ ( minus_836160335le_a_o @ A_163 @ B_93 ) ) ) ) ) ).
thf(fact_405_insert__Diff__if,axiom,
! [A_163: int > $o,X_71: int,B_93: int > $o] :
( ( ( member_int @ X_71 @ B_93 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_71 @ A_163 ) @ B_93 )
= ( minus_minus_int_o @ A_163 @ B_93 ) ) )
& ( ~ ( member_int @ X_71 @ B_93 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_71 @ A_163 ) @ B_93 )
= ( insert_int @ X_71 @ ( minus_minus_int_o @ A_163 @ B_93 ) ) ) ) ) ).
thf(fact_406_insert__Diff1,axiom,
! [A_162: nat > $o,X_70: nat,B_92: nat > $o] :
( ( member_nat @ X_70 @ B_92 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_70 @ A_162 ) @ B_92 )
= ( minus_minus_nat_o @ A_162 @ B_92 ) ) ) ).
thf(fact_407_insert__Diff1,axiom,
! [A_162: hoare_2091234717iple_a > $o,X_70: hoare_2091234717iple_a,B_92: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_70 @ B_92 )
=> ( ( minus_836160335le_a_o @ ( insert1597628439iple_a @ X_70 @ A_162 ) @ B_92 )
= ( minus_836160335le_a_o @ A_162 @ B_92 ) ) ) ).
thf(fact_408_insert__Diff1,axiom,
! [A_162: int > $o,X_70: int,B_92: int > $o] :
( ( member_int @ X_70 @ B_92 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_70 @ A_162 ) @ B_92 )
= ( minus_minus_int_o @ A_162 @ B_92 ) ) ) ).
thf(fact_409_fold__graph__permute__diff,axiom,
! [A_161: nat,B_91: nat,A_160: nat > $o,X_69: nat] :
( ( finite929467206at_nat @ times_times_nat @ B_91 @ A_160 @ X_69 )
=> ( ( member_nat @ A_161 @ A_160 )
=> ( ~ ( member_nat @ B_91 @ A_160 )
=> ( finite929467206at_nat @ times_times_nat @ A_161 @ ( insert_nat @ B_91 @ ( minus_minus_nat_o @ A_160 @ ( insert_nat @ A_161 @ bot_bot_nat_o ) ) ) @ X_69 ) ) ) ) ).
thf(fact_410_fold__graph__permute__diff,axiom,
! [A_161: int,B_91: int,A_160: int > $o,X_69: int] :
( ( finite772772422nt_int @ times_times_int @ B_91 @ A_160 @ X_69 )
=> ( ( member_int @ A_161 @ A_160 )
=> ( ~ ( member_int @ B_91 @ A_160 )
=> ( finite772772422nt_int @ times_times_int @ A_161 @ ( insert_int @ B_91 @ ( minus_minus_int_o @ A_160 @ ( insert_int @ A_161 @ bot_bot_int_o ) ) ) @ X_69 ) ) ) ) ).
thf(fact_411_Diff__insert,axiom,
! [A_159: nat > $o,A_158: nat,B_90: nat > $o] :
( ( minus_minus_nat_o @ A_159 @ ( insert_nat @ A_158 @ B_90 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_159 @ B_90 ) @ ( insert_nat @ A_158 @ bot_bot_nat_o ) ) ) ).
thf(fact_412_Diff__insert,axiom,
! [A_159: hoare_2091234717iple_a > $o,A_158: hoare_2091234717iple_a,B_90: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_159 @ ( insert1597628439iple_a @ A_158 @ B_90 ) )
= ( minus_836160335le_a_o @ ( minus_836160335le_a_o @ A_159 @ B_90 ) @ ( insert1597628439iple_a @ A_158 @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_413_Diff__insert,axiom,
! [A_159: int > $o,A_158: int,B_90: int > $o] :
( ( minus_minus_int_o @ A_159 @ ( insert_int @ A_158 @ B_90 ) )
= ( minus_minus_int_o @ ( minus_minus_int_o @ A_159 @ B_90 ) @ ( insert_int @ A_158 @ bot_bot_int_o ) ) ) ).
thf(fact_414_Diff__insert2,axiom,
! [A_157: nat > $o,A_156: nat,B_89: nat > $o] :
( ( minus_minus_nat_o @ A_157 @ ( insert_nat @ A_156 @ B_89 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_157 @ ( insert_nat @ A_156 @ bot_bot_nat_o ) ) @ B_89 ) ) ).
thf(fact_415_Diff__insert2,axiom,
! [A_157: hoare_2091234717iple_a > $o,A_156: hoare_2091234717iple_a,B_89: hoare_2091234717iple_a > $o] :
( ( minus_836160335le_a_o @ A_157 @ ( insert1597628439iple_a @ A_156 @ B_89 ) )
= ( minus_836160335le_a_o @ ( minus_836160335le_a_o @ A_157 @ ( insert1597628439iple_a @ A_156 @ bot_bo1791335050le_a_o ) ) @ B_89 ) ) ).
thf(fact_416_Diff__insert2,axiom,
! [A_157: int > $o,A_156: int,B_89: int > $o] :
( ( minus_minus_int_o @ A_157 @ ( insert_int @ A_156 @ B_89 ) )
= ( minus_minus_int_o @ ( minus_minus_int_o @ A_157 @ ( insert_int @ A_156 @ bot_bot_int_o ) ) @ B_89 ) ) ).
thf(fact_417_insert__Diff__single,axiom,
! [A_155: nat,A_154: nat > $o] :
( ( insert_nat @ A_155 @ ( minus_minus_nat_o @ A_154 @ ( insert_nat @ A_155 @ bot_bot_nat_o ) ) )
= ( insert_nat @ A_155 @ A_154 ) ) ).
thf(fact_418_insert__Diff__single,axiom,
! [A_155: hoare_2091234717iple_a,A_154: hoare_2091234717iple_a > $o] :
( ( insert1597628439iple_a @ A_155 @ ( minus_836160335le_a_o @ A_154 @ ( insert1597628439iple_a @ A_155 @ bot_bo1791335050le_a_o ) ) )
= ( insert1597628439iple_a @ A_155 @ A_154 ) ) ).
thf(fact_419_insert__Diff__single,axiom,
! [A_155: int,A_154: int > $o] :
( ( insert_int @ A_155 @ ( minus_minus_int_o @ A_154 @ ( insert_int @ A_155 @ bot_bot_int_o ) ) )
= ( insert_int @ A_155 @ A_154 ) ) ).
thf(fact_420_Diff__insert__absorb,axiom,
! [X_68: nat,A_153: nat > $o] :
( ~ ( member_nat @ X_68 @ A_153 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_153 ) @ ( insert_nat @ X_68 @ bot_bot_nat_o ) )
= A_153 ) ) ).
thf(fact_421_Diff__insert__absorb,axiom,
! [X_68: hoare_2091234717iple_a,A_153: hoare_2091234717iple_a > $o] :
( ~ ( member290856304iple_a @ X_68 @ A_153 )
=> ( ( minus_836160335le_a_o @ ( insert1597628439iple_a @ X_68 @ A_153 ) @ ( insert1597628439iple_a @ X_68 @ bot_bo1791335050le_a_o ) )
= A_153 ) ) ).
thf(fact_422_Diff__insert__absorb,axiom,
! [X_68: int,A_153: int > $o] :
( ~ ( member_int @ X_68 @ A_153 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_68 @ A_153 ) @ ( insert_int @ X_68 @ bot_bot_int_o ) )
= A_153 ) ) ).
thf(fact_423_insert__Diff,axiom,
! [A_152: nat,A_151: nat > $o] :
( ( member_nat @ A_152 @ A_151 )
=> ( ( insert_nat @ A_152 @ ( minus_minus_nat_o @ A_151 @ ( insert_nat @ A_152 @ bot_bot_nat_o ) ) )
= A_151 ) ) ).
thf(fact_424_insert__Diff,axiom,
! [A_152: hoare_2091234717iple_a,A_151: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ A_152 @ A_151 )
=> ( ( insert1597628439iple_a @ A_152 @ ( minus_836160335le_a_o @ A_151 @ ( insert1597628439iple_a @ A_152 @ bot_bo1791335050le_a_o ) ) )
= A_151 ) ) ).
thf(fact_425_insert__Diff,axiom,
! [A_152: int,A_151: int > $o] :
( ( member_int @ A_152 @ A_151 )
=> ( ( insert_int @ A_152 @ ( minus_minus_int_o @ A_151 @ ( insert_int @ A_152 @ bot_bot_int_o ) ) )
= A_151 ) ) ).
thf(fact_426_finite__Diff__insert,axiom,
! [A_150: nat > $o,A_149: nat,B_88: nat > $o] :
( ( finite_finite_nat @ ( minus_minus_nat_o @ A_150 @ ( insert_nat @ A_149 @ B_88 ) ) )
<=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_150 @ B_88 ) ) ) ).
thf(fact_427_finite__Diff__insert,axiom,
! [A_150: hoare_2091234717iple_a > $o,A_149: hoare_2091234717iple_a,B_88: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ ( minus_836160335le_a_o @ A_150 @ ( insert1597628439iple_a @ A_149 @ B_88 ) ) )
<=> ( finite232261744iple_a @ ( minus_836160335le_a_o @ A_150 @ B_88 ) ) ) ).
thf(fact_428_finite__Diff__insert,axiom,
! [A_150: int > $o,A_149: int,B_88: int > $o] :
( ( finite_finite_int @ ( minus_minus_int_o @ A_150 @ ( insert_int @ A_149 @ B_88 ) ) )
<=> ( finite_finite_int @ ( minus_minus_int_o @ A_150 @ B_88 ) ) ) ).
thf(fact_429_fold__graph__insert__swap,axiom,
! [Z_30: nat,B_87: nat,A_148: nat > $o,Y_34: nat] :
( ( finite929467206at_nat @ times_times_nat @ B_87 @ A_148 @ Y_34 )
=> ( ~ ( member_nat @ B_87 @ A_148 )
=> ( finite929467206at_nat @ times_times_nat @ Z_30 @ ( insert_nat @ B_87 @ A_148 ) @ ( times_times_nat @ Z_30 @ Y_34 ) ) ) ) ).
thf(fact_430_fold__graph__insert__swap,axiom,
! [Z_30: int,B_87: int,A_148: int > $o,Y_34: int] :
( ( finite772772422nt_int @ times_times_int @ B_87 @ A_148 @ Y_34 )
=> ( ~ ( member_int @ B_87 @ A_148 )
=> ( finite772772422nt_int @ times_times_int @ Z_30 @ ( insert_int @ B_87 @ A_148 ) @ ( times_times_int @ Z_30 @ Y_34 ) ) ) ) ).
thf(fact_431_folding__one__idem_Oin__idem,axiom,
! [X_67: hoare_2091234717iple_a,A_147: hoare_2091234717iple_a > $o,F_56: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_55: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_56 @ F_55 )
=> ( ( finite232261744iple_a @ A_147 )
=> ( ( member290856304iple_a @ X_67 @ A_147 )
=> ( ( F_56 @ X_67 @ ( F_55 @ A_147 ) )
= ( F_55 @ A_147 ) ) ) ) ) ).
thf(fact_432_folding__one__idem_Oin__idem,axiom,
! [X_67: nat,A_147: nat > $o,F_56: nat > nat > nat,F_55: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_56 @ F_55 )
=> ( ( finite_finite_nat @ A_147 )
=> ( ( member_nat @ X_67 @ A_147 )
=> ( ( F_56 @ X_67 @ ( F_55 @ A_147 ) )
= ( F_55 @ A_147 ) ) ) ) ) ).
thf(fact_433_folding__one__idem_Oin__idem,axiom,
! [X_67: int,A_147: int > $o,F_56: int > int > int,F_55: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_56 @ F_55 )
=> ( ( finite_finite_int @ A_147 )
=> ( ( member_int @ X_67 @ A_147 )
=> ( ( F_56 @ X_67 @ ( F_55 @ A_147 ) )
= ( F_55 @ A_147 ) ) ) ) ) ).
thf(fact_434_fold1__in,axiom,
! [A_146: nat > $o] :
( ( finite_finite_nat @ A_146 )
=> ( ( A_146 != bot_bot_nat_o )
=> ( ! [X_1: nat,Y_28: nat] : ( member_nat @ ( times_times_nat @ X_1 @ Y_28 ) @ ( insert_nat @ X_1 @ ( insert_nat @ Y_28 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( finite_fold1_nat @ times_times_nat @ A_146 ) @ A_146 ) ) ) ) ).
thf(fact_435_fold1__in,axiom,
! [A_146: int > $o] :
( ( finite_finite_int @ A_146 )
=> ( ( A_146 != bot_bot_int_o )
=> ( ! [X_1: int,Y_28: int] : ( member_int @ ( times_times_int @ X_1 @ Y_28 ) @ ( insert_int @ X_1 @ ( insert_int @ Y_28 @ bot_bot_int_o ) ) )
=> ( member_int @ ( finite_fold1_int @ times_times_int @ A_146 ) @ A_146 ) ) ) ) ).
thf(fact_436_folding__one__idem_Ohom__commute,axiom,
! [N_5: hoare_2091234717iple_a > $o,H_4: hoare_2091234717iple_a > hoare_2091234717iple_a,F_54: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_53: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_54 @ F_53 )
=> ( ! [X_1: hoare_2091234717iple_a,Y_28: hoare_2091234717iple_a] :
( ( H_4 @ ( F_54 @ X_1 @ Y_28 ) )
= ( F_54 @ ( H_4 @ X_1 ) @ ( H_4 @ Y_28 ) ) )
=> ( ( finite232261744iple_a @ N_5 )
=> ( ( N_5 != bot_bo1791335050le_a_o )
=> ( ( H_4 @ ( F_53 @ N_5 ) )
= ( F_53 @ ( image_1661191109iple_a @ H_4 @ N_5 ) ) ) ) ) ) ) ).
thf(fact_437_folding__one__idem_Ohom__commute,axiom,
! [N_5: int > $o,H_4: int > int,F_54: int > int > int,F_53: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_54 @ F_53 )
=> ( ! [X_1: int,Y_28: int] :
( ( H_4 @ ( F_54 @ X_1 @ Y_28 ) )
= ( F_54 @ ( H_4 @ X_1 ) @ ( H_4 @ Y_28 ) ) )
=> ( ( finite_finite_int @ N_5 )
=> ( ( N_5 != bot_bot_int_o )
=> ( ( H_4 @ ( F_53 @ N_5 ) )
= ( F_53 @ ( image_int_int @ H_4 @ N_5 ) ) ) ) ) ) ) ).
thf(fact_438_folding__one__idem_Ohom__commute,axiom,
! [N_5: nat > $o,H_4: nat > nat,F_54: nat > nat > nat,F_53: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_54 @ F_53 )
=> ( ! [X_1: nat,Y_28: nat] :
( ( H_4 @ ( F_54 @ X_1 @ Y_28 ) )
= ( F_54 @ ( H_4 @ X_1 ) @ ( H_4 @ Y_28 ) ) )
=> ( ( finite_finite_nat @ N_5 )
=> ( ( N_5 != bot_bot_nat_o )
=> ( ( H_4 @ ( F_53 @ N_5 ) )
= ( F_53 @ ( image_nat_nat @ H_4 @ N_5 ) ) ) ) ) ) ) ).
thf(fact_439_finite__empty__induct,axiom,
! [P_6: ( nat > $o ) > $o,A_145: nat > $o] :
( ( finite_finite_nat @ A_145 )
=> ( ( P_6 @ A_145 )
=> ( ! [A_19: nat,A_134: nat > $o] :
( ( finite_finite_nat @ A_134 )
=> ( ( member_nat @ A_19 @ A_134 )
=> ( ( P_6 @ A_134 )
=> ( P_6 @ ( minus_minus_nat_o @ A_134 @ ( insert_nat @ A_19 @ bot_bot_nat_o ) ) ) ) ) )
=> ( P_6 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_440_finite__empty__induct,axiom,
! [P_6: ( hoare_2091234717iple_a > $o ) > $o,A_145: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_145 )
=> ( ( P_6 @ A_145 )
=> ( ! [A_19: hoare_2091234717iple_a,A_134: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_134 )
=> ( ( member290856304iple_a @ A_19 @ A_134 )
=> ( ( P_6 @ A_134 )
=> ( P_6 @ ( minus_836160335le_a_o @ A_134 @ ( insert1597628439iple_a @ A_19 @ bot_bo1791335050le_a_o ) ) ) ) ) )
=> ( P_6 @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_441_finite__empty__induct,axiom,
! [P_6: ( int > $o ) > $o,A_145: int > $o] :
( ( finite_finite_int @ A_145 )
=> ( ( P_6 @ A_145 )
=> ( ! [A_19: int,A_134: int > $o] :
( ( finite_finite_int @ A_134 )
=> ( ( member_int @ A_19 @ A_134 )
=> ( ( P_6 @ A_134 )
=> ( P_6 @ ( minus_minus_int_o @ A_134 @ ( insert_int @ A_19 @ bot_bot_int_o ) ) ) ) ) )
=> ( P_6 @ bot_bot_int_o ) ) ) ) ).
thf(fact_442_minus__apply,axiom,
! [A_144: int > $o,B_86: int > $o,X_66: int] :
( ( minus_minus_int_o @ A_144 @ B_86 @ X_66 )
<=> ( minus_minus_o @ ( A_144 @ X_66 ) @ ( B_86 @ X_66 ) ) ) ).
thf(fact_443_minus__apply,axiom,
! [A_144: hoare_2091234717iple_a > $o,B_86: hoare_2091234717iple_a > $o,X_66: hoare_2091234717iple_a] :
( ( minus_836160335le_a_o @ A_144 @ B_86 @ X_66 )
<=> ( minus_minus_o @ ( A_144 @ X_66 ) @ ( B_86 @ X_66 ) ) ) ).
thf(fact_444_minus__apply,axiom,
! [A_144: nat > $o,B_86: nat > $o,X_66: nat] :
( ( minus_minus_nat_o @ A_144 @ B_86 @ X_66 )
<=> ( minus_minus_o @ ( A_144 @ X_66 ) @ ( B_86 @ X_66 ) ) ) ).
thf(fact_445_fun__diff__def,axiom,
! [A_143: int > $o,B_85: int > $o,X_1: int] :
( ( minus_minus_int_o @ A_143 @ B_85 @ X_1 )
<=> ( minus_minus_o @ ( A_143 @ X_1 ) @ ( B_85 @ X_1 ) ) ) ).
thf(fact_446_fun__diff__def,axiom,
! [A_143: hoare_2091234717iple_a > $o,B_85: hoare_2091234717iple_a > $o,X_1: hoare_2091234717iple_a] :
( ( minus_836160335le_a_o @ A_143 @ B_85 @ X_1 )
<=> ( minus_minus_o @ ( A_143 @ X_1 ) @ ( B_85 @ X_1 ) ) ) ).
thf(fact_447_fun__diff__def,axiom,
! [A_143: nat > $o,B_85: nat > $o,X_1: nat] :
( ( minus_minus_nat_o @ A_143 @ B_85 @ X_1 )
<=> ( minus_minus_o @ ( A_143 @ X_1 ) @ ( B_85 @ X_1 ) ) ) ).
thf(fact_448_comp__fun__idem__remove,axiom,
( finite389864113_nat_o
@ ^ [X_1: nat,A_134: nat > $o] : ( minus_minus_nat_o @ A_134 @ ( insert_nat @ X_1 @ bot_bot_nat_o ) ) ) ).
thf(fact_449_comp__fun__idem__remove,axiom,
( finite438582129le_a_o
@ ^ [X_1: hoare_2091234717iple_a,A_134: hoare_2091234717iple_a > $o] : ( minus_836160335le_a_o @ A_134 @ ( insert1597628439iple_a @ X_1 @ bot_bo1791335050le_a_o ) ) ) ).
thf(fact_450_comp__fun__idem__remove,axiom,
( finite175163825_int_o
@ ^ [X_1: int,A_134: int > $o] : ( minus_minus_int_o @ A_134 @ ( insert_int @ X_1 @ bot_bot_int_o ) ) ) ).
thf(fact_451_comp__fun__idem_Ofun__left__idem,axiom,
! [X_65: int,Z_29: int > $o,F_52: int > ( int > $o ) > int > $o] :
( ( finite175163825_int_o @ F_52 )
=> ( ( F_52 @ X_65 @ ( F_52 @ X_65 @ Z_29 ) )
= ( F_52 @ X_65 @ Z_29 ) ) ) ).
thf(fact_452_comp__fun__idem_Ofun__left__idem,axiom,
! [X_65: hoare_2091234717iple_a,Z_29: hoare_2091234717iple_a > $o,F_52: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite438582129le_a_o @ F_52 )
=> ( ( F_52 @ X_65 @ ( F_52 @ X_65 @ Z_29 ) )
= ( F_52 @ X_65 @ Z_29 ) ) ) ).
thf(fact_453_comp__fun__idem_Ofun__left__idem,axiom,
! [X_65: nat,Z_29: nat > $o,F_52: nat > ( nat > $o ) > nat > $o] :
( ( finite389864113_nat_o @ F_52 )
=> ( ( F_52 @ X_65 @ ( F_52 @ X_65 @ Z_29 ) )
= ( F_52 @ X_65 @ Z_29 ) ) ) ).
thf(fact_454_comp__fun__idem__insert,axiom,
finite389864113_nat_o @ insert_nat ).
thf(fact_455_comp__fun__idem__insert,axiom,
finite438582129le_a_o @ insert1597628439iple_a ).
thf(fact_456_comp__fun__idem__insert,axiom,
finite175163825_int_o @ insert_int ).
thf(fact_457_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_142: nat,Z_28: nat,A_141: nat > $o,Y_32: nat,F_51: nat > nat > nat] :
( ( finite2130160977at_nat @ F_51 )
=> ( ( finite929467206at_nat @ F_51 @ Z_28 @ A_141 @ Y_32 )
=> ( ( member_nat @ A_142 @ A_141 )
=> ? [Y_33: nat] :
( ( Y_32
= ( F_51 @ A_142 @ Y_33 ) )
& ( finite929467206at_nat @ F_51 @ Z_28 @ ( minus_minus_nat_o @ A_141 @ ( insert_nat @ A_142 @ bot_bot_nat_o ) ) @ Y_33 ) ) ) ) ) ).
thf(fact_458_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_142: hoare_2091234717iple_a,Z_28: hoare_2091234717iple_a,A_141: hoare_2091234717iple_a > $o,Y_32: hoare_2091234717iple_a,F_51: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_51 )
=> ( ( finite1218641926iple_a @ F_51 @ Z_28 @ A_141 @ Y_32 )
=> ( ( member290856304iple_a @ A_142 @ A_141 )
=> ? [Y_33: hoare_2091234717iple_a] :
( ( Y_32
= ( F_51 @ A_142 @ Y_33 ) )
& ( finite1218641926iple_a @ F_51 @ Z_28 @ ( minus_836160335le_a_o @ A_141 @ ( insert1597628439iple_a @ A_142 @ bot_bo1791335050le_a_o ) ) @ Y_33 ) ) ) ) ) ).
thf(fact_459_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [A_142: int,Z_28: int,A_141: int > $o,Y_32: int,F_51: int > int > int] :
( ( finite1973466193nt_int @ F_51 )
=> ( ( finite772772422nt_int @ F_51 @ Z_28 @ A_141 @ Y_32 )
=> ( ( member_int @ A_142 @ A_141 )
=> ? [Y_33: int] :
( ( Y_32
= ( F_51 @ A_142 @ Y_33 ) )
& ( finite772772422nt_int @ F_51 @ Z_28 @ ( minus_minus_int_o @ A_141 @ ( insert_int @ A_142 @ bot_bot_int_o ) ) @ Y_33 ) ) ) ) ) ).
thf(fact_460_setsum__diff1,axiom,
! [F_50: nat > int,A_140: nat,A_139: nat > $o] :
( ( finite_finite_nat @ A_139 )
=> ( ( ( member_nat @ A_140 @ A_139 )
=> ( ( big_co1024481617at_int @ F_50 @ ( minus_minus_nat_o @ A_139 @ ( insert_nat @ A_140 @ bot_bot_nat_o ) ) )
= ( minus_minus_int @ ( big_co1024481617at_int @ F_50 @ A_139 ) @ ( F_50 @ A_140 ) ) ) )
& ( ~ ( member_nat @ A_140 @ A_139 )
=> ( ( big_co1024481617at_int @ F_50 @ ( minus_minus_nat_o @ A_139 @ ( insert_nat @ A_140 @ bot_bot_nat_o ) ) )
= ( big_co1024481617at_int @ F_50 @ A_139 ) ) ) ) ) ).
thf(fact_461_setsum__diff1,axiom,
! [F_50: hoare_2091234717iple_a > int,A_140: hoare_2091234717iple_a,A_139: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_139 )
=> ( ( ( member290856304iple_a @ A_140 @ A_139 )
=> ( ( big_co1555037566_a_int @ F_50 @ ( minus_836160335le_a_o @ A_139 @ ( insert1597628439iple_a @ A_140 @ bot_bo1791335050le_a_o ) ) )
= ( minus_minus_int @ ( big_co1555037566_a_int @ F_50 @ A_139 ) @ ( F_50 @ A_140 ) ) ) )
& ( ~ ( member290856304iple_a @ A_140 @ A_139 )
=> ( ( big_co1555037566_a_int @ F_50 @ ( minus_836160335le_a_o @ A_139 @ ( insert1597628439iple_a @ A_140 @ bot_bo1791335050le_a_o ) ) )
= ( big_co1555037566_a_int @ F_50 @ A_139 ) ) ) ) ) ).
thf(fact_462_setsum__diff1,axiom,
! [F_50: int > int,A_140: int,A_139: int > $o] :
( ( finite_finite_int @ A_139 )
=> ( ( ( member_int @ A_140 @ A_139 )
=> ( ( big_co230513141nt_int @ F_50 @ ( minus_minus_int_o @ A_139 @ ( insert_int @ A_140 @ bot_bot_int_o ) ) )
= ( minus_minus_int @ ( big_co230513141nt_int @ F_50 @ A_139 ) @ ( F_50 @ A_140 ) ) ) )
& ( ~ ( member_int @ A_140 @ A_139 )
=> ( ( big_co230513141nt_int @ F_50 @ ( minus_minus_int_o @ A_139 @ ( insert_int @ A_140 @ bot_bot_int_o ) ) )
= ( big_co230513141nt_int @ F_50 @ A_139 ) ) ) ) ) ).
thf(fact_463_setsum__diff1__ring,axiom,
! [F_49: nat > int,A_138: nat,A_137: nat > $o] :
( ( finite_finite_nat @ A_137 )
=> ( ( member_nat @ A_138 @ A_137 )
=> ( ( big_co1024481617at_int @ F_49 @ ( minus_minus_nat_o @ A_137 @ ( insert_nat @ A_138 @ bot_bot_nat_o ) ) )
= ( minus_minus_int @ ( big_co1024481617at_int @ F_49 @ A_137 ) @ ( F_49 @ A_138 ) ) ) ) ) ).
thf(fact_464_setsum__diff1__ring,axiom,
! [F_49: hoare_2091234717iple_a > int,A_138: hoare_2091234717iple_a,A_137: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_137 )
=> ( ( member290856304iple_a @ A_138 @ A_137 )
=> ( ( big_co1555037566_a_int @ F_49 @ ( minus_836160335le_a_o @ A_137 @ ( insert1597628439iple_a @ A_138 @ bot_bo1791335050le_a_o ) ) )
= ( minus_minus_int @ ( big_co1555037566_a_int @ F_49 @ A_137 ) @ ( F_49 @ A_138 ) ) ) ) ) ).
thf(fact_465_setsum__diff1__ring,axiom,
! [F_49: int > int,A_138: int,A_137: int > $o] :
( ( finite_finite_int @ A_137 )
=> ( ( member_int @ A_138 @ A_137 )
=> ( ( big_co230513141nt_int @ F_49 @ ( minus_minus_int_o @ A_137 @ ( insert_int @ A_138 @ bot_bot_int_o ) ) )
= ( minus_minus_int @ ( big_co230513141nt_int @ F_49 @ A_137 ) @ ( F_49 @ A_138 ) ) ) ) ) ).
thf(fact_466_setprod_Oinsert__remove,axiom,
! [G_21: nat > nat,X_64: nat,A_136: nat > $o] :
( ( finite_finite_nat @ A_136 )
=> ( ( big_co1705425894at_nat @ G_21 @ ( insert_nat @ X_64 @ A_136 ) )
= ( times_times_nat @ ( G_21 @ X_64 ) @ ( big_co1705425894at_nat @ G_21 @ ( minus_minus_nat_o @ A_136 @ ( insert_nat @ X_64 @ bot_bot_nat_o ) ) ) ) ) ) ).
thf(fact_467_setprod_Oinsert__remove,axiom,
! [G_21: hoare_2091234717iple_a > nat,X_64: hoare_2091234717iple_a,A_136: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_136 )
=> ( ( big_co1393145363_a_nat @ G_21 @ ( insert1597628439iple_a @ X_64 @ A_136 ) )
= ( times_times_nat @ ( G_21 @ X_64 ) @ ( big_co1393145363_a_nat @ G_21 @ ( minus_836160335le_a_o @ A_136 @ ( insert1597628439iple_a @ X_64 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ).
thf(fact_468_setprod_Oinsert__remove,axiom,
! [G_21: int > nat,X_64: int,A_136: int > $o] :
( ( finite_finite_int @ A_136 )
=> ( ( big_co911457418nt_nat @ G_21 @ ( insert_int @ X_64 @ A_136 ) )
= ( times_times_nat @ ( G_21 @ X_64 ) @ ( big_co911457418nt_nat @ G_21 @ ( minus_minus_int_o @ A_136 @ ( insert_int @ X_64 @ bot_bot_int_o ) ) ) ) ) ) ).
thf(fact_469_setprod_Oinsert__remove,axiom,
! [G_21: nat > int,X_64: nat,A_136: nat > $o] :
( ( finite_finite_nat @ A_136 )
=> ( ( big_co195215938at_int @ G_21 @ ( insert_nat @ X_64 @ A_136 ) )
= ( times_times_int @ ( G_21 @ X_64 ) @ ( big_co195215938at_int @ G_21 @ ( minus_minus_nat_o @ A_136 @ ( insert_nat @ X_64 @ bot_bot_nat_o ) ) ) ) ) ) ).
thf(fact_470_setprod_Oinsert__remove,axiom,
! [G_21: hoare_2091234717iple_a > int,X_64: hoare_2091234717iple_a,A_136: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_136 )
=> ( ( big_co2030419055_a_int @ G_21 @ ( insert1597628439iple_a @ X_64 @ A_136 ) )
= ( times_times_int @ ( G_21 @ X_64 ) @ ( big_co2030419055_a_int @ G_21 @ ( minus_836160335le_a_o @ A_136 @ ( insert1597628439iple_a @ X_64 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ).
thf(fact_471_setprod_Oinsert__remove,axiom,
! [G_21: int > int,X_64: int,A_136: int > $o] :
( ( finite_finite_int @ A_136 )
=> ( ( big_co1548731110nt_int @ G_21 @ ( insert_int @ X_64 @ A_136 ) )
= ( times_times_int @ ( G_21 @ X_64 ) @ ( big_co1548731110nt_int @ G_21 @ ( minus_minus_int_o @ A_136 @ ( insert_int @ X_64 @ bot_bot_int_o ) ) ) ) ) ) ).
thf(fact_472_setprod_Oremove,axiom,
! [G_20: nat > nat,X_63: nat,A_135: nat > $o] :
( ( finite_finite_nat @ A_135 )
=> ( ( member_nat @ X_63 @ A_135 )
=> ( ( big_co1705425894at_nat @ G_20 @ A_135 )
= ( times_times_nat @ ( G_20 @ X_63 ) @ ( big_co1705425894at_nat @ G_20 @ ( minus_minus_nat_o @ A_135 @ ( insert_nat @ X_63 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_473_setprod_Oremove,axiom,
! [G_20: hoare_2091234717iple_a > nat,X_63: hoare_2091234717iple_a,A_135: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_135 )
=> ( ( member290856304iple_a @ X_63 @ A_135 )
=> ( ( big_co1393145363_a_nat @ G_20 @ A_135 )
= ( times_times_nat @ ( G_20 @ X_63 ) @ ( big_co1393145363_a_nat @ G_20 @ ( minus_836160335le_a_o @ A_135 @ ( insert1597628439iple_a @ X_63 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_474_setprod_Oremove,axiom,
! [G_20: int > nat,X_63: int,A_135: int > $o] :
( ( finite_finite_int @ A_135 )
=> ( ( member_int @ X_63 @ A_135 )
=> ( ( big_co911457418nt_nat @ G_20 @ A_135 )
= ( times_times_nat @ ( G_20 @ X_63 ) @ ( big_co911457418nt_nat @ G_20 @ ( minus_minus_int_o @ A_135 @ ( insert_int @ X_63 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_475_setprod_Oremove,axiom,
! [G_20: nat > int,X_63: nat,A_135: nat > $o] :
( ( finite_finite_nat @ A_135 )
=> ( ( member_nat @ X_63 @ A_135 )
=> ( ( big_co195215938at_int @ G_20 @ A_135 )
= ( times_times_int @ ( G_20 @ X_63 ) @ ( big_co195215938at_int @ G_20 @ ( minus_minus_nat_o @ A_135 @ ( insert_nat @ X_63 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_476_setprod_Oremove,axiom,
! [G_20: hoare_2091234717iple_a > int,X_63: hoare_2091234717iple_a,A_135: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_135 )
=> ( ( member290856304iple_a @ X_63 @ A_135 )
=> ( ( big_co2030419055_a_int @ G_20 @ A_135 )
= ( times_times_int @ ( G_20 @ X_63 ) @ ( big_co2030419055_a_int @ G_20 @ ( minus_836160335le_a_o @ A_135 @ ( insert1597628439iple_a @ X_63 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_477_setprod_Oremove,axiom,
! [G_20: int > int,X_63: int,A_135: int > $o] :
( ( finite_finite_int @ A_135 )
=> ( ( member_int @ X_63 @ A_135 )
=> ( ( big_co1548731110nt_int @ G_20 @ A_135 )
= ( times_times_int @ ( G_20 @ X_63 ) @ ( big_co1548731110nt_int @ G_20 @ ( minus_minus_int_o @ A_135 @ ( insert_int @ X_63 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_478_minus__fold__remove,axiom,
! [B_84: nat > $o,A_133: nat > $o] :
( ( finite_finite_nat @ A_133 )
=> ( ( minus_minus_nat_o @ B_84 @ A_133 )
= ( finite326637109_nat_o
@ ^ [X_1: nat,A_134: nat > $o] : ( minus_minus_nat_o @ A_134 @ ( insert_nat @ X_1 @ bot_bot_nat_o ) )
@ B_84
@ A_133 ) ) ) ).
thf(fact_479_minus__fold__remove,axiom,
! [B_84: hoare_2091234717iple_a > $o,A_133: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_133 )
=> ( ( minus_836160335le_a_o @ B_84 @ A_133 )
= ( finite2010064629le_a_o
@ ^ [X_1: hoare_2091234717iple_a,A_134: hoare_2091234717iple_a > $o] : ( minus_836160335le_a_o @ A_134 @ ( insert1597628439iple_a @ X_1 @ bot_bo1791335050le_a_o ) )
@ B_84
@ A_133 ) ) ) ).
thf(fact_480_minus__fold__remove,axiom,
! [B_84: int > $o,A_133: int > $o] :
( ( finite_finite_int @ A_133 )
=> ( ( minus_minus_int_o @ B_84 @ A_133 )
= ( finite111936821_int_o
@ ^ [X_1: int,A_134: int > $o] : ( minus_minus_int_o @ A_134 @ ( insert_int @ X_1 @ bot_bot_int_o ) )
@ B_84
@ A_133 ) ) ) ).
thf(fact_481_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_62: hoare_2091234717iple_a,Z_27: hoare_2091234717iple_a,A_132: hoare_2091234717iple_a > $o,F_48: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_48 )
=> ( ( finite232261744iple_a @ A_132 )
=> ( ( F_48 @ X_62 @ ( finite186236040iple_a @ F_48 @ Z_27 @ A_132 ) )
= ( finite186236040iple_a @ F_48 @ ( F_48 @ X_62 @ Z_27 ) @ A_132 ) ) ) ) ).
thf(fact_482_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_62: hoare_2091234717iple_a,Z_27: hoare_2091234717iple_a > $o,A_132: hoare_2091234717iple_a > $o,F_48: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite2078358188le_a_o @ F_48 )
=> ( ( finite232261744iple_a @ A_132 )
=> ( ( F_48 @ X_62 @ ( finite2010064629le_a_o @ F_48 @ Z_27 @ A_132 ) )
= ( finite2010064629le_a_o @ F_48 @ ( F_48 @ X_62 @ Z_27 ) @ A_132 ) ) ) ) ).
thf(fact_483_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_62: nat,Z_27: nat,A_132: nat > $o,F_48: nat > nat > nat] :
( ( finite2130160977at_nat @ F_48 )
=> ( ( finite_finite_nat @ A_132 )
=> ( ( F_48 @ X_62 @ ( finite_fold_nat_nat @ F_48 @ Z_27 @ A_132 ) )
= ( finite_fold_nat_nat @ F_48 @ ( F_48 @ X_62 @ Z_27 ) @ A_132 ) ) ) ) ).
thf(fact_484_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_62: nat,Z_27: nat > $o,A_132: nat > $o,F_48: nat > ( nat > $o ) > nat > $o] :
( ( finite1690695148_nat_o @ F_48 )
=> ( ( finite_finite_nat @ A_132 )
=> ( ( F_48 @ X_62 @ ( finite326637109_nat_o @ F_48 @ Z_27 @ A_132 ) )
= ( finite326637109_nat_o @ F_48 @ ( F_48 @ X_62 @ Z_27 ) @ A_132 ) ) ) ) ).
thf(fact_485_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_62: int,Z_27: int,A_132: int > $o,F_48: int > int > int] :
( ( finite1973466193nt_int @ F_48 )
=> ( ( finite_finite_int @ A_132 )
=> ( ( F_48 @ X_62 @ ( finite_fold_int_int @ F_48 @ Z_27 @ A_132 ) )
= ( finite_fold_int_int @ F_48 @ ( F_48 @ X_62 @ Z_27 ) @ A_132 ) ) ) ) ).
thf(fact_486_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_62: int,Z_27: int > $o,A_132: int > $o,F_48: int > ( int > $o ) > int > $o] :
( ( finite1475994860_int_o @ F_48 )
=> ( ( finite_finite_int @ A_132 )
=> ( ( F_48 @ X_62 @ ( finite111936821_int_o @ F_48 @ Z_27 @ A_132 ) )
= ( finite111936821_int_o @ F_48 @ ( F_48 @ X_62 @ Z_27 ) @ A_132 ) ) ) ) ).
thf(fact_487_comp__fun__commute_Ofun__left__comm,axiom,
! [X_61: int,Y_31: int,Z_26: int,F_47: int > int > int] :
( ( finite1973466193nt_int @ F_47 )
=> ( ( F_47 @ X_61 @ ( F_47 @ Y_31 @ Z_26 ) )
= ( F_47 @ Y_31 @ ( F_47 @ X_61 @ Z_26 ) ) ) ) ).
thf(fact_488_comp__fun__commute_Ofun__left__comm,axiom,
! [X_61: nat,Y_31: nat,Z_26: nat,F_47: nat > nat > nat] :
( ( finite2130160977at_nat @ F_47 )
=> ( ( F_47 @ X_61 @ ( F_47 @ Y_31 @ Z_26 ) )
= ( F_47 @ Y_31 @ ( F_47 @ X_61 @ Z_26 ) ) ) ) ).
thf(fact_489_setsum__commute,axiom,
! [F_46: nat > int > nat,B_83: int > $o,A_131: nat > $o] :
( ( big_co387207925at_nat
@ ^ [I_1: nat] : ( big_co1740723097nt_nat @ ( F_46 @ I_1 ) @ B_83 )
@ A_131 )
= ( big_co1740723097nt_nat
@ ^ [J_1: int] :
( big_co387207925at_nat
@ ^ [I_1: nat] : ( F_46 @ I_1 @ J_1 )
@ A_131 )
@ B_83 ) ) ).
thf(fact_490_setsum__commute,axiom,
! [F_46: nat > hoare_2091234717iple_a > nat,B_83: hoare_2091234717iple_a > $o,A_131: nat > $o] :
( ( big_co387207925at_nat
@ ^ [I_1: nat] : ( big_co917763874_a_nat @ ( F_46 @ I_1 ) @ B_83 )
@ A_131 )
= ( big_co917763874_a_nat
@ ^ [J_1: hoare_2091234717iple_a] :
( big_co387207925at_nat
@ ^ [I_1: nat] : ( F_46 @ I_1 @ J_1 )
@ A_131 )
@ B_83 ) ) ).
thf(fact_491_setsum__commute,axiom,
! [F_46: int > nat > nat,B_83: nat > $o,A_131: int > $o] :
( ( big_co1740723097nt_nat
@ ^ [I_1: int] : ( big_co387207925at_nat @ ( F_46 @ I_1 ) @ B_83 )
@ A_131 )
= ( big_co387207925at_nat
@ ^ [J_1: nat] :
( big_co1740723097nt_nat
@ ^ [I_1: int] : ( F_46 @ I_1 @ J_1 )
@ A_131 )
@ B_83 ) ) ).
thf(fact_492_setsum__commute,axiom,
! [F_46: hoare_2091234717iple_a > nat > nat,B_83: nat > $o,A_131: hoare_2091234717iple_a > $o] :
( ( big_co917763874_a_nat
@ ^ [I_1: hoare_2091234717iple_a] : ( big_co387207925at_nat @ ( F_46 @ I_1 ) @ B_83 )
@ A_131 )
= ( big_co387207925at_nat
@ ^ [J_1: nat] :
( big_co917763874_a_nat
@ ^ [I_1: hoare_2091234717iple_a] : ( F_46 @ I_1 @ J_1 )
@ A_131 )
@ B_83 ) ) ).
thf(fact_493_comp__fun__commute_Ofold__equality,axiom,
! [Z_25: int,A_130: int > $o,Y_30: int,F_45: int > int > int] :
( ( finite1973466193nt_int @ F_45 )
=> ( ( finite772772422nt_int @ F_45 @ Z_25 @ A_130 @ Y_30 )
=> ( ( finite_fold_int_int @ F_45 @ Z_25 @ A_130 )
= Y_30 ) ) ) ).
thf(fact_494_comp__fun__commute_Ofold__equality,axiom,
! [Z_25: nat,A_130: nat > $o,Y_30: nat,F_45: nat > nat > nat] :
( ( finite2130160977at_nat @ F_45 )
=> ( ( finite929467206at_nat @ F_45 @ Z_25 @ A_130 @ Y_30 )
=> ( ( finite_fold_nat_nat @ F_45 @ Z_25 @ A_130 )
= Y_30 ) ) ) ).
thf(fact_495_comp__fun__commute_Ofold__equality,axiom,
! [Z_25: hoare_2091234717iple_a,A_130: hoare_2091234717iple_a > $o,Y_30: hoare_2091234717iple_a,F_45: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_45 )
=> ( ( finite1218641926iple_a @ F_45 @ Z_25 @ A_130 @ Y_30 )
=> ( ( finite186236040iple_a @ F_45 @ Z_25 @ A_130 )
= Y_30 ) ) ) ).
thf(fact_496_comp__fun__commute_Ofold__equality,axiom,
! [Z_25: int > $o,A_130: int > $o,Y_30: int > $o,F_45: int > ( int > $o ) > int > $o] :
( ( finite1475994860_int_o @ F_45 )
=> ( ( finite2015036215_int_o @ F_45 @ Z_25 @ A_130 @ Y_30 )
=> ( ( finite111936821_int_o @ F_45 @ Z_25 @ A_130 )
= Y_30 ) ) ) ).
thf(fact_497_comp__fun__commute_Ofold__equality,axiom,
! [Z_25: hoare_2091234717iple_a > $o,A_130: hoare_2091234717iple_a > $o,Y_30: hoare_2091234717iple_a > $o,F_45: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite2078358188le_a_o @ F_45 )
=> ( ( finite1003551991le_a_o @ F_45 @ Z_25 @ A_130 @ Y_30 )
=> ( ( finite2010064629le_a_o @ F_45 @ Z_25 @ A_130 )
= Y_30 ) ) ) ).
thf(fact_498_comp__fun__commute_Ofold__equality,axiom,
! [Z_25: nat > $o,A_130: nat > $o,Y_30: nat > $o,F_45: nat > ( nat > $o ) > nat > $o] :
( ( finite1690695148_nat_o @ F_45 )
=> ( ( finite82252855_nat_o @ F_45 @ Z_25 @ A_130 @ Y_30 )
=> ( ( finite326637109_nat_o @ F_45 @ Z_25 @ A_130 )
= Y_30 ) ) ) ).
thf(fact_499_fold__empty,axiom,
! [F_44: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,Z_24: hoare_2091234717iple_a] :
( ( finite186236040iple_a @ F_44 @ Z_24 @ bot_bo1791335050le_a_o )
= Z_24 ) ).
thf(fact_500_fold__empty,axiom,
! [F_44: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,Z_24: hoare_2091234717iple_a > $o] :
( ( finite2010064629le_a_o @ F_44 @ Z_24 @ bot_bo1791335050le_a_o )
= Z_24 ) ).
thf(fact_501_fold__empty,axiom,
! [F_44: int > int > int,Z_24: int] :
( ( finite_fold_int_int @ F_44 @ Z_24 @ bot_bot_int_o )
= Z_24 ) ).
thf(fact_502_fold__empty,axiom,
! [F_44: int > ( int > $o ) > int > $o,Z_24: int > $o] :
( ( finite111936821_int_o @ F_44 @ Z_24 @ bot_bot_int_o )
= Z_24 ) ).
thf(fact_503_fold__empty,axiom,
! [F_44: nat > nat > nat,Z_24: nat] :
( ( finite_fold_nat_nat @ F_44 @ Z_24 @ bot_bot_nat_o )
= Z_24 ) ).
thf(fact_504_fold__empty,axiom,
! [F_44: nat > ( nat > $o ) > nat > $o,Z_24: nat > $o] :
( ( finite326637109_nat_o @ F_44 @ Z_24 @ bot_bot_nat_o )
= Z_24 ) ).
thf(fact_505_comp__fun__commute_Ofold__insert2,axiom,
! [Z_23: nat,X_60: nat,A_129: nat > $o,F_43: nat > nat > nat] :
( ( finite2130160977at_nat @ F_43 )
=> ( ( finite_finite_nat @ A_129 )
=> ( ~ ( member_nat @ X_60 @ A_129 )
=> ( ( finite_fold_nat_nat @ F_43 @ Z_23 @ ( insert_nat @ X_60 @ A_129 ) )
= ( finite_fold_nat_nat @ F_43 @ ( F_43 @ X_60 @ Z_23 ) @ A_129 ) ) ) ) ) ).
thf(fact_506_comp__fun__commute_Ofold__insert2,axiom,
! [Z_23: nat > $o,X_60: nat,A_129: nat > $o,F_43: nat > ( nat > $o ) > nat > $o] :
( ( finite1690695148_nat_o @ F_43 )
=> ( ( finite_finite_nat @ A_129 )
=> ( ~ ( member_nat @ X_60 @ A_129 )
=> ( ( finite326637109_nat_o @ F_43 @ Z_23 @ ( insert_nat @ X_60 @ A_129 ) )
= ( finite326637109_nat_o @ F_43 @ ( F_43 @ X_60 @ Z_23 ) @ A_129 ) ) ) ) ) ).
thf(fact_507_comp__fun__commute_Ofold__insert2,axiom,
! [Z_23: hoare_2091234717iple_a,X_60: hoare_2091234717iple_a,A_129: hoare_2091234717iple_a > $o,F_43: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_43 )
=> ( ( finite232261744iple_a @ A_129 )
=> ( ~ ( member290856304iple_a @ X_60 @ A_129 )
=> ( ( finite186236040iple_a @ F_43 @ Z_23 @ ( insert1597628439iple_a @ X_60 @ A_129 ) )
= ( finite186236040iple_a @ F_43 @ ( F_43 @ X_60 @ Z_23 ) @ A_129 ) ) ) ) ) ).
thf(fact_508_comp__fun__commute_Ofold__insert2,axiom,
! [Z_23: hoare_2091234717iple_a > $o,X_60: hoare_2091234717iple_a,A_129: hoare_2091234717iple_a > $o,F_43: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite2078358188le_a_o @ F_43 )
=> ( ( finite232261744iple_a @ A_129 )
=> ( ~ ( member290856304iple_a @ X_60 @ A_129 )
=> ( ( finite2010064629le_a_o @ F_43 @ Z_23 @ ( insert1597628439iple_a @ X_60 @ A_129 ) )
= ( finite2010064629le_a_o @ F_43 @ ( F_43 @ X_60 @ Z_23 ) @ A_129 ) ) ) ) ) ).
thf(fact_509_comp__fun__commute_Ofold__insert2,axiom,
! [Z_23: int,X_60: int,A_129: int > $o,F_43: int > int > int] :
( ( finite1973466193nt_int @ F_43 )
=> ( ( finite_finite_int @ A_129 )
=> ( ~ ( member_int @ X_60 @ A_129 )
=> ( ( finite_fold_int_int @ F_43 @ Z_23 @ ( insert_int @ X_60 @ A_129 ) )
= ( finite_fold_int_int @ F_43 @ ( F_43 @ X_60 @ Z_23 ) @ A_129 ) ) ) ) ) ).
thf(fact_510_comp__fun__commute_Ofold__insert2,axiom,
! [Z_23: int > $o,X_60: int,A_129: int > $o,F_43: int > ( int > $o ) > int > $o] :
( ( finite1475994860_int_o @ F_43 )
=> ( ( finite_finite_int @ A_129 )
=> ( ~ ( member_int @ X_60 @ A_129 )
=> ( ( finite111936821_int_o @ F_43 @ Z_23 @ ( insert_int @ X_60 @ A_129 ) )
= ( finite111936821_int_o @ F_43 @ ( F_43 @ X_60 @ Z_23 ) @ A_129 ) ) ) ) ) ).
thf(fact_511_comp__fun__commute_Ofold__insert,axiom,
! [Z_22: nat,X_59: nat,A_128: nat > $o,F_42: nat > nat > nat] :
( ( finite2130160977at_nat @ F_42 )
=> ( ( finite_finite_nat @ A_128 )
=> ( ~ ( member_nat @ X_59 @ A_128 )
=> ( ( finite_fold_nat_nat @ F_42 @ Z_22 @ ( insert_nat @ X_59 @ A_128 ) )
= ( F_42 @ X_59 @ ( finite_fold_nat_nat @ F_42 @ Z_22 @ A_128 ) ) ) ) ) ) ).
thf(fact_512_comp__fun__commute_Ofold__insert,axiom,
! [Z_22: nat > $o,X_59: nat,A_128: nat > $o,F_42: nat > ( nat > $o ) > nat > $o] :
( ( finite1690695148_nat_o @ F_42 )
=> ( ( finite_finite_nat @ A_128 )
=> ( ~ ( member_nat @ X_59 @ A_128 )
=> ( ( finite326637109_nat_o @ F_42 @ Z_22 @ ( insert_nat @ X_59 @ A_128 ) )
= ( F_42 @ X_59 @ ( finite326637109_nat_o @ F_42 @ Z_22 @ A_128 ) ) ) ) ) ) ).
thf(fact_513_comp__fun__commute_Ofold__insert,axiom,
! [Z_22: hoare_2091234717iple_a,X_59: hoare_2091234717iple_a,A_128: hoare_2091234717iple_a > $o,F_42: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_42 )
=> ( ( finite232261744iple_a @ A_128 )
=> ( ~ ( member290856304iple_a @ X_59 @ A_128 )
=> ( ( finite186236040iple_a @ F_42 @ Z_22 @ ( insert1597628439iple_a @ X_59 @ A_128 ) )
= ( F_42 @ X_59 @ ( finite186236040iple_a @ F_42 @ Z_22 @ A_128 ) ) ) ) ) ) ).
thf(fact_514_comp__fun__commute_Ofold__insert,axiom,
! [Z_22: hoare_2091234717iple_a > $o,X_59: hoare_2091234717iple_a,A_128: hoare_2091234717iple_a > $o,F_42: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite2078358188le_a_o @ F_42 )
=> ( ( finite232261744iple_a @ A_128 )
=> ( ~ ( member290856304iple_a @ X_59 @ A_128 )
=> ( ( finite2010064629le_a_o @ F_42 @ Z_22 @ ( insert1597628439iple_a @ X_59 @ A_128 ) )
= ( F_42 @ X_59 @ ( finite2010064629le_a_o @ F_42 @ Z_22 @ A_128 ) ) ) ) ) ) ).
thf(fact_515_comp__fun__commute_Ofold__insert,axiom,
! [Z_22: int,X_59: int,A_128: int > $o,F_42: int > int > int] :
( ( finite1973466193nt_int @ F_42 )
=> ( ( finite_finite_int @ A_128 )
=> ( ~ ( member_int @ X_59 @ A_128 )
=> ( ( finite_fold_int_int @ F_42 @ Z_22 @ ( insert_int @ X_59 @ A_128 ) )
= ( F_42 @ X_59 @ ( finite_fold_int_int @ F_42 @ Z_22 @ A_128 ) ) ) ) ) ) ).
thf(fact_516_comp__fun__commute_Ofold__insert,axiom,
! [Z_22: int > $o,X_59: int,A_128: int > $o,F_42: int > ( int > $o ) > int > $o] :
( ( finite1475994860_int_o @ F_42 )
=> ( ( finite_finite_int @ A_128 )
=> ( ~ ( member_int @ X_59 @ A_128 )
=> ( ( finite111936821_int_o @ F_42 @ Z_22 @ ( insert_int @ X_59 @ A_128 ) )
= ( F_42 @ X_59 @ ( finite111936821_int_o @ F_42 @ Z_22 @ A_128 ) ) ) ) ) ) ).
thf(fact_517_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_21: hoare_2091234717iple_a,A_127: hoare_2091234717iple_a > $o,F_41: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_41 )
=> ( ( finite232261744iple_a @ A_127 )
=> ( finite1218641926iple_a @ F_41 @ Z_21 @ A_127 @ ( finite186236040iple_a @ F_41 @ Z_21 @ A_127 ) ) ) ) ).
thf(fact_518_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_21: hoare_2091234717iple_a > $o,A_127: hoare_2091234717iple_a > $o,F_41: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite2078358188le_a_o @ F_41 )
=> ( ( finite232261744iple_a @ A_127 )
=> ( finite1003551991le_a_o @ F_41 @ Z_21 @ A_127 @ ( finite2010064629le_a_o @ F_41 @ Z_21 @ A_127 ) ) ) ) ).
thf(fact_519_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_21: nat,A_127: nat > $o,F_41: nat > nat > nat] :
( ( finite2130160977at_nat @ F_41 )
=> ( ( finite_finite_nat @ A_127 )
=> ( finite929467206at_nat @ F_41 @ Z_21 @ A_127 @ ( finite_fold_nat_nat @ F_41 @ Z_21 @ A_127 ) ) ) ) ).
thf(fact_520_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_21: nat > $o,A_127: nat > $o,F_41: nat > ( nat > $o ) > nat > $o] :
( ( finite1690695148_nat_o @ F_41 )
=> ( ( finite_finite_nat @ A_127 )
=> ( finite82252855_nat_o @ F_41 @ Z_21 @ A_127 @ ( finite326637109_nat_o @ F_41 @ Z_21 @ A_127 ) ) ) ) ).
thf(fact_521_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_21: int,A_127: int > $o,F_41: int > int > int] :
( ( finite1973466193nt_int @ F_41 )
=> ( ( finite_finite_int @ A_127 )
=> ( finite772772422nt_int @ F_41 @ Z_21 @ A_127 @ ( finite_fold_int_int @ F_41 @ Z_21 @ A_127 ) ) ) ) ).
thf(fact_522_comp__fun__commute_Ofold__graph__fold,axiom,
! [Z_21: int > $o,A_127: int > $o,F_41: int > ( int > $o ) > int > $o] :
( ( finite1475994860_int_o @ F_41 )
=> ( ( finite_finite_int @ A_127 )
=> ( finite2015036215_int_o @ F_41 @ Z_21 @ A_127 @ ( finite111936821_int_o @ F_41 @ Z_21 @ A_127 ) ) ) ) ).
thf(fact_523_setsum__product,axiom,
! [F_40: nat > nat,A_126: nat > $o,G_19: int > nat,B_82: int > $o] :
( ( times_times_nat @ ( big_co387207925at_nat @ F_40 @ A_126 ) @ ( big_co1740723097nt_nat @ G_19 @ B_82 ) )
= ( big_co387207925at_nat
@ ^ [I_1: nat] :
( big_co1740723097nt_nat
@ ^ [J_1: int] : ( times_times_nat @ ( F_40 @ I_1 ) @ ( G_19 @ J_1 ) )
@ B_82 )
@ A_126 ) ) ).
thf(fact_524_setsum__product,axiom,
! [F_40: nat > nat,A_126: nat > $o,G_19: hoare_2091234717iple_a > nat,B_82: hoare_2091234717iple_a > $o] :
( ( times_times_nat @ ( big_co387207925at_nat @ F_40 @ A_126 ) @ ( big_co917763874_a_nat @ G_19 @ B_82 ) )
= ( big_co387207925at_nat
@ ^ [I_1: nat] :
( big_co917763874_a_nat
@ ^ [J_1: hoare_2091234717iple_a] : ( times_times_nat @ ( F_40 @ I_1 ) @ ( G_19 @ J_1 ) )
@ B_82 )
@ A_126 ) ) ).
thf(fact_525_setsum__product,axiom,
! [F_40: int > nat,A_126: int > $o,G_19: nat > nat,B_82: nat > $o] :
( ( times_times_nat @ ( big_co1740723097nt_nat @ F_40 @ A_126 ) @ ( big_co387207925at_nat @ G_19 @ B_82 ) )
= ( big_co1740723097nt_nat
@ ^ [I_1: int] :
( big_co387207925at_nat
@ ^ [J_1: nat] : ( times_times_nat @ ( F_40 @ I_1 ) @ ( G_19 @ J_1 ) )
@ B_82 )
@ A_126 ) ) ).
thf(fact_526_setsum__product,axiom,
! [F_40: hoare_2091234717iple_a > nat,A_126: hoare_2091234717iple_a > $o,G_19: nat > nat,B_82: nat > $o] :
( ( times_times_nat @ ( big_co917763874_a_nat @ F_40 @ A_126 ) @ ( big_co387207925at_nat @ G_19 @ B_82 ) )
= ( big_co917763874_a_nat
@ ^ [I_1: hoare_2091234717iple_a] :
( big_co387207925at_nat
@ ^ [J_1: nat] : ( times_times_nat @ ( F_40 @ I_1 ) @ ( G_19 @ J_1 ) )
@ B_82 )
@ A_126 ) ) ).
thf(fact_527_setsum__left__distrib,axiom,
! [F_39: nat > nat,A_125: nat > $o,R_3: nat] :
( ( times_times_nat @ ( big_co387207925at_nat @ F_39 @ A_125 ) @ R_3 )
= ( big_co387207925at_nat
@ ^ [N_1: nat] : ( times_times_nat @ ( F_39 @ N_1 ) @ R_3 )
@ A_125 ) ) ).
thf(fact_528_setsum__left__distrib,axiom,
! [F_39: int > nat,A_125: int > $o,R_3: nat] :
( ( times_times_nat @ ( big_co1740723097nt_nat @ F_39 @ A_125 ) @ R_3 )
= ( big_co1740723097nt_nat
@ ^ [N_1: int] : ( times_times_nat @ ( F_39 @ N_1 ) @ R_3 )
@ A_125 ) ) ).
thf(fact_529_setsum__left__distrib,axiom,
! [F_39: hoare_2091234717iple_a > nat,A_125: hoare_2091234717iple_a > $o,R_3: nat] :
( ( times_times_nat @ ( big_co917763874_a_nat @ F_39 @ A_125 ) @ R_3 )
= ( big_co917763874_a_nat
@ ^ [N_1: hoare_2091234717iple_a] : ( times_times_nat @ ( F_39 @ N_1 ) @ R_3 )
@ A_125 ) ) ).
thf(fact_530_setsum__right__distrib,axiom,
! [R_2: nat,F_38: nat > nat,A_124: nat > $o] :
( ( times_times_nat @ R_2 @ ( big_co387207925at_nat @ F_38 @ A_124 ) )
= ( big_co387207925at_nat
@ ^ [N_1: nat] : ( times_times_nat @ R_2 @ ( F_38 @ N_1 ) )
@ A_124 ) ) ).
thf(fact_531_setsum__right__distrib,axiom,
! [R_2: nat,F_38: int > nat,A_124: int > $o] :
( ( times_times_nat @ R_2 @ ( big_co1740723097nt_nat @ F_38 @ A_124 ) )
= ( big_co1740723097nt_nat
@ ^ [N_1: int] : ( times_times_nat @ R_2 @ ( F_38 @ N_1 ) )
@ A_124 ) ) ).
thf(fact_532_setsum__right__distrib,axiom,
! [R_2: nat,F_38: hoare_2091234717iple_a > nat,A_124: hoare_2091234717iple_a > $o] :
( ( times_times_nat @ R_2 @ ( big_co917763874_a_nat @ F_38 @ A_124 ) )
= ( big_co917763874_a_nat
@ ^ [N_1: hoare_2091234717iple_a] : ( times_times_nat @ R_2 @ ( F_38 @ N_1 ) )
@ A_124 ) ) ).
thf(fact_533_comp__fun__commute,axiom,
finite2130160977at_nat @ times_times_nat ).
thf(fact_534_comp__fun__commute,axiom,
finite1973466193nt_int @ times_times_int ).
thf(fact_535_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_29: int,Z_20: int,A_123: int > $o,X_58: int,F_37: int > int > int] :
( ( finite1973466193nt_int @ F_37 )
=> ( ( finite772772422nt_int @ F_37 @ Z_20 @ A_123 @ X_58 )
=> ( ( finite772772422nt_int @ F_37 @ Z_20 @ A_123 @ Y_29 )
=> ( Y_29 = X_58 ) ) ) ) ).
thf(fact_536_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_29: nat,Z_20: nat,A_123: nat > $o,X_58: nat,F_37: nat > nat > nat] :
( ( finite2130160977at_nat @ F_37 )
=> ( ( finite929467206at_nat @ F_37 @ Z_20 @ A_123 @ X_58 )
=> ( ( finite929467206at_nat @ F_37 @ Z_20 @ A_123 @ Y_29 )
=> ( Y_29 = X_58 ) ) ) ) ).
thf(fact_537_comp__fun__commute_Ofold__graph__determ,axiom,
! [Y_29: hoare_2091234717iple_a,Z_20: hoare_2091234717iple_a,A_123: hoare_2091234717iple_a > $o,X_58: hoare_2091234717iple_a,F_37: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_37 )
=> ( ( finite1218641926iple_a @ F_37 @ Z_20 @ A_123 @ X_58 )
=> ( ( finite1218641926iple_a @ F_37 @ Z_20 @ A_123 @ Y_29 )
=> ( Y_29 = X_58 ) ) ) ) ).
thf(fact_538_setprod_Odistrib,axiom,
! [G_18: hoare_2091234717iple_a > nat,H_3: hoare_2091234717iple_a > nat,A_122: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_122 )
=> ( ( big_co1393145363_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : ( times_times_nat @ ( G_18 @ X_1 ) @ ( H_3 @ X_1 ) )
@ A_122 )
= ( times_times_nat @ ( big_co1393145363_a_nat @ G_18 @ A_122 ) @ ( big_co1393145363_a_nat @ H_3 @ A_122 ) ) ) ) ).
thf(fact_539_setprod_Odistrib,axiom,
! [G_18: hoare_2091234717iple_a > int,H_3: hoare_2091234717iple_a > int,A_122: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_122 )
=> ( ( big_co2030419055_a_int
@ ^ [X_1: hoare_2091234717iple_a] : ( times_times_int @ ( G_18 @ X_1 ) @ ( H_3 @ X_1 ) )
@ A_122 )
= ( times_times_int @ ( big_co2030419055_a_int @ G_18 @ A_122 ) @ ( big_co2030419055_a_int @ H_3 @ A_122 ) ) ) ) ).
thf(fact_540_comp__fun__commute_Ofold__rec,axiom,
! [Z_19: nat,X_57: nat,A_121: nat > $o,F_36: nat > nat > nat] :
( ( finite2130160977at_nat @ F_36 )
=> ( ( finite_finite_nat @ A_121 )
=> ( ( member_nat @ X_57 @ A_121 )
=> ( ( finite_fold_nat_nat @ F_36 @ Z_19 @ A_121 )
= ( F_36 @ X_57 @ ( finite_fold_nat_nat @ F_36 @ Z_19 @ ( minus_minus_nat_o @ A_121 @ ( insert_nat @ X_57 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ).
thf(fact_541_comp__fun__commute_Ofold__rec,axiom,
! [Z_19: nat > $o,X_57: nat,A_121: nat > $o,F_36: nat > ( nat > $o ) > nat > $o] :
( ( finite1690695148_nat_o @ F_36 )
=> ( ( finite_finite_nat @ A_121 )
=> ( ( member_nat @ X_57 @ A_121 )
=> ( ( finite326637109_nat_o @ F_36 @ Z_19 @ A_121 )
= ( F_36 @ X_57 @ ( finite326637109_nat_o @ F_36 @ Z_19 @ ( minus_minus_nat_o @ A_121 @ ( insert_nat @ X_57 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ).
thf(fact_542_comp__fun__commute_Ofold__rec,axiom,
! [Z_19: hoare_2091234717iple_a,X_57: hoare_2091234717iple_a,A_121: hoare_2091234717iple_a > $o,F_36: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_36 )
=> ( ( finite232261744iple_a @ A_121 )
=> ( ( member290856304iple_a @ X_57 @ A_121 )
=> ( ( finite186236040iple_a @ F_36 @ Z_19 @ A_121 )
= ( F_36 @ X_57 @ ( finite186236040iple_a @ F_36 @ Z_19 @ ( minus_836160335le_a_o @ A_121 @ ( insert1597628439iple_a @ X_57 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ) ).
thf(fact_543_comp__fun__commute_Ofold__rec,axiom,
! [Z_19: hoare_2091234717iple_a > $o,X_57: hoare_2091234717iple_a,A_121: hoare_2091234717iple_a > $o,F_36: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite2078358188le_a_o @ F_36 )
=> ( ( finite232261744iple_a @ A_121 )
=> ( ( member290856304iple_a @ X_57 @ A_121 )
=> ( ( finite2010064629le_a_o @ F_36 @ Z_19 @ A_121 )
= ( F_36 @ X_57 @ ( finite2010064629le_a_o @ F_36 @ Z_19 @ ( minus_836160335le_a_o @ A_121 @ ( insert1597628439iple_a @ X_57 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ) ).
thf(fact_544_comp__fun__commute_Ofold__rec,axiom,
! [Z_19: int,X_57: int,A_121: int > $o,F_36: int > int > int] :
( ( finite1973466193nt_int @ F_36 )
=> ( ( finite_finite_int @ A_121 )
=> ( ( member_int @ X_57 @ A_121 )
=> ( ( finite_fold_int_int @ F_36 @ Z_19 @ A_121 )
= ( F_36 @ X_57 @ ( finite_fold_int_int @ F_36 @ Z_19 @ ( minus_minus_int_o @ A_121 @ ( insert_int @ X_57 @ bot_bot_int_o ) ) ) ) ) ) ) ) ).
thf(fact_545_comp__fun__commute_Ofold__rec,axiom,
! [Z_19: int > $o,X_57: int,A_121: int > $o,F_36: int > ( int > $o ) > int > $o] :
( ( finite1475994860_int_o @ F_36 )
=> ( ( finite_finite_int @ A_121 )
=> ( ( member_int @ X_57 @ A_121 )
=> ( ( finite111936821_int_o @ F_36 @ Z_19 @ A_121 )
= ( F_36 @ X_57 @ ( finite111936821_int_o @ F_36 @ Z_19 @ ( minus_minus_int_o @ A_121 @ ( insert_int @ X_57 @ bot_bot_int_o ) ) ) ) ) ) ) ) ).
thf(fact_546_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_18: nat,X_56: nat,A_120: nat > $o,F_35: nat > nat > nat] :
( ( finite2130160977at_nat @ F_35 )
=> ( ( finite_finite_nat @ A_120 )
=> ( ( finite_fold_nat_nat @ F_35 @ Z_18 @ ( insert_nat @ X_56 @ A_120 ) )
= ( F_35 @ X_56 @ ( finite_fold_nat_nat @ F_35 @ Z_18 @ ( minus_minus_nat_o @ A_120 @ ( insert_nat @ X_56 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_547_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_18: nat > $o,X_56: nat,A_120: nat > $o,F_35: nat > ( nat > $o ) > nat > $o] :
( ( finite1690695148_nat_o @ F_35 )
=> ( ( finite_finite_nat @ A_120 )
=> ( ( finite326637109_nat_o @ F_35 @ Z_18 @ ( insert_nat @ X_56 @ A_120 ) )
= ( F_35 @ X_56 @ ( finite326637109_nat_o @ F_35 @ Z_18 @ ( minus_minus_nat_o @ A_120 @ ( insert_nat @ X_56 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_548_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_18: hoare_2091234717iple_a,X_56: hoare_2091234717iple_a,A_120: hoare_2091234717iple_a > $o,F_35: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_35 )
=> ( ( finite232261744iple_a @ A_120 )
=> ( ( finite186236040iple_a @ F_35 @ Z_18 @ ( insert1597628439iple_a @ X_56 @ A_120 ) )
= ( F_35 @ X_56 @ ( finite186236040iple_a @ F_35 @ Z_18 @ ( minus_836160335le_a_o @ A_120 @ ( insert1597628439iple_a @ X_56 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_549_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_18: hoare_2091234717iple_a > $o,X_56: hoare_2091234717iple_a,A_120: hoare_2091234717iple_a > $o,F_35: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite2078358188le_a_o @ F_35 )
=> ( ( finite232261744iple_a @ A_120 )
=> ( ( finite2010064629le_a_o @ F_35 @ Z_18 @ ( insert1597628439iple_a @ X_56 @ A_120 ) )
= ( F_35 @ X_56 @ ( finite2010064629le_a_o @ F_35 @ Z_18 @ ( minus_836160335le_a_o @ A_120 @ ( insert1597628439iple_a @ X_56 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_550_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_18: int,X_56: int,A_120: int > $o,F_35: int > int > int] :
( ( finite1973466193nt_int @ F_35 )
=> ( ( finite_finite_int @ A_120 )
=> ( ( finite_fold_int_int @ F_35 @ Z_18 @ ( insert_int @ X_56 @ A_120 ) )
= ( F_35 @ X_56 @ ( finite_fold_int_int @ F_35 @ Z_18 @ ( minus_minus_int_o @ A_120 @ ( insert_int @ X_56 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_551_comp__fun__commute_Ofold__insert__remove,axiom,
! [Z_18: int > $o,X_56: int,A_120: int > $o,F_35: int > ( int > $o ) > int > $o] :
( ( finite1475994860_int_o @ F_35 )
=> ( ( finite_finite_int @ A_120 )
=> ( ( finite111936821_int_o @ F_35 @ Z_18 @ ( insert_int @ X_56 @ A_120 ) )
= ( F_35 @ X_56 @ ( finite111936821_int_o @ F_35 @ Z_18 @ ( minus_minus_int_o @ A_120 @ ( insert_int @ X_56 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_552_setprod__insert,axiom,
! [F_34: nat > nat,A_119: nat,A_118: nat > $o] :
( ( finite_finite_nat @ A_118 )
=> ( ~ ( member_nat @ A_119 @ A_118 )
=> ( ( big_co1705425894at_nat @ F_34 @ ( insert_nat @ A_119 @ A_118 ) )
= ( times_times_nat @ ( F_34 @ A_119 ) @ ( big_co1705425894at_nat @ F_34 @ A_118 ) ) ) ) ) ).
thf(fact_553_setprod__insert,axiom,
! [F_34: hoare_2091234717iple_a > nat,A_119: hoare_2091234717iple_a,A_118: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_118 )
=> ( ~ ( member290856304iple_a @ A_119 @ A_118 )
=> ( ( big_co1393145363_a_nat @ F_34 @ ( insert1597628439iple_a @ A_119 @ A_118 ) )
= ( times_times_nat @ ( F_34 @ A_119 ) @ ( big_co1393145363_a_nat @ F_34 @ A_118 ) ) ) ) ) ).
thf(fact_554_setprod__insert,axiom,
! [F_34: nat > int,A_119: nat,A_118: nat > $o] :
( ( finite_finite_nat @ A_118 )
=> ( ~ ( member_nat @ A_119 @ A_118 )
=> ( ( big_co195215938at_int @ F_34 @ ( insert_nat @ A_119 @ A_118 ) )
= ( times_times_int @ ( F_34 @ A_119 ) @ ( big_co195215938at_int @ F_34 @ A_118 ) ) ) ) ) ).
thf(fact_555_setprod__insert,axiom,
! [F_34: hoare_2091234717iple_a > int,A_119: hoare_2091234717iple_a,A_118: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_118 )
=> ( ~ ( member290856304iple_a @ A_119 @ A_118 )
=> ( ( big_co2030419055_a_int @ F_34 @ ( insert1597628439iple_a @ A_119 @ A_118 ) )
= ( times_times_int @ ( F_34 @ A_119 ) @ ( big_co2030419055_a_int @ F_34 @ A_118 ) ) ) ) ) ).
thf(fact_556_setprod_Oinsert,axiom,
! [G_17: nat > nat,X_55: nat,A_117: nat > $o] :
( ( finite_finite_nat @ A_117 )
=> ( ~ ( member_nat @ X_55 @ A_117 )
=> ( ( big_co1705425894at_nat @ G_17 @ ( insert_nat @ X_55 @ A_117 ) )
= ( times_times_nat @ ( G_17 @ X_55 ) @ ( big_co1705425894at_nat @ G_17 @ A_117 ) ) ) ) ) ).
thf(fact_557_setprod_Oinsert,axiom,
! [G_17: hoare_2091234717iple_a > nat,X_55: hoare_2091234717iple_a,A_117: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_117 )
=> ( ~ ( member290856304iple_a @ X_55 @ A_117 )
=> ( ( big_co1393145363_a_nat @ G_17 @ ( insert1597628439iple_a @ X_55 @ A_117 ) )
= ( times_times_nat @ ( G_17 @ X_55 ) @ ( big_co1393145363_a_nat @ G_17 @ A_117 ) ) ) ) ) ).
thf(fact_558_setprod_Oinsert,axiom,
! [G_17: nat > int,X_55: nat,A_117: nat > $o] :
( ( finite_finite_nat @ A_117 )
=> ( ~ ( member_nat @ X_55 @ A_117 )
=> ( ( big_co195215938at_int @ G_17 @ ( insert_nat @ X_55 @ A_117 ) )
= ( times_times_int @ ( G_17 @ X_55 ) @ ( big_co195215938at_int @ G_17 @ A_117 ) ) ) ) ) ).
thf(fact_559_setprod_Oinsert,axiom,
! [G_17: hoare_2091234717iple_a > int,X_55: hoare_2091234717iple_a,A_117: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_117 )
=> ( ~ ( member290856304iple_a @ X_55 @ A_117 )
=> ( ( big_co2030419055_a_int @ G_17 @ ( insert1597628439iple_a @ X_55 @ A_117 ) )
= ( times_times_int @ ( G_17 @ X_55 ) @ ( big_co2030419055_a_int @ G_17 @ A_117 ) ) ) ) ) ).
thf(fact_560_setsum__diff1__nat,axiom,
! [F_33: nat > nat,A_116: nat,A_115: nat > $o] :
( ( ( member_nat @ A_116 @ A_115 )
=> ( ( big_co387207925at_nat @ F_33 @ ( minus_minus_nat_o @ A_115 @ ( insert_nat @ A_116 @ bot_bot_nat_o ) ) )
= ( minus_minus_nat @ ( big_co387207925at_nat @ F_33 @ A_115 ) @ ( F_33 @ A_116 ) ) ) )
& ( ~ ( member_nat @ A_116 @ A_115 )
=> ( ( big_co387207925at_nat @ F_33 @ ( minus_minus_nat_o @ A_115 @ ( insert_nat @ A_116 @ bot_bot_nat_o ) ) )
= ( big_co387207925at_nat @ F_33 @ A_115 ) ) ) ) ).
thf(fact_561_setsum__diff1__nat,axiom,
! [F_33: hoare_2091234717iple_a > nat,A_116: hoare_2091234717iple_a,A_115: hoare_2091234717iple_a > $o] :
( ( ( member290856304iple_a @ A_116 @ A_115 )
=> ( ( big_co917763874_a_nat @ F_33 @ ( minus_836160335le_a_o @ A_115 @ ( insert1597628439iple_a @ A_116 @ bot_bo1791335050le_a_o ) ) )
= ( minus_minus_nat @ ( big_co917763874_a_nat @ F_33 @ A_115 ) @ ( F_33 @ A_116 ) ) ) )
& ( ~ ( member290856304iple_a @ A_116 @ A_115 )
=> ( ( big_co917763874_a_nat @ F_33 @ ( minus_836160335le_a_o @ A_115 @ ( insert1597628439iple_a @ A_116 @ bot_bo1791335050le_a_o ) ) )
= ( big_co917763874_a_nat @ F_33 @ A_115 ) ) ) ) ).
thf(fact_562_setsum__diff1__nat,axiom,
! [F_33: int > nat,A_116: int,A_115: int > $o] :
( ( ( member_int @ A_116 @ A_115 )
=> ( ( big_co1740723097nt_nat @ F_33 @ ( minus_minus_int_o @ A_115 @ ( insert_int @ A_116 @ bot_bot_int_o ) ) )
= ( minus_minus_nat @ ( big_co1740723097nt_nat @ F_33 @ A_115 ) @ ( F_33 @ A_116 ) ) ) )
& ( ~ ( member_int @ A_116 @ A_115 )
=> ( ( big_co1740723097nt_nat @ F_33 @ ( minus_minus_int_o @ A_115 @ ( insert_int @ A_116 @ bot_bot_int_o ) ) )
= ( big_co1740723097nt_nat @ F_33 @ A_115 ) ) ) ) ).
thf(fact_563_fold__def,axiom,
! [F_32: int > int > int,Z_17: int,A_114: int > $o] :
( ( finite_fold_int_int @ F_32 @ Z_17 @ A_114 )
= ( the_int @ ( finite772772422nt_int @ F_32 @ Z_17 @ A_114 ) ) ) ).
thf(fact_564_fold__def,axiom,
! [F_32: nat > nat > nat,Z_17: nat,A_114: nat > $o] :
( ( finite_fold_nat_nat @ F_32 @ Z_17 @ A_114 )
= ( the_nat @ ( finite929467206at_nat @ F_32 @ Z_17 @ A_114 ) ) ) ).
thf(fact_565_fold__def,axiom,
! [F_32: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,Z_17: hoare_2091234717iple_a,A_114: hoare_2091234717iple_a > $o] :
( ( finite186236040iple_a @ F_32 @ Z_17 @ A_114 )
= ( the_Ho1471183438iple_a @ ( finite1218641926iple_a @ F_32 @ Z_17 @ A_114 ) ) ) ).
thf(fact_566_fold__def,axiom,
! [F_32: int > ( int > $o ) > int > $o,Z_17: int > $o,A_114: int > $o] :
( ( finite111936821_int_o @ F_32 @ Z_17 @ A_114 )
= ( the_int_o @ ( finite2015036215_int_o @ F_32 @ Z_17 @ A_114 ) ) ) ).
thf(fact_567_fold__def,axiom,
! [F_32: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o,Z_17: hoare_2091234717iple_a > $o,A_114: hoare_2091234717iple_a > $o] :
( ( finite2010064629le_a_o @ F_32 @ Z_17 @ A_114 )
= ( the_Ho2077879471le_a_o @ ( finite1003551991le_a_o @ F_32 @ Z_17 @ A_114 ) ) ) ).
thf(fact_568_fold__def,axiom,
! [F_32: nat > ( nat > $o ) > nat > $o,Z_17: nat > $o,A_114: nat > $o] :
( ( finite326637109_nat_o @ F_32 @ Z_17 @ A_114 )
= ( the_nat_o @ ( finite82252855_nat_o @ F_32 @ Z_17 @ A_114 ) ) ) ).
thf(fact_569_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_16: nat,X_54: nat,A_113: nat > $o,F_31: nat > nat > nat] :
( ( finite1860950092at_nat @ F_31 )
=> ( ( finite_finite_nat @ A_113 )
=> ( ( finite_fold_nat_nat @ F_31 @ Z_16 @ ( insert_nat @ X_54 @ A_113 ) )
= ( finite_fold_nat_nat @ F_31 @ ( F_31 @ X_54 @ Z_16 ) @ A_113 ) ) ) ) ).
thf(fact_570_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_16: nat > $o,X_54: nat,A_113: nat > $o,F_31: nat > ( nat > $o ) > nat > $o] :
( ( finite389864113_nat_o @ F_31 )
=> ( ( finite_finite_nat @ A_113 )
=> ( ( finite326637109_nat_o @ F_31 @ Z_16 @ ( insert_nat @ X_54 @ A_113 ) )
= ( finite326637109_nat_o @ F_31 @ ( F_31 @ X_54 @ Z_16 ) @ A_113 ) ) ) ) ).
thf(fact_571_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_16: hoare_2091234717iple_a,X_54: hoare_2091234717iple_a,A_113: hoare_2091234717iple_a > $o,F_31: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite1776604428iple_a @ F_31 )
=> ( ( finite232261744iple_a @ A_113 )
=> ( ( finite186236040iple_a @ F_31 @ Z_16 @ ( insert1597628439iple_a @ X_54 @ A_113 ) )
= ( finite186236040iple_a @ F_31 @ ( F_31 @ X_54 @ Z_16 ) @ A_113 ) ) ) ) ).
thf(fact_572_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_16: hoare_2091234717iple_a > $o,X_54: hoare_2091234717iple_a,A_113: hoare_2091234717iple_a > $o,F_31: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite438582129le_a_o @ F_31 )
=> ( ( finite232261744iple_a @ A_113 )
=> ( ( finite2010064629le_a_o @ F_31 @ Z_16 @ ( insert1597628439iple_a @ X_54 @ A_113 ) )
= ( finite2010064629le_a_o @ F_31 @ ( F_31 @ X_54 @ Z_16 ) @ A_113 ) ) ) ) ).
thf(fact_573_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_16: int,X_54: int,A_113: int > $o,F_31: int > int > int] :
( ( finite1704255308nt_int @ F_31 )
=> ( ( finite_finite_int @ A_113 )
=> ( ( finite_fold_int_int @ F_31 @ Z_16 @ ( insert_int @ X_54 @ A_113 ) )
= ( finite_fold_int_int @ F_31 @ ( F_31 @ X_54 @ Z_16 ) @ A_113 ) ) ) ) ).
thf(fact_574_comp__fun__idem_Ofold__insert__idem2,axiom,
! [Z_16: int > $o,X_54: int,A_113: int > $o,F_31: int > ( int > $o ) > int > $o] :
( ( finite175163825_int_o @ F_31 )
=> ( ( finite_finite_int @ A_113 )
=> ( ( finite111936821_int_o @ F_31 @ Z_16 @ ( insert_int @ X_54 @ A_113 ) )
= ( finite111936821_int_o @ F_31 @ ( F_31 @ X_54 @ Z_16 ) @ A_113 ) ) ) ) ).
thf(fact_575_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_15: nat,X_53: nat,A_112: nat > $o,F_30: nat > nat > nat] :
( ( finite1860950092at_nat @ F_30 )
=> ( ( finite_finite_nat @ A_112 )
=> ( ( finite_fold_nat_nat @ F_30 @ Z_15 @ ( insert_nat @ X_53 @ A_112 ) )
= ( F_30 @ X_53 @ ( finite_fold_nat_nat @ F_30 @ Z_15 @ A_112 ) ) ) ) ) ).
thf(fact_576_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_15: nat > $o,X_53: nat,A_112: nat > $o,F_30: nat > ( nat > $o ) > nat > $o] :
( ( finite389864113_nat_o @ F_30 )
=> ( ( finite_finite_nat @ A_112 )
=> ( ( finite326637109_nat_o @ F_30 @ Z_15 @ ( insert_nat @ X_53 @ A_112 ) )
= ( F_30 @ X_53 @ ( finite326637109_nat_o @ F_30 @ Z_15 @ A_112 ) ) ) ) ) ).
thf(fact_577_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_15: hoare_2091234717iple_a,X_53: hoare_2091234717iple_a,A_112: hoare_2091234717iple_a > $o,F_30: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite1776604428iple_a @ F_30 )
=> ( ( finite232261744iple_a @ A_112 )
=> ( ( finite186236040iple_a @ F_30 @ Z_15 @ ( insert1597628439iple_a @ X_53 @ A_112 ) )
= ( F_30 @ X_53 @ ( finite186236040iple_a @ F_30 @ Z_15 @ A_112 ) ) ) ) ) ).
thf(fact_578_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_15: hoare_2091234717iple_a > $o,X_53: hoare_2091234717iple_a,A_112: hoare_2091234717iple_a > $o,F_30: hoare_2091234717iple_a > ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a > $o] :
( ( finite438582129le_a_o @ F_30 )
=> ( ( finite232261744iple_a @ A_112 )
=> ( ( finite2010064629le_a_o @ F_30 @ Z_15 @ ( insert1597628439iple_a @ X_53 @ A_112 ) )
= ( F_30 @ X_53 @ ( finite2010064629le_a_o @ F_30 @ Z_15 @ A_112 ) ) ) ) ) ).
thf(fact_579_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_15: int,X_53: int,A_112: int > $o,F_30: int > int > int] :
( ( finite1704255308nt_int @ F_30 )
=> ( ( finite_finite_int @ A_112 )
=> ( ( finite_fold_int_int @ F_30 @ Z_15 @ ( insert_int @ X_53 @ A_112 ) )
= ( F_30 @ X_53 @ ( finite_fold_int_int @ F_30 @ Z_15 @ A_112 ) ) ) ) ) ).
thf(fact_580_comp__fun__idem_Ofold__insert__idem,axiom,
! [Z_15: int > $o,X_53: int,A_112: int > $o,F_30: int > ( int > $o ) > int > $o] :
( ( finite175163825_int_o @ F_30 )
=> ( ( finite_finite_int @ A_112 )
=> ( ( finite111936821_int_o @ F_30 @ Z_15 @ ( insert_int @ X_53 @ A_112 ) )
= ( F_30 @ X_53 @ ( finite111936821_int_o @ F_30 @ Z_15 @ A_112 ) ) ) ) ) ).
thf(fact_581_folding__one_Oeq__fold_H,axiom,
! [X_52: nat,A_111: nat > $o,F_29: nat > nat > nat,F_28: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_29 @ F_28 )
=> ( ( finite_finite_nat @ A_111 )
=> ( ~ ( member_nat @ X_52 @ A_111 )
=> ( ( F_28 @ ( insert_nat @ X_52 @ A_111 ) )
= ( finite_fold_nat_nat @ F_29 @ X_52 @ A_111 ) ) ) ) ) ).
thf(fact_582_folding__one_Oeq__fold_H,axiom,
! [X_52: hoare_2091234717iple_a,A_111: hoare_2091234717iple_a > $o,F_29: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_28: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite247037978iple_a @ F_29 @ F_28 )
=> ( ( finite232261744iple_a @ A_111 )
=> ( ~ ( member290856304iple_a @ X_52 @ A_111 )
=> ( ( F_28 @ ( insert1597628439iple_a @ X_52 @ A_111 ) )
= ( finite186236040iple_a @ F_29 @ X_52 @ A_111 ) ) ) ) ) ).
thf(fact_583_folding__one_Oeq__fold_H,axiom,
! [X_52: int,A_111: int > $o,F_29: int > int > int,F_28: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_29 @ F_28 )
=> ( ( finite_finite_int @ A_111 )
=> ( ~ ( member_int @ X_52 @ A_111 )
=> ( ( F_28 @ ( insert_int @ X_52 @ A_111 ) )
= ( finite_fold_int_int @ F_29 @ X_52 @ A_111 ) ) ) ) ) ).
thf(fact_584_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_110: nat,A_109: nat > $o,F_27: nat > nat > nat,F_26: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_27 @ F_26 )
=> ( ( finite_finite_nat @ A_109 )
=> ( ( F_26 @ ( insert_nat @ A_110 @ A_109 ) )
= ( finite_fold_nat_nat @ F_27 @ A_110 @ A_109 ) ) ) ) ).
thf(fact_585_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_110: hoare_2091234717iple_a,A_109: hoare_2091234717iple_a > $o,F_27: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_26: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_27 @ F_26 )
=> ( ( finite232261744iple_a @ A_109 )
=> ( ( F_26 @ ( insert1597628439iple_a @ A_110 @ A_109 ) )
= ( finite186236040iple_a @ F_27 @ A_110 @ A_109 ) ) ) ) ).
thf(fact_586_folding__one__idem_Oeq__fold__idem_H,axiom,
! [A_110: int,A_109: int > $o,F_27: int > int > int,F_26: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_27 @ F_26 )
=> ( ( finite_finite_int @ A_109 )
=> ( ( F_26 @ ( insert_int @ A_110 @ A_109 ) )
= ( finite_fold_int_int @ F_27 @ A_110 @ A_109 ) ) ) ) ).
thf(fact_587_fold1__eq__fold,axiom,
! [A_108: nat,A_107: nat > $o] :
( ( finite_finite_nat @ A_107 )
=> ( ~ ( member_nat @ A_108 @ A_107 )
=> ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ A_108 @ A_107 ) )
= ( finite_fold_nat_nat @ times_times_nat @ A_108 @ A_107 ) ) ) ) ).
thf(fact_588_fold1__eq__fold,axiom,
! [A_108: int,A_107: int > $o] :
( ( finite_finite_int @ A_107 )
=> ( ~ ( member_int @ A_108 @ A_107 )
=> ( ( finite_fold1_int @ times_times_int @ ( insert_int @ A_108 @ A_107 ) )
= ( finite_fold_int_int @ times_times_int @ A_108 @ A_107 ) ) ) ) ).
thf(fact_589_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_14: nat,X_51: nat,A_106: nat > $o,V: nat,F_25: nat > nat > nat] :
( ( finite2130160977at_nat @ F_25 )
=> ( ( finite929467206at_nat @ F_25 @ Z_14 @ ( insert_nat @ X_51 @ A_106 ) @ V )
=> ( ~ ( member_nat @ X_51 @ A_106 )
=> ~ ! [Y_28: nat] :
( ( V
= ( F_25 @ X_51 @ Y_28 ) )
=> ~ ( finite929467206at_nat @ F_25 @ Z_14 @ A_106 @ Y_28 ) ) ) ) ) ).
thf(fact_590_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_14: hoare_2091234717iple_a,X_51: hoare_2091234717iple_a,A_106: hoare_2091234717iple_a > $o,V: hoare_2091234717iple_a,F_25: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a] :
( ( finite408405521iple_a @ F_25 )
=> ( ( finite1218641926iple_a @ F_25 @ Z_14 @ ( insert1597628439iple_a @ X_51 @ A_106 ) @ V )
=> ( ~ ( member290856304iple_a @ X_51 @ A_106 )
=> ~ ! [Y_28: hoare_2091234717iple_a] :
( ( V
= ( F_25 @ X_51 @ Y_28 ) )
=> ~ ( finite1218641926iple_a @ F_25 @ Z_14 @ A_106 @ Y_28 ) ) ) ) ) ).
thf(fact_591_comp__fun__commute_Ofold__graph__insertE,axiom,
! [Z_14: int,X_51: int,A_106: int > $o,V: int,F_25: int > int > int] :
( ( finite1973466193nt_int @ F_25 )
=> ( ( finite772772422nt_int @ F_25 @ Z_14 @ ( insert_int @ X_51 @ A_106 ) @ V )
=> ( ~ ( member_int @ X_51 @ A_106 )
=> ~ ! [Y_28: int] :
( ( V
= ( F_25 @ X_51 @ Y_28 ) )
=> ~ ( finite772772422nt_int @ F_25 @ Z_14 @ A_106 @ Y_28 ) ) ) ) ) ).
thf(fact_592_setsum__image__gen,axiom,
! [G_16: hoare_2091234717iple_a > nat,F_24: hoare_2091234717iple_a > nat,S_4: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ S_4 )
=> ( ( big_co917763874_a_nat @ G_16 @ S_4 )
= ( big_co387207925at_nat
@ ^ [Y_28: nat] :
( big_co917763874_a_nat @ G_16
@ ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] :
( (&) @ ( member290856304iple_a @ X_1 @ S_4 )
@ ( ( F_24 @ X_1 )
= Y_28 ) ) ) )
@ ( image_1773322034_a_nat @ F_24 @ S_4 ) ) ) ) ).
thf(fact_593_setsum__image__gen,axiom,
! [G_16: int > nat,F_24: int > nat,S_4: int > $o] :
( ( finite_finite_int @ S_4 )
=> ( ( big_co1740723097nt_nat @ G_16 @ S_4 )
= ( big_co387207925at_nat
@ ^ [Y_28: nat] :
( big_co1740723097nt_nat @ G_16
@ ( collect_int
@ ^ [X_1: int] :
( (&) @ ( member_int @ X_1 @ S_4 )
@ ( ( F_24 @ X_1 )
= Y_28 ) ) ) )
@ ( image_int_nat @ F_24 @ S_4 ) ) ) ) ).
thf(fact_594_setsum__image__gen,axiom,
! [G_16: nat > nat,F_24: nat > hoare_2091234717iple_a,S_4: nat > $o] :
( ( finite_finite_nat @ S_4 )
=> ( ( big_co387207925at_nat @ G_16 @ S_4 )
= ( big_co917763874_a_nat
@ ^ [Y_28: hoare_2091234717iple_a] :
( big_co387207925at_nat @ G_16
@ ( collect_nat
@ ^ [X_1: nat] :
( (&) @ ( member_nat @ X_1 @ S_4 )
@ ( ( F_24 @ X_1 )
= Y_28 ) ) ) )
@ ( image_359186840iple_a @ F_24 @ S_4 ) ) ) ) ).
thf(fact_595_setsum__image__gen,axiom,
! [G_16: int > nat,F_24: int > int,S_4: int > $o] :
( ( finite_finite_int @ S_4 )
=> ( ( big_co1740723097nt_nat @ G_16 @ S_4 )
= ( big_co1740723097nt_nat
@ ^ [Y_28: int] :
( big_co1740723097nt_nat @ G_16
@ ( collect_int
@ ^ [X_1: int] :
( (&) @ ( member_int @ X_1 @ S_4 )
@ ( ( F_24 @ X_1 )
= Y_28 ) ) ) )
@ ( image_int_int @ F_24 @ S_4 ) ) ) ) ).
thf(fact_596_setsum__image__gen,axiom,
! [G_16: int > nat,F_24: int > hoare_2091234717iple_a,S_4: int > $o] :
( ( finite_finite_int @ S_4 )
=> ( ( big_co1740723097nt_nat @ G_16 @ S_4 )
= ( big_co917763874_a_nat
@ ^ [Y_28: hoare_2091234717iple_a] :
( big_co1740723097nt_nat @ G_16
@ ( collect_int
@ ^ [X_1: int] :
( (&) @ ( member_int @ X_1 @ S_4 )
@ ( ( F_24 @ X_1 )
= Y_28 ) ) ) )
@ ( image_338319932iple_a @ F_24 @ S_4 ) ) ) ) ).
thf(fact_597_setsum__image__gen,axiom,
! [G_16: nat > nat,F_24: nat > int,S_4: nat > $o] :
( ( finite_finite_nat @ S_4 )
=> ( ( big_co387207925at_nat @ G_16 @ S_4 )
= ( big_co1740723097nt_nat
@ ^ [Y_28: int] :
( big_co387207925at_nat @ G_16
@ ( collect_nat
@ ^ [X_1: nat] :
( (&) @ ( member_nat @ X_1 @ S_4 )
@ ( ( F_24 @ X_1 )
= Y_28 ) ) ) )
@ ( image_nat_int @ F_24 @ S_4 ) ) ) ) ).
thf(fact_598_setsum_Ocong,axiom,
! [G_15: hoare_2091234717iple_a > nat,H_2: hoare_2091234717iple_a > nat,A_105: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_105 )
=> ( ! [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_105 )
=> ( ( G_15 @ X_1 )
= ( H_2 @ X_1 ) ) )
=> ( ( big_co917763874_a_nat @ G_15 @ A_105 )
= ( big_co917763874_a_nat @ H_2 @ A_105 ) ) ) ) ).
thf(fact_599_setsum_Ocong,axiom,
! [G_15: nat > nat,H_2: nat > nat,A_105: nat > $o] :
( ( finite_finite_nat @ A_105 )
=> ( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_105 )
=> ( ( G_15 @ X_1 )
= ( H_2 @ X_1 ) ) )
=> ( ( big_co387207925at_nat @ G_15 @ A_105 )
= ( big_co387207925at_nat @ H_2 @ A_105 ) ) ) ) ).
thf(fact_600_setsum_Ocong,axiom,
! [G_15: int > nat,H_2: int > nat,A_105: int > $o] :
( ( finite_finite_int @ A_105 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ A_105 )
=> ( ( G_15 @ X_1 )
= ( H_2 @ X_1 ) ) )
=> ( ( big_co1740723097nt_nat @ G_15 @ A_105 )
= ( big_co1740723097nt_nat @ H_2 @ A_105 ) ) ) ) ).
thf(fact_601_setsum__setsum__restrict,axiom,
! [F_23: int > nat > nat,R_1: int > nat > $o,T_1: nat > $o,S_3: int > $o] :
( ( finite_finite_int @ S_3 )
=> ( ( finite_finite_nat @ T_1 )
=> ( ( big_co1740723097nt_nat
@ ^ [X_1: int] :
( big_co387207925at_nat @ ( F_23 @ X_1 )
@ ( collect_nat
@ ^ [Y_28: nat] : ( (&) @ ( member_nat @ Y_28 @ T_1 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ S_3 )
= ( big_co387207925at_nat
@ ^ [Y_28: nat] :
( big_co1740723097nt_nat
@ ^ [X_1: int] : ( F_23 @ X_1 @ Y_28 )
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ S_3 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ T_1 ) ) ) ) ).
thf(fact_602_setsum__setsum__restrict,axiom,
! [F_23: hoare_2091234717iple_a > nat > nat,R_1: hoare_2091234717iple_a > nat > $o,T_1: nat > $o,S_3: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ S_3 )
=> ( ( finite_finite_nat @ T_1 )
=> ( ( big_co917763874_a_nat
@ ^ [X_1: hoare_2091234717iple_a] :
( big_co387207925at_nat @ ( F_23 @ X_1 )
@ ( collect_nat
@ ^ [Y_28: nat] : ( (&) @ ( member_nat @ Y_28 @ T_1 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ S_3 )
= ( big_co387207925at_nat
@ ^ [Y_28: nat] :
( big_co917763874_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : ( F_23 @ X_1 @ Y_28 )
@ ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( member290856304iple_a @ X_1 @ S_3 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ T_1 ) ) ) ) ).
thf(fact_603_setsum__setsum__restrict,axiom,
! [F_23: hoare_2091234717iple_a > int > nat,R_1: hoare_2091234717iple_a > int > $o,T_1: int > $o,S_3: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ S_3 )
=> ( ( finite_finite_int @ T_1 )
=> ( ( big_co917763874_a_nat
@ ^ [X_1: hoare_2091234717iple_a] :
( big_co1740723097nt_nat @ ( F_23 @ X_1 )
@ ( collect_int
@ ^ [Y_28: int] : ( (&) @ ( member_int @ Y_28 @ T_1 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ S_3 )
= ( big_co1740723097nt_nat
@ ^ [Y_28: int] :
( big_co917763874_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : ( F_23 @ X_1 @ Y_28 )
@ ( collec992574898iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( (&) @ ( member290856304iple_a @ X_1 @ S_3 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ T_1 ) ) ) ) ).
thf(fact_604_setsum__setsum__restrict,axiom,
! [F_23: nat > int > nat,R_1: nat > int > $o,T_1: int > $o,S_3: nat > $o] :
( ( finite_finite_nat @ S_3 )
=> ( ( finite_finite_int @ T_1 )
=> ( ( big_co387207925at_nat
@ ^ [X_1: nat] :
( big_co1740723097nt_nat @ ( F_23 @ X_1 )
@ ( collect_int
@ ^ [Y_28: int] : ( (&) @ ( member_int @ Y_28 @ T_1 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ S_3 )
= ( big_co1740723097nt_nat
@ ^ [Y_28: int] :
( big_co387207925at_nat
@ ^ [X_1: nat] : ( F_23 @ X_1 @ Y_28 )
@ ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ S_3 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ T_1 ) ) ) ) ).
thf(fact_605_setsum__setsum__restrict,axiom,
! [F_23: nat > hoare_2091234717iple_a > nat,R_1: nat > hoare_2091234717iple_a > $o,T_1: hoare_2091234717iple_a > $o,S_3: nat > $o] :
( ( finite_finite_nat @ S_3 )
=> ( ( finite232261744iple_a @ T_1 )
=> ( ( big_co387207925at_nat
@ ^ [X_1: nat] :
( big_co917763874_a_nat @ ( F_23 @ X_1 )
@ ( collec992574898iple_a
@ ^ [Y_28: hoare_2091234717iple_a] : ( (&) @ ( member290856304iple_a @ Y_28 @ T_1 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ S_3 )
= ( big_co917763874_a_nat
@ ^ [Y_28: hoare_2091234717iple_a] :
( big_co387207925at_nat
@ ^ [X_1: nat] : ( F_23 @ X_1 @ Y_28 )
@ ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ S_3 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ T_1 ) ) ) ) ).
thf(fact_606_setsum__setsum__restrict,axiom,
! [F_23: int > int > nat,R_1: int > int > $o,T_1: int > $o,S_3: int > $o] :
( ( finite_finite_int @ S_3 )
=> ( ( finite_finite_int @ T_1 )
=> ( ( big_co1740723097nt_nat
@ ^ [X_1: int] :
( big_co1740723097nt_nat @ ( F_23 @ X_1 )
@ ( collect_int
@ ^ [Y_28: int] : ( (&) @ ( member_int @ Y_28 @ T_1 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ S_3 )
= ( big_co1740723097nt_nat
@ ^ [Y_28: int] :
( big_co1740723097nt_nat
@ ^ [X_1: int] : ( F_23 @ X_1 @ Y_28 )
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ S_3 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ T_1 ) ) ) ) ).
thf(fact_607_setsum__setsum__restrict,axiom,
! [F_23: int > hoare_2091234717iple_a > nat,R_1: int > hoare_2091234717iple_a > $o,T_1: hoare_2091234717iple_a > $o,S_3: int > $o] :
( ( finite_finite_int @ S_3 )
=> ( ( finite232261744iple_a @ T_1 )
=> ( ( big_co1740723097nt_nat
@ ^ [X_1: int] :
( big_co917763874_a_nat @ ( F_23 @ X_1 )
@ ( collec992574898iple_a
@ ^ [Y_28: hoare_2091234717iple_a] : ( (&) @ ( member290856304iple_a @ Y_28 @ T_1 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ S_3 )
= ( big_co917763874_a_nat
@ ^ [Y_28: hoare_2091234717iple_a] :
( big_co1740723097nt_nat
@ ^ [X_1: int] : ( F_23 @ X_1 @ Y_28 )
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ S_3 ) @ ( R_1 @ X_1 @ Y_28 ) ) ) )
@ T_1 ) ) ) ) ).
thf(fact_608_setsum_Oremove,axiom,
! [G_14: nat > nat,X_50: nat,A_104: nat > $o] :
( ( finite_finite_nat @ A_104 )
=> ( ( member_nat @ X_50 @ A_104 )
=> ( ( big_co387207925at_nat @ G_14 @ A_104 )
= ( plus_plus_nat @ ( G_14 @ X_50 ) @ ( big_co387207925at_nat @ G_14 @ ( minus_minus_nat_o @ A_104 @ ( insert_nat @ X_50 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_609_setsum_Oremove,axiom,
! [G_14: nat > int,X_50: nat,A_104: nat > $o] :
( ( finite_finite_nat @ A_104 )
=> ( ( member_nat @ X_50 @ A_104 )
=> ( ( big_co1024481617at_int @ G_14 @ A_104 )
= ( plus_plus_int @ ( G_14 @ X_50 ) @ ( big_co1024481617at_int @ G_14 @ ( minus_minus_nat_o @ A_104 @ ( insert_nat @ X_50 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_610_setsum_Oremove,axiom,
! [G_14: hoare_2091234717iple_a > int,X_50: hoare_2091234717iple_a,A_104: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_104 )
=> ( ( member290856304iple_a @ X_50 @ A_104 )
=> ( ( big_co1555037566_a_int @ G_14 @ A_104 )
= ( plus_plus_int @ ( G_14 @ X_50 ) @ ( big_co1555037566_a_int @ G_14 @ ( minus_836160335le_a_o @ A_104 @ ( insert1597628439iple_a @ X_50 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_611_setsum_Oremove,axiom,
! [G_14: int > int,X_50: int,A_104: int > $o] :
( ( finite_finite_int @ A_104 )
=> ( ( member_int @ X_50 @ A_104 )
=> ( ( big_co230513141nt_int @ G_14 @ A_104 )
= ( plus_plus_int @ ( G_14 @ X_50 ) @ ( big_co230513141nt_int @ G_14 @ ( minus_minus_int_o @ A_104 @ ( insert_int @ X_50 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_612_setsum_Oremove,axiom,
! [G_14: hoare_2091234717iple_a > nat,X_50: hoare_2091234717iple_a,A_104: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_104 )
=> ( ( member290856304iple_a @ X_50 @ A_104 )
=> ( ( big_co917763874_a_nat @ G_14 @ A_104 )
= ( plus_plus_nat @ ( G_14 @ X_50 ) @ ( big_co917763874_a_nat @ G_14 @ ( minus_836160335le_a_o @ A_104 @ ( insert1597628439iple_a @ X_50 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_613_setsum_Oremove,axiom,
! [G_14: int > nat,X_50: int,A_104: int > $o] :
( ( finite_finite_int @ A_104 )
=> ( ( member_int @ X_50 @ A_104 )
=> ( ( big_co1740723097nt_nat @ G_14 @ A_104 )
= ( plus_plus_nat @ ( G_14 @ X_50 ) @ ( big_co1740723097nt_nat @ G_14 @ ( minus_minus_int_o @ A_104 @ ( insert_int @ X_50 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_614_setsum__diff1_H,axiom,
! [F_22: nat > nat,A_103: nat,A_102: nat > $o] :
( ( finite_finite_nat @ A_102 )
=> ( ( member_nat @ A_103 @ A_102 )
=> ( ( big_co387207925at_nat @ F_22 @ A_102 )
= ( plus_plus_nat @ ( F_22 @ A_103 ) @ ( big_co387207925at_nat @ F_22 @ ( minus_minus_nat_o @ A_102 @ ( insert_nat @ A_103 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_615_setsum__diff1_H,axiom,
! [F_22: nat > int,A_103: nat,A_102: nat > $o] :
( ( finite_finite_nat @ A_102 )
=> ( ( member_nat @ A_103 @ A_102 )
=> ( ( big_co1024481617at_int @ F_22 @ A_102 )
= ( plus_plus_int @ ( F_22 @ A_103 ) @ ( big_co1024481617at_int @ F_22 @ ( minus_minus_nat_o @ A_102 @ ( insert_nat @ A_103 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_616_setsum__diff1_H,axiom,
! [F_22: hoare_2091234717iple_a > int,A_103: hoare_2091234717iple_a,A_102: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_102 )
=> ( ( member290856304iple_a @ A_103 @ A_102 )
=> ( ( big_co1555037566_a_int @ F_22 @ A_102 )
= ( plus_plus_int @ ( F_22 @ A_103 ) @ ( big_co1555037566_a_int @ F_22 @ ( minus_836160335le_a_o @ A_102 @ ( insert1597628439iple_a @ A_103 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_617_setsum__diff1_H,axiom,
! [F_22: int > int,A_103: int,A_102: int > $o] :
( ( finite_finite_int @ A_102 )
=> ( ( member_int @ A_103 @ A_102 )
=> ( ( big_co230513141nt_int @ F_22 @ A_102 )
= ( plus_plus_int @ ( F_22 @ A_103 ) @ ( big_co230513141nt_int @ F_22 @ ( minus_minus_int_o @ A_102 @ ( insert_int @ A_103 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_618_setsum__diff1_H,axiom,
! [F_22: hoare_2091234717iple_a > nat,A_103: hoare_2091234717iple_a,A_102: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_102 )
=> ( ( member290856304iple_a @ A_103 @ A_102 )
=> ( ( big_co917763874_a_nat @ F_22 @ A_102 )
= ( plus_plus_nat @ ( F_22 @ A_103 ) @ ( big_co917763874_a_nat @ F_22 @ ( minus_836160335le_a_o @ A_102 @ ( insert1597628439iple_a @ A_103 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_619_setsum__diff1_H,axiom,
! [F_22: int > nat,A_103: int,A_102: int > $o] :
( ( finite_finite_int @ A_102 )
=> ( ( member_int @ A_103 @ A_102 )
=> ( ( big_co1740723097nt_nat @ F_22 @ A_102 )
= ( plus_plus_nat @ ( F_22 @ A_103 ) @ ( big_co1740723097nt_nat @ F_22 @ ( minus_minus_int_o @ A_102 @ ( insert_int @ A_103 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_620_setsum_Oinsert__remove,axiom,
! [G_13: nat > nat,X_49: nat,A_101: nat > $o] :
( ( finite_finite_nat @ A_101 )
=> ( ( big_co387207925at_nat @ G_13 @ ( insert_nat @ X_49 @ A_101 ) )
= ( plus_plus_nat @ ( G_13 @ X_49 ) @ ( big_co387207925at_nat @ G_13 @ ( minus_minus_nat_o @ A_101 @ ( insert_nat @ X_49 @ bot_bot_nat_o ) ) ) ) ) ) ).
thf(fact_621_setsum_Oinsert__remove,axiom,
! [G_13: nat > int,X_49: nat,A_101: nat > $o] :
( ( finite_finite_nat @ A_101 )
=> ( ( big_co1024481617at_int @ G_13 @ ( insert_nat @ X_49 @ A_101 ) )
= ( plus_plus_int @ ( G_13 @ X_49 ) @ ( big_co1024481617at_int @ G_13 @ ( minus_minus_nat_o @ A_101 @ ( insert_nat @ X_49 @ bot_bot_nat_o ) ) ) ) ) ) ).
thf(fact_622_setsum_Oinsert__remove,axiom,
! [G_13: hoare_2091234717iple_a > int,X_49: hoare_2091234717iple_a,A_101: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_101 )
=> ( ( big_co1555037566_a_int @ G_13 @ ( insert1597628439iple_a @ X_49 @ A_101 ) )
= ( plus_plus_int @ ( G_13 @ X_49 ) @ ( big_co1555037566_a_int @ G_13 @ ( minus_836160335le_a_o @ A_101 @ ( insert1597628439iple_a @ X_49 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ).
thf(fact_623_setsum_Oinsert__remove,axiom,
! [G_13: int > int,X_49: int,A_101: int > $o] :
( ( finite_finite_int @ A_101 )
=> ( ( big_co230513141nt_int @ G_13 @ ( insert_int @ X_49 @ A_101 ) )
= ( plus_plus_int @ ( G_13 @ X_49 ) @ ( big_co230513141nt_int @ G_13 @ ( minus_minus_int_o @ A_101 @ ( insert_int @ X_49 @ bot_bot_int_o ) ) ) ) ) ) ).
thf(fact_624_setsum_Oinsert__remove,axiom,
! [G_13: hoare_2091234717iple_a > nat,X_49: hoare_2091234717iple_a,A_101: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_101 )
=> ( ( big_co917763874_a_nat @ G_13 @ ( insert1597628439iple_a @ X_49 @ A_101 ) )
= ( plus_plus_nat @ ( G_13 @ X_49 ) @ ( big_co917763874_a_nat @ G_13 @ ( minus_836160335le_a_o @ A_101 @ ( insert1597628439iple_a @ X_49 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ).
thf(fact_625_setsum_Oinsert__remove,axiom,
! [G_13: int > nat,X_49: int,A_101: int > $o] :
( ( finite_finite_int @ A_101 )
=> ( ( big_co1740723097nt_nat @ G_13 @ ( insert_int @ X_49 @ A_101 ) )
= ( plus_plus_nat @ ( G_13 @ X_49 ) @ ( big_co1740723097nt_nat @ G_13 @ ( minus_minus_int_o @ A_101 @ ( insert_int @ X_49 @ bot_bot_int_o ) ) ) ) ) ) ).
thf(fact_626_setsum__addf,axiom,
! [F_21: nat > nat,G_12: nat > nat,A_100: nat > $o] :
( ( big_co387207925at_nat
@ ^ [X_1: nat] : ( plus_plus_nat @ ( F_21 @ X_1 ) @ ( G_12 @ X_1 ) )
@ A_100 )
= ( plus_plus_nat @ ( big_co387207925at_nat @ F_21 @ A_100 ) @ ( big_co387207925at_nat @ G_12 @ A_100 ) ) ) ).
thf(fact_627_setsum__addf,axiom,
! [F_21: int > nat,G_12: int > nat,A_100: int > $o] :
( ( big_co1740723097nt_nat
@ ^ [X_1: int] : ( plus_plus_nat @ ( F_21 @ X_1 ) @ ( G_12 @ X_1 ) )
@ A_100 )
= ( plus_plus_nat @ ( big_co1740723097nt_nat @ F_21 @ A_100 ) @ ( big_co1740723097nt_nat @ G_12 @ A_100 ) ) ) ).
thf(fact_628_setsum__addf,axiom,
! [F_21: hoare_2091234717iple_a > nat,G_12: hoare_2091234717iple_a > nat,A_100: hoare_2091234717iple_a > $o] :
( ( big_co917763874_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : ( plus_plus_nat @ ( F_21 @ X_1 ) @ ( G_12 @ X_1 ) )
@ A_100 )
= ( plus_plus_nat @ ( big_co917763874_a_nat @ F_21 @ A_100 ) @ ( big_co917763874_a_nat @ G_12 @ A_100 ) ) ) ).
thf(fact_629_setsum_Odistrib,axiom,
! [G_11: nat > nat,H_1: nat > nat,A_99: nat > $o] :
( ( finite_finite_nat @ A_99 )
=> ( ( big_co387207925at_nat
@ ^ [X_1: nat] : ( plus_plus_nat @ ( G_11 @ X_1 ) @ ( H_1 @ X_1 ) )
@ A_99 )
= ( plus_plus_nat @ ( big_co387207925at_nat @ G_11 @ A_99 ) @ ( big_co387207925at_nat @ H_1 @ A_99 ) ) ) ) ).
thf(fact_630_setsum_Odistrib,axiom,
! [G_11: hoare_2091234717iple_a > nat,H_1: hoare_2091234717iple_a > nat,A_99: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_99 )
=> ( ( big_co917763874_a_nat
@ ^ [X_1: hoare_2091234717iple_a] : ( plus_plus_nat @ ( G_11 @ X_1 ) @ ( H_1 @ X_1 ) )
@ A_99 )
= ( plus_plus_nat @ ( big_co917763874_a_nat @ G_11 @ A_99 ) @ ( big_co917763874_a_nat @ H_1 @ A_99 ) ) ) ) ).
thf(fact_631_setsum_Odistrib,axiom,
! [G_11: hoare_2091234717iple_a > int,H_1: hoare_2091234717iple_a > int,A_99: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_99 )
=> ( ( big_co1555037566_a_int
@ ^ [X_1: hoare_2091234717iple_a] : ( plus_plus_int @ ( G_11 @ X_1 ) @ ( H_1 @ X_1 ) )
@ A_99 )
= ( plus_plus_int @ ( big_co1555037566_a_int @ G_11 @ A_99 ) @ ( big_co1555037566_a_int @ H_1 @ A_99 ) ) ) ) ).
thf(fact_632_setsum_Odistrib,axiom,
! [G_11: int > nat,H_1: int > nat,A_99: int > $o] :
( ( finite_finite_int @ A_99 )
=> ( ( big_co1740723097nt_nat
@ ^ [X_1: int] : ( plus_plus_nat @ ( G_11 @ X_1 ) @ ( H_1 @ X_1 ) )
@ A_99 )
= ( plus_plus_nat @ ( big_co1740723097nt_nat @ G_11 @ A_99 ) @ ( big_co1740723097nt_nat @ H_1 @ A_99 ) ) ) ) ).
thf(fact_633_setsum__insert,axiom,
! [F_20: nat > nat,A_98: nat,F_19: nat > $o] :
( ( finite_finite_nat @ F_19 )
=> ( ~ ( member_nat @ A_98 @ F_19 )
=> ( ( big_co387207925at_nat @ F_20 @ ( insert_nat @ A_98 @ F_19 ) )
= ( plus_plus_nat @ ( F_20 @ A_98 ) @ ( big_co387207925at_nat @ F_20 @ F_19 ) ) ) ) ) ).
thf(fact_634_setsum__insert,axiom,
! [F_20: nat > int,A_98: nat,F_19: nat > $o] :
( ( finite_finite_nat @ F_19 )
=> ( ~ ( member_nat @ A_98 @ F_19 )
=> ( ( big_co1024481617at_int @ F_20 @ ( insert_nat @ A_98 @ F_19 ) )
= ( plus_plus_int @ ( F_20 @ A_98 ) @ ( big_co1024481617at_int @ F_20 @ F_19 ) ) ) ) ) ).
thf(fact_635_setsum__insert,axiom,
! [F_20: hoare_2091234717iple_a > int,A_98: hoare_2091234717iple_a,F_19: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_19 )
=> ( ~ ( member290856304iple_a @ A_98 @ F_19 )
=> ( ( big_co1555037566_a_int @ F_20 @ ( insert1597628439iple_a @ A_98 @ F_19 ) )
= ( plus_plus_int @ ( F_20 @ A_98 ) @ ( big_co1555037566_a_int @ F_20 @ F_19 ) ) ) ) ) ).
thf(fact_636_setsum__insert,axiom,
! [F_20: hoare_2091234717iple_a > nat,A_98: hoare_2091234717iple_a,F_19: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_19 )
=> ( ~ ( member290856304iple_a @ A_98 @ F_19 )
=> ( ( big_co917763874_a_nat @ F_20 @ ( insert1597628439iple_a @ A_98 @ F_19 ) )
= ( plus_plus_nat @ ( F_20 @ A_98 ) @ ( big_co917763874_a_nat @ F_20 @ F_19 ) ) ) ) ) ).
thf(fact_637_setsum__insert,axiom,
! [F_20: int > nat,A_98: int,F_19: int > $o] :
( ( finite_finite_int @ F_19 )
=> ( ~ ( member_int @ A_98 @ F_19 )
=> ( ( big_co1740723097nt_nat @ F_20 @ ( insert_int @ A_98 @ F_19 ) )
= ( plus_plus_nat @ ( F_20 @ A_98 ) @ ( big_co1740723097nt_nat @ F_20 @ F_19 ) ) ) ) ) ).
thf(fact_638_setsum_Oinsert,axiom,
! [G_10: nat > nat,X_48: nat,A_97: nat > $o] :
( ( finite_finite_nat @ A_97 )
=> ( ~ ( member_nat @ X_48 @ A_97 )
=> ( ( big_co387207925at_nat @ G_10 @ ( insert_nat @ X_48 @ A_97 ) )
= ( plus_plus_nat @ ( G_10 @ X_48 ) @ ( big_co387207925at_nat @ G_10 @ A_97 ) ) ) ) ) ).
thf(fact_639_setsum_Oinsert,axiom,
! [G_10: nat > int,X_48: nat,A_97: nat > $o] :
( ( finite_finite_nat @ A_97 )
=> ( ~ ( member_nat @ X_48 @ A_97 )
=> ( ( big_co1024481617at_int @ G_10 @ ( insert_nat @ X_48 @ A_97 ) )
= ( plus_plus_int @ ( G_10 @ X_48 ) @ ( big_co1024481617at_int @ G_10 @ A_97 ) ) ) ) ) ).
thf(fact_640_setsum_Oinsert,axiom,
! [G_10: hoare_2091234717iple_a > int,X_48: hoare_2091234717iple_a,A_97: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_97 )
=> ( ~ ( member290856304iple_a @ X_48 @ A_97 )
=> ( ( big_co1555037566_a_int @ G_10 @ ( insert1597628439iple_a @ X_48 @ A_97 ) )
= ( plus_plus_int @ ( G_10 @ X_48 ) @ ( big_co1555037566_a_int @ G_10 @ A_97 ) ) ) ) ) ).
thf(fact_641_setsum_Oinsert,axiom,
! [G_10: hoare_2091234717iple_a > nat,X_48: hoare_2091234717iple_a,A_97: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_97 )
=> ( ~ ( member290856304iple_a @ X_48 @ A_97 )
=> ( ( big_co917763874_a_nat @ G_10 @ ( insert1597628439iple_a @ X_48 @ A_97 ) )
= ( plus_plus_nat @ ( G_10 @ X_48 ) @ ( big_co917763874_a_nat @ G_10 @ A_97 ) ) ) ) ) ).
thf(fact_642_setsum_Oinsert,axiom,
! [G_10: int > nat,X_48: int,A_97: int > $o] :
( ( finite_finite_int @ A_97 )
=> ( ~ ( member_int @ X_48 @ A_97 )
=> ( ( big_co1740723097nt_nat @ G_10 @ ( insert_int @ X_48 @ A_97 ) )
= ( plus_plus_nat @ ( G_10 @ X_48 ) @ ( big_co1740723097nt_nat @ G_10 @ A_97 ) ) ) ) ) ).
thf(fact_643_eq__add__iff2,axiom,
! [A_96: int,E_4: int,C_43: int,B_81: int,D_13: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_96 @ E_4 ) @ C_43 )
= ( plus_plus_int @ ( times_times_int @ B_81 @ E_4 ) @ D_13 ) )
<=> ( C_43
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_81 @ A_96 ) @ E_4 ) @ D_13 ) ) ) ).
thf(fact_644_eq__add__iff1,axiom,
! [A_95: int,E_3: int,C_42: int,B_80: int,D_12: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_95 @ E_3 ) @ C_42 )
= ( plus_plus_int @ ( times_times_int @ B_80 @ E_3 ) @ D_12 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_95 @ B_80 ) @ E_3 ) @ C_42 )
= D_12 ) ) ).
thf(fact_645_diff__add__cancel,axiom,
! [A_94: int,B_79: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_94 @ B_79 ) @ B_79 )
= A_94 ) ).
thf(fact_646_add__diff__cancel,axiom,
! [A_93: int,B_78: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_93 @ B_78 ) @ B_78 )
= A_93 ) ).
thf(fact_647_crossproduct__eq,axiom,
! [W_2: nat,Y_27: nat,X_47: nat,Z_13: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_2 @ Y_27 ) @ ( times_times_nat @ X_47 @ Z_13 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_2 @ Z_13 ) @ ( times_times_nat @ X_47 @ Y_27 ) ) )
<=> ( ( W_2 = X_47 )
| ( Y_27 = Z_13 ) ) ) ).
thf(fact_648_crossproduct__eq,axiom,
! [W_2: int,Y_27: int,X_47: int,Z_13: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_2 @ Y_27 ) @ ( times_times_int @ X_47 @ Z_13 ) )
= ( plus_plus_int @ ( times_times_int @ W_2 @ Z_13 ) @ ( times_times_int @ X_47 @ Y_27 ) ) )
<=> ( ( W_2 = X_47 )
| ( Y_27 = Z_13 ) ) ) ).
thf(fact_649_combine__common__factor,axiom,
! [A_92: nat,E_2: nat,B_77: nat,C_41: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_92 @ E_2 ) @ ( plus_plus_nat @ ( times_times_nat @ B_77 @ E_2 ) @ C_41 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_92 @ B_77 ) @ E_2 ) @ C_41 ) ) ).
thf(fact_650_combine__common__factor,axiom,
! [A_92: int,E_2: int,B_77: int,C_41: int] :
( ( plus_plus_int @ ( times_times_int @ A_92 @ E_2 ) @ ( plus_plus_int @ ( times_times_int @ B_77 @ E_2 ) @ C_41 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_92 @ B_77 ) @ E_2 ) @ C_41 ) ) ).
thf(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_91: nat,M_4: nat,B_76: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_91 @ M_4 ) @ ( times_times_nat @ B_76 @ M_4 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_91 @ B_76 ) @ M_4 ) ) ).
thf(fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_91: int,M_4: int,B_76: int] :
( ( plus_plus_int @ ( times_times_int @ A_91 @ M_4 ) @ ( times_times_int @ B_76 @ M_4 ) )
= ( times_times_int @ ( plus_plus_int @ A_91 @ B_76 ) @ M_4 ) ) ).
thf(fact_653_add__right__imp__eq,axiom,
! [B_75: nat,A_90: nat,C_40: nat] :
( ( ( plus_plus_nat @ B_75 @ A_90 )
= ( plus_plus_nat @ C_40 @ A_90 ) )
=> ( B_75 = C_40 ) ) ).
thf(fact_654_add__right__imp__eq,axiom,
! [B_75: int,A_90: int,C_40: int] :
( ( ( plus_plus_int @ B_75 @ A_90 )
= ( plus_plus_int @ C_40 @ A_90 ) )
=> ( B_75 = C_40 ) ) ).
thf(fact_655_add__imp__eq,axiom,
! [A_89: nat,B_74: nat,C_39: nat] :
( ( ( plus_plus_nat @ A_89 @ B_74 )
= ( plus_plus_nat @ A_89 @ C_39 ) )
=> ( B_74 = C_39 ) ) ).
thf(fact_656_add__imp__eq,axiom,
! [A_89: int,B_74: int,C_39: int] :
( ( ( plus_plus_int @ A_89 @ B_74 )
= ( plus_plus_int @ A_89 @ C_39 ) )
=> ( B_74 = C_39 ) ) ).
thf(fact_657_add__left__imp__eq,axiom,
! [A_88: nat,B_73: nat,C_38: nat] :
( ( ( plus_plus_nat @ A_88 @ B_73 )
= ( plus_plus_nat @ A_88 @ C_38 ) )
=> ( B_73 = C_38 ) ) ).
thf(fact_658_add__left__imp__eq,axiom,
! [A_88: int,B_73: int,C_38: int] :
( ( ( plus_plus_int @ A_88 @ B_73 )
= ( plus_plus_int @ A_88 @ C_38 ) )
=> ( B_73 = C_38 ) ) ).
thf(fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_87: nat,B_72: nat,C_37: nat,D_11: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_87 @ B_72 ) @ ( plus_plus_nat @ C_37 @ D_11 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_87 @ C_37 ) @ ( plus_plus_nat @ B_72 @ D_11 ) ) ) ).
thf(fact_660_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_87: int,B_72: int,C_37: int,D_11: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_87 @ B_72 ) @ ( plus_plus_int @ C_37 @ D_11 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_87 @ C_37 ) @ ( plus_plus_int @ B_72 @ D_11 ) ) ) ).
thf(fact_661_add__right__cancel,axiom,
! [B_71: nat,A_86: nat,C_36: nat] :
( ( ( plus_plus_nat @ B_71 @ A_86 )
= ( plus_plus_nat @ C_36 @ A_86 ) )
<=> ( B_71 = C_36 ) ) ).
thf(fact_662_add__right__cancel,axiom,
! [B_71: int,A_86: int,C_36: int] :
( ( ( plus_plus_int @ B_71 @ A_86 )
= ( plus_plus_int @ C_36 @ A_86 ) )
<=> ( B_71 = C_36 ) ) ).
thf(fact_663_add__left__cancel,axiom,
! [A_85: nat,B_70: nat,C_35: nat] :
( ( ( plus_plus_nat @ A_85 @ B_70 )
= ( plus_plus_nat @ A_85 @ C_35 ) )
<=> ( B_70 = C_35 ) ) ).
thf(fact_664_add__left__cancel,axiom,
! [A_85: int,B_70: int,C_35: int] :
( ( ( plus_plus_int @ A_85 @ B_70 )
= ( plus_plus_int @ A_85 @ C_35 ) )
<=> ( B_70 = C_35 ) ) ).
thf(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_84: nat,B_69: nat,C_34: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_84 @ B_69 ) @ C_34 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_84 @ C_34 ) @ B_69 ) ) ).
thf(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_84: int,B_69: int,C_34: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_84 @ B_69 ) @ C_34 )
= ( plus_plus_int @ ( plus_plus_int @ A_84 @ C_34 ) @ B_69 ) ) ).
thf(fact_667_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_83: nat,B_68: nat,C_33: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_83 @ B_68 ) @ C_33 )
= ( plus_plus_nat @ A_83 @ ( plus_plus_nat @ B_68 @ C_33 ) ) ) ).
thf(fact_668_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_83: int,B_68: int,C_33: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_83 @ B_68 ) @ C_33 )
= ( plus_plus_int @ A_83 @ ( plus_plus_int @ B_68 @ C_33 ) ) ) ).
thf(fact_669_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_82: nat,B_67: nat,C_32: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_82 @ B_67 ) @ C_32 )
= ( plus_plus_nat @ A_82 @ ( plus_plus_nat @ B_67 @ C_32 ) ) ) ).
thf(fact_670_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_82: int,B_67: int,C_32: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_82 @ B_67 ) @ C_32 )
= ( plus_plus_int @ A_82 @ ( plus_plus_int @ B_67 @ C_32 ) ) ) ).
thf(fact_671_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_81: nat,C_31: nat,D_10: nat] :
( ( plus_plus_nat @ A_81 @ ( plus_plus_nat @ C_31 @ D_10 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_81 @ C_31 ) @ D_10 ) ) ).
thf(fact_672_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_81: int,C_31: int,D_10: int] :
( ( plus_plus_int @ A_81 @ ( plus_plus_int @ C_31 @ D_10 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_81 @ C_31 ) @ D_10 ) ) ).
thf(fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_80: nat,C_30: nat,D_9: nat] :
( ( plus_plus_nat @ A_80 @ ( plus_plus_nat @ C_30 @ D_9 ) )
= ( plus_plus_nat @ C_30 @ ( plus_plus_nat @ A_80 @ D_9 ) ) ) ).
thf(fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_80: int,C_30: int,D_9: int] :
( ( plus_plus_int @ A_80 @ ( plus_plus_int @ C_30 @ D_9 ) )
= ( plus_plus_int @ C_30 @ ( plus_plus_int @ A_80 @ D_9 ) ) ) ).
thf(fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_79: nat,C_29: nat] :
( ( plus_plus_nat @ A_79 @ C_29 )
= ( plus_plus_nat @ C_29 @ A_79 ) ) ).
thf(fact_676_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_79: int,C_29: int] :
( ( plus_plus_int @ A_79 @ C_29 )
= ( plus_plus_int @ C_29 @ A_79 ) ) ).
thf(fact_677_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_6 @ Ly_4 ) @ ( times_times_nat @ Rx_6 @ Ry_4 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_6 @ Rx_6 ) @ ( times_times_nat @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_678_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_6 @ Ly_4 ) @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ ( times_times_int @ Lx_6 @ Rx_6 ) @ ( times_times_int @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_679_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ ( times_times_nat @ Rx_5 @ Ry_3 ) )
= ( times_times_nat @ Rx_5 @ ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_680_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ Rx_5 @ ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_4 @ Ly_2 ) @ ( times_times_nat @ Rx_4 @ Ry_2 ) )
= ( times_times_nat @ Lx_4 @ ( times_times_nat @ Ly_2 @ ( times_times_nat @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_682_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_2 ) @ ( times_times_int @ Rx_4 @ Ry_2 ) )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_683_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_1: nat,Rx_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_nat @ ( times_times_nat @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_684_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_1: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_685_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: nat,Ly: nat,Rx_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_nat @ Lx_2 @ ( times_times_nat @ Ly @ Rx_2 ) ) ) ).
thf(fact_686_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: int,Ly: int,Rx_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly @ Rx_2 ) ) ) ).
thf(fact_687_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_78: nat,B_66: nat,C_28: nat] :
( ( times_times_nat @ ( times_times_nat @ A_78 @ B_66 ) @ C_28 )
= ( times_times_nat @ A_78 @ ( times_times_nat @ B_66 @ C_28 ) ) ) ).
thf(fact_688_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_78: int,B_66: int,C_28: int] :
( ( times_times_int @ ( times_times_int @ A_78 @ B_66 ) @ C_28 )
= ( times_times_int @ A_78 @ ( times_times_int @ B_66 @ C_28 ) ) ) ).
thf(fact_689_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: nat,Rx_1: nat,Ry_1: nat] :
( ( times_times_nat @ Lx_1 @ ( times_times_nat @ Rx_1 @ Ry_1 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_690_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ Lx_1 @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ ( times_times_int @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: nat,Rx: nat,Ry: nat] :
( ( times_times_nat @ Lx @ ( times_times_nat @ Rx @ Ry ) )
= ( times_times_nat @ Rx @ ( times_times_nat @ Lx @ Ry ) ) ) ).
thf(fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: int,Rx: int,Ry: int] :
( ( times_times_int @ Lx @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ Rx @ ( times_times_int @ Lx @ Ry ) ) ) ).
thf(fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_77: nat,B_65: nat] :
( ( times_times_nat @ A_77 @ B_65 )
= ( times_times_nat @ B_65 @ A_77 ) ) ).
thf(fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_77: int,B_65: int] :
( ( times_times_int @ A_77 @ B_65 )
= ( times_times_int @ B_65 @ A_77 ) ) ).
thf(fact_695_diff__eq__diff__eq,axiom,
! [A_76: int,B_64: int,C_27: int,D_8: int] :
( ( ( minus_minus_int @ A_76 @ B_64 )
= ( minus_minus_int @ C_27 @ D_8 ) )
=> ( ( A_76 = B_64 )
<=> ( C_27 = D_8 ) ) ) ).
thf(fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_46: nat,Y_26: nat,Z_12: nat] :
( ( times_times_nat @ X_46 @ ( plus_plus_nat @ Y_26 @ Z_12 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_46 @ Y_26 ) @ ( times_times_nat @ X_46 @ Z_12 ) ) ) ).
thf(fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_46: int,Y_26: int,Z_12: int] :
( ( times_times_int @ X_46 @ ( plus_plus_int @ Y_26 @ Z_12 ) )
= ( plus_plus_int @ ( times_times_int @ X_46 @ Y_26 ) @ ( times_times_int @ X_46 @ Z_12 ) ) ) ).
thf(fact_698_crossproduct__noteq,axiom,
! [C_26: nat,D_7: nat,A_75: nat,B_63: nat] :
( ( ( A_75 != B_63 )
& ( C_26 != D_7 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_75 @ C_26 ) @ ( times_times_nat @ B_63 @ D_7 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_75 @ D_7 ) @ ( times_times_nat @ B_63 @ C_26 ) ) ) ) ).
thf(fact_699_crossproduct__noteq,axiom,
! [C_26: int,D_7: int,A_75: int,B_63: int] :
( ( ( A_75 != B_63 )
& ( C_26 != D_7 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_75 @ C_26 ) @ ( times_times_int @ B_63 @ D_7 ) )
!= ( plus_plus_int @ ( times_times_int @ A_75 @ D_7 ) @ ( times_times_int @ B_63 @ C_26 ) ) ) ) ).
thf(fact_700_comm__semiring__class_Odistrib,axiom,
! [A_74: int,B_62: int,C_25: int] :
( ( times_times_int @ ( plus_plus_int @ A_74 @ B_62 ) @ C_25 )
= ( plus_plus_int @ ( times_times_int @ A_74 @ C_25 ) @ ( times_times_int @ B_62 @ C_25 ) ) ) ).
thf(fact_701_comm__semiring__class_Odistrib,axiom,
! [A_74: nat,B_62: nat,C_25: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_74 @ B_62 ) @ C_25 )
= ( plus_plus_nat @ ( times_times_nat @ A_74 @ C_25 ) @ ( times_times_nat @ B_62 @ C_25 ) ) ) ).
thf(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_73: int,B_61: int,C_24: int] :
( ( times_times_int @ ( plus_plus_int @ A_73 @ B_61 ) @ C_24 )
= ( plus_plus_int @ ( times_times_int @ A_73 @ C_24 ) @ ( times_times_int @ B_61 @ C_24 ) ) ) ).
thf(fact_703_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_73: nat,B_61: nat,C_24: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_73 @ B_61 ) @ C_24 )
= ( plus_plus_nat @ ( times_times_nat @ A_73 @ C_24 ) @ ( times_times_nat @ B_61 @ C_24 ) ) ) ).
thf(fact_704_setsum__cong2,axiom,
! [F_18: int > nat,G_9: int > nat,A_72: int > $o] :
( ! [X_1: int] :
( ( member_int @ X_1 @ A_72 )
=> ( ( F_18 @ X_1 )
= ( G_9 @ X_1 ) ) )
=> ( ( big_co1740723097nt_nat @ F_18 @ A_72 )
= ( big_co1740723097nt_nat @ G_9 @ A_72 ) ) ) ).
thf(fact_705_setsum__cong2,axiom,
! [F_18: hoare_2091234717iple_a > nat,G_9: hoare_2091234717iple_a > nat,A_72: hoare_2091234717iple_a > $o] :
( ! [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_72 )
=> ( ( F_18 @ X_1 )
= ( G_9 @ X_1 ) ) )
=> ( ( big_co917763874_a_nat @ F_18 @ A_72 )
= ( big_co917763874_a_nat @ G_9 @ A_72 ) ) ) ).
thf(fact_706_setsum__cong2,axiom,
! [F_18: nat > nat,G_9: nat > nat,A_72: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_72 )
=> ( ( F_18 @ X_1 )
= ( G_9 @ X_1 ) ) )
=> ( ( big_co387207925at_nat @ F_18 @ A_72 )
= ( big_co387207925at_nat @ G_9 @ A_72 ) ) ) ).
thf(fact_707_setsum__cong,axiom,
! [F_17: int > nat,G_8: int > nat,A_71: int > $o,B_60: int > $o] :
( ( A_71 = B_60 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ B_60 )
=> ( ( F_17 @ X_1 )
= ( G_8 @ X_1 ) ) )
=> ( ( big_co1740723097nt_nat @ F_17 @ A_71 )
= ( big_co1740723097nt_nat @ G_8 @ B_60 ) ) ) ) ).
thf(fact_708_setsum__cong,axiom,
! [F_17: hoare_2091234717iple_a > nat,G_8: hoare_2091234717iple_a > nat,A_71: hoare_2091234717iple_a > $o,B_60: hoare_2091234717iple_a > $o] :
( ( A_71 = B_60 )
=> ( ! [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ B_60 )
=> ( ( F_17 @ X_1 )
= ( G_8 @ X_1 ) ) )
=> ( ( big_co917763874_a_nat @ F_17 @ A_71 )
= ( big_co917763874_a_nat @ G_8 @ B_60 ) ) ) ) ).
thf(fact_709_setsum__cong,axiom,
! [F_17: nat > nat,G_8: nat > nat,A_71: nat > $o,B_60: nat > $o] :
( ( A_71 = B_60 )
=> ( ! [X_1: nat] :
( ( member_nat @ X_1 @ B_60 )
=> ( ( F_17 @ X_1 )
= ( G_8 @ X_1 ) ) )
=> ( ( big_co387207925at_nat @ F_17 @ A_71 )
= ( big_co387207925at_nat @ G_8 @ B_60 ) ) ) ) ).
thf(fact_710_setsum_OF__cong,axiom,
! [H: int > nat,G_7: int > nat,A_70: int > $o,B_59: int > $o] :
( ( A_70 = B_59 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ B_59 )
=> ( ( H @ X_1 )
= ( G_7 @ X_1 ) ) )
=> ( ( big_co1740723097nt_nat @ H @ A_70 )
= ( big_co1740723097nt_nat @ G_7 @ B_59 ) ) ) ) ).
thf(fact_711_setsum_OF__cong,axiom,
! [H: hoare_2091234717iple_a > nat,G_7: hoare_2091234717iple_a > nat,A_70: hoare_2091234717iple_a > $o,B_59: hoare_2091234717iple_a > $o] :
( ( A_70 = B_59 )
=> ( ! [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ B_59 )
=> ( ( H @ X_1 )
= ( G_7 @ X_1 ) ) )
=> ( ( big_co917763874_a_nat @ H @ A_70 )
= ( big_co917763874_a_nat @ G_7 @ B_59 ) ) ) ) ).
thf(fact_712_setsum_OF__cong,axiom,
! [H: nat > nat,G_7: nat > nat,A_70: nat > $o,B_59: nat > $o] :
( ( A_70 = B_59 )
=> ( ! [X_1: nat] :
( ( member_nat @ X_1 @ B_59 )
=> ( ( H @ X_1 )
= ( G_7 @ X_1 ) ) )
=> ( ( big_co387207925at_nat @ H @ A_70 )
= ( big_co387207925at_nat @ G_7 @ B_59 ) ) ) ) ).
thf(fact_713_single__Diff__lessThan,axiom,
! [K_3: nat] :
( ( minus_minus_nat_o @ ( insert_nat @ K_3 @ bot_bot_nat_o ) @ ( ord_lessThan_nat @ K_3 ) )
= ( insert_nat @ K_3 @ bot_bot_nat_o ) ) ).
thf(fact_714_single__Diff__lessThan,axiom,
! [K_3: int] :
( ( minus_minus_int_o @ ( insert_int @ K_3 @ bot_bot_int_o ) @ ( ord_lessThan_int @ K_3 ) )
= ( insert_int @ K_3 @ bot_bot_int_o ) ) ).
thf(fact_715_setsum__diff__nat,axiom,
! [F_16: hoare_2091234717iple_a > nat,A_69: hoare_2091234717iple_a > $o,B_58: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ B_58 )
=> ( ( ord_le35180118le_a_o @ B_58 @ A_69 )
=> ( ( big_co917763874_a_nat @ F_16 @ ( minus_836160335le_a_o @ A_69 @ B_58 ) )
= ( minus_minus_nat @ ( big_co917763874_a_nat @ F_16 @ A_69 ) @ ( big_co917763874_a_nat @ F_16 @ B_58 ) ) ) ) ) ).
thf(fact_716_setsum__diff__nat,axiom,
! [F_16: int > nat,A_69: int > $o,B_58: int > $o] :
( ( finite_finite_int @ B_58 )
=> ( ( ord_less_eq_int_o @ B_58 @ A_69 )
=> ( ( big_co1740723097nt_nat @ F_16 @ ( minus_minus_int_o @ A_69 @ B_58 ) )
= ( minus_minus_nat @ ( big_co1740723097nt_nat @ F_16 @ A_69 ) @ ( big_co1740723097nt_nat @ F_16 @ B_58 ) ) ) ) ) ).
thf(fact_717_setsum__diff__nat,axiom,
! [F_16: nat > nat,A_69: nat > $o,B_58: nat > $o] :
( ( finite_finite_nat @ B_58 )
=> ( ( ord_less_eq_nat_o @ B_58 @ A_69 )
=> ( ( big_co387207925at_nat @ F_16 @ ( minus_minus_nat_o @ A_69 @ B_58 ) )
= ( minus_minus_nat @ ( big_co387207925at_nat @ F_16 @ A_69 ) @ ( big_co387207925at_nat @ F_16 @ B_58 ) ) ) ) ) ).
thf(fact_718_order__refl,axiom,
! [X_45: int > $o] : ( ord_less_eq_int_o @ X_45 @ X_45 ) ).
thf(fact_719_order__refl,axiom,
! [X_45: nat > $o] : ( ord_less_eq_nat_o @ X_45 @ X_45 ) ).
thf(fact_720_order__refl,axiom,
! [X_45: int] : ( ord_less_eq_int @ X_45 @ X_45 ) ).
thf(fact_721_order__refl,axiom,
! [X_45: nat] : ( ord_less_eq_nat @ X_45 @ X_45 ) ).
thf(fact_722_equalityI,axiom,
! [A_68: int > $o,B_57: int > $o] :
( ( ord_less_eq_int_o @ A_68 @ B_57 )
=> ( ( ord_less_eq_int_o @ B_57 @ A_68 )
=> ( A_68 = B_57 ) ) ) ).
thf(fact_723_equalityI,axiom,
! [A_68: nat > $o,B_57: nat > $o] :
( ( ord_less_eq_nat_o @ A_68 @ B_57 )
=> ( ( ord_less_eq_nat_o @ B_57 @ A_68 )
=> ( A_68 = B_57 ) ) ) ).
thf(fact_724_subsetD,axiom,
! [C_23: hoare_2091234717iple_a,A_67: hoare_2091234717iple_a > $o,B_56: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_67 @ B_56 )
=> ( ( member290856304iple_a @ C_23 @ A_67 )
=> ( member290856304iple_a @ C_23 @ B_56 ) ) ) ).
thf(fact_725_subsetD,axiom,
! [C_23: int,A_67: int > $o,B_56: int > $o] :
( ( ord_less_eq_int_o @ A_67 @ B_56 )
=> ( ( member_int @ C_23 @ A_67 )
=> ( member_int @ C_23 @ B_56 ) ) ) ).
thf(fact_726_subsetD,axiom,
! [C_23: nat,A_67: nat > $o,B_56: nat > $o] :
( ( ord_less_eq_nat_o @ A_67 @ B_56 )
=> ( ( member_nat @ C_23 @ A_67 )
=> ( member_nat @ C_23 @ B_56 ) ) ) ).
thf(fact_727_empty__subsetI,axiom,
! [A_66: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_66 ) ).
thf(fact_728_empty__subsetI,axiom,
! [A_66: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_66 ) ).
thf(fact_729_empty__subsetI,axiom,
! [A_66: hoare_2091234717iple_a > $o] : ( ord_le35180118le_a_o @ bot_bo1791335050le_a_o @ A_66 ) ).
thf(fact_730_finite__Collect__subsets,axiom,
! [A_65: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_65 )
=> ( finite1829014797le_a_o
@ ( collec1008234059le_a_o
@ ^ [B_55: hoare_2091234717iple_a > $o] : ( ord_le35180118le_a_o @ B_55 @ A_65 ) ) ) ) ).
thf(fact_731_finite__Collect__subsets,axiom,
! [A_65: int > $o] :
( ( finite_finite_int @ A_65 )
=> ( finite_finite_int_o
@ ( collect_int_o
@ ^ [B_55: int > $o] : ( ord_less_eq_int_o @ B_55 @ A_65 ) ) ) ) ).
thf(fact_732_finite__Collect__subsets,axiom,
! [A_65: nat > $o] :
( ( finite_finite_nat @ A_65 )
=> ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [B_55: nat > $o] : ( ord_less_eq_nat_o @ B_55 @ A_65 ) ) ) ) ).
thf(fact_733_pred__subset__eq,axiom,
! [R: hoare_2091234717iple_a > $o,S_2: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o
@ ^ [X_1: hoare_2091234717iple_a] : ( member290856304iple_a @ X_1 @ R )
@ ^ [X_1: hoare_2091234717iple_a] : ( member290856304iple_a @ X_1 @ S_2 ) )
<=> ( ord_le35180118le_a_o @ R @ S_2 ) ) ).
thf(fact_734_pred__subset__eq,axiom,
! [R: int > $o,S_2: int > $o] :
( ( ord_less_eq_int_o
@ ^ [X_1: int] : ( member_int @ X_1 @ R )
@ ^ [X_1: int] : ( member_int @ X_1 @ S_2 ) )
<=> ( ord_less_eq_int_o @ R @ S_2 ) ) ).
thf(fact_735_pred__subset__eq,axiom,
! [R: nat > $o,S_2: nat > $o] :
( ( ord_less_eq_nat_o
@ ^ [X_1: nat] : ( member_nat @ X_1 @ R )
@ ^ [X_1: nat] : ( member_nat @ X_1 @ S_2 ) )
<=> ( ord_less_eq_nat_o @ R @ S_2 ) ) ).
thf(fact_736_linorder__le__cases,axiom,
! [X_44: int,Y_25: int] :
( ~ ( ord_less_eq_int @ X_44 @ Y_25 )
=> ( ord_less_eq_int @ Y_25 @ X_44 ) ) ).
thf(fact_737_linorder__le__cases,axiom,
! [X_44: nat,Y_25: nat] :
( ~ ( ord_less_eq_nat @ X_44 @ Y_25 )
=> ( ord_less_eq_nat @ Y_25 @ X_44 ) ) ).
thf(fact_738_le__funE,axiom,
! [X_43: int,F_15: int > $o,G_6: int > $o] :
( ( ord_less_eq_int_o @ F_15 @ G_6 )
=> ( ord_less_eq_o @ ( F_15 @ X_43 ) @ ( G_6 @ X_43 ) ) ) ).
thf(fact_739_le__funE,axiom,
! [X_43: nat,F_15: nat > $o,G_6: nat > $o] :
( ( ord_less_eq_nat_o @ F_15 @ G_6 )
=> ( ord_less_eq_o @ ( F_15 @ X_43 ) @ ( G_6 @ X_43 ) ) ) ).
thf(fact_740_xt1_I6_J,axiom,
! [Z_11: int > $o,Y_24: int > $o,X_42: int > $o] :
( ( ord_less_eq_int_o @ Y_24 @ X_42 )
=> ( ( ord_less_eq_int_o @ Z_11 @ Y_24 )
=> ( ord_less_eq_int_o @ Z_11 @ X_42 ) ) ) ).
thf(fact_741_xt1_I6_J,axiom,
! [Z_11: nat > $o,Y_24: nat > $o,X_42: nat > $o] :
( ( ord_less_eq_nat_o @ Y_24 @ X_42 )
=> ( ( ord_less_eq_nat_o @ Z_11 @ Y_24 )
=> ( ord_less_eq_nat_o @ Z_11 @ X_42 ) ) ) ).
thf(fact_742_xt1_I6_J,axiom,
! [Z_11: int,Y_24: int,X_42: int] :
( ( ord_less_eq_int @ Y_24 @ X_42 )
=> ( ( ord_less_eq_int @ Z_11 @ Y_24 )
=> ( ord_less_eq_int @ Z_11 @ X_42 ) ) ) ).
thf(fact_743_xt1_I6_J,axiom,
! [Z_11: nat,Y_24: nat,X_42: nat] :
( ( ord_less_eq_nat @ Y_24 @ X_42 )
=> ( ( ord_less_eq_nat @ Z_11 @ Y_24 )
=> ( ord_less_eq_nat @ Z_11 @ X_42 ) ) ) ).
thf(fact_744_xt1_I5_J,axiom,
! [Y_23: int > $o,X_41: int > $o] :
( ( ord_less_eq_int_o @ Y_23 @ X_41 )
=> ( ( ord_less_eq_int_o @ X_41 @ Y_23 )
=> ( X_41 = Y_23 ) ) ) ).
thf(fact_745_xt1_I5_J,axiom,
! [Y_23: nat > $o,X_41: nat > $o] :
( ( ord_less_eq_nat_o @ Y_23 @ X_41 )
=> ( ( ord_less_eq_nat_o @ X_41 @ Y_23 )
=> ( X_41 = Y_23 ) ) ) ).
thf(fact_746_xt1_I5_J,axiom,
! [Y_23: int,X_41: int] :
( ( ord_less_eq_int @ Y_23 @ X_41 )
=> ( ( ord_less_eq_int @ X_41 @ Y_23 )
=> ( X_41 = Y_23 ) ) ) ).
thf(fact_747_xt1_I5_J,axiom,
! [Y_23: nat,X_41: nat] :
( ( ord_less_eq_nat @ Y_23 @ X_41 )
=> ( ( ord_less_eq_nat @ X_41 @ Y_23 )
=> ( X_41 = Y_23 ) ) ) ).
thf(fact_748_order__trans,axiom,
! [Z_10: int > $o,X_40: int > $o,Y_22: int > $o] :
( ( ord_less_eq_int_o @ X_40 @ Y_22 )
=> ( ( ord_less_eq_int_o @ Y_22 @ Z_10 )
=> ( ord_less_eq_int_o @ X_40 @ Z_10 ) ) ) ).
thf(fact_749_order__trans,axiom,
! [Z_10: nat > $o,X_40: nat > $o,Y_22: nat > $o] :
( ( ord_less_eq_nat_o @ X_40 @ Y_22 )
=> ( ( ord_less_eq_nat_o @ Y_22 @ Z_10 )
=> ( ord_less_eq_nat_o @ X_40 @ Z_10 ) ) ) ).
thf(fact_750_order__trans,axiom,
! [Z_10: int,X_40: int,Y_22: int] :
( ( ord_less_eq_int @ X_40 @ Y_22 )
=> ( ( ord_less_eq_int @ Y_22 @ Z_10 )
=> ( ord_less_eq_int @ X_40 @ Z_10 ) ) ) ).
thf(fact_751_order__trans,axiom,
! [Z_10: nat,X_40: nat,Y_22: nat] :
( ( ord_less_eq_nat @ X_40 @ Y_22 )
=> ( ( ord_less_eq_nat @ Y_22 @ Z_10 )
=> ( ord_less_eq_nat @ X_40 @ Z_10 ) ) ) ).
thf(fact_752_order__antisym,axiom,
! [X_39: int > $o,Y_21: int > $o] :
( ( ord_less_eq_int_o @ X_39 @ Y_21 )
=> ( ( ord_less_eq_int_o @ Y_21 @ X_39 )
=> ( X_39 = Y_21 ) ) ) ).
thf(fact_753_order__antisym,axiom,
! [X_39: nat > $o,Y_21: nat > $o] :
( ( ord_less_eq_nat_o @ X_39 @ Y_21 )
=> ( ( ord_less_eq_nat_o @ Y_21 @ X_39 )
=> ( X_39 = Y_21 ) ) ) ).
thf(fact_754_order__antisym,axiom,
! [X_39: int,Y_21: int] :
( ( ord_less_eq_int @ X_39 @ Y_21 )
=> ( ( ord_less_eq_int @ Y_21 @ X_39 )
=> ( X_39 = Y_21 ) ) ) ).
thf(fact_755_order__antisym,axiom,
! [X_39: nat,Y_21: nat] :
( ( ord_less_eq_nat @ X_39 @ Y_21 )
=> ( ( ord_less_eq_nat @ Y_21 @ X_39 )
=> ( X_39 = Y_21 ) ) ) ).
thf(fact_756_xt1_I4_J,axiom,
! [C_22: int > $o,B_54: int > $o,A_64: int > $o] :
( ( ord_less_eq_int_o @ B_54 @ A_64 )
=> ( ( B_54 = C_22 )
=> ( ord_less_eq_int_o @ C_22 @ A_64 ) ) ) ).
thf(fact_757_xt1_I4_J,axiom,
! [C_22: nat > $o,B_54: nat > $o,A_64: nat > $o] :
( ( ord_less_eq_nat_o @ B_54 @ A_64 )
=> ( ( B_54 = C_22 )
=> ( ord_less_eq_nat_o @ C_22 @ A_64 ) ) ) ).
thf(fact_758_xt1_I4_J,axiom,
! [C_22: int,B_54: int,A_64: int] :
( ( ord_less_eq_int @ B_54 @ A_64 )
=> ( ( B_54 = C_22 )
=> ( ord_less_eq_int @ C_22 @ A_64 ) ) ) ).
thf(fact_759_xt1_I4_J,axiom,
! [C_22: nat,B_54: nat,A_64: nat] :
( ( ord_less_eq_nat @ B_54 @ A_64 )
=> ( ( B_54 = C_22 )
=> ( ord_less_eq_nat @ C_22 @ A_64 ) ) ) ).
thf(fact_760_ord__le__eq__trans,axiom,
! [C_21: int > $o,A_63: int > $o,B_53: int > $o] :
( ( ord_less_eq_int_o @ A_63 @ B_53 )
=> ( ( B_53 = C_21 )
=> ( ord_less_eq_int_o @ A_63 @ C_21 ) ) ) ).
thf(fact_761_ord__le__eq__trans,axiom,
! [C_21: nat > $o,A_63: nat > $o,B_53: nat > $o] :
( ( ord_less_eq_nat_o @ A_63 @ B_53 )
=> ( ( B_53 = C_21 )
=> ( ord_less_eq_nat_o @ A_63 @ C_21 ) ) ) ).
thf(fact_762_ord__le__eq__trans,axiom,
! [C_21: int,A_63: int,B_53: int] :
( ( ord_less_eq_int @ A_63 @ B_53 )
=> ( ( B_53 = C_21 )
=> ( ord_less_eq_int @ A_63 @ C_21 ) ) ) ).
thf(fact_763_ord__le__eq__trans,axiom,
! [C_21: nat,A_63: nat,B_53: nat] :
( ( ord_less_eq_nat @ A_63 @ B_53 )
=> ( ( B_53 = C_21 )
=> ( ord_less_eq_nat @ A_63 @ C_21 ) ) ) ).
thf(fact_764_xt1_I3_J,axiom,
! [C_20: int > $o,A_62: int > $o,B_52: int > $o] :
( ( A_62 = B_52 )
=> ( ( ord_less_eq_int_o @ C_20 @ B_52 )
=> ( ord_less_eq_int_o @ C_20 @ A_62 ) ) ) ).
thf(fact_765_xt1_I3_J,axiom,
! [C_20: nat > $o,A_62: nat > $o,B_52: nat > $o] :
( ( A_62 = B_52 )
=> ( ( ord_less_eq_nat_o @ C_20 @ B_52 )
=> ( ord_less_eq_nat_o @ C_20 @ A_62 ) ) ) ).
thf(fact_766_xt1_I3_J,axiom,
! [C_20: int,A_62: int,B_52: int] :
( ( A_62 = B_52 )
=> ( ( ord_less_eq_int @ C_20 @ B_52 )
=> ( ord_less_eq_int @ C_20 @ A_62 ) ) ) ).
thf(fact_767_xt1_I3_J,axiom,
! [C_20: nat,A_62: nat,B_52: nat] :
( ( A_62 = B_52 )
=> ( ( ord_less_eq_nat @ C_20 @ B_52 )
=> ( ord_less_eq_nat @ C_20 @ A_62 ) ) ) ).
thf(fact_768_ord__eq__le__trans,axiom,
! [C_19: int > $o,A_61: int > $o,B_51: int > $o] :
( ( A_61 = B_51 )
=> ( ( ord_less_eq_int_o @ B_51 @ C_19 )
=> ( ord_less_eq_int_o @ A_61 @ C_19 ) ) ) ).
thf(fact_769_ord__eq__le__trans,axiom,
! [C_19: nat > $o,A_61: nat > $o,B_51: nat > $o] :
( ( A_61 = B_51 )
=> ( ( ord_less_eq_nat_o @ B_51 @ C_19 )
=> ( ord_less_eq_nat_o @ A_61 @ C_19 ) ) ) ).
thf(fact_770_ord__eq__le__trans,axiom,
! [C_19: int,A_61: int,B_51: int] :
( ( A_61 = B_51 )
=> ( ( ord_less_eq_int @ B_51 @ C_19 )
=> ( ord_less_eq_int @ A_61 @ C_19 ) ) ) ).
thf(fact_771_ord__eq__le__trans,axiom,
! [C_19: nat,A_61: nat,B_51: nat] :
( ( A_61 = B_51 )
=> ( ( ord_less_eq_nat @ B_51 @ C_19 )
=> ( ord_less_eq_nat @ A_61 @ C_19 ) ) ) ).
thf(fact_772_order__antisym__conv,axiom,
! [Y_20: int > $o,X_38: int > $o] :
( ( ord_less_eq_int_o @ Y_20 @ X_38 )
=> ( ( ord_less_eq_int_o @ X_38 @ Y_20 )
<=> ( X_38 = Y_20 ) ) ) ).
thf(fact_773_order__antisym__conv,axiom,
! [Y_20: nat > $o,X_38: nat > $o] :
( ( ord_less_eq_nat_o @ Y_20 @ X_38 )
=> ( ( ord_less_eq_nat_o @ X_38 @ Y_20 )
<=> ( X_38 = Y_20 ) ) ) ).
thf(fact_774_order__antisym__conv,axiom,
! [Y_20: int,X_38: int] :
( ( ord_less_eq_int @ Y_20 @ X_38 )
=> ( ( ord_less_eq_int @ X_38 @ Y_20 )
<=> ( X_38 = Y_20 ) ) ) ).
thf(fact_775_order__antisym__conv,axiom,
! [Y_20: nat,X_38: nat] :
( ( ord_less_eq_nat @ Y_20 @ X_38 )
=> ( ( ord_less_eq_nat @ X_38 @ Y_20 )
<=> ( X_38 = Y_20 ) ) ) ).
thf(fact_776_le__funD,axiom,
! [X_37: int,F_14: int > $o,G_5: int > $o] :
( ( ord_less_eq_int_o @ F_14 @ G_5 )
=> ( ord_less_eq_o @ ( F_14 @ X_37 ) @ ( G_5 @ X_37 ) ) ) ).
thf(fact_777_le__funD,axiom,
! [X_37: nat,F_14: nat > $o,G_5: nat > $o] :
( ( ord_less_eq_nat_o @ F_14 @ G_5 )
=> ( ord_less_eq_o @ ( F_14 @ X_37 ) @ ( G_5 @ X_37 ) ) ) ).
thf(fact_778_order__eq__refl,axiom,
! [X_36: int > $o,Y_19: int > $o] :
( ( X_36 = Y_19 )
=> ( ord_less_eq_int_o @ X_36 @ Y_19 ) ) ).
thf(fact_779_order__eq__refl,axiom,
! [X_36: nat > $o,Y_19: nat > $o] :
( ( X_36 = Y_19 )
=> ( ord_less_eq_nat_o @ X_36 @ Y_19 ) ) ).
thf(fact_780_order__eq__refl,axiom,
! [X_36: int,Y_19: int] :
( ( X_36 = Y_19 )
=> ( ord_less_eq_int @ X_36 @ Y_19 ) ) ).
thf(fact_781_order__eq__refl,axiom,
! [X_36: nat,Y_19: nat] :
( ( X_36 = Y_19 )
=> ( ord_less_eq_nat @ X_36 @ Y_19 ) ) ).
thf(fact_782_order__eq__iff,axiom,
! [X_35: int > $o,Y_18: int > $o] :
( ( X_35 = Y_18 )
<=> ( ( ord_less_eq_int_o @ X_35 @ Y_18 )
& ( ord_less_eq_int_o @ Y_18 @ X_35 ) ) ) ).
thf(fact_783_order__eq__iff,axiom,
! [X_35: nat > $o,Y_18: nat > $o] :
( ( X_35 = Y_18 )
<=> ( ( ord_less_eq_nat_o @ X_35 @ Y_18 )
& ( ord_less_eq_nat_o @ Y_18 @ X_35 ) ) ) ).
thf(fact_784_order__eq__iff,axiom,
! [X_35: int,Y_18: int] :
( ( X_35 = Y_18 )
<=> ( ( ord_less_eq_int @ X_35 @ Y_18 )
& ( ord_less_eq_int @ Y_18 @ X_35 ) ) ) ).
thf(fact_785_order__eq__iff,axiom,
! [X_35: nat,Y_18: nat] :
( ( X_35 = Y_18 )
<=> ( ( ord_less_eq_nat @ X_35 @ Y_18 )
& ( ord_less_eq_nat @ Y_18 @ X_35 ) ) ) ).
thf(fact_786_linorder__linear,axiom,
! [X_34: int,Y_17: int] :
( ( ord_less_eq_int @ X_34 @ Y_17 )
| ( ord_less_eq_int @ Y_17 @ X_34 ) ) ).
thf(fact_787_linorder__linear,axiom,
! [X_34: nat,Y_17: nat] :
( ( ord_less_eq_nat @ X_34 @ Y_17 )
| ( ord_less_eq_nat @ Y_17 @ X_34 ) ) ).
thf(fact_788_le__fun__def,axiom,
! [F_13: int > $o,G_4: int > $o] :
( ( ord_less_eq_int_o @ F_13 @ G_4 )
<=> ! [X_1: int] : ( ord_less_eq_o @ ( F_13 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
thf(fact_789_le__fun__def,axiom,
! [F_13: nat > $o,G_4: nat > $o] :
( ( ord_less_eq_nat_o @ F_13 @ G_4 )
<=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_13 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
thf(fact_790_equalityE,axiom,
! [A_60: int > $o,B_50: int > $o] :
( ( A_60 = B_50 )
=> ~ ( ( ord_less_eq_int_o @ A_60 @ B_50 )
=> ~ ( ord_less_eq_int_o @ B_50 @ A_60 ) ) ) ).
thf(fact_791_equalityE,axiom,
! [A_60: nat > $o,B_50: nat > $o] :
( ( A_60 = B_50 )
=> ~ ( ( ord_less_eq_nat_o @ A_60 @ B_50 )
=> ~ ( ord_less_eq_nat_o @ B_50 @ A_60 ) ) ) ).
thf(fact_792_subset__trans,axiom,
! [C_18: int > $o,A_59: int > $o,B_49: int > $o] :
( ( ord_less_eq_int_o @ A_59 @ B_49 )
=> ( ( ord_less_eq_int_o @ B_49 @ C_18 )
=> ( ord_less_eq_int_o @ A_59 @ C_18 ) ) ) ).
thf(fact_793_subset__trans,axiom,
! [C_18: nat > $o,A_59: nat > $o,B_49: nat > $o] :
( ( ord_less_eq_nat_o @ A_59 @ B_49 )
=> ( ( ord_less_eq_nat_o @ B_49 @ C_18 )
=> ( ord_less_eq_nat_o @ A_59 @ C_18 ) ) ) ).
thf(fact_794_set__mp,axiom,
! [X_33: hoare_2091234717iple_a,A_58: hoare_2091234717iple_a > $o,B_48: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_58 @ B_48 )
=> ( ( member290856304iple_a @ X_33 @ A_58 )
=> ( member290856304iple_a @ X_33 @ B_48 ) ) ) ).
thf(fact_795_set__mp,axiom,
! [X_33: int,A_58: int > $o,B_48: int > $o] :
( ( ord_less_eq_int_o @ A_58 @ B_48 )
=> ( ( member_int @ X_33 @ A_58 )
=> ( member_int @ X_33 @ B_48 ) ) ) ).
thf(fact_796_set__mp,axiom,
! [X_33: nat,A_58: nat > $o,B_48: nat > $o] :
( ( ord_less_eq_nat_o @ A_58 @ B_48 )
=> ( ( member_nat @ X_33 @ A_58 )
=> ( member_nat @ X_33 @ B_48 ) ) ) ).
thf(fact_797_set__rev__mp,axiom,
! [B_47: hoare_2091234717iple_a > $o,X_32: hoare_2091234717iple_a,A_57: hoare_2091234717iple_a > $o] :
( ( member290856304iple_a @ X_32 @ A_57 )
=> ( ( ord_le35180118le_a_o @ A_57 @ B_47 )
=> ( member290856304iple_a @ X_32 @ B_47 ) ) ) ).
thf(fact_798_set__rev__mp,axiom,
! [B_47: int > $o,X_32: int,A_57: int > $o] :
( ( member_int @ X_32 @ A_57 )
=> ( ( ord_less_eq_int_o @ A_57 @ B_47 )
=> ( member_int @ X_32 @ B_47 ) ) ) ).
thf(fact_799_set__rev__mp,axiom,
! [B_47: nat > $o,X_32: nat,A_57: nat > $o] :
( ( member_nat @ X_32 @ A_57 )
=> ( ( ord_less_eq_nat_o @ A_57 @ B_47 )
=> ( member_nat @ X_32 @ B_47 ) ) ) ).
thf(fact_800_in__mono,axiom,
! [X_31: hoare_2091234717iple_a,A_56: hoare_2091234717iple_a > $o,B_46: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_56 @ B_46 )
=> ( ( member290856304iple_a @ X_31 @ A_56 )
=> ( member290856304iple_a @ X_31 @ B_46 ) ) ) ).
thf(fact_801_in__mono,axiom,
! [X_31: int,A_56: int > $o,B_46: int > $o] :
( ( ord_less_eq_int_o @ A_56 @ B_46 )
=> ( ( member_int @ X_31 @ A_56 )
=> ( member_int @ X_31 @ B_46 ) ) ) ).
thf(fact_802_in__mono,axiom,
! [X_31: nat,A_56: nat > $o,B_46: nat > $o] :
( ( ord_less_eq_nat_o @ A_56 @ B_46 )
=> ( ( member_nat @ X_31 @ A_56 )
=> ( member_nat @ X_31 @ B_46 ) ) ) ).
thf(fact_803_equalityD2,axiom,
! [A_55: int > $o,B_45: int > $o] :
( ( A_55 = B_45 )
=> ( ord_less_eq_int_o @ B_45 @ A_55 ) ) ).
thf(fact_804_equalityD2,axiom,
! [A_55: nat > $o,B_45: nat > $o] :
( ( A_55 = B_45 )
=> ( ord_less_eq_nat_o @ B_45 @ A_55 ) ) ).
thf(fact_805_equalityD1,axiom,
! [A_54: int > $o,B_44: int > $o] :
( ( A_54 = B_44 )
=> ( ord_less_eq_int_o @ A_54 @ B_44 ) ) ).
thf(fact_806_equalityD1,axiom,
! [A_54: nat > $o,B_44: nat > $o] :
( ( A_54 = B_44 )
=> ( ord_less_eq_nat_o @ A_54 @ B_44 ) ) ).
thf(fact_807_set__eq__subset,axiom,
! [A_53: int > $o,B_43: int > $o] :
( ( A_53 = B_43 )
<=> ( ( ord_less_eq_int_o @ A_53 @ B_43 )
& ( ord_less_eq_int_o @ B_43 @ A_53 ) ) ) ).
thf(fact_808_set__eq__subset,axiom,
! [A_53: nat > $o,B_43: nat > $o] :
( ( A_53 = B_43 )
<=> ( ( ord_less_eq_nat_o @ A_53 @ B_43 )
& ( ord_less_eq_nat_o @ B_43 @ A_53 ) ) ) ).
thf(fact_809_subset__refl,axiom,
! [A_52: int > $o] : ( ord_less_eq_int_o @ A_52 @ A_52 ) ).
thf(fact_810_subset__refl,axiom,
! [A_52: nat > $o] : ( ord_less_eq_nat_o @ A_52 @ A_52 ) ).
thf(fact_811_predicate1D,axiom,
! [X_30: int,P_5: int > $o,Q_4: int > $o] :
( ( ord_less_eq_int_o @ P_5 @ Q_4 )
=> ( ( P_5 @ X_30 )
=> ( Q_4 @ X_30 ) ) ) ).
thf(fact_812_predicate1D,axiom,
! [X_30: nat,P_5: nat > $o,Q_4: nat > $o] :
( ( ord_less_eq_nat_o @ P_5 @ Q_4 )
=> ( ( P_5 @ X_30 )
=> ( Q_4 @ X_30 ) ) ) ).
thf(fact_813_rev__predicate1D,axiom,
! [Q_3: int > $o,P_4: int > $o,X_29: int] :
( ( P_4 @ X_29 )
=> ( ( ord_less_eq_int_o @ P_4 @ Q_3 )
=> ( Q_3 @ X_29 ) ) ) ).
thf(fact_814_rev__predicate1D,axiom,
! [Q_3: nat > $o,P_4: nat > $o,X_29: nat] :
( ( P_4 @ X_29 )
=> ( ( ord_less_eq_nat_o @ P_4 @ Q_3 )
=> ( Q_3 @ X_29 ) ) ) ).
thf(fact_815_lessThan__subset__iff,axiom,
! [X_28: int,Y_16: int] :
( ( ord_less_eq_int_o @ ( ord_lessThan_int @ X_28 ) @ ( ord_lessThan_int @ Y_16 ) )
<=> ( ord_less_eq_int @ X_28 @ Y_16 ) ) ).
thf(fact_816_lessThan__subset__iff,axiom,
! [X_28: nat,Y_16: nat] :
( ( ord_less_eq_nat_o @ ( ord_lessThan_nat @ X_28 ) @ ( ord_lessThan_nat @ Y_16 ) )
<=> ( ord_less_eq_nat @ X_28 @ Y_16 ) ) ).
thf(fact_817_lessThan__eq__iff,axiom,
! [X_27: nat,Y_15: nat] :
( ( ( ord_lessThan_nat @ X_27 )
= ( ord_lessThan_nat @ Y_15 ) )
<=> ( X_27 = Y_15 ) ) ).
thf(fact_818_add__le__imp__le__left,axiom,
! [C_17: int,A_51: int,B_42: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_17 @ A_51 ) @ ( plus_plus_int @ C_17 @ B_42 ) )
=> ( ord_less_eq_int @ A_51 @ B_42 ) ) ).
thf(fact_819_add__le__imp__le__left,axiom,
! [C_17: nat,A_51: nat,B_42: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_17 @ A_51 ) @ ( plus_plus_nat @ C_17 @ B_42 ) )
=> ( ord_less_eq_nat @ A_51 @ B_42 ) ) ).
thf(fact_820_add__le__imp__le__right,axiom,
! [A_50: int,C_16: int,B_41: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_50 @ C_16 ) @ ( plus_plus_int @ B_41 @ C_16 ) )
=> ( ord_less_eq_int @ A_50 @ B_41 ) ) ).
thf(fact_821_add__le__imp__le__right,axiom,
! [A_50: nat,C_16: nat,B_41: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_50 @ C_16 ) @ ( plus_plus_nat @ B_41 @ C_16 ) )
=> ( ord_less_eq_nat @ A_50 @ B_41 ) ) ).
thf(fact_822_add__mono,axiom,
! [C_15: int,D_6: int,A_49: int,B_40: int] :
( ( ord_less_eq_int @ A_49 @ B_40 )
=> ( ( ord_less_eq_int @ C_15 @ D_6 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_49 @ C_15 ) @ ( plus_plus_int @ B_40 @ D_6 ) ) ) ) ).
thf(fact_823_add__mono,axiom,
! [C_15: nat,D_6: nat,A_49: nat,B_40: nat] :
( ( ord_less_eq_nat @ A_49 @ B_40 )
=> ( ( ord_less_eq_nat @ C_15 @ D_6 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_49 @ C_15 ) @ ( plus_plus_nat @ B_40 @ D_6 ) ) ) ) ).
thf(fact_824_add__left__mono,axiom,
! [C_14: int,A_48: int,B_39: int] :
( ( ord_less_eq_int @ A_48 @ B_39 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_14 @ A_48 ) @ ( plus_plus_int @ C_14 @ B_39 ) ) ) ).
thf(fact_825_add__left__mono,axiom,
! [C_14: nat,A_48: nat,B_39: nat] :
( ( ord_less_eq_nat @ A_48 @ B_39 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_14 @ A_48 ) @ ( plus_plus_nat @ C_14 @ B_39 ) ) ) ).
thf(fact_826_add__right__mono,axiom,
! [C_13: int,A_47: int,B_38: int] :
( ( ord_less_eq_int @ A_47 @ B_38 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_47 @ C_13 ) @ ( plus_plus_int @ B_38 @ C_13 ) ) ) ).
thf(fact_827_add__right__mono,axiom,
! [C_13: nat,A_47: nat,B_38: nat] :
( ( ord_less_eq_nat @ A_47 @ B_38 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_47 @ C_13 ) @ ( plus_plus_nat @ B_38 @ C_13 ) ) ) ).
thf(fact_828_add__le__cancel__left,axiom,
! [C_12: int,A_46: int,B_37: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_12 @ A_46 ) @ ( plus_plus_int @ C_12 @ B_37 ) )
<=> ( ord_less_eq_int @ A_46 @ B_37 ) ) ).
thf(fact_829_add__le__cancel__left,axiom,
! [C_12: nat,A_46: nat,B_37: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_12 @ A_46 ) @ ( plus_plus_nat @ C_12 @ B_37 ) )
<=> ( ord_less_eq_nat @ A_46 @ B_37 ) ) ).
thf(fact_830_add__le__cancel__right,axiom,
! [A_45: int,C_11: int,B_36: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_45 @ C_11 ) @ ( plus_plus_int @ B_36 @ C_11 ) )
<=> ( ord_less_eq_int @ A_45 @ B_36 ) ) ).
thf(fact_831_add__le__cancel__right,axiom,
! [A_45: nat,C_11: nat,B_36: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_45 @ C_11 ) @ ( plus_plus_nat @ B_36 @ C_11 ) )
<=> ( ord_less_eq_nat @ A_45 @ B_36 ) ) ).
thf(fact_832_le__bot,axiom,
! [A_44: $o] :
( ( ord_less_eq_o @ A_44 @ bot_bot_o )
=> ( A_44
<=> bot_bot_o ) ) ).
thf(fact_833_le__bot,axiom,
! [A_44: nat > $o] :
( ( ord_less_eq_nat_o @ A_44 @ bot_bot_nat_o )
=> ( A_44 = bot_bot_nat_o ) ) ).
thf(fact_834_le__bot,axiom,
! [A_44: int > $o] :
( ( ord_less_eq_int_o @ A_44 @ bot_bot_int_o )
=> ( A_44 = bot_bot_int_o ) ) ).
thf(fact_835_le__bot,axiom,
! [A_44: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_44 @ bot_bo1791335050le_a_o )
=> ( A_44 = bot_bo1791335050le_a_o ) ) ).
thf(fact_836_le__bot,axiom,
! [A_44: nat] :
( ( ord_less_eq_nat @ A_44 @ bot_bot_nat )
=> ( A_44 = bot_bot_nat ) ) ).
thf(fact_837_bot__unique,axiom,
! [A_43: $o] :
( ( ord_less_eq_o @ A_43 @ bot_bot_o )
<=> ( A_43
<=> bot_bot_o ) ) ).
thf(fact_838_bot__unique,axiom,
! [A_43: nat > $o] :
( ( ord_less_eq_nat_o @ A_43 @ bot_bot_nat_o )
<=> ( A_43 = bot_bot_nat_o ) ) ).
thf(fact_839_bot__unique,axiom,
! [A_43: int > $o] :
( ( ord_less_eq_int_o @ A_43 @ bot_bot_int_o )
<=> ( A_43 = bot_bot_int_o ) ) ).
thf(fact_840_bot__unique,axiom,
! [A_43: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_43 @ bot_bo1791335050le_a_o )
<=> ( A_43 = bot_bo1791335050le_a_o ) ) ).
thf(fact_841_bot__unique,axiom,
! [A_43: nat] :
( ( ord_less_eq_nat @ A_43 @ bot_bot_nat )
<=> ( A_43 = bot_bot_nat ) ) ).
thf(fact_842_bot__least,axiom,
! [A_42: $o] : ( ord_less_eq_o @ bot_bot_o @ A_42 ) ).
thf(fact_843_bot__least,axiom,
! [A_42: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_42 ) ).
thf(fact_844_bot__least,axiom,
! [A_42: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_42 ) ).
thf(fact_845_bot__least,axiom,
! [A_42: hoare_2091234717iple_a > $o] : ( ord_le35180118le_a_o @ bot_bo1791335050le_a_o @ A_42 ) ).
thf(fact_846_bot__least,axiom,
! [A_42: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_42 ) ).
thf(fact_847_diff__eq__diff__less__eq,axiom,
! [A_41: int,B_35: int,C_10: int,D_5: int] :
( ( ( minus_minus_int @ A_41 @ B_35 )
= ( minus_minus_int @ C_10 @ D_5 ) )
=> ( ( ord_less_eq_int @ A_41 @ B_35 )
<=> ( ord_less_eq_int @ C_10 @ D_5 ) ) ) ).
thf(fact_848_subset__empty,axiom,
! [A_40: nat > $o] :
( ( ord_less_eq_nat_o @ A_40 @ bot_bot_nat_o )
<=> ( A_40 = bot_bot_nat_o ) ) ).
thf(fact_849_subset__empty,axiom,
! [A_40: int > $o] :
( ( ord_less_eq_int_o @ A_40 @ bot_bot_int_o )
<=> ( A_40 = bot_bot_int_o ) ) ).
thf(fact_850_subset__empty,axiom,
! [A_40: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_40 @ bot_bo1791335050le_a_o )
<=> ( A_40 = bot_bo1791335050le_a_o ) ) ).
thf(fact_851_finite__subset,axiom,
! [A_39: hoare_2091234717iple_a > $o,B_34: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_39 @ B_34 )
=> ( ( finite232261744iple_a @ B_34 )
=> ( finite232261744iple_a @ A_39 ) ) ) ).
thf(fact_852_finite__subset,axiom,
! [A_39: int > $o,B_34: int > $o] :
( ( ord_less_eq_int_o @ A_39 @ B_34 )
=> ( ( finite_finite_int @ B_34 )
=> ( finite_finite_int @ A_39 ) ) ) ).
thf(fact_853_finite__subset,axiom,
! [A_39: nat > $o,B_34: nat > $o] :
( ( ord_less_eq_nat_o @ A_39 @ B_34 )
=> ( ( finite_finite_nat @ B_34 )
=> ( finite_finite_nat @ A_39 ) ) ) ).
thf(fact_854_rev__finite__subset,axiom,
! [A_38: hoare_2091234717iple_a > $o,B_33: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ B_33 )
=> ( ( ord_le35180118le_a_o @ A_38 @ B_33 )
=> ( finite232261744iple_a @ A_38 ) ) ) ).
thf(fact_855_rev__finite__subset,axiom,
! [A_38: int > $o,B_33: int > $o] :
( ( finite_finite_int @ B_33 )
=> ( ( ord_less_eq_int_o @ A_38 @ B_33 )
=> ( finite_finite_int @ A_38 ) ) ) ).
thf(fact_856_rev__finite__subset,axiom,
! [A_38: nat > $o,B_33: nat > $o] :
( ( finite_finite_nat @ B_33 )
=> ( ( ord_less_eq_nat_o @ A_38 @ B_33 )
=> ( finite_finite_nat @ A_38 ) ) ) ).
thf(fact_857_insert__mono,axiom,
! [A_37: nat,C_9: nat > $o,D_4: nat > $o] :
( ( ord_less_eq_nat_o @ C_9 @ D_4 )
=> ( ord_less_eq_nat_o @ ( insert_nat @ A_37 @ C_9 ) @ ( insert_nat @ A_37 @ D_4 ) ) ) ).
thf(fact_858_insert__mono,axiom,
! [A_37: int,C_9: int > $o,D_4: int > $o] :
( ( ord_less_eq_int_o @ C_9 @ D_4 )
=> ( ord_less_eq_int_o @ ( insert_int @ A_37 @ C_9 ) @ ( insert_int @ A_37 @ D_4 ) ) ) ).
thf(fact_859_insert__mono,axiom,
! [A_37: hoare_2091234717iple_a,C_9: hoare_2091234717iple_a > $o,D_4: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ C_9 @ D_4 )
=> ( ord_le35180118le_a_o @ ( insert1597628439iple_a @ A_37 @ C_9 ) @ ( insert1597628439iple_a @ A_37 @ D_4 ) ) ) ).
thf(fact_860_subset__insertI2,axiom,
! [B_32: nat,A_36: nat > $o,B_31: nat > $o] :
( ( ord_less_eq_nat_o @ A_36 @ B_31 )
=> ( ord_less_eq_nat_o @ A_36 @ ( insert_nat @ B_32 @ B_31 ) ) ) ).
thf(fact_861_subset__insertI2,axiom,
! [B_32: int,A_36: int > $o,B_31: int > $o] :
( ( ord_less_eq_int_o @ A_36 @ B_31 )
=> ( ord_less_eq_int_o @ A_36 @ ( insert_int @ B_32 @ B_31 ) ) ) ).
thf(fact_862_subset__insertI2,axiom,
! [B_32: hoare_2091234717iple_a,A_36: hoare_2091234717iple_a > $o,B_31: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_36 @ B_31 )
=> ( ord_le35180118le_a_o @ A_36 @ ( insert1597628439iple_a @ B_32 @ B_31 ) ) ) ).
thf(fact_863_subset__insert,axiom,
! [B_30: nat > $o,X_26: nat,A_35: nat > $o] :
( ~ ( member_nat @ X_26 @ A_35 )
=> ( ( ord_less_eq_nat_o @ A_35 @ ( insert_nat @ X_26 @ B_30 ) )
<=> ( ord_less_eq_nat_o @ A_35 @ B_30 ) ) ) ).
thf(fact_864_subset__insert,axiom,
! [B_30: int > $o,X_26: int,A_35: int > $o] :
( ~ ( member_int @ X_26 @ A_35 )
=> ( ( ord_less_eq_int_o @ A_35 @ ( insert_int @ X_26 @ B_30 ) )
<=> ( ord_less_eq_int_o @ A_35 @ B_30 ) ) ) ).
thf(fact_865_subset__insert,axiom,
! [B_30: hoare_2091234717iple_a > $o,X_26: hoare_2091234717iple_a,A_35: hoare_2091234717iple_a > $o] :
( ~ ( member290856304iple_a @ X_26 @ A_35 )
=> ( ( ord_le35180118le_a_o @ A_35 @ ( insert1597628439iple_a @ X_26 @ B_30 ) )
<=> ( ord_le35180118le_a_o @ A_35 @ B_30 ) ) ) ).
thf(fact_866_insert__subset,axiom,
! [X_25: nat,A_34: nat > $o,B_29: nat > $o] :
( ( ord_less_eq_nat_o @ ( insert_nat @ X_25 @ A_34 ) @ B_29 )
<=> ( ( member_nat @ X_25 @ B_29 )
& ( ord_less_eq_nat_o @ A_34 @ B_29 ) ) ) ).
thf(fact_867_insert__subset,axiom,
! [X_25: int,A_34: int > $o,B_29: int > $o] :
( ( ord_less_eq_int_o @ ( insert_int @ X_25 @ A_34 ) @ B_29 )
<=> ( ( member_int @ X_25 @ B_29 )
& ( ord_less_eq_int_o @ A_34 @ B_29 ) ) ) ).
thf(fact_868_insert__subset,axiom,
! [X_25: hoare_2091234717iple_a,A_34: hoare_2091234717iple_a > $o,B_29: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ ( insert1597628439iple_a @ X_25 @ A_34 ) @ B_29 )
<=> ( ( member290856304iple_a @ X_25 @ B_29 )
& ( ord_le35180118le_a_o @ A_34 @ B_29 ) ) ) ).
thf(fact_869_subset__insertI,axiom,
! [B_28: nat > $o,A_33: nat] : ( ord_less_eq_nat_o @ B_28 @ ( insert_nat @ A_33 @ B_28 ) ) ).
thf(fact_870_subset__insertI,axiom,
! [B_28: int > $o,A_33: int] : ( ord_less_eq_int_o @ B_28 @ ( insert_int @ A_33 @ B_28 ) ) ).
thf(fact_871_subset__insertI,axiom,
! [B_28: hoare_2091234717iple_a > $o,A_33: hoare_2091234717iple_a] : ( ord_le35180118le_a_o @ B_28 @ ( insert1597628439iple_a @ A_33 @ B_28 ) ) ).
thf(fact_872_image__mono,axiom,
! [F_12: int > int,A_32: int > $o,B_27: int > $o] :
( ( ord_less_eq_int_o @ A_32 @ B_27 )
=> ( ord_less_eq_int_o @ ( image_int_int @ F_12 @ A_32 ) @ ( image_int_int @ F_12 @ B_27 ) ) ) ).
thf(fact_873_image__mono,axiom,
! [F_12: hoare_2091234717iple_a > hoare_2091234717iple_a,A_32: hoare_2091234717iple_a > $o,B_27: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_32 @ B_27 )
=> ( ord_le35180118le_a_o @ ( image_1661191109iple_a @ F_12 @ A_32 ) @ ( image_1661191109iple_a @ F_12 @ B_27 ) ) ) ).
thf(fact_874_image__mono,axiom,
! [F_12: nat > int,A_32: nat > $o,B_27: nat > $o] :
( ( ord_less_eq_nat_o @ A_32 @ B_27 )
=> ( ord_less_eq_int_o @ ( image_nat_int @ F_12 @ A_32 ) @ ( image_nat_int @ F_12 @ B_27 ) ) ) ).
thf(fact_875_image__mono,axiom,
! [F_12: nat > nat,A_32: nat > $o,B_27: nat > $o] :
( ( ord_less_eq_nat_o @ A_32 @ B_27 )
=> ( ord_less_eq_nat_o @ ( image_nat_nat @ F_12 @ A_32 ) @ ( image_nat_nat @ F_12 @ B_27 ) ) ) ).
thf(fact_876_subset__image__iff,axiom,
! [B_26: int > $o,F_11: int > int,A_31: int > $o] :
( ( ord_less_eq_int_o @ B_26 @ ( image_int_int @ F_11 @ A_31 ) )
<=> ? [AA: int > $o] :
( ( ord_less_eq_int_o @ AA @ A_31 )
& ( B_26
= ( image_int_int @ F_11 @ AA ) ) ) ) ).
thf(fact_877_subset__image__iff,axiom,
! [B_26: hoare_2091234717iple_a > $o,F_11: hoare_2091234717iple_a > hoare_2091234717iple_a,A_31: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ B_26 @ ( image_1661191109iple_a @ F_11 @ A_31 ) )
<=> ? [AA: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ AA @ A_31 )
& ( B_26
= ( image_1661191109iple_a @ F_11 @ AA ) ) ) ) ).
thf(fact_878_subset__image__iff,axiom,
! [B_26: int > $o,F_11: nat > int,A_31: nat > $o] :
( ( ord_less_eq_int_o @ B_26 @ ( image_nat_int @ F_11 @ A_31 ) )
<=> ? [AA: nat > $o] :
( ( ord_less_eq_nat_o @ AA @ A_31 )
& ( B_26
= ( image_nat_int @ F_11 @ AA ) ) ) ) ).
thf(fact_879_subset__image__iff,axiom,
! [B_26: nat > $o,F_11: nat > nat,A_31: nat > $o] :
( ( ord_less_eq_nat_o @ B_26 @ ( image_nat_nat @ F_11 @ A_31 ) )
<=> ? [AA: nat > $o] :
( ( ord_less_eq_nat_o @ AA @ A_31 )
& ( B_26
= ( image_nat_nat @ F_11 @ AA ) ) ) ) ).
thf(fact_880_double__diff,axiom,
! [C_8: int > $o,A_30: int > $o,B_25: int > $o] :
( ( ord_less_eq_int_o @ A_30 @ B_25 )
=> ( ( ord_less_eq_int_o @ B_25 @ C_8 )
=> ( ( minus_minus_int_o @ B_25 @ ( minus_minus_int_o @ C_8 @ A_30 ) )
= A_30 ) ) ) ).
thf(fact_881_double__diff,axiom,
! [C_8: hoare_2091234717iple_a > $o,A_30: hoare_2091234717iple_a > $o,B_25: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_30 @ B_25 )
=> ( ( ord_le35180118le_a_o @ B_25 @ C_8 )
=> ( ( minus_836160335le_a_o @ B_25 @ ( minus_836160335le_a_o @ C_8 @ A_30 ) )
= A_30 ) ) ) ).
thf(fact_882_double__diff,axiom,
! [C_8: nat > $o,A_30: nat > $o,B_25: nat > $o] :
( ( ord_less_eq_nat_o @ A_30 @ B_25 )
=> ( ( ord_less_eq_nat_o @ B_25 @ C_8 )
=> ( ( minus_minus_nat_o @ B_25 @ ( minus_minus_nat_o @ C_8 @ A_30 ) )
= A_30 ) ) ) ).
thf(fact_883_Diff__mono,axiom,
! [D_3: int > $o,B_24: int > $o,A_29: int > $o,C_7: int > $o] :
( ( ord_less_eq_int_o @ A_29 @ C_7 )
=> ( ( ord_less_eq_int_o @ D_3 @ B_24 )
=> ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_29 @ B_24 ) @ ( minus_minus_int_o @ C_7 @ D_3 ) ) ) ) ).
thf(fact_884_Diff__mono,axiom,
! [D_3: hoare_2091234717iple_a > $o,B_24: hoare_2091234717iple_a > $o,A_29: hoare_2091234717iple_a > $o,C_7: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_29 @ C_7 )
=> ( ( ord_le35180118le_a_o @ D_3 @ B_24 )
=> ( ord_le35180118le_a_o @ ( minus_836160335le_a_o @ A_29 @ B_24 ) @ ( minus_836160335le_a_o @ C_7 @ D_3 ) ) ) ) ).
thf(fact_885_Diff__mono,axiom,
! [D_3: nat > $o,B_24: nat > $o,A_29: nat > $o,C_7: nat > $o] :
( ( ord_less_eq_nat_o @ A_29 @ C_7 )
=> ( ( ord_less_eq_nat_o @ D_3 @ B_24 )
=> ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_29 @ B_24 ) @ ( minus_minus_nat_o @ C_7 @ D_3 ) ) ) ) ).
thf(fact_886_Diff__subset,axiom,
! [A_28: int > $o,B_23: int > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_28 @ B_23 ) @ A_28 ) ).
thf(fact_887_Diff__subset,axiom,
! [A_28: hoare_2091234717iple_a > $o,B_23: hoare_2091234717iple_a > $o] : ( ord_le35180118le_a_o @ ( minus_836160335le_a_o @ A_28 @ B_23 ) @ A_28 ) ).
thf(fact_888_Diff__subset,axiom,
! [A_28: nat > $o,B_23: nat > $o] : ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_28 @ B_23 ) @ A_28 ) ).
thf(fact_889_weaken,axiom,
! [Ts_2: hoare_2091234717iple_a > $o,G_3: hoare_2091234717iple_a > $o,Ts_1: hoare_2091234717iple_a > $o] :
( ( hoare_1467856363rivs_a @ G_3 @ Ts_1 )
=> ( ( ord_le35180118le_a_o @ Ts_2 @ Ts_1 )
=> ( hoare_1467856363rivs_a @ G_3 @ Ts_2 ) ) ) ).
thf(fact_890_asm,axiom,
! [Ts: hoare_2091234717iple_a > $o,G_2: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ Ts @ G_2 )
=> ( hoare_1467856363rivs_a @ G_2 @ Ts ) ) ).
thf(fact_891_subset__singletonD,axiom,
! [A_27: nat > $o,X_24: nat] :
( ( ord_less_eq_nat_o @ A_27 @ ( insert_nat @ X_24 @ bot_bot_nat_o ) )
=> ( ( A_27 = bot_bot_nat_o )
| ( A_27
= ( insert_nat @ X_24 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_892_subset__singletonD,axiom,
! [A_27: int > $o,X_24: int] :
( ( ord_less_eq_int_o @ A_27 @ ( insert_int @ X_24 @ bot_bot_int_o ) )
=> ( ( A_27 = bot_bot_int_o )
| ( A_27
= ( insert_int @ X_24 @ bot_bot_int_o ) ) ) ) ).
thf(fact_893_subset__singletonD,axiom,
! [A_27: hoare_2091234717iple_a > $o,X_24: hoare_2091234717iple_a] :
( ( ord_le35180118le_a_o @ A_27 @ ( insert1597628439iple_a @ X_24 @ bot_bo1791335050le_a_o ) )
=> ( ( A_27 = bot_bo1791335050le_a_o )
| ( A_27
= ( insert1597628439iple_a @ X_24 @ bot_bo1791335050le_a_o ) ) ) ) ).
thf(fact_894_finite__surj,axiom,
! [B_22: int > $o,F_10: int > int,A_26: int > $o] :
( ( finite_finite_int @ A_26 )
=> ( ( ord_less_eq_int_o @ B_22 @ ( image_int_int @ F_10 @ A_26 ) )
=> ( finite_finite_int @ B_22 ) ) ) ).
thf(fact_895_finite__surj,axiom,
! [B_22: hoare_2091234717iple_a > $o,F_10: hoare_2091234717iple_a > hoare_2091234717iple_a,A_26: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ A_26 )
=> ( ( ord_le35180118le_a_o @ B_22 @ ( image_1661191109iple_a @ F_10 @ A_26 ) )
=> ( finite232261744iple_a @ B_22 ) ) ) ).
thf(fact_896_finite__surj,axiom,
! [B_22: int > $o,F_10: nat > int,A_26: nat > $o] :
( ( finite_finite_nat @ A_26 )
=> ( ( ord_less_eq_int_o @ B_22 @ ( image_nat_int @ F_10 @ A_26 ) )
=> ( finite_finite_int @ B_22 ) ) ) ).
thf(fact_897_finite__surj,axiom,
! [B_22: nat > $o,F_10: nat > nat,A_26: nat > $o] :
( ( finite_finite_nat @ A_26 )
=> ( ( ord_less_eq_nat_o @ B_22 @ ( image_nat_nat @ F_10 @ A_26 ) )
=> ( finite_finite_nat @ B_22 ) ) ) ).
thf(fact_898_image__diff__subset,axiom,
! [F_9: int > int,A_25: int > $o,B_21: int > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ ( image_int_int @ F_9 @ A_25 ) @ ( image_int_int @ F_9 @ B_21 ) ) @ ( image_int_int @ F_9 @ ( minus_minus_int_o @ A_25 @ B_21 ) ) ) ).
thf(fact_899_image__diff__subset,axiom,
! [F_9: hoare_2091234717iple_a > hoare_2091234717iple_a,A_25: hoare_2091234717iple_a > $o,B_21: hoare_2091234717iple_a > $o] : ( ord_le35180118le_a_o @ ( minus_836160335le_a_o @ ( image_1661191109iple_a @ F_9 @ A_25 ) @ ( image_1661191109iple_a @ F_9 @ B_21 ) ) @ ( image_1661191109iple_a @ F_9 @ ( minus_836160335le_a_o @ A_25 @ B_21 ) ) ) ).
thf(fact_900_image__diff__subset,axiom,
! [F_9: nat > int,A_25: nat > $o,B_21: nat > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ ( image_nat_int @ F_9 @ A_25 ) @ ( image_nat_int @ F_9 @ B_21 ) ) @ ( image_nat_int @ F_9 @ ( minus_minus_nat_o @ A_25 @ B_21 ) ) ) ).
thf(fact_901_image__diff__subset,axiom,
! [F_9: nat > nat,A_25: nat > $o,B_21: nat > $o] : ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ ( image_nat_nat @ F_9 @ A_25 ) @ ( image_nat_nat @ F_9 @ B_21 ) ) @ ( image_nat_nat @ F_9 @ ( minus_minus_nat_o @ A_25 @ B_21 ) ) ) ).
thf(fact_902_le__add__iff1,axiom,
! [A_24: int,E_1: int,C_6: int,B_20: int,D_2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_24 @ E_1 ) @ C_6 ) @ ( plus_plus_int @ ( times_times_int @ B_20 @ E_1 ) @ D_2 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_24 @ B_20 ) @ E_1 ) @ C_6 ) @ D_2 ) ) ).
thf(fact_903_le__add__iff2,axiom,
! [A_23: int,E: int,C_5: int,B_19: int,D_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_23 @ E ) @ C_5 ) @ ( plus_plus_int @ ( times_times_int @ B_19 @ E ) @ D_1 ) )
<=> ( ord_less_eq_int @ C_5 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_19 @ A_23 ) @ E ) @ D_1 ) ) ) ).
thf(fact_904_diff__single__insert,axiom,
! [A_22: nat > $o,X_23: nat,B_18: nat > $o] :
( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_22 @ ( insert_nat @ X_23 @ bot_bot_nat_o ) ) @ B_18 )
=> ( ( member_nat @ X_23 @ A_22 )
=> ( ord_less_eq_nat_o @ A_22 @ ( insert_nat @ X_23 @ B_18 ) ) ) ) ).
thf(fact_905_diff__single__insert,axiom,
! [A_22: int > $o,X_23: int,B_18: int > $o] :
( ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_22 @ ( insert_int @ X_23 @ bot_bot_int_o ) ) @ B_18 )
=> ( ( member_int @ X_23 @ A_22 )
=> ( ord_less_eq_int_o @ A_22 @ ( insert_int @ X_23 @ B_18 ) ) ) ) ).
thf(fact_906_diff__single__insert,axiom,
! [A_22: hoare_2091234717iple_a > $o,X_23: hoare_2091234717iple_a,B_18: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ ( minus_836160335le_a_o @ A_22 @ ( insert1597628439iple_a @ X_23 @ bot_bo1791335050le_a_o ) ) @ B_18 )
=> ( ( member290856304iple_a @ X_23 @ A_22 )
=> ( ord_le35180118le_a_o @ A_22 @ ( insert1597628439iple_a @ X_23 @ B_18 ) ) ) ) ).
thf(fact_907_subset__insert__iff,axiom,
! [A_21: nat > $o,X_22: nat,B_17: nat > $o] :
( ( ord_less_eq_nat_o @ A_21 @ ( insert_nat @ X_22 @ B_17 ) )
<=> ( ( ( member_nat @ X_22 @ A_21 )
=> ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_21 @ ( insert_nat @ X_22 @ bot_bot_nat_o ) ) @ B_17 ) )
& ( ~ ( member_nat @ X_22 @ A_21 )
=> ( ord_less_eq_nat_o @ A_21 @ B_17 ) ) ) ) ).
thf(fact_908_subset__insert__iff,axiom,
! [A_21: int > $o,X_22: int,B_17: int > $o] :
( ( ord_less_eq_int_o @ A_21 @ ( insert_int @ X_22 @ B_17 ) )
<=> ( ( ( member_int @ X_22 @ A_21 )
=> ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_21 @ ( insert_int @ X_22 @ bot_bot_int_o ) ) @ B_17 ) )
& ( ~ ( member_int @ X_22 @ A_21 )
=> ( ord_less_eq_int_o @ A_21 @ B_17 ) ) ) ) ).
thf(fact_909_subset__insert__iff,axiom,
! [A_21: hoare_2091234717iple_a > $o,X_22: hoare_2091234717iple_a,B_17: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ A_21 @ ( insert1597628439iple_a @ X_22 @ B_17 ) )
<=> ( ( ( member290856304iple_a @ X_22 @ A_21 )
=> ( ord_le35180118le_a_o @ ( minus_836160335le_a_o @ A_21 @ ( insert1597628439iple_a @ X_22 @ bot_bo1791335050le_a_o ) ) @ B_17 ) )
& ( ~ ( member290856304iple_a @ X_22 @ A_21 )
=> ( ord_le35180118le_a_o @ A_21 @ B_17 ) ) ) ) ).
thf(fact_910_folding__one__idem_Osubset__idem,axiom,
! [B_16: hoare_2091234717iple_a > $o,A_20: hoare_2091234717iple_a > $o,F_8: hoare_2091234717iple_a > hoare_2091234717iple_a > hoare_2091234717iple_a,F_7: ( hoare_2091234717iple_a > $o ) > hoare_2091234717iple_a] :
( ( finite1674555159iple_a @ F_8 @ F_7 )
=> ( ( finite232261744iple_a @ A_20 )
=> ( ( B_16 != bot_bo1791335050le_a_o )
=> ( ( ord_le35180118le_a_o @ B_16 @ A_20 )
=> ( ( F_8 @ ( F_7 @ B_16 ) @ ( F_7 @ A_20 ) )
= ( F_7 @ A_20 ) ) ) ) ) ) ).
thf(fact_911_folding__one__idem_Osubset__idem,axiom,
! [B_16: int > $o,A_20: int > $o,F_8: int > int > int,F_7: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_8 @ F_7 )
=> ( ( finite_finite_int @ A_20 )
=> ( ( B_16 != bot_bot_int_o )
=> ( ( ord_less_eq_int_o @ B_16 @ A_20 )
=> ( ( F_8 @ ( F_7 @ B_16 ) @ ( F_7 @ A_20 ) )
= ( F_7 @ A_20 ) ) ) ) ) ) ).
thf(fact_912_folding__one__idem_Osubset__idem,axiom,
! [B_16: nat > $o,A_20: nat > $o,F_8: nat > nat > nat,F_7: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_8 @ F_7 )
=> ( ( finite_finite_nat @ A_20 )
=> ( ( B_16 != bot_bot_nat_o )
=> ( ( ord_less_eq_nat_o @ B_16 @ A_20 )
=> ( ( F_8 @ ( F_7 @ B_16 ) @ ( F_7 @ A_20 ) )
= ( F_7 @ A_20 ) ) ) ) ) ) ).
thf(fact_913_finite__subset__induct,axiom,
! [P_3: ( nat > $o ) > $o,A_18: nat > $o,F_5: nat > $o] :
( ( finite_finite_nat @ F_5 )
=> ( ( ord_less_eq_nat_o @ F_5 @ A_18 )
=> ( ( P_3 @ bot_bot_nat_o )
=> ( ! [A_19: nat,F_6: nat > $o] :
( ( finite_finite_nat @ F_6 )
=> ( ( member_nat @ A_19 @ A_18 )
=> ( ~ ( member_nat @ A_19 @ F_6 )
=> ( ( P_3 @ F_6 )
=> ( P_3 @ ( insert_nat @ A_19 @ F_6 ) ) ) ) ) )
=> ( P_3 @ F_5 ) ) ) ) ) ).
thf(fact_914_finite__subset__induct,axiom,
! [P_3: ( int > $o ) > $o,A_18: int > $o,F_5: int > $o] :
( ( finite_finite_int @ F_5 )
=> ( ( ord_less_eq_int_o @ F_5 @ A_18 )
=> ( ( P_3 @ bot_bot_int_o )
=> ( ! [A_19: int,F_6: int > $o] :
( ( finite_finite_int @ F_6 )
=> ( ( member_int @ A_19 @ A_18 )
=> ( ~ ( member_int @ A_19 @ F_6 )
=> ( ( P_3 @ F_6 )
=> ( P_3 @ ( insert_int @ A_19 @ F_6 ) ) ) ) ) )
=> ( P_3 @ F_5 ) ) ) ) ) ).
thf(fact_915_finite__subset__induct,axiom,
! [P_3: ( hoare_2091234717iple_a > $o ) > $o,A_18: hoare_2091234717iple_a > $o,F_5: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_5 )
=> ( ( ord_le35180118le_a_o @ F_5 @ A_18 )
=> ( ( P_3 @ bot_bo1791335050le_a_o )
=> ( ! [A_19: hoare_2091234717iple_a,F_6: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ F_6 )
=> ( ( member290856304iple_a @ A_19 @ A_18 )
=> ( ~ ( member290856304iple_a @ A_19 @ F_6 )
=> ( ( P_3 @ F_6 )
=> ( P_3 @ ( insert1597628439iple_a @ A_19 @ F_6 ) ) ) ) ) )
=> ( P_3 @ F_5 ) ) ) ) ) ).
thf(fact_916_subsetI,axiom,
! [B_15: hoare_2091234717iple_a > $o,A_17: hoare_2091234717iple_a > $o] :
( ! [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_17 )
=> ( member290856304iple_a @ X_1 @ B_15 ) )
=> ( ord_le35180118le_a_o @ A_17 @ B_15 ) ) ).
thf(fact_917_subsetI,axiom,
! [B_15: int > $o,A_17: int > $o] :
( ! [X_1: int] :
( ( member_int @ X_1 @ A_17 )
=> ( member_int @ X_1 @ B_15 ) )
=> ( ord_less_eq_int_o @ A_17 @ B_15 ) ) ).
thf(fact_918_subsetI,axiom,
! [B_15: nat > $o,A_17: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_17 )
=> ( member_nat @ X_1 @ B_15 ) )
=> ( ord_less_eq_nat_o @ A_17 @ B_15 ) ) ).
thf(fact_919_finite__subset__image,axiom,
! [F_4: int > int,A_16: int > $o,B_14: int > $o] :
( ( finite_finite_int @ B_14 )
=> ( ( ord_less_eq_int_o @ B_14 @ ( image_int_int @ F_4 @ A_16 ) )
=> ? [C_4: int > $o] :
( ( ord_less_eq_int_o @ C_4 @ A_16 )
& ( finite_finite_int @ C_4 )
& ( B_14
= ( image_int_int @ F_4 @ C_4 ) ) ) ) ) ).
thf(fact_920_finite__subset__image,axiom,
! [F_4: hoare_2091234717iple_a > hoare_2091234717iple_a,A_16: hoare_2091234717iple_a > $o,B_14: hoare_2091234717iple_a > $o] :
( ( finite232261744iple_a @ B_14 )
=> ( ( ord_le35180118le_a_o @ B_14 @ ( image_1661191109iple_a @ F_4 @ A_16 ) )
=> ? [C_4: hoare_2091234717iple_a > $o] :
( ( ord_le35180118le_a_o @ C_4 @ A_16 )
& ( finite232261744iple_a @ C_4 )
& ( B_14
= ( image_1661191109iple_a @ F_4 @ C_4 ) ) ) ) ) ).
thf(fact_921_finite__subset__image,axiom,
! [F_4: nat > int,A_16: nat > $o,B_14: int > $o] :
( ( finite_finite_int @ B_14 )
=> ( ( ord_less_eq_int_o @ B_14 @ ( image_nat_int @ F_4 @ A_16 ) )
=> ? [C_4: nat > $o] :
( ( ord_less_eq_nat_o @ C_4 @ A_16 )
& ( finite_finite_nat @ C_4 )
& ( B_14
= ( image_nat_int @ F_4 @ C_4 ) ) ) ) ) ).
thf(fact_922_finite__subset__image,axiom,
! [F_4: nat > nat,A_16: nat > $o,B_14: nat > $o] :
( ( finite_finite_nat @ B_14 )
=> ( ( ord_less_eq_nat_o @ B_14 @ ( image_nat_nat @ F_4 @ A_16 ) )
=> ? [C_4: nat > $o] :
( ( ord_less_eq_nat_o @ C_4 @ A_16 )
& ( finite_finite_nat @ C_4 )
& ( B_14
= ( image_nat_nat @ F_4 @ C_4 ) ) ) ) ) ).
thf(fact_923_setsum__mono,axiom,
! [F_3: int > nat,G_1: int > nat,K_2: int > $o] :
( ! [I_1: int] :
( ( member_int @ I_1 @ K_2 )
=> ( ord_less_eq_nat @ ( F_3 @ I_1 ) @ ( G_1 @ I_1 ) ) )
=> ( ord_less_eq_nat @ ( big_co1740723097nt_nat @ F_3 @ K_2 ) @ ( big_co1740723097nt_nat @ G_1 @ K_2 ) ) ) ).
thf(fact_924_setsum__mono,axiom,
! [F_3: hoare_2091234717iple_a > nat,G_1: hoare_2091234717iple_a > nat,K_2: hoare_2091234717iple_a > $o] :
( ! [I_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ I_1 @ K_2 )
=> ( ord_less_eq_nat @ ( F_3 @ I_1 ) @ ( G_1 @ I_1 ) ) )
=> ( ord_less_eq_nat @ ( big_co917763874_a_nat @ F_3 @ K_2 ) @ ( big_co917763874_a_nat @ G_1 @ K_2 ) ) ) ).
thf(fact_925_setsum__mono,axiom,
! [F_3: nat > nat,G_1: nat > nat,K_2: nat > $o] :
( ! [I_1: nat] :
( ( member_nat @ I_1 @ K_2 )
=> ( ord_less_eq_nat @ ( F_3 @ I_1 ) @ ( G_1 @ I_1 ) ) )
=> ( ord_less_eq_nat @ ( big_co387207925at_nat @ F_3 @ K_2 ) @ ( big_co387207925at_nat @ G_1 @ K_2 ) ) ) ).
thf(fact_926_image__subsetI,axiom,
! [F_2: int > int,B_13: int > $o,A_15: int > $o] :
( ! [X_1: int] :
( ( member_int @ X_1 @ A_15 )
=> ( member_int @ ( F_2 @ X_1 ) @ B_13 ) )
=> ( ord_less_eq_int_o @ ( image_int_int @ F_2 @ A_15 ) @ B_13 ) ) ).
thf(fact_927_image__subsetI,axiom,
! [F_2: hoare_2091234717iple_a > hoare_2091234717iple_a,B_13: hoare_2091234717iple_a > $o,A_15: hoare_2091234717iple_a > $o] :
( ! [X_1: hoare_2091234717iple_a] :
( ( member290856304iple_a @ X_1 @ A_15 )
=> ( member290856304iple_a @ ( F_2 @ X_1 ) @ B_13 ) )
=> ( ord_le35180118le_a_o @ ( image_1661191109iple_a @ F_2 @ A_15 ) @ B_13 ) ) ).
thf(fact_928_image__subsetI,axiom,
! [F_2: nat > int,B_13: int > $o,A_15: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_15 )
=> ( member_int @ ( F_2 @ X_1 ) @ B_13 ) )
=> ( ord_less_eq_int_o @ ( image_nat_int @ F_2 @ A_15 ) @ B_13 ) ) ).
thf(fact_929_image__subsetI,axiom,
! [F_2: nat > nat,B_13: nat > $o,A_15: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_15 )
=> ( member_nat @ ( F_2 @ X_1 ) @ B_13 ) )
=> ( ord_less_eq_nat_o @ ( image_nat_nat @ F_2 @ A_15 ) @ B_13 ) ) ).
thf(fact_930_le__funI,axiom,
! [F_1: int > $o,G: int > $o] :
( ! [X_1: int] : ( ord_less_eq_o @ ( F_1 @ X_1 ) @ ( G @ X_1 ) )
=> ( ord_less_eq_int_o @ F_1 @ G ) ) ).
thf(fact_931_le__funI,axiom,
! [F_1: nat > $o,G: nat > $o] :
( ! [X_1: nat] : ( ord_less_eq_o @ ( F_1 @ X_1 ) @ ( G @ X_1 ) )
=> ( ord_less_eq_nat_o @ F_1 @ G ) ) ).
thf(fact_932_flat__lub__def,axiom,
! [A_14: int > $o,B_12: int] :
( ( ( ord_less_eq_int_o @ A_14 @ ( insert_int @ B_12 @ bot_bot_int_o ) )
=> ( ( partial_flat_lub_int @ B_12 @ A_14 )
= B_12 ) )
& ( ~ ( ord_less_eq_int_o @ A_14 @ ( insert_int @ B_12 @ bot_bot_int_o ) )
=> ( ( partial_flat_lub_int @ B_12 @ A_14 )
= ( the_int
@ ^ [X_1: int] : ( member_int @ X_1 @ ( minus_minus_int_o @ A_14 @ ( insert_int @ B_12 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_933_flat__lub__def,axiom,
! [A_14: hoare_2091234717iple_a > $o,B_12: hoare_2091234717iple_a] :
( ( ( ord_le35180118le_a_o @ A_14 @ ( insert1597628439iple_a @ B_12 @ bot_bo1791335050le_a_o ) )
=> ( ( partia443170835iple_a @ B_12 @ A_14 )
= B_12 ) )
& ( ~ ( ord_le35180118le_a_o @ A_14 @ ( insert1597628439iple_a @ B_12 @ bot_bo1791335050le_a_o ) )
=> ( ( partia443170835iple_a @ B_12 @ A_14 )
= ( the_Ho1471183438iple_a
@ ^ [X_1: hoare_2091234717iple_a] : ( member290856304iple_a @ X_1 @ ( minus_836160335le_a_o @ A_14 @ ( insert1597628439iple_a @ B_12 @ bot_bo1791335050le_a_o ) ) ) ) ) ) ) ).
thf(fact_934_flat__lub__def,axiom,
! [A_14: nat > $o,B_12: nat] :
( ( ( ord_less_eq_nat_o @ A_14 @ ( insert_nat @ B_12 @ bot_bot_nat_o ) )
=> ( ( partial_flat_lub_nat @ B_12 @ A_14 )
= B_12 ) )
& ( ~ ( ord_less_eq_nat_o @ A_14 @ ( insert_nat @ B_12 @ bot_bot_nat_o ) )
=> ( ( partial_flat_lub_nat @ B_12 @ A_14 )
= ( the_nat
@ ^ [X_1: nat] : ( member_nat @ X_1 @ ( minus_minus_nat_o @ A_14 @ ( insert_nat @ B_12 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_935_sum__diff__distrib,axiom,
! [N: nat,Q_2: nat > nat,P: nat > nat] :
( ! [X_1: nat] : ( ord_less_eq_nat @ ( Q_2 @ X_1 ) @ ( P @ X_1 ) )
=> ( ( minus_minus_nat @ ( big_co387207925at_nat @ P @ ( ord_lessThan_nat @ N ) ) @ ( big_co387207925at_nat @ Q_2 @ ( ord_lessThan_nat @ N ) ) )
= ( big_co387207925at_nat
@ ^ [X_1: nat] : ( minus_minus_nat @ ( P @ X_1 ) @ ( Q_2 @ X_1 ) )
@ ( ord_lessThan_nat @ N ) ) ) ) ).
thf(fact_936_fold1__antimono,axiom,
! [B_11: int > $o,A_13: int > $o] :
( ( A_13 != bot_bot_int_o )
=> ( ( ord_less_eq_int_o @ A_13 @ B_11 )
=> ( ( finite_finite_int @ B_11 )
=> ( ord_less_eq_int @ ( finite_fold1_int @ ord_min_int @ B_11 ) @ ( finite_fold1_int @ ord_min_int @ A_13 ) ) ) ) ) ).
thf(fact_937_fold1__antimono,axiom,
! [B_11: nat > $o,A_13: nat > $o] :
( ( A_13 != bot_bot_nat_o )
=> ( ( ord_less_eq_nat_o @ A_13 @ B_11 )
=> ( ( finite_finite_nat @ B_11 )
=> ( ord_less_eq_nat @ ( finite_fold1_nat @ ord_min_nat @ B_11 ) @ ( finite_fold1_nat @ ord_min_nat @ A_13 ) ) ) ) ) ).
thf(fact_938_predicate1I,axiom,
! [Q_1: int > $o,P_2: int > $o] :
( ! [X_1: int] :
( ( P_2 @ X_1 )
=> ( Q_1 @ X_1 ) )
=> ( ord_less_eq_int_o @ P_2 @ Q_1 ) ) ).
thf(fact_939_predicate1I,axiom,
! [Q_1: nat > $o,P_2: nat > $o] :
( ! [X_1: nat] :
( ( P_2 @ X_1 )
=> ( Q_1 @ X_1 ) )
=> ( ord_less_eq_nat_o @ P_2 @ Q_1 ) ) ).
thf(fact_940_Collect__mono,axiom,
! [Q: int > $o,P_1: int > $o] :
( ! [X_1: int] :
( ( P_1 @ X_1 )
=> ( Q @ X_1 ) )
=> ( ord_less_eq_int_o @ ( collect_int @ P_1 ) @ ( collect_int @ Q ) ) ) ).
thf(fact_941_Collect__mono,axiom,
! [Q: hoare_2091234717iple_a > $o,P_1: hoare_2091234717iple_a > $o] :
( ! [X_1: hoare_2091234717iple_a] :
( ( P_1 @ X_1 )
=> ( Q @ X_1 ) )
=> ( ord_le35180118le_a_o @ ( collec992574898iple_a @ P_1 ) @ ( collec992574898iple_a @ Q ) ) ) ).
thf(fact_942_Collect__mono,axiom,
! [Q: nat > $o,P_1: nat > $o] :
( ! [X_1: nat] :
( ( P_1 @ X_1 )
=> ( Q @ X_1 ) )
=> ( ord_less_eq_nat_o @ ( collect_nat @ P_1 ) @ ( collect_nat @ Q ) ) ) ).
thf(fact_943_Min__antimono,axiom,
! [M_3: int > $o,N_4: int > $o] :
( ( ord_less_eq_int_o @ M_3 @ N_4 )
=> ( ( M_3 != bot_bot_int_o )
=> ( ( finite_finite_int @ N_4 )
=> ( ord_less_eq_int @ ( big_linorder_Min_int @ N_4 ) @ ( big_linorder_Min_int @ M_3 ) ) ) ) ) ).
thf(fact_944_Min__antimono,axiom,
! [M_3: nat > $o,N_4: nat > $o] :
( ( ord_less_eq_nat_o @ M_3 @ N_4 )
=> ( ( M_3 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ N_4 )
=> ( ord_less_eq_nat @ ( big_linorder_Min_nat @ N_4 ) @ ( big_linorder_Min_nat @ M_3 ) ) ) ) ) ).
thf(fact_945_Max__mono,axiom,
! [M_2: int > $o,N_3: int > $o] :
( ( ord_less_eq_int_o @ M_2 @ N_3 )
=> ( ( M_2 != bot_bot_int_o )
=> ( ( finite_finite_int @ N_3 )
=> ( ord_less_eq_int @ ( big_linorder_Max_int @ M_2 ) @ ( big_linorder_Max_int @ N_3 ) ) ) ) ) ).
thf(fact_946_Max__mono,axiom,
! [M_2: nat > $o,N_3: nat > $o] :
( ( ord_less_eq_nat_o @ M_2 @ N_3 )
=> ( ( M_2 != bot_bot_nat_o )
=> ( ( finite_finite_nat @ N_3 )
=> ( ord_less_eq_nat @ ( big_linorder_Max_nat @ M_2 ) @ ( big_linorder_Max_nat @ N_3 ) ) ) ) ) ).
thf(fact_947_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( ord_lessThan_nat @ K ) ) ).
thf(fact_948_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
thf(fact_949_finite__nat__set__iff__bounded__le,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M_1: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_eq_nat @ X_1 @ M_1 ) ) ) ).
thf(fact_950_Min_Oin__idem,axiom,
! [X_21: int,A_12: int > $o] :
( ( finite_finite_int @ A_12 )
=> ( ( member_int @ X_21 @ A_12 )
=> ( ( ord_min_int @ X_21 @ ( big_linorder_Min_int @ A_12 ) )
= ( big_linorder_Min_int @ A_12 ) ) ) ) ).
thf(fact_951_Min_Oin__idem,axiom,
! [X_21: nat,A_12: nat > $o] :
( ( finite_finite_nat @ A_12 )
=> ( ( member_nat @ X_21 @ A_12 )
=> ( ( ord_min_nat @ X_21 @ ( big_linorder_Min_nat @ A_12 ) )
= ( big_linorder_Min_nat @ A_12 ) ) ) ) ).
thf(fact_952_Min_Oidem,axiom,
! [X_20: nat] :
( ( ord_min_nat @ X_20 @ X_20 )
= X_20 ) ).
thf(fact_953_min__max_Oinf__assoc,axiom,
! [X_19: nat,Y_14: nat,Z_9: nat] :
( ( ord_min_nat @ ( ord_min_nat @ X_19 @ Y_14 ) @ Z_9 )
= ( ord_min_nat @ X_19 @ ( ord_min_nat @ Y_14 @ Z_9 ) ) ) ).
thf(fact_954_min__max_Oinf_Oassoc,axiom,
! [A_11: nat,B_10: nat,C_3: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A_11 @ B_10 ) @ C_3 )
= ( ord_min_nat @ A_11 @ ( ord_min_nat @ B_10 @ C_3 ) ) ) ).
thf(fact_955_min__max_Oinf__left__commute,axiom,
! [X_18: nat,Y_13: nat,Z_8: nat] :
( ( ord_min_nat @ X_18 @ ( ord_min_nat @ Y_13 @ Z_8 ) )
= ( ord_min_nat @ Y_13 @ ( ord_min_nat @ X_18 @ Z_8 ) ) ) ).
thf(fact_956_min__max_Oinf_Oleft__commute,axiom,
! [B_9: nat,A_10: nat,C_2: nat] :
( ( ord_min_nat @ B_9 @ ( ord_min_nat @ A_10 @ C_2 ) )
= ( ord_min_nat @ A_10 @ ( ord_min_nat @ B_9 @ C_2 ) ) ) ).
thf(fact_957_min__max_Oinf__left__idem,axiom,
! [X_17: nat,Y_12: nat] :
( ( ord_min_nat @ X_17 @ ( ord_min_nat @ X_17 @ Y_12 ) )
= ( ord_min_nat @ X_17 @ Y_12 ) ) ).
thf(fact_958_min__max_Oinf_Oleft__idem,axiom,
! [A_9: nat,B_8: nat] :
( ( ord_min_nat @ A_9 @ ( ord_min_nat @ A_9 @ B_8 ) )
= ( ord_min_nat @ A_9 @ B_8 ) ) ).
thf(fact_959_min__max_Oinf__commute,axiom,
! [X_16: nat,Y_11: nat] :
( ( ord_min_nat @ X_16 @ Y_11 )
= ( ord_min_nat @ Y_11 @ X_16 ) ) ).
thf(fact_960_min__max_Oinf_Ocommute,axiom,
! [A_8: nat,B_7: nat] :
( ( ord_min_nat @ A_8 @ B_7 )
= ( ord_min_nat @ B_7 @ A_8 ) ) ).
thf(fact_961_min__max_Oinf_Oidem,axiom,
! [A_7: nat] :
( ( ord_min_nat @ A_7 @ A_7 )
= A_7 ) ).
thf(fact_962_min__le__iff__disj,axiom,
! [X_15: nat,Y_10: nat,Z_7: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X_15 @ Y_10 ) @ Z_7 )
<=> ( ( ord_less_eq_nat @ X_15 @ Z_7 )
| ( ord_less_eq_nat @ Y_10 @ Z_7 ) ) ) ).
thf(fact_963_min__le__iff__disj,axiom,
! [X_15: int,Y_10: int,Z_7: int] :
( ( ord_less_eq_int @ ( ord_min_int @ X_15 @ Y_10 ) @ Z_7 )
<=> ( ( ord_less_eq_int @ X_15 @ Z_7 )
| ( ord_less_eq_int @ Y_10 @ Z_7 ) ) ) ).
thf(fact_964_min__max_Ole__infE,axiom,
! [X_14: nat,A_6: nat,B_6: nat] :
( ( ord_less_eq_nat @ X_14 @ ( ord_min_nat @ A_6 @ B_6 ) )
=> ~ ( ( ord_less_eq_nat @ X_14 @ A_6 )
=> ~ ( ord_less_eq_nat @ X_14 @ B_6 ) ) ) ).
thf(fact_965_min__max_Ole__infE,axiom,
! [X_14: int,A_6: int,B_6: int] :
( ( ord_less_eq_int @ X_14 @ ( ord_min_int @ A_6 @ B_6 ) )
=> ~ ( ( ord_less_eq_int @ X_14 @ A_6 )
=> ~ ( ord_less_eq_int @ X_14 @ B_6 ) ) ) ).
thf(fact_966_min__max_Oinf__mono,axiom,
! [B_5: nat,D: nat,A_5: nat,C_1: nat] :
( ( ord_less_eq_nat @ A_5 @ C_1 )
=> ( ( ord_less_eq_nat @ B_5 @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A_5 @ B_5 ) @ ( ord_min_nat @ C_1 @ D ) ) ) ) ).
thf(fact_967_min__max_Oinf__mono,axiom,
! [B_5: int,D: int,A_5: int,C_1: int] :
( ( ord_less_eq_int @ A_5 @ C_1 )
=> ( ( ord_less_eq_int @ B_5 @ D )
=> ( ord_less_eq_int @ ( ord_min_int @ A_5 @ B_5 ) @ ( ord_min_int @ C_1 @ D ) ) ) ) ).
thf(fact_968_min__max_Oinf__greatest,axiom,
! [Z_6: nat,X_13: nat,Y_9: nat] :
( ( ord_less_eq_nat @ X_13 @ Y_9 )
=> ( ( ord_less_eq_nat @ X_13 @ Z_6 )
=> ( ord_less_eq_nat @ X_13 @ ( ord_min_nat @ Y_9 @ Z_6 ) ) ) ) ).
thf(fact_969_min__max_Oinf__greatest,axiom,
! [Z_6: int,X_13: int,Y_9: int] :
( ( ord_less_eq_int @ X_13 @ Y_9 )
=> ( ( ord_less_eq_int @ X_13 @ Z_6 )
=> ( ord_less_eq_int @ X_13 @ ( ord_min_int @ Y_9 @ Z_6 ) ) ) ) ).
thf(fact_970_min__max_Ole__infI,axiom,
! [B_4: nat,X_12: nat,A_4: nat] :
( ( ord_less_eq_nat @ X_12 @ A_4 )
=> ( ( ord_less_eq_nat @ X_12 @ B_4 )
=> ( ord_less_eq_nat @ X_12 @ ( ord_min_nat @ A_4 @ B_4 ) ) ) ) ).
thf(fact_971_min__max_Ole__infI,axiom,
! [B_4: int,X_12: int,A_4: int] :
( ( ord_less_eq_int @ X_12 @ A_4 )
=> ( ( ord_less_eq_int @ X_12 @ B_4 )
=> ( ord_less_eq_int @ X_12 @ ( ord_min_int @ A_4 @ B_4 ) ) ) ) ).
thf(fact_972_min__max_Oinf__absorb2,axiom,
! [Y_8: nat,X_11: nat] :
( ( ord_less_eq_nat @ Y_8 @ X_11 )
=> ( ( ord_min_nat @ X_11 @ Y_8 )
= Y_8 ) ) ).
thf(fact_973_min__max_Oinf__absorb2,axiom,
! [Y_8: int,X_11: int] :
( ( ord_less_eq_int @ Y_8 @ X_11 )
=> ( ( ord_min_int @ X_11 @ Y_8 )
= Y_8 ) ) ).
thf(fact_974_min__max_Oinf__absorb1,axiom,
! [X_10: nat,Y_7: nat] :
( ( ord_less_eq_nat @ X_10 @ Y_7 )
=> ( ( ord_min_nat @ X_10 @ Y_7 )
= X_10 ) ) ).
thf(fact_975_min__max_Oinf__absorb1,axiom,
! [X_10: int,Y_7: int] :
( ( ord_less_eq_int @ X_10 @ Y_7 )
=> ( ( ord_min_int @ X_10 @ Y_7 )
= X_10 ) ) ).
thf(fact_976_min__max_Ole__infI2,axiom,
! [A_3: nat,B_3: nat,X_9: nat] :
( ( ord_less_eq_nat @ B_3 @ X_9 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A_3 @ B_3 ) @ X_9 ) ) ).
thf(fact_977_min__max_Ole__infI2,axiom,
! [A_3: int,B_3: int,X_9: int] :
( ( ord_less_eq_int @ B_3 @ X_9 )
=> ( ord_less_eq_int @ ( ord_min_int @ A_3 @ B_3 ) @ X_9 ) ) ).
thf(fact_978_min__max_Ole__infI1,axiom,
! [B_2: nat,A_2: nat,X_8: nat] :
( ( ord_less_eq_nat @ A_2 @ X_8 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A_2 @ B_2 ) @ X_8 ) ) ).
thf(fact_979_min__max_Ole__infI1,axiom,
! [B_2: int,A_2: int,X_8: int] :
( ( ord_less_eq_int @ A_2 @ X_8 )
=> ( ord_less_eq_int @ ( ord_min_int @ A_2 @ B_2 ) @ X_8 ) ) ).
thf(fact_980_min__max_Ole__inf__iff,axiom,
! [X_7: nat,Y_6: nat,Z_5: nat] :
( ( ord_less_eq_nat @ X_7 @ ( ord_min_nat @ Y_6 @ Z_5 ) )
<=> ( ( ord_less_eq_nat @ X_7 @ Y_6 )
& ( ord_less_eq_nat @ X_7 @ Z_5 ) ) ) ).
thf(fact_981_min__max_Ole__inf__iff,axiom,
! [X_7: int,Y_6: int,Z_5: int] :
( ( ord_less_eq_int @ X_7 @ ( ord_min_int @ Y_6 @ Z_5 ) )
<=> ( ( ord_less_eq_int @ X_7 @ Y_6 )
& ( ord_less_eq_int @ X_7 @ Z_5 ) ) ) ).
thf(fact_982_min__max_Ole__iff__inf,axiom,
! [X_6: nat,Y_5: nat] :
( ( ord_less_eq_nat @ X_6 @ Y_5 )
<=> ( ( ord_min_nat @ X_6 @ Y_5 )
= X_6 ) ) ).
thf(fact_983_min__max_Ole__iff__inf,axiom,
! [X_6: int,Y_5: int] :
( ( ord_less_eq_int @ X_6 @ Y_5 )
<=> ( ( ord_min_int @ X_6 @ Y_5 )
= X_6 ) ) ).
thf(fact_984_min__max_Oinf__le2,axiom,
! [X_5: nat,Y_4: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ X_5 @ Y_4 ) @ Y_4 ) ).
thf(fact_985_min__max_Oinf__le2,axiom,
! [X_5: int,Y_4: int] : ( ord_less_eq_int @ ( ord_min_int @ X_5 @ Y_4 ) @ Y_4 ) ).
thf(fact_986_min__max_Oinf__le1,axiom,
! [X_4: nat,Y_3: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ X_4 @ Y_3 ) @ X_4 ) ).
thf(fact_987_min__max_Oinf__le1,axiom,
! [X_4: int,Y_3: int] : ( ord_less_eq_int @ ( ord_min_int @ X_4 @ Y_3 ) @ X_4 ) ).
thf(fact_988_min__add__distrib__left,axiom,
! [X_3: int,Y_2: int,Z_4: int] :
( ( plus_plus_int @ ( ord_min_int @ X_3 @ Y_2 ) @ Z_4 )
= ( ord_min_int @ ( plus_plus_int @ X_3 @ Z_4 ) @ ( plus_plus_int @ Y_2 @ Z_4 ) ) ) ).
thf(fact_989_min__add__distrib__left,axiom,
! [X_3: nat,Y_2: nat,Z_4: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X_3 @ Y_2 ) @ Z_4 )
= ( ord_min_nat @ ( plus_plus_nat @ X_3 @ Z_4 ) @ ( plus_plus_nat @ Y_2 @ Z_4 ) ) ) ).
thf(fact_990_min__diff__distrib__left,axiom,
! [X_2: int,Y_1: int,Z_3: int] :
( ( minus_minus_int @ ( ord_min_int @ X_2 @ Y_1 ) @ Z_3 )
= ( ord_min_int @ ( minus_minus_int @ X_2 @ Z_3 ) @ ( minus_minus_int @ Y_1 @ Z_3 ) ) ) ).
thf(fact_991_finite__less__ub,axiom,
! [U: nat,F: nat > nat] :
( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F @ N_1 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F @ N_1 ) @ U ) ) ) ) ).
thf(fact_992_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_993_mult__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_994_mult__le__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
thf(fact_995_mult__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
thf(fact_996_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_997_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_998_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_999_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_1000_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1001_le__trans,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
thf(fact_1002_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_1003_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_1004_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_1005_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_1006_add__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_1007_add__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_1008_trans__le__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_1009_trans__le__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_1010_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_1011_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_1012_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_1013_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_1014_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
thf(fact_1015_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_1016_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_1017_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_1018_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_1019_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
thf(fact_1020_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_1021_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_1022_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_1023_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_1024_diff__diff__right,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_1025_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
thf(fact_1026_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_1027_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_1028_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_1029_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
thf(fact_1030_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_1031_add__diff__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
thf(fact_1032_le__diff__conv2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_1033_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_1034_le__imp__diff__is__add,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
<=> ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
thf(fact_1035_diff__add__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
thf(fact_1036_add__diff__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
thf(fact_1037_diff__add__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
thf(fact_1038_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_1039_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_1040_nat__eq__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1041_nat__diff__add__eq2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1042_nat__le__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1043_nat__eq__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
thf(fact_1044_nat__diff__add__eq1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1045_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_1046_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_1047_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_1048_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_1049_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_1050_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_1: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z_1 ) ) ) ).
thf(fact_1051_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_1052_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_1053_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_1054_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
thf(fact_1055_nat__le__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1056_termination__basic__simps_I3_J,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_1057_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( ( ord_less_eq_nat @ X @ Z_1 )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_1058_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
thf(fact_1059_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( ord_lessThan_nat @ U ) )
= U ) ).
thf(fact_1060_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_1061_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_1062_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_1063_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_1064_zadd__int__left,axiom,
! [M: nat,N: nat,Z_1: int] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N ) @ Z_1 ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) @ Z_1 ) ) ).
thf(fact_1065_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_1066_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_1067_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_1068_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_1069_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_1070_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_1071_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) @ ( semiri1621563631at_int @ M ) ) ).
thf(fact_1072_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at238088361st_nat @ L @ U ) ) ).
thf(fact_1073_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K ) ) ) ).
thf(fact_1074_zle__iff__zadd,axiom,
! [W: int,Z_1: int] :
( ( ord_less_eq_int @ W @ Z_1 )
<=> ? [N_1: nat] :
( Z_1
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_1075_int__power,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1621563631at_int @ M ) @ N ) ) ).
thf(fact_1076_zpower__int,axiom,
! [M: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ M ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) ) ) ).
thf(fact_1077_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N ) )
<=> ( M = N ) ) ).
thf(fact_1078_zmult__zminus,axiom,
! [Z_1: int,W: int] :
( ( times_times_int @ ( uminus_uminus_int @ Z_1 ) @ W )
= ( uminus_uminus_int @ ( times_times_int @ Z_1 @ W ) ) ) ).
thf(fact_1079_diff__int__def__symmetric,axiom,
! [Z_1: int,W: int] :
( ( plus_plus_int @ Z_1 @ ( uminus_uminus_int @ W ) )
= ( minus_minus_int @ Z_1 @ W ) ) ).
thf(fact_1080_diff__int__def,axiom,
! [Z_1: int,W: int] :
( ( minus_minus_int @ Z_1 @ W )
= ( plus_plus_int @ Z_1 @ ( uminus_uminus_int @ W ) ) ) ).
thf(fact_1081_zminus__zadd__distrib,axiom,
! [Z_1: int,W: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ Z_1 @ W ) )
= ( plus_plus_int @ ( uminus_uminus_int @ Z_1 ) @ ( uminus_uminus_int @ W ) ) ) ).
thf(fact_1082_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_1083_zle__linear,axiom,
! [Z_1: int,W: int] :
( ( ord_less_eq_int @ Z_1 @ W )
| ( ord_less_eq_int @ W @ Z_1 ) ) ).
thf(fact_1084_zle__trans,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I @ K ) ) ) ).
thf(fact_1085_zle__antisym,axiom,
! [Z_1: int,W: int] :
( ( ord_less_eq_int @ Z_1 @ W )
=> ( ( ord_less_eq_int @ W @ Z_1 )
=> ( Z_1 = W ) ) ) ).
thf(fact_1086_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_1087_zpower__zpower,axiom,
! [X: int,Y: nat,Z_1: nat] :
( ( power_power_int @ ( power_power_int @ X @ Y ) @ Z_1 )
= ( power_power_int @ X @ ( times_times_nat @ Y @ Z_1 ) ) ) ).
thf(fact_1088_zmult__commute,axiom,
! [Z_1: int,W: int] :
( ( times_times_int @ Z_1 @ W )
= ( times_times_int @ W @ Z_1 ) ) ).
thf(fact_1089_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_1090_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_1091_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_1092_zadd__left__commute,axiom,
! [X: int,Y: int,Z_1: int] :
( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z_1 ) )
= ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z_1 ) ) ) ).
thf(fact_1093_zadd__commute,axiom,
! [Z_1: int,W: int] :
( ( plus_plus_int @ Z_1 @ W )
= ( plus_plus_int @ W @ Z_1 ) ) ).
thf(fact_1094_zpower__zadd__distrib,axiom,
! [X: int,Y: nat,Z_1: nat] :
( ( power_power_int @ X @ ( plus_plus_nat @ Y @ Z_1 ) )
= ( times_times_int @ ( power_power_int @ X @ Y ) @ ( power_power_int @ X @ Z_1 ) ) ) ).
thf(fact_1095_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_1096_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_1097_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_1098_zmult__1__right,axiom,
! [Z_1: int] :
( ( times_times_int @ Z_1 @ one_one_int )
= Z_1 ) ).
thf(fact_1099_zmult__1,axiom,
! [Z_1: int] :
( ( times_times_int @ one_one_int @ Z_1 )
= Z_1 ) ).
thf(fact_1100_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
thf(fact_1101_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_1102_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_1103_add__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_1104_trans__less__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_1105_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_1106_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_1107_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
thf(fact_1108_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
thf(fact_1109_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_1110_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_1111_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1112_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1113_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_1114_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_1115_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_1116_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_1117_less__not__refl3,axiom,
! [S_1: nat,T: nat] :
( ( ord_less_nat @ S_1 @ T )
=> ( S_1 != T ) ) ).
thf(fact_1118_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_1119_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_1120_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_1121_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_1122_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_1123_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_1124_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( ( ord_less_nat @ X @ Z_1 )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_1125_termination__basic__simps_I1_J,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_1126_image__add__atLeastAtMost,axiom,
! [K: nat,I: nat,J: nat] :
( ( image_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K )
@ ( ord_at238088361st_nat @ I @ J ) )
= ( ord_at238088361st_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_1127_finite__nat__set__iff__bounded,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M_1: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_nat @ X_1 @ M_1 ) ) ) ).
thf(fact_1128_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( P @ K_1 ) @ ( ord_less_nat @ K_1 @ I ) ) ) ) ).
thf(fact_1129_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_nat @ I_1 @ N ) ) )
= N ) ).
thf(fact_1130_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_1131_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
thf(fact_1132_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_1133_diff__less__mono,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( ord_less_nat @ A_1 @ B_1 )
=> ( ( ord_less_eq_nat @ C @ A_1 )
=> ( ord_less_nat @ ( minus_minus_nat @ A_1 @ C ) @ ( minus_minus_nat @ B_1 @ C ) ) ) ) ).
thf(fact_1134_nat__less__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1135_nat__less__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1136_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ N ) @ ( uminus_uminus_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_1137_zless__le,axiom,
! [Z_1: int,W: int] :
( ( ord_less_int @ Z_1 @ W )
<=> ( ( ord_less_eq_int @ Z_1 @ W )
& ( Z_1 != W ) ) ) ).
thf(fact_1138_zadd__strict__right__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
thf(fact_1139_zminus__zminus,axiom,
! [Z_1: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ Z_1 ) )
= Z_1 ) ).
thf(fact_1140_zadd__zless__mono,axiom,
! [Z_2: int,Z_1: int,W_1: int,W: int] :
( ( ord_less_int @ W_1 @ W )
=> ( ( ord_less_eq_int @ Z_2 @ Z_1 )
=> ( ord_less_int @ ( plus_plus_int @ W_1 @ Z_2 ) @ ( plus_plus_int @ W @ Z_1 ) ) ) ) ).
thf(fact_1141_zless__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1142_zless__add1__eq,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z_1 @ one_one_int ) )
<=> ( ( ord_less_int @ W @ Z_1 )
| ( W = Z_1 ) ) ) ).
thf(fact_1143_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
=> ( ( ord_at875362053st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( ord_at875362053st_int @ M @ N ) ) ) ) ).
thf(fact_1144_zless__imp__add1__zle,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ W @ Z_1 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z_1 ) ) ).
thf(fact_1145_add1__zle__eq,axiom,
! [W: int,Z_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z_1 )
<=> ( ord_less_int @ W @ Z_1 ) ) ).
thf(fact_1146_zle__add1__eq__le,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z_1 @ one_one_int ) )
<=> ( ord_less_eq_int @ W @ Z_1 ) ) ).
thf(fact_1147_zle__diff1__eq,axiom,
! [W: int,Z_1: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z_1 @ one_one_int ) )
<=> ( ord_less_int @ W @ Z_1 ) ) ).
thf(fact_1148_int__le__induct,axiom,
! [P: int > $o,I: int,K: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1149_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_at875362053st_int @ L @ U ) ) ).
thf(fact_1150_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_1151_simp__from__to,axiom,
! [J: int,I: int] :
( ( ( ord_less_int @ J @ I )
=> ( ( ord_at875362053st_int @ I @ J )
= bot_bot_int_o ) )
& ( ~ ( ord_less_int @ J @ I )
=> ( ( ord_at875362053st_int @ I @ J )
= ( insert_int @ I @ ( ord_at875362053st_int @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ) ).
thf(fact_1152_int__less__induct,axiom,
! [P: int > $o,I: int,K: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1153_transfer__nat__int__set__relations_I1_J,axiom,
! [A: nat > $o] :
( ( finite_finite_nat @ A )
<=> ( finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_1154_transfer__nat__int__set__relations_I5_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
<=> ( ord_less_eq_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_1155_transfer__nat__int__set__relations_I2_J,axiom,
! [X: nat,A: nat > $o] :
( ( member_nat @ X @ A )
<=> ( member_int @ ( semiri1621563631at_int @ X ) @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_1156_transfer__nat__int__set__relations_I3_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( A = B )
<=> ( ( image_nat_int @ semiri1621563631at_int @ A )
= ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_1157_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_1158_transfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_1159_transfer__nat__int__set__relations_I4_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_nat_o @ A @ B )
<=> ( ord_less_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_1160_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A: nat > $o] :
( ( finite_card_nat @ A )
= ( finite_card_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_1161_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
( bot_bot_int_o
= ( image_nat_int @ semiri1621563631at_int @ bot_bot_nat_o ) ) ).
thf(fact_1162_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_1163_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_1164_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_1165_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_1166_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_1167_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ X ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ X @ N ) ) ) ).
thf(fact_1168_int__gr__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1169_int__ge__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1170_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_1171_bounded__nat__set__is__finite,axiom,
! [N: nat,N_2: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_nat @ X_1 @ N ) )
=> ( finite_finite_nat @ N_2 ) ) ).
thf(fact_1172_int__diff__cases,axiom,
! [Z_1: int] :
~ ! [M_1: nat,N_1: nat] :
( Z_1
!= ( minus_minus_int @ ( semiri1621563631at_int @ M_1 ) @ ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_1173_less__mono__imp__le__mono,axiom,
! [I: nat,J: nat,F: nat > nat] :
( ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ ( F @ I_1 ) @ ( F @ J_1 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
thf(fact_1174_int__induct,axiom,
! [I: int,P: int > $o,K: int] :
( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1175_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_1176_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_1177_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_gr375877188st_nat @ L @ U ) ) ).
thf(fact_1178_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N: nat,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_1179_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_1180_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_1181_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z_1: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z_1 ) ) ).
thf(fact_1182_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_1183_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_1184_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_1185_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_1186_int__less__0__conv,axiom,
! [K: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
thf(fact_1187_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_1188_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_1189_zmult__zless__mono2,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
thf(fact_1190_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_1191_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_1192_nat__power__less__imp__less,axiom,
! [M: nat,N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1193_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_1194_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_1195_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_1196_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_1197_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
%----Helper facts (9)
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 ) ) ).
thf(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ~ ( fequal_int @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( X != Y )
| ( fequal_int @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ ( fequal_nat @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| ( fequal_nat @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_,axiom,
! [X: hoare_2091234717iple_a,Y: hoare_2091234717iple_a] :
( ~ ( fequal1604381340iple_a @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____nqhfsdfvyv__Otriple_It__a_J_,axiom,
! [X: hoare_2091234717iple_a,Y: hoare_2091234717iple_a] :
( ( X != Y )
| ( fequal1604381340iple_a @ X @ Y ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( hoare_1467856363rivs_a @ g
@ ( insert1597628439iple_a
@ ( hoare_657976383iple_a
@ ^ [Z: x_a,S: state] : $false
@ c
@ ^ [Z: x_a,S: state] : ( (&) @ ( p @ Z @ S ) @ ( (~) @ ( b @ S ) ) ) )
@ bot_bo1791335050le_a_o ) ) ).
%------------------------------------------------------------------------------