TPTP Problem File: SLH0882^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00217_008167__14727830_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1494 ( 658 unt; 226 typ; 0 def)
% Number of atoms : 3425 (1394 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10183 ( 245 ~; 48 |; 246 &;8339 @)
% ( 0 <=>;1305 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 1111 (1111 >; 0 *; 0 +; 0 <<)
% Number of symbols : 207 ( 204 usr; 13 con; 0-5 aty)
% Number of variables : 3141 ( 107 ^;2934 !; 100 ?;3141 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:44:25.435
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
thf(ty_n_t__Multiset__Omultiset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
multiset_nat_b_nat_b: $tType ).
thf(ty_n_t__Option__Ooption_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
option_nat_b_nat_b: $tType ).
thf(ty_n_t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
list_nat_b_nat_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
set_nat_b_nat_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Real__Oreal_J,type,
multiset_real: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
multiset_int: $tType ).
thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
option_real: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
option_int: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__b_J,type,
multiset_b: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (204)
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
finite1660923644538309244_nat_b: set_nat_b_nat_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Fun_Oid_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
id_nat_b_nat_b: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_Fun_Oid_001_062_It__Nat__Onat_Mtf__b_J,type,
id_nat_b: ( nat > b ) > nat > b ).
thf(sy_c_Fun_Oid_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
id_list_nat_b_nat_b: list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_Fun_Oid_001t__List__Olist_It__Nat__Onat_J,type,
id_list_nat: list_nat > list_nat ).
thf(sy_c_Fun_Oid_001t__Multiset__Omultiset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
id_mul5310361620946034820_nat_b: multiset_nat_b_nat_b > multiset_nat_b_nat_b ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
minus_2952184192894347252_nat_b: set_nat_b_nat_b > set_nat_b_nat_b > set_nat_b_nat_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__b_J,type,
plus_plus_multiset_b: multiset_b > multiset_b > multiset_b ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Int__Oint_J,type,
zero_z3170743180189231877et_int: multiset_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__b_J,type,
zero_zero_multiset_b: multiset_b ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Int__Oint,type,
groups3087159172733796685_b_int: ( ( ( nat > b ) > nat > b ) > int ) > set_nat_b_nat_b > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Nat__Onat,type,
groups3089649643242846961_b_nat: ( ( ( nat > b ) > nat > b ) > nat ) > set_nat_b_nat_b > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Real__Oreal,type,
groups8666044050912730445b_real: ( ( ( nat > b ) > nat > b ) > real ) > set_nat_b_nat_b > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Harmonic__Numbers_Oharm_001t__Real__Oreal,type,
harmonic_harm_real: nat > real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_OListMem_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
listMem_nat_b_nat_b: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b > $o ).
thf(sy_c_List_OListMem_001t__Int__Oint,type,
listMem_int: int > list_int > $o ).
thf(sy_c_List_OListMem_001t__Nat__Onat,type,
listMem_nat: nat > list_nat > $o ).
thf(sy_c_List_OListMem_001t__Real__Oreal,type,
listMem_real: real > list_real > $o ).
thf(sy_c_List_Oall__interval__nat,type,
all_interval_nat: ( nat > $o ) > nat > nat > $o ).
thf(sy_c_List_Ocan__select_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
can_se6628294081300001064_nat_b: ( ( ( nat > b ) > nat > b ) > $o ) > set_nat_b_nat_b > $o ).
thf(sy_c_List_Ocan__select_001t__Int__Oint,type,
can_select_int: ( int > $o ) > set_int > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Real__Oreal,type,
can_select_real: ( real > $o ) > set_real > $o ).
thf(sy_c_List_Ocoset_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
coset_nat_b_nat_b: list_nat_b_nat_b > set_nat_b_nat_b ).
thf(sy_c_List_Ocoset_001t__Int__Oint,type,
coset_int: list_int > set_int ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Real__Oreal,type,
coset_real: list_real > set_real ).
thf(sy_c_List_Ocount__list_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
count_6092015575532387177_nat_b: list_nat_b_nat_b > ( ( nat > b ) > nat > b ) > nat ).
thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
count_list_int: list_int > int > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
count_list_real: list_real > real > nat ).
thf(sy_c_List_Odistinct_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
distinct_nat_b_nat_b: list_nat_b_nat_b > $o ).
thf(sy_c_List_Odistinct_001t__Int__Oint,type,
distinct_int: list_int > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Real__Oreal,type,
distinct_real: list_real > $o ).
thf(sy_c_List_Ofind_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
find_nat_b_nat_b: ( ( ( nat > b ) > nat > b ) > $o ) > list_nat_b_nat_b > option_nat_b_nat_b ).
thf(sy_c_List_Ofind_001t__Int__Oint,type,
find_int: ( int > $o ) > list_int > option_int ).
thf(sy_c_List_Ofind_001t__Nat__Onat,type,
find_nat: ( nat > $o ) > list_nat > option_nat ).
thf(sy_c_List_Ofind_001t__Real__Oreal,type,
find_real: ( real > $o ) > list_real > option_real ).
thf(sy_c_List_Ofold_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
fold_n1883935874045088215_nat_b: ( ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_nat_b_nat_b > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__b_J,type,
fold_n5624292861071574516_nat_b: ( ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_nat_b_nat_b > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
fold_n8547691366255251559_nat_b: ( ( ( nat > b ) > nat > b ) > list_nat_b_nat_b > list_nat_b_nat_b ) > list_nat_b_nat_b > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Ofold_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
fold_n5526574726712445965_nat_b: ( ( ( nat > b ) > nat > b ) > set_nat_b_nat_b > set_nat_b_nat_b ) > list_nat_b_nat_b > set_nat_b_nat_b > set_nat_b_nat_b ).
thf(sy_c_List_Ofold_001t__Int__Oint_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
fold_int_nat_b_nat_b: ( int > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_int > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001t__Int__Oint_001t__List__Olist_It__Int__Oint_J,type,
fold_int_list_int: ( int > list_int > list_int ) > list_int > list_int > list_int ).
thf(sy_c_List_Ofold_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
fold_int_set_int: ( int > set_int > set_int ) > list_int > set_int > set_int ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
fold_nat_nat_b_nat_b: ( nat > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_nat > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
fold_nat_list_nat: ( nat > list_nat > list_nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fold_nat_set_nat: ( nat > set_nat > set_nat ) > list_nat > set_nat > set_nat ).
thf(sy_c_List_Ofold_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
fold_r4529934814194841502_nat_b: ( real > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_real > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001t__Real__Oreal_001t__List__Olist_It__Real__Oreal_J,type,
fold_real_list_real: ( real > list_real > list_real ) > list_real > list_real > list_real ).
thf(sy_c_List_Ofold_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
fold_real_set_real: ( real > set_real > set_real ) > list_real > set_real > set_real ).
thf(sy_c_List_Oinsert_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
insert_nat_b_nat_b: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
insert_real: real > list_real > list_real ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Int__Oint,type,
linord2612477271533052124et_int: set_int > list_int ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Real__Oreal,type,
linord4252657396651189596t_real: set_real > list_real ).
thf(sy_c_List_Olist_Oset_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
set_nat_b_nat_b2: list_nat_b_nat_b > set_nat_b_nat_b ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist__ex1_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
list_ex1_nat_b_nat_b: ( ( ( nat > b ) > nat > b ) > $o ) > list_nat_b_nat_b > $o ).
thf(sy_c_List_Olist__ex1_001t__Int__Oint,type,
list_ex1_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olist__ex_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
list_ex_nat_b_nat_b: ( ( ( nat > b ) > nat > b ) > $o ) > list_nat_b_nat_b > $o ).
thf(sy_c_List_Olist__ex_001t__Int__Oint,type,
list_ex_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Real__Oreal,type,
list_ex_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Omember_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
member_nat_b_nat_b: list_nat_b_nat_b > ( ( nat > b ) > nat > b ) > $o ).
thf(sy_c_List_Omember_001t__Int__Oint,type,
member_int: list_int > int > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Real__Oreal,type,
member_real: list_real > real > $o ).
thf(sy_c_List_OremoveAll_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
remove7346898498583989977_nat_b: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
removeAll_int: int > list_int > list_int ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
removeAll_real: real > list_real > list_real ).
thf(sy_c_List_Orotate1_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
rotate1_nat_b_nat_b: list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Orotate1_001t__Int__Oint,type,
rotate1_int: list_int > list_int ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Real__Oreal,type,
rotate1_real: list_real > list_real ).
thf(sy_c_List_Orotate_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
rotate_nat_b_nat_b: nat > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Orotate_001t__Int__Oint,type,
rotate_int: nat > list_int > list_int ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Real__Oreal,type,
rotate_real: nat > list_real > list_real ).
thf(sy_c_List_Ounion_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
union_nat_b_nat_b: list_nat_b_nat_b > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Ounion_001t__Int__Oint,type,
union_int: list_int > list_int > list_int ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Real__Oreal,type,
union_real: list_real > list_real > list_real ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Multiset_Oimage__mset_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
image_8551226173605001479_nat_b: ( ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > multiset_nat_b_nat_b > multiset_nat_b_nat_b ).
thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001tf__b,type,
image_mset_nat_b: ( nat > b ) > multiset_nat > multiset_b ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
linord3047872887403683810et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset_Omset_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
mset_nat_b_nat_b: list_nat_b_nat_b > multiset_nat_b_nat_b ).
thf(sy_c_Multiset_Omset_001t__Int__Oint,type,
mset_int: list_int > multiset_int ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Real__Oreal,type,
mset_real: list_real > multiset_real ).
thf(sy_c_Multiset_Omset__set_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
mset_set_nat_b_nat_b: set_nat_b_nat_b > multiset_nat_b_nat_b ).
thf(sy_c_Multiset_Omset__set_001t__Int__Oint,type,
mset_set_int: set_int > multiset_int ).
thf(sy_c_Multiset_Omset__set_001t__Nat__Onat,type,
mset_set_nat: set_nat > multiset_nat ).
thf(sy_c_Multiset_Omset__set_001t__Real__Oreal,type,
mset_set_real: set_real > multiset_real ).
thf(sy_c_Multiset_Omultiset_Ocount_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
count_nat_b_nat_b: multiset_nat_b_nat_b > ( ( nat > b ) > nat > b ) > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Int__Oint,type,
count_int: multiset_int > int > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Nat__Onat,type,
count_nat: multiset_nat > nat > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Real__Oreal,type,
count_real: multiset_real > real > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
ord_less_nat_b_nat_b: ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
ord_le4513651211808907375_nat_b: set_nat_b_nat_b > set_nat_b_nat_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
ord_le535081054522138693_nat_b: ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
ord_le9047053354294502011_nat_b: set_nat_b_nat_b > set_nat_b_nat_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
collect_nat_b_nat_b: ( ( ( nat > b ) > nat > b ) > $o ) > set_nat_b_nat_b ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oremove_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
remove_nat_b_nat_b: ( ( nat > b ) > nat > b ) > set_nat_b_nat_b > set_nat_b_nat_b ).
thf(sy_c_Set_Oremove_001t__Int__Oint,type,
remove_int: int > set_int > set_int ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Real__Oreal,type,
remove_real: real > set_real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
set_or4963087151523820308_nat_b: ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > set_nat_b_nat_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
set_or5633471081860273246_nat_b: ( ( nat > b ) > nat > b ) > set_nat_b_nat_b ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
set_or7136064339709882498_nat_b: ( ( nat > b ) > nat > b ) > set_nat_b_nat_b ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
member_nat_b_nat_b2: ( ( nat > b ) > nat > b ) > set_nat_b_nat_b > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int2: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real2: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_f,type,
f: nat > b ).
thf(sy_v_is__swap____,type,
is_swap: ( ( nat > b ) > nat > b ) > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_t____,type,
t: list_nat_b_nat_b ).
thf(sy_v_x,type,
x: ( nat > b ) > nat > b ).
% Relevant facts (1263)
thf(fact_0_in__set__member,axiom,
! [X: real,Xs: list_real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
= ( member_real @ Xs @ X ) ) ).
% in_set_member
thf(fact_1_in__set__member,axiom,
! [X: int,Xs: list_int] :
( ( member_int2 @ X @ ( set_int2 @ Xs ) )
= ( member_int @ Xs @ X ) ) ).
% in_set_member
thf(fact_2_in__set__member,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ X @ ( set_nat_b_nat_b2 @ Xs ) )
= ( member_nat_b_nat_b @ Xs @ X ) ) ).
% in_set_member
thf(fact_3_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_4_in__set__insert,axiom,
! [X: real,Xs: list_real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_5_in__set__insert,axiom,
! [X: int,Xs: list_int] :
( ( member_int2 @ X @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_6_in__set__insert,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( insert_nat_b_nat_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_7_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_8_list__ex1__iff,axiom,
( list_ex1_real
= ( ^ [P: real > $o,Xs2: list_real] :
? [X2: real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: real] :
( ( ( member_real2 @ Y @ ( set_real2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_9_list__ex1__iff,axiom,
( list_ex1_int
= ( ^ [P: int > $o,Xs2: list_int] :
? [X2: int] :
( ( member_int2 @ X2 @ ( set_int2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: int] :
( ( ( member_int2 @ Y @ ( set_int2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_10_list__ex1__iff,axiom,
( list_ex1_nat_b_nat_b
= ( ^ [P: ( ( nat > b ) > nat > b ) > $o,Xs2: list_nat_b_nat_b] :
? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X2 @ ( set_nat_b_nat_b2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: ( nat > b ) > nat > b] :
( ( ( member_nat_b_nat_b2 @ Y @ ( set_nat_b_nat_b2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_11_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: nat] :
( ( ( member_nat2 @ Y @ ( set_nat2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_12_calculation,axiom,
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ t ) )
=> ( is_swap @ X3 ) )
=> ( ( image_mset_nat_b @ ( fold_n5624292861071574516_nat_b @ id_nat_b_nat_b @ t @ f ) @ ( mset_nat @ ( upt @ zero_zero_nat @ n ) ) )
= ( image_mset_nat_b @ f @ ( mset_nat @ ( upt @ zero_zero_nat @ n ) ) ) ) ) ).
% calculation
thf(fact_13_ListMem__iff,axiom,
( listMem_real
= ( ^ [X2: real,Xs2: list_real] : ( member_real2 @ X2 @ ( set_real2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_14_ListMem__iff,axiom,
( listMem_int
= ( ^ [X2: int,Xs2: list_int] : ( member_int2 @ X2 @ ( set_int2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_15_ListMem__iff,axiom,
( listMem_nat_b_nat_b
= ( ^ [X2: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] : ( member_nat_b_nat_b2 @ X2 @ ( set_nat_b_nat_b2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_16_ListMem__iff,axiom,
( listMem_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% ListMem_iff
thf(fact_17_set__rotate1,axiom,
! [Xs: list_int] :
( ( set_int2 @ ( rotate1_int @ Xs ) )
= ( set_int2 @ Xs ) ) ).
% set_rotate1
thf(fact_18_set__rotate1,axiom,
! [Xs: list_real] :
( ( set_real2 @ ( rotate1_real @ Xs ) )
= ( set_real2 @ Xs ) ) ).
% set_rotate1
thf(fact_19_set__rotate1,axiom,
! [Xs: list_nat_b_nat_b] :
( ( set_nat_b_nat_b2 @ ( rotate1_nat_b_nat_b @ Xs ) )
= ( set_nat_b_nat_b2 @ Xs ) ) ).
% set_rotate1
thf(fact_20_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_21_removeAll__id,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) )
=> ( ( removeAll_real @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_22_removeAll__id,axiom,
! [X: int,Xs: list_int] :
( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
=> ( ( removeAll_int @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_23_removeAll__id,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ~ ( member_nat_b_nat_b2 @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( remove7346898498583989977_nat_b @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_24_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_25_set__rotate,axiom,
! [N: nat,Xs: list_int] :
( ( set_int2 @ ( rotate_int @ N @ Xs ) )
= ( set_int2 @ Xs ) ) ).
% set_rotate
thf(fact_26_set__rotate,axiom,
! [N: nat,Xs: list_real] :
( ( set_real2 @ ( rotate_real @ N @ Xs ) )
= ( set_real2 @ Xs ) ) ).
% set_rotate
thf(fact_27_set__rotate,axiom,
! [N: nat,Xs: list_nat_b_nat_b] :
( ( set_nat_b_nat_b2 @ ( rotate_nat_b_nat_b @ N @ Xs ) )
= ( set_nat_b_nat_b2 @ Xs ) ) ).
% set_rotate
thf(fact_28_set__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( set_nat2 @ ( rotate_nat @ N @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate
thf(fact_29_find__cong,axiom,
! [Xs: list_real,Ys: list_real,P2: real > $o,Q: real > $o] :
( ( Xs = Ys )
=> ( ! [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Ys ) )
=> ( ( P2 @ X3 )
= ( Q @ X3 ) ) )
=> ( ( find_real @ P2 @ Xs )
= ( find_real @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_30_find__cong,axiom,
! [Xs: list_int,Ys: list_int,P2: int > $o,Q: int > $o] :
( ( Xs = Ys )
=> ( ! [X3: int] :
( ( member_int2 @ X3 @ ( set_int2 @ Ys ) )
=> ( ( P2 @ X3 )
= ( Q @ X3 ) ) )
=> ( ( find_int @ P2 @ Xs )
= ( find_int @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_31_find__cong,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,P2: ( ( nat > b ) > nat > b ) > $o,Q: ( ( nat > b ) > nat > b ) > $o] :
( ( Xs = Ys )
=> ( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Ys ) )
=> ( ( P2 @ X3 )
= ( Q @ X3 ) ) )
=> ( ( find_nat_b_nat_b @ P2 @ Xs )
= ( find_nat_b_nat_b @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_32_find__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P2: nat > $o,Q: nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( P2 @ X3 )
= ( Q @ X3 ) ) )
=> ( ( find_nat @ P2 @ Xs )
= ( find_nat @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_33_list__ex__cong,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > $o,G: ( ( nat > b ) > nat > b ) > $o] :
( ( Xs = Ys )
=> ( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_nat_b_nat_b @ F @ Xs )
= ( list_ex_nat_b_nat_b @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_34_list__ex__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_nat @ F @ Xs )
= ( list_ex_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_35_list__ex__cong,axiom,
! [Xs: list_int,Ys: list_int,F: int > $o,G: int > $o] :
( ( Xs = Ys )
=> ( ! [X3: int] :
( ( member_int2 @ X3 @ ( set_int2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_int @ F @ Xs )
= ( list_ex_int @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_36_list__ex__cong,axiom,
! [Xs: list_real,Ys: list_real,F: real > $o,G: real > $o] :
( ( Xs = Ys )
=> ( ! [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex_real @ F @ Xs )
= ( list_ex_real @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_37_fold__id,axiom,
! [Xs: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b_nat_b ) )
=> ( ( fold_n1883935874045088215_nat_b @ F @ Xs )
= id_nat_b_nat_b ) ) ).
% fold_id
thf(fact_38_fold__id,axiom,
! [Xs: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b ) )
=> ( ( fold_n5624292861071574516_nat_b @ F @ Xs )
= id_nat_b ) ) ).
% fold_id
thf(fact_39_fold__id,axiom,
! [Xs: list_nat,F: nat > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b_nat_b ) )
=> ( ( fold_nat_nat_b_nat_b @ F @ Xs )
= id_nat_b_nat_b ) ) ).
% fold_id
thf(fact_40_fold__id,axiom,
! [Xs: list_int,F: int > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: int] :
( ( member_int2 @ X3 @ ( set_int2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b_nat_b ) )
=> ( ( fold_int_nat_b_nat_b @ F @ Xs )
= id_nat_b_nat_b ) ) ).
% fold_id
thf(fact_41_fold__id,axiom,
! [Xs: list_real,F: real > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b_nat_b ) )
=> ( ( fold_r4529934814194841502_nat_b @ F @ Xs )
= id_nat_b_nat_b ) ) ).
% fold_id
thf(fact_42_rotate1__rotate__swap,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate1_nat @ ( rotate_nat @ N @ Xs ) )
= ( rotate_nat @ N @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_43_rotate1__rotate__swap,axiom,
! [N: nat,Xs: list_nat_b_nat_b] :
( ( rotate1_nat_b_nat_b @ ( rotate_nat_b_nat_b @ N @ Xs ) )
= ( rotate_nat_b_nat_b @ N @ ( rotate1_nat_b_nat_b @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_44_fold__invariant,axiom,
! [Xs: list_nat_b_nat_b,Q: ( ( nat > b ) > nat > b ) > $o,P2: ( nat > b ) > $o,S: nat > b,F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P2 @ S )
=> ( ! [X3: ( nat > b ) > nat > b,S2: nat > b] :
( ( Q @ X3 )
=> ( ( P2 @ S2 )
=> ( P2 @ ( F @ X3 @ S2 ) ) ) )
=> ( P2 @ ( fold_n5624292861071574516_nat_b @ F @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_45_List_Ofold__cong,axiom,
! [A: nat > b,B: nat > b,Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b,G: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( A = B )
=> ( ( Xs = Ys )
=> ( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_n5624292861071574516_nat_b @ F @ Xs @ A )
= ( fold_n5624292861071574516_nat_b @ G @ Ys @ B ) ) ) ) ) ).
% List.fold_cong
thf(fact_46_multiset_Omap__id,axiom,
! [T: multiset_nat_b_nat_b] :
( ( image_8551226173605001479_nat_b @ id_nat_b_nat_b @ T )
= T ) ).
% multiset.map_id
thf(fact_47_multiset_Omap__id0,axiom,
( ( image_8551226173605001479_nat_b @ id_nat_b_nat_b )
= id_mul5310361620946034820_nat_b ) ).
% multiset.map_id0
thf(fact_48_image__mset__id,axiom,
! [X: multiset_nat_b_nat_b] :
( ( image_8551226173605001479_nat_b @ id_nat_b_nat_b @ X )
= X ) ).
% image_mset_id
thf(fact_49_fold__permuted__eq,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,P2: ( nat > b ) > $o,Z: nat > b,F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( ( mset_nat_b_nat_b @ Xs )
= ( mset_nat_b_nat_b @ Ys ) )
=> ( ( P2 @ Z )
=> ( ! [X3: ( nat > b ) > nat > b,Z2: nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ ( F @ X3 @ Z2 ) ) ) )
=> ( ! [X3: ( nat > b ) > nat > b,Y2: ( nat > b ) > nat > b,Z2: nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( member_nat_b_nat_b2 @ Y2 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( P2 @ Z2 )
=> ( ( F @ X3 @ ( F @ Y2 @ Z2 ) )
= ( F @ Y2 @ ( F @ X3 @ Z2 ) ) ) ) ) )
=> ( ( fold_n5624292861071574516_nat_b @ F @ Xs @ Z )
= ( fold_n5624292861071574516_nat_b @ F @ Ys @ Z ) ) ) ) ) ) ).
% fold_permuted_eq
thf(fact_50_can__select__set__list__ex1,axiom,
! [P2: nat > $o,A2: list_nat] :
( ( can_select_nat @ P2 @ ( set_nat2 @ A2 ) )
= ( list_ex1_nat @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_51_can__select__set__list__ex1,axiom,
! [P2: ( ( nat > b ) > nat > b ) > $o,A2: list_nat_b_nat_b] :
( ( can_se6628294081300001064_nat_b @ P2 @ ( set_nat_b_nat_b2 @ A2 ) )
= ( list_ex1_nat_b_nat_b @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_52_can__select__set__list__ex1,axiom,
! [P2: int > $o,A2: list_int] :
( ( can_select_int @ P2 @ ( set_int2 @ A2 ) )
= ( list_ex1_int @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_53_can__select__set__list__ex1,axiom,
! [P2: real > $o,A2: list_real] :
( ( can_select_real @ P2 @ ( set_real2 @ A2 ) )
= ( list_ex1_real @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_54_id__apply,axiom,
( id_nat_b_nat_b
= ( ^ [X2: ( nat > b ) > nat > b] : X2 ) ) ).
% id_apply
thf(fact_55_mset__eq__setD,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( ( mset_nat_b_nat_b @ Xs )
= ( mset_nat_b_nat_b @ Ys ) )
=> ( ( set_nat_b_nat_b2 @ Xs )
= ( set_nat_b_nat_b2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_56_mset__eq__setD,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( mset_int @ Xs )
= ( mset_int @ Ys ) )
=> ( ( set_int2 @ Xs )
= ( set_int2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_57_mset__eq__setD,axiom,
! [Xs: list_real,Ys: list_real] :
( ( ( mset_real @ Xs )
= ( mset_real @ Ys ) )
=> ( ( set_real2 @ Xs )
= ( set_real2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_58_mset__eq__setD,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_59_a,axiom,
! [X: ( nat > b ) > nat > b,F: nat > b] :
( ( is_swap @ X )
=> ( ( image_mset_nat_b @ ( X @ F ) @ ( mset_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n ) ) )
= ( image_mset_nat_b @ F @ ( mset_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ n ) ) ) ) ) ).
% a
thf(fact_60_remove__code_I1_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_61_remove__code_I1_J,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( remove_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
= ( set_nat_b_nat_b2 @ ( remove7346898498583989977_nat_b @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_62_remove__code_I1_J,axiom,
! [X: int,Xs: list_int] :
( ( remove_int @ X @ ( set_int2 @ Xs ) )
= ( set_int2 @ ( removeAll_int @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_63_remove__code_I1_J,axiom,
! [X: real,Xs: list_real] :
( ( remove_real @ X @ ( set_real2 @ Xs ) )
= ( set_real2 @ ( removeAll_real @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_64_count__notin,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ~ ( member_nat_b_nat_b2 @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( count_6092015575532387177_nat_b @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_65_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_66_count__notin,axiom,
! [X: int,Xs: list_int] :
( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
=> ( ( count_list_int @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_67_count__notin,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_68_ex__mset,axiom,
! [X4: multiset_nat] :
? [Xs3: list_nat] :
( ( mset_nat @ Xs3 )
= X4 ) ).
% ex_mset
thf(fact_69_image__mset__is__empty__iff,axiom,
! [F: nat > b,M: multiset_nat] :
( ( ( image_mset_nat_b @ F @ M )
= zero_zero_multiset_b )
= ( M = zero_z7348594199698428585et_nat ) ) ).
% image_mset_is_empty_iff
thf(fact_70_image__mset__empty,axiom,
! [F: nat > b] :
( ( image_mset_nat_b @ F @ zero_z7348594199698428585et_nat )
= zero_zero_multiset_b ) ).
% image_mset_empty
thf(fact_71_rotate0,axiom,
( ( rotate_nat @ zero_zero_nat )
= id_list_nat ) ).
% rotate0
thf(fact_72_rotate0,axiom,
( ( rotate_nat_b_nat_b @ zero_zero_nat )
= id_list_nat_b_nat_b ) ).
% rotate0
thf(fact_73_mset__upt,axiom,
! [M2: nat,N: nat] :
( ( mset_nat @ ( upt @ M2 @ N ) )
= ( mset_set_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ).
% mset_upt
thf(fact_74_can__select__def,axiom,
( can_se6628294081300001064_nat_b
= ( ^ [P: ( ( nat > b ) > nat > b ) > $o,A3: set_nat_b_nat_b] :
? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: ( nat > b ) > nat > b] :
( ( ( member_nat_b_nat_b2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_75_can__select__def,axiom,
( can_select_nat
= ( ^ [P: nat > $o,A3: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: nat] :
( ( ( member_nat2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_76_can__select__def,axiom,
( can_select_real
= ( ^ [P: real > $o,A3: set_real] :
? [X2: real] :
( ( member_real2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: real] :
( ( ( member_real2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_77_can__select__def,axiom,
( can_select_int
= ( ^ [P: int > $o,A3: set_int] :
? [X2: int] :
( ( member_int2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: int] :
( ( ( member_int2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_78_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I: nat,J: nat] : ( set_nat2 @ ( upt @ I @ J ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_79_count__list__0__iff,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( ( count_6092015575532387177_nat_b @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_nat_b_nat_b2 @ X @ ( set_nat_b_nat_b2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_80_count__list__0__iff,axiom,
! [Xs: list_nat,X: nat] :
( ( ( count_list_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_81_count__list__0__iff,axiom,
! [Xs: list_int,X: int] :
( ( ( count_list_int @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_82_count__list__0__iff,axiom,
! [Xs: list_real,X: real] :
( ( ( count_list_real @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_83_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_84_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_85_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_86_eq__id__iff,axiom,
! [F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( ! [X2: ( nat > b ) > nat > b] :
( ( F @ X2 )
= X2 ) )
= ( F = id_nat_b_nat_b ) ) ).
% eq_id_iff
thf(fact_87_id__def,axiom,
( id_nat_b_nat_b
= ( ^ [X2: ( nat > b ) > nat > b] : X2 ) ) ).
% id_def
thf(fact_88_member__remove,axiom,
! [X: ( nat > b ) > nat > b,Y3: ( nat > b ) > nat > b,A2: set_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ X @ ( remove_nat_b_nat_b @ Y3 @ A2 ) )
= ( ( member_nat_b_nat_b2 @ X @ A2 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_89_member__remove,axiom,
! [X: nat,Y3: nat,A2: set_nat] :
( ( member_nat2 @ X @ ( remove_nat @ Y3 @ A2 ) )
= ( ( member_nat2 @ X @ A2 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_90_member__remove,axiom,
! [X: real,Y3: real,A2: set_real] :
( ( member_real2 @ X @ ( remove_real @ Y3 @ A2 ) )
= ( ( member_real2 @ X @ A2 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_91_member__remove,axiom,
! [X: int,Y3: int,A2: set_int] :
( ( member_int2 @ X @ ( remove_int @ Y3 @ A2 ) )
= ( ( member_int2 @ X @ A2 )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_92_mset__set__upto__eq__mset__upto,axiom,
! [N: nat] :
( ( mset_set_nat @ ( set_ord_lessThan_nat @ N ) )
= ( mset_nat @ ( upt @ zero_zero_nat @ N ) ) ) ).
% mset_set_upto_eq_mset_upto
thf(fact_93_List_Ounion__def,axiom,
( union_nat
= ( fold_nat_list_nat @ insert_nat ) ) ).
% List.union_def
thf(fact_94_List_Ounion__def,axiom,
( union_nat_b_nat_b
= ( fold_n8547691366255251559_nat_b @ insert_nat_b_nat_b ) ) ).
% List.union_def
thf(fact_95_List_Ounion__def,axiom,
( union_int
= ( fold_int_list_int @ insert_int ) ) ).
% List.union_def
thf(fact_96_List_Ounion__def,axiom,
( union_real
= ( fold_real_list_real @ insert_real ) ) ).
% List.union_def
thf(fact_97_remove__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( insert_nat @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_98_remove__code_I2_J,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( remove_nat_b_nat_b @ X @ ( coset_nat_b_nat_b @ Xs ) )
= ( coset_nat_b_nat_b @ ( insert_nat_b_nat_b @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_99_remove__code_I2_J,axiom,
! [X: int,Xs: list_int] :
( ( remove_int @ X @ ( coset_int @ Xs ) )
= ( coset_int @ ( insert_int @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_100_remove__code_I2_J,axiom,
! [X: real,Xs: list_real] :
( ( remove_real @ X @ ( coset_real @ Xs ) )
= ( coset_real @ ( insert_real @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_101_all__interval__nat__def,axiom,
( all_interval_nat
= ( ^ [P: nat > $o,I: nat,J: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ I @ J ) )
=> ( P @ X2 ) ) ) ) ).
% all_interval_nat_def
thf(fact_102_mset__sorted__list__of__multiset,axiom,
! [M: multiset_nat] :
( ( mset_nat @ ( linord3047872887403683810et_nat @ M ) )
= M ) ).
% mset_sorted_list_of_multiset
thf(fact_103_count__mset__0__iff,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( ( count_nat_b_nat_b @ ( mset_nat_b_nat_b @ Xs ) @ X )
= zero_zero_nat )
= ( ~ ( member_nat_b_nat_b2 @ X @ ( set_nat_b_nat_b2 @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_104_count__mset__0__iff,axiom,
! [Xs: list_int,X: int] :
( ( ( count_int @ ( mset_int @ Xs ) @ X )
= zero_zero_nat )
= ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_105_count__mset__0__iff,axiom,
! [Xs: list_real,X: real] :
( ( ( count_real @ ( mset_real @ Xs ) @ X )
= zero_zero_nat )
= ( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_106_count__mset__0__iff,axiom,
! [Xs: list_nat,X: nat] :
( ( ( count_nat @ ( mset_nat @ Xs ) @ X )
= zero_zero_nat )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_107_mem__Collect__eq,axiom,
! [A: ( nat > b ) > nat > b,P2: ( ( nat > b ) > nat > b ) > $o] :
( ( member_nat_b_nat_b2 @ A @ ( collect_nat_b_nat_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A: real,P2: real > $o] :
( ( member_real2 @ A @ ( collect_real @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int2 @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A2: set_nat_b_nat_b] :
( ( collect_nat_b_nat_b
@ ^ [X2: ( nat > b ) > nat > b] : ( member_nat_b_nat_b2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_115_sorted__list__of__set__range,axiom,
! [M2: nat,N: nat] :
( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sorted_list_of_set_range
thf(fact_116_rotate__Suc,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate_nat @ ( suc @ N ) @ Xs )
= ( rotate1_nat @ ( rotate_nat @ N @ Xs ) ) ) ).
% rotate_Suc
thf(fact_117_rotate__Suc,axiom,
! [N: nat,Xs: list_nat_b_nat_b] :
( ( rotate_nat_b_nat_b @ ( suc @ N ) @ Xs )
= ( rotate1_nat_b_nat_b @ ( rotate_nat_b_nat_b @ N @ Xs ) ) ) ).
% rotate_Suc
thf(fact_118_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_119_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_120_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_121_count__mset__set_I3_J,axiom,
! [X: ( nat > b ) > nat > b,A2: set_nat_b_nat_b] :
( ~ ( member_nat_b_nat_b2 @ X @ A2 )
=> ( ( count_nat_b_nat_b @ ( mset_set_nat_b_nat_b @ A2 ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_122_count__mset__set_I3_J,axiom,
! [X: real,A2: set_real] :
( ~ ( member_real2 @ X @ A2 )
=> ( ( count_real @ ( mset_set_real @ A2 ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_123_count__mset__set_I3_J,axiom,
! [X: int,A2: set_int] :
( ~ ( member_int2 @ X @ A2 )
=> ( ( count_int @ ( mset_set_int @ A2 ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_124_count__mset__set_I3_J,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ( count_nat @ ( mset_set_nat @ A2 ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(3)
thf(fact_125_sorted__list__of__mset__set,axiom,
! [A2: set_nat] :
( ( linord3047872887403683810et_nat @ ( mset_set_nat @ A2 ) )
= ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_mset_set
thf(fact_126_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_127_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_128_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_129_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_130_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_131_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_132_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_133_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_134_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P2 @ X3 @ Y2 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_135_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_136_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_137_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_138_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_139_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_140_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_141_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_142_lessThan__eq__iff,axiom,
! [X: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_143_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_144_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N2: nat] :
( ~ ( P2 @ N2 )
& ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_145_count__mset__gt__0,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_nat_b_nat_b @ ( mset_nat_b_nat_b @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_146_count__mset__gt__0,axiom,
! [X: int,Xs: list_int] :
( ( member_int2 @ X @ ( set_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_int @ ( mset_int @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_147_count__mset__gt__0,axiom,
! [X: real,Xs: list_real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_real @ ( mset_real @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_148_count__mset__gt__0,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_nat @ ( mset_nat @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_149_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_150_count__mset__set_I2_J,axiom,
! [A2: set_int,X: int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( count_int @ ( mset_set_int @ A2 ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(2)
thf(fact_151_count__mset__set_I2_J,axiom,
! [A2: set_nat,X: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( count_nat @ ( mset_set_nat @ A2 ) @ X )
= zero_zero_nat ) ) ).
% count_mset_set(2)
thf(fact_152_atMost__eq__iff,axiom,
! [X: nat,Y3: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y3 ) )
= ( X = Y3 ) ) ).
% atMost_eq_iff
thf(fact_153_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_154_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_155_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_156_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_157_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_158_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_159_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_160_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_161_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_162_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_163_lessThan__iff,axiom,
! [I2: ( nat > b ) > nat > b,K: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ I2 @ ( set_or7136064339709882498_nat_b @ K ) )
= ( ord_less_nat_b_nat_b @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_164_lessThan__iff,axiom,
! [I2: int,K: int] :
( ( member_int2 @ I2 @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_165_lessThan__iff,axiom,
! [I2: real,K: real] :
( ( member_real2 @ I2 @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_166_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat2 @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_167_List_Ofinite__set,axiom,
! [Xs: list_nat_b_nat_b] : ( finite1660923644538309244_nat_b @ ( set_nat_b_nat_b2 @ Xs ) ) ).
% List.finite_set
thf(fact_168_List_Ofinite__set,axiom,
! [Xs: list_real] : ( finite_finite_real @ ( set_real2 @ Xs ) ) ).
% List.finite_set
thf(fact_169_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_170_List_Ofinite__set,axiom,
! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% List.finite_set
thf(fact_171_mset__set__eq__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( mset_set_int @ A2 )
= ( mset_set_int @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mset_set_eq_iff
thf(fact_172_mset__set__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( mset_set_nat @ A2 )
= ( mset_set_nat @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mset_set_eq_iff
thf(fact_173_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_174_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_175_infinite__Ico__iff,axiom,
! [A: real,B: real] :
( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) )
= ( ord_less_real @ A @ B ) ) ).
% infinite_Ico_iff
thf(fact_176_mset__set_Oinfinite,axiom,
! [A2: set_int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( mset_set_int @ A2 )
= zero_z3170743180189231877et_int ) ) ).
% mset_set.infinite
thf(fact_177_mset__set_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( mset_set_nat @ A2 )
= zero_z7348594199698428585et_nat ) ) ).
% mset_set.infinite
thf(fact_178_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( set_real2 @ ( linord4252657396651189596t_real @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_179_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( set_int2 @ ( linord2612477271533052124et_int @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_180_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_181_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_182_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_183_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_184_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_185_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_186_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P2 @ M4 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_187_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P2 @ M4 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_188_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_189_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_190_lessThan__strict__subset__iff,axiom,
! [M2: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_191_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_192_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M5: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_193_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_194_infinite__Iic,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).
% infinite_Iic
thf(fact_195_infinite__Ico,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) ) ).
% infinite_Ico
thf(fact_196_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_197_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_198_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_199_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_200_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_201_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_202_atLeastLessThan__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_203_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_204_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_205_Ico__eq__Ico,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or66887138388493659n_real @ L @ H )
= ( set_or66887138388493659n_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_real @ L @ H )
& ~ ( ord_less_real @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_206_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_207_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_208_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_209_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_210_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_211_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_212_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_213_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_214_infinite__Iio,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% infinite_Iio
thf(fact_215_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_216_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_217_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_218_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_219_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P2 @ M4 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_220_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_221_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_222_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_223_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_224_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_225_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_226_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_227_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_228_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P2 @ I3 @ J3 )
=> ( ( P2 @ J3 @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_229_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_230_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_231_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_232_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_233_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_234_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_235_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_236_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_237_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_238_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_239_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_240_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_241_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_242_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_243_finite__list,axiom,
! [A2: set_nat_b_nat_b] :
( ( finite1660923644538309244_nat_b @ A2 )
=> ? [Xs3: list_nat_b_nat_b] :
( ( set_nat_b_nat_b2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_244_finite__list,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ? [Xs3: list_real] :
( ( set_real2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_245_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_246_finite__list,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ? [Xs3: list_int] :
( ( set_int2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_247_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_248_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_int,B2: set_int] :
( ( ( linord2612477271533052124et_int @ A2 )
= ( linord2612477271533052124et_int @ B2 ) )
=> ( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_249_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B2 ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_250_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_251_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_252_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_253_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_254_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_255_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P2 @ M5 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_256_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ( P2 @ M5 ) ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P2 @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_257_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_258_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_259_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_260_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_261_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_262_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_263_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_264_count__mset__set_H,axiom,
! [A2: set_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( ( ( finite1660923644538309244_nat_b @ A2 )
& ( member_nat_b_nat_b2 @ X @ A2 ) )
=> ( ( count_nat_b_nat_b @ ( mset_set_nat_b_nat_b @ A2 ) @ X )
= one_one_nat ) )
& ( ~ ( ( finite1660923644538309244_nat_b @ A2 )
& ( member_nat_b_nat_b2 @ X @ A2 ) )
=> ( ( count_nat_b_nat_b @ ( mset_set_nat_b_nat_b @ A2 ) @ X )
= zero_zero_nat ) ) ) ).
% count_mset_set'
thf(fact_265_count__mset__set_H,axiom,
! [A2: set_real,X: real] :
( ( ( ( finite_finite_real @ A2 )
& ( member_real2 @ X @ A2 ) )
=> ( ( count_real @ ( mset_set_real @ A2 ) @ X )
= one_one_nat ) )
& ( ~ ( ( finite_finite_real @ A2 )
& ( member_real2 @ X @ A2 ) )
=> ( ( count_real @ ( mset_set_real @ A2 ) @ X )
= zero_zero_nat ) ) ) ).
% count_mset_set'
thf(fact_266_count__mset__set_H,axiom,
! [A2: set_int,X: int] :
( ( ( ( finite_finite_int @ A2 )
& ( member_int2 @ X @ A2 ) )
=> ( ( count_int @ ( mset_set_int @ A2 ) @ X )
= one_one_nat ) )
& ( ~ ( ( finite_finite_int @ A2 )
& ( member_int2 @ X @ A2 ) )
=> ( ( count_int @ ( mset_set_int @ A2 ) @ X )
= zero_zero_nat ) ) ) ).
% count_mset_set'
thf(fact_267_count__mset__set_H,axiom,
! [A2: set_nat,X: nat] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat2 @ X @ A2 ) )
=> ( ( count_nat @ ( mset_set_nat @ A2 ) @ X )
= one_one_nat ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat2 @ X @ A2 ) )
=> ( ( count_nat @ ( mset_set_nat @ A2 ) @ X )
= zero_zero_nat ) ) ) ).
% count_mset_set'
thf(fact_268_count__mset__set_I1_J,axiom,
! [A2: set_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( finite1660923644538309244_nat_b @ A2 )
=> ( ( member_nat_b_nat_b2 @ X @ A2 )
=> ( ( count_nat_b_nat_b @ ( mset_set_nat_b_nat_b @ A2 ) @ X )
= one_one_nat ) ) ) ).
% count_mset_set(1)
thf(fact_269_count__mset__set_I1_J,axiom,
! [A2: set_real,X: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real2 @ X @ A2 )
=> ( ( count_real @ ( mset_set_real @ A2 ) @ X )
= one_one_nat ) ) ) ).
% count_mset_set(1)
thf(fact_270_count__mset__set_I1_J,axiom,
! [A2: set_int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int2 @ X @ A2 )
=> ( ( count_int @ ( mset_set_int @ A2 ) @ X )
= one_one_nat ) ) ) ).
% count_mset_set(1)
thf(fact_271_count__mset__set_I1_J,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( count_nat @ ( mset_set_nat @ A2 ) @ X )
= one_one_nat ) ) ) ).
% count_mset_set(1)
thf(fact_272_harm__pos__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% harm_pos_iff
thf(fact_273_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_274_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > int,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_275_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > real,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_276_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_277_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_278_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_279_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_280_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_281_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_282_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_283_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_284_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_285_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_286_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_287_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_288_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_289_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_290_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_291_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_292_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_293_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_294_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_295_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_296_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_297_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_298_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_299_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_300_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_301_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_302_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_303_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_304_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_305_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_306_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_307_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_308_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_309_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_310_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_311_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_312_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_313_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_314_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_315_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_316_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_317_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_318_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_319_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_320_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_321_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_322_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_323_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_324_psubsetD,axiom,
! [A2: set_nat_b_nat_b,B2: set_nat_b_nat_b,C: ( nat > b ) > nat > b] :
( ( ord_le4513651211808907375_nat_b @ A2 @ B2 )
=> ( ( member_nat_b_nat_b2 @ C @ A2 )
=> ( member_nat_b_nat_b2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_325_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_326_psubsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ( ( member_real2 @ C @ A2 )
=> ( member_real2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_327_psubsetD,axiom,
! [A2: set_int,B2: set_int,C: int] :
( ( ord_less_set_int @ A2 @ B2 )
=> ( ( member_int2 @ C @ A2 )
=> ( member_int2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_328_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_329_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_330_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_331_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_332_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_333_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_334_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_335_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_336_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_337_diff__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_338_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_339_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_340_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_341_finite__psubset__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B3: set_nat] :
( ( ord_less_set_nat @ B3 @ A4 )
=> ( P2 @ B3 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_342_finite__psubset__induct,axiom,
! [A2: set_int,P2: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ! [B3: set_int] :
( ( ord_less_set_int @ B3 @ A4 )
=> ( P2 @ B3 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_343_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_344_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_345_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_346_harm__expand_I2_J,axiom,
( ( harmonic_harm_real @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
% harm_expand(2)
thf(fact_347_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_348_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_349_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A5: real,B4: real] :
( ( minus_minus_real @ A5 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_350_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_351_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_352_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_353_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_354_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_355_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_356_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_357_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_358_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_359_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_360_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I2: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_361_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_362_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_363_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_364_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_365_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_366_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_367_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_368_harm__expand_I1_J,axiom,
( ( harmonic_harm_real @ zero_zero_nat )
= zero_zero_real ) ).
% harm_expand(1)
thf(fact_369_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_370_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_371_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_372_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_373_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_374_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_375_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_376_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_377_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_378_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_379_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_380_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_381_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_382_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_383_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_384_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_385_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_386_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_387_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_388_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_389_harm__pos,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N ) ) ) ).
% harm_pos
thf(fact_390_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ F2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_391_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > nat] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X2: int] :
( ( member_int2 @ X2 @ F2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_392_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_393_sum_Oinfinite,axiom,
! [A2: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_394_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_395_sum_Oinfinite,axiom,
! [A2: set_int,G: int > int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_396_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_397_sum_Oinfinite,axiom,
! [A2: set_int,G: int > real] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups8778361861064173332t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_398_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_399_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_int @ one_one_int @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_400_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_401_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_402_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_403_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_404_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_405_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_406_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_407_unbounded__k__infinite,axiom,
! [K: nat,S3: set_nat] :
( ! [M3: nat] :
( ( ord_less_nat @ K @ M3 )
=> ? [N7: nat] :
( ( ord_less_nat @ M3 @ N7 )
& ( member_nat2 @ N7 @ S3 ) ) )
=> ~ ( finite_finite_nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_408_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_409_finite__Diff2,axiom,
! [B2: set_int,A2: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
= ( finite_finite_int @ A2 ) ) ) ).
% finite_Diff2
thf(fact_410_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_411_finite__Diff,axiom,
! [A2: set_int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_412_lessThan__minus__lessThan,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M2 ) )
= ( set_or4665077453230672383an_nat @ M2 @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_413_lessThan__minus__lessThan,axiom,
! [N: int,M2: int] :
( ( minus_minus_set_int @ ( set_ord_lessThan_int @ N ) @ ( set_ord_lessThan_int @ M2 ) )
= ( set_or4662586982721622107an_int @ M2 @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_414_Diff__infinite__finite,axiom,
! [T2: set_nat,S3: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_415_Diff__infinite__finite,axiom,
! [T2: set_int,S3: set_int] :
( ( finite_finite_int @ T2 )
=> ( ~ ( finite_finite_int @ S3 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_416_psubset__imp__ex__mem,axiom,
! [A2: set_nat_b_nat_b,B2: set_nat_b_nat_b] :
( ( ord_le4513651211808907375_nat_b @ A2 @ B2 )
=> ? [B5: ( nat > b ) > nat > b] : ( member_nat_b_nat_b2 @ B5 @ ( minus_2952184192894347252_nat_b @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_417_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B5: nat] : ( member_nat2 @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_418_psubset__imp__ex__mem,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ? [B5: real] : ( member_real2 @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_419_psubset__imp__ex__mem,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_set_int @ A2 @ B2 )
=> ? [B5: int] : ( member_int2 @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_420_sum__eq__Suc0__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( ( F @ X2 )
= ( suc @ zero_zero_nat ) )
& ! [Y: nat] :
( ( member_nat2 @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_421_sum__eq__Suc0__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: int] :
( ( member_int2 @ X2 @ A2 )
& ( ( F @ X2 )
= ( suc @ zero_zero_nat ) )
& ! [Y: int] :
( ( member_int2 @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_422_sum__eq__1__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= one_one_nat )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y: nat] :
( ( member_nat2 @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_423_sum__eq__1__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= one_one_nat )
= ( ? [X2: int] :
( ( member_int2 @ X2 @ A2 )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y: int] :
( ( member_int2 @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_424_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_real,S3: set_real,I2: real > real,J2: real > real,T2: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_real @ T3 )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( member_real2 @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T2 @ T3 ) ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( member_real2 @ ( I2 @ B5 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S3 )
= ( groups1935376822645274424al_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_425_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_nat,S3: set_real,I2: nat > real,J2: real > nat,T2: set_nat,G: real > nat,H: nat > nat] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( member_nat2 @ ( J2 @ A6 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_real2 @ ( I2 @ B5 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S3 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_426_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_int,S3: set_real,I2: int > real,J2: real > int,T2: set_int,G: real > nat,H: int > nat] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( member_int2 @ ( J2 @ A6 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_real2 @ ( I2 @ B5 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S3 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_427_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_real,S3: set_nat,I2: real > nat,J2: nat > real,T2: set_real,G: nat > nat,H: real > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_real @ T3 )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member_real2 @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T2 @ T3 ) ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( member_nat2 @ ( I2 @ B5 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S3 )
= ( groups1935376822645274424al_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_428_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_nat,S3: set_nat,I2: nat > nat,J2: nat > nat,T2: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member_nat2 @ ( J2 @ A6 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_nat2 @ ( I2 @ B5 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S3 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_429_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T3: set_int,S3: set_nat,I2: int > nat,J2: nat > int,T2: set_int,G: nat > nat,H: int > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member_int2 @ ( J2 @ A6 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_nat2 @ ( I2 @ B5 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: nat] :
( ( member_nat2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S3 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_430_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_real,S3: set_int,I2: real > int,J2: int > real,T2: set_real,G: int > nat,H: real > nat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_real @ T3 )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( member_real2 @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T2 @ T3 ) ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( member_int2 @ ( I2 @ B5 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S3 )
= ( groups1935376822645274424al_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_431_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_nat,S3: set_int,I2: nat > int,J2: int > nat,T2: set_nat,G: int > nat,H: nat > nat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( member_nat2 @ ( J2 @ A6 ) @ ( minus_minus_set_nat @ T2 @ T3 ) ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ T2 @ T3 ) )
=> ( member_int2 @ ( I2 @ B5 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S3 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_432_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_int,S3: set_int,I2: int > int,J2: int > int,T2: set_int,G: int > nat,H: int > nat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( member_int2 @ ( J2 @ A6 ) @ ( minus_minus_set_int @ T2 @ T3 ) ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ T2 @ T3 ) )
=> ( member_int2 @ ( I2 @ B5 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_nat ) )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_nat ) )
=> ( ! [A6: int] :
( ( member_int2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G @ S3 )
= ( groups4541462559716669496nt_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_433_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_real,S3: set_real,I2: real > real,J2: real > real,T2: set_real,G: real > int,H: real > int] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_real @ T3 )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( ( I2 @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( member_real2 @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T2 @ T3 ) ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ T2 @ T3 ) )
=> ( member_real2 @ ( I2 @ B5 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S4 )
=> ( ( G @ A6 )
= zero_zero_int ) )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ T3 )
=> ( ( H @ B5 )
= zero_zero_int ) )
=> ( ! [A6: real] :
( ( member_real2 @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ S3 )
= ( groups1932886352136224148al_int @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_434_minus__set__fold,axiom,
! [A2: set_nat_b_nat_b,Xs: list_nat_b_nat_b] :
( ( minus_2952184192894347252_nat_b @ A2 @ ( set_nat_b_nat_b2 @ Xs ) )
= ( fold_n5526574726712445965_nat_b @ remove_nat_b_nat_b @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_435_minus__set__fold,axiom,
! [A2: set_nat,Xs: list_nat] :
( ( minus_minus_set_nat @ A2 @ ( set_nat2 @ Xs ) )
= ( fold_nat_set_nat @ remove_nat @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_436_minus__set__fold,axiom,
! [A2: set_int,Xs: list_int] :
( ( minus_minus_set_int @ A2 @ ( set_int2 @ Xs ) )
= ( fold_int_set_int @ remove_int @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_437_minus__set__fold,axiom,
! [A2: set_real,Xs: list_real] :
( ( minus_minus_set_real @ A2 @ ( set_real2 @ Xs ) )
= ( fold_real_set_real @ remove_real @ Xs @ A2 ) ) ).
% minus_set_fold
thf(fact_438_linorder__neqE__linordered__idom,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
=> ( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_439_linorder__neqE__linordered__idom,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
=> ( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_440_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_441_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_442_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_443_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_444_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A6: real] :
( ( member_real2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_445_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A2: set_int] :
( ( ( groups4541462559716669496nt_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A6: int] :
( ( member_int2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_446_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A2: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_447_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A2: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A6: real] :
( ( member_real2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_448_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > int,A2: set_int] :
( ( ( groups4538972089207619220nt_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A6: int] :
( ( member_int2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_449_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_450_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A6: real] :
( ( member_real2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_451_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A2: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A6: int] :
( ( member_int2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_452_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: ( ( nat > b ) > nat > b ) > nat,A2: set_nat_b_nat_b] :
( ( ( groups3089649643242846961_b_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A6: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ A6 @ A2 )
=> ( ( G @ A6 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_453_infinite__nat__iff__unbounded,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M5: nat] :
? [N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ( member_nat2 @ N3 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_454_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_455_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_456_sum_Oop__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_457_sum_Oop__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_458_sum_Oop__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_459_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_460_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_461_last__upt,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( last_nat @ ( upt @ I2 @ J2 ) )
= ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).
% last_upt
thf(fact_462_nat__approx__posE,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_463_distinct__count__atmost__1,axiom,
( distinct_nat_b_nat_b
= ( ^ [X2: list_nat_b_nat_b] :
! [A5: ( nat > b ) > nat > b] :
( ( ( member_nat_b_nat_b2 @ A5 @ ( set_nat_b_nat_b2 @ X2 ) )
=> ( ( count_nat_b_nat_b @ ( mset_nat_b_nat_b @ X2 ) @ A5 )
= one_one_nat ) )
& ( ~ ( member_nat_b_nat_b2 @ A5 @ ( set_nat_b_nat_b2 @ X2 ) )
=> ( ( count_nat_b_nat_b @ ( mset_nat_b_nat_b @ X2 ) @ A5 )
= zero_zero_nat ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_464_distinct__count__atmost__1,axiom,
( distinct_int
= ( ^ [X2: list_int] :
! [A5: int] :
( ( ( member_int2 @ A5 @ ( set_int2 @ X2 ) )
=> ( ( count_int @ ( mset_int @ X2 ) @ A5 )
= one_one_nat ) )
& ( ~ ( member_int2 @ A5 @ ( set_int2 @ X2 ) )
=> ( ( count_int @ ( mset_int @ X2 ) @ A5 )
= zero_zero_nat ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_465_distinct__count__atmost__1,axiom,
( distinct_real
= ( ^ [X2: list_real] :
! [A5: real] :
( ( ( member_real2 @ A5 @ ( set_real2 @ X2 ) )
=> ( ( count_real @ ( mset_real @ X2 ) @ A5 )
= one_one_nat ) )
& ( ~ ( member_real2 @ A5 @ ( set_real2 @ X2 ) )
=> ( ( count_real @ ( mset_real @ X2 ) @ A5 )
= zero_zero_nat ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_466_distinct__count__atmost__1,axiom,
( distinct_nat
= ( ^ [X2: list_nat] :
! [A5: nat] :
( ( ( member_nat2 @ A5 @ ( set_nat2 @ X2 ) )
=> ( ( count_nat @ ( mset_nat @ X2 ) @ A5 )
= one_one_nat ) )
& ( ~ ( member_nat2 @ A5 @ ( set_nat2 @ X2 ) )
=> ( ( count_nat @ ( mset_nat @ X2 ) @ A5 )
= zero_zero_nat ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_467_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_468_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_469_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_470_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_471_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_472_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_473_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_474_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_475_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_476_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_477_Diff__iff,axiom,
! [C: ( nat > b ) > nat > b,A2: set_nat_b_nat_b,B2: set_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ C @ ( minus_2952184192894347252_nat_b @ A2 @ B2 ) )
= ( ( member_nat_b_nat_b2 @ C @ A2 )
& ~ ( member_nat_b_nat_b2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_478_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat2 @ C @ A2 )
& ~ ( member_nat2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_479_Diff__iff,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( ( member_real2 @ C @ A2 )
& ~ ( member_real2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_480_Diff__iff,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int2 @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
= ( ( member_int2 @ C @ A2 )
& ~ ( member_int2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_481_DiffI,axiom,
! [C: ( nat > b ) > nat > b,A2: set_nat_b_nat_b,B2: set_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ C @ A2 )
=> ( ~ ( member_nat_b_nat_b2 @ C @ B2 )
=> ( member_nat_b_nat_b2 @ C @ ( minus_2952184192894347252_nat_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_482_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ A2 )
=> ( ~ ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_483_DiffI,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real2 @ C @ A2 )
=> ( ~ ( member_real2 @ C @ B2 )
=> ( member_real2 @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_484_DiffI,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int2 @ C @ A2 )
=> ( ~ ( member_int2 @ C @ B2 )
=> ( member_int2 @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_485_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_486_image__mset__union,axiom,
! [F: nat > b,M: multiset_nat,N5: multiset_nat] :
( ( image_mset_nat_b @ F @ ( plus_p6334493942879108393et_nat @ M @ N5 ) )
= ( plus_plus_multiset_b @ ( image_mset_nat_b @ F @ M ) @ ( image_mset_nat_b @ F @ N5 ) ) ) ).
% image_mset_union
thf(fact_487_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_488_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_489_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_490_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_491_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_492_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_493_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_494_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_495_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_496_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_497_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_498_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_499_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_500_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_501_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_502_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_503_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_504_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_505_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_506_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_507_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_508_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_509_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_510_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_511_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_512_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_513_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_514_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_515_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_516_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_517_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_518_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_519_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_520_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_521_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_522_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_523_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_524_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_525_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_526_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_527_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_528_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_529_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_530_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_531_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_532_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_533_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_534_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_535_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_536_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_537_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_538_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_539_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_540_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_541_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_542_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_543_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_544_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_545_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_546_distinct__rotate,axiom,
! [N: nat,Xs: list_nat_b_nat_b] :
( ( distinct_nat_b_nat_b @ ( rotate_nat_b_nat_b @ N @ Xs ) )
= ( distinct_nat_b_nat_b @ Xs ) ) ).
% distinct_rotate
thf(fact_547_distinct__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( distinct_nat @ ( rotate_nat @ N @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_rotate
thf(fact_548_distinct1__rotate,axiom,
! [Xs: list_nat_b_nat_b] :
( ( distinct_nat_b_nat_b @ ( rotate1_nat_b_nat_b @ Xs ) )
= ( distinct_nat_b_nat_b @ Xs ) ) ).
% distinct1_rotate
thf(fact_549_distinct1__rotate,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct1_rotate
thf(fact_550_distinct__insert,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( distinct_nat_b_nat_b @ ( insert_nat_b_nat_b @ X @ Xs ) )
= ( distinct_nat_b_nat_b @ Xs ) ) ).
% distinct_insert
thf(fact_551_distinct__insert,axiom,
! [X: int,Xs: list_int] :
( ( distinct_int @ ( insert_int @ X @ Xs ) )
= ( distinct_int @ Xs ) ) ).
% distinct_insert
thf(fact_552_distinct__insert,axiom,
! [X: real,Xs: list_real] :
( ( distinct_real @ ( insert_real @ X @ Xs ) )
= ( distinct_real @ Xs ) ) ).
% distinct_insert
thf(fact_553_distinct__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( insert_nat @ X @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_insert
thf(fact_554_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_555_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_556_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_557_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_558_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_559_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_560_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_561_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_562_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_563_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_564_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_565_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_566_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_567_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_568_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_569_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_570_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_571_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_572_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_573_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_574_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_575_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_576_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_577_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_578_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_579_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_580_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_581_sum_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_582_sum_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_583_sum_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_584_sum_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_585_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_586_int__less__induct,axiom,
! [I2: int,K: int,P2: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_587_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_588_int__gr__induct,axiom,
! [K: int,I2: int,P2: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_589_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_590_DiffD2,axiom,
! [C: ( nat > b ) > nat > b,A2: set_nat_b_nat_b,B2: set_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ C @ ( minus_2952184192894347252_nat_b @ A2 @ B2 ) )
=> ~ ( member_nat_b_nat_b2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_591_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_592_DiffD2,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( member_real2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_593_DiffD2,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int2 @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ~ ( member_int2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_594_DiffD1,axiom,
! [C: ( nat > b ) > nat > b,A2: set_nat_b_nat_b,B2: set_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ C @ ( minus_2952184192894347252_nat_b @ A2 @ B2 ) )
=> ( member_nat_b_nat_b2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_595_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_596_DiffD1,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ( member_real2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_597_DiffD1,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int2 @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ( member_int2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_598_DiffE,axiom,
! [C: ( nat > b ) > nat > b,A2: set_nat_b_nat_b,B2: set_nat_b_nat_b] :
( ( member_nat_b_nat_b2 @ C @ ( minus_2952184192894347252_nat_b @ A2 @ B2 ) )
=> ~ ( ( member_nat_b_nat_b2 @ C @ A2 )
=> ( member_nat_b_nat_b2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_599_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_600_DiffE,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real2 @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real2 @ C @ A2 )
=> ( member_real2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_601_DiffE,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int2 @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ~ ( ( member_int2 @ C @ A2 )
=> ( member_int2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_602_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_603_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_604_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_605_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_606_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_607_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_608_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_609_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_610_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_611_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_612_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_613_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_614_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_615_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_616_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_617_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_618_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_619_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_620_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_621_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_622_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_623_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_624_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_625_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_626_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_627_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_628_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_629_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_630_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_631_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_632_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_633_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_634_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_635_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_636_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_637_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_638_zadd__int__left,axiom,
! [M2: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_639_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_640_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_641_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_642_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_643_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_644_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_645_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_646_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_647_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_648_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_649_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_650_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_651_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_652_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_653_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_654_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_655_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_656_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_657_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_658_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_659_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_660_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_661_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_662_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_663_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_664_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_665_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_666_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_667_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_668_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_669_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_670_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_671_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_672_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( I2 = J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_673_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_674_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_675_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_676_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_677_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_678_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_679_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_680_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_681_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_682_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_683_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_684_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_685_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_686_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_687_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_688_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_689_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_690_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_691_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_692_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_693_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_694_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_695_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_696_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_697_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_698_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_699_mset__eq__imp__distinct__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
= ( distinct_nat @ Ys ) ) ) ).
% mset_eq_imp_distinct_iff
thf(fact_700_distinct__upt,axiom,
! [I2: nat,J2: nat] : ( distinct_nat @ ( upt @ I2 @ J2 ) ) ).
% distinct_upt
thf(fact_701_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_702_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_703_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_704_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_705_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_706_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_707_trans__less__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_708_trans__less__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_709_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_710_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_711_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_712_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_713_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_714_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_715_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_716_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_717_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_718_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( distinct_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_719_image__mset__eq__plusD,axiom,
! [F: nat > b,A2: multiset_nat,B2: multiset_b,C2: multiset_b] :
( ( ( image_mset_nat_b @ F @ A2 )
= ( plus_plus_multiset_b @ B2 @ C2 ) )
=> ? [B6: multiset_nat,C3: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ B6 @ C3 ) )
& ( B2
= ( image_mset_nat_b @ F @ B6 ) )
& ( C2
= ( image_mset_nat_b @ F @ C3 ) ) ) ) ).
% image_mset_eq_plusD
thf(fact_720_distinct__removeAll,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_721_distinct__removeAll,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( distinct_nat_b_nat_b @ Xs )
=> ( distinct_nat_b_nat_b @ ( remove7346898498583989977_nat_b @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_722_distinct__removeAll,axiom,
! [Xs: list_int,X: int] :
( ( distinct_int @ Xs )
=> ( distinct_int @ ( removeAll_int @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_723_distinct__removeAll,axiom,
! [Xs: list_real,X: real] :
( ( distinct_real @ Xs )
=> ( distinct_real @ ( removeAll_real @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_724_rotate__rotate,axiom,
! [M2: nat,N: nat,Xs: list_nat] :
( ( rotate_nat @ M2 @ ( rotate_nat @ N @ Xs ) )
= ( rotate_nat @ ( plus_plus_nat @ M2 @ N ) @ Xs ) ) ).
% rotate_rotate
thf(fact_725_rotate__rotate,axiom,
! [M2: nat,N: nat,Xs: list_nat_b_nat_b] :
( ( rotate_nat_b_nat_b @ M2 @ ( rotate_nat_b_nat_b @ N @ Xs ) )
= ( rotate_nat_b_nat_b @ ( plus_plus_nat @ M2 @ N ) @ Xs ) ) ).
% rotate_rotate
thf(fact_726_finite__distinct__list,axiom,
! [A2: set_nat_b_nat_b] :
( ( finite1660923644538309244_nat_b @ A2 )
=> ? [Xs3: list_nat_b_nat_b] :
( ( ( set_nat_b_nat_b2 @ Xs3 )
= A2 )
& ( distinct_nat_b_nat_b @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_727_finite__distinct__list,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ? [Xs3: list_real] :
( ( ( set_real2 @ Xs3 )
= A2 )
& ( distinct_real @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_728_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A2 )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_729_finite__distinct__list,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ? [Xs3: list_int] :
( ( ( set_int2 @ Xs3 )
= A2 )
& ( distinct_int @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_730_set__eq__iff__mset__eq__distinct,axiom,
! [X: list_nat_b_nat_b,Y3: list_nat_b_nat_b] :
( ( distinct_nat_b_nat_b @ X )
=> ( ( distinct_nat_b_nat_b @ Y3 )
=> ( ( ( set_nat_b_nat_b2 @ X )
= ( set_nat_b_nat_b2 @ Y3 ) )
= ( ( mset_nat_b_nat_b @ X )
= ( mset_nat_b_nat_b @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_731_set__eq__iff__mset__eq__distinct,axiom,
! [X: list_int,Y3: list_int] :
( ( distinct_int @ X )
=> ( ( distinct_int @ Y3 )
=> ( ( ( set_int2 @ X )
= ( set_int2 @ Y3 ) )
= ( ( mset_int @ X )
= ( mset_int @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_732_set__eq__iff__mset__eq__distinct,axiom,
! [X: list_real,Y3: list_real] :
( ( distinct_real @ X )
=> ( ( distinct_real @ Y3 )
=> ( ( ( set_real2 @ X )
= ( set_real2 @ Y3 ) )
= ( ( mset_real @ X )
= ( mset_real @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_733_set__eq__iff__mset__eq__distinct,axiom,
! [X: list_nat,Y3: list_nat] :
( ( distinct_nat @ X )
=> ( ( distinct_nat @ Y3 )
=> ( ( ( set_nat2 @ X )
= ( set_nat2 @ Y3 ) )
= ( ( mset_nat @ X )
= ( mset_nat @ Y3 ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_734_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_735_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_736_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_737_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_738_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_739_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_740_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_741_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_742_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_743_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_744_add__less__zeroD,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y3 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_745_add__less__zeroD,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_746_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_747_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_748_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_749_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_750_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_751_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_752_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_753_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_754_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_755_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_756_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_757_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_758_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_759_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_760_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_761_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_762_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_763_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_764_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_765_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_766_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_767_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_768_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_769_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_770_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_771_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_772_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_773_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_774_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_775_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_776_sum_Orelated,axiom,
! [R: nat > nat > $o,S3: set_nat,H: nat > nat,G: nat > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ S3 )
=> ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3542108847815614940at_nat @ H @ S3 ) @ ( groups3542108847815614940at_nat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_777_sum_Orelated,axiom,
! [R: nat > nat > $o,S3: set_int,H: int > nat,G: int > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int2 @ X3 @ S3 )
=> ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups4541462559716669496nt_nat @ H @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_778_sum_Orelated,axiom,
! [R: int > int > $o,S3: set_nat,H: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X23: int,Y23: int] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ S3 )
=> ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_779_sum_Orelated,axiom,
! [R: int > int > $o,S3: set_int,H: int > int,G: int > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X23: int,Y23: int] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int2 @ X3 @ S3 )
=> ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups4538972089207619220nt_int @ H @ S3 ) @ ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_780_sum_Orelated,axiom,
! [R: real > real > $o,S3: set_nat,H: nat > real,G: nat > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X23: real,Y23: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ S3 )
=> ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups6591440286371151544t_real @ H @ S3 ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_781_sum_Orelated,axiom,
! [R: real > real > $o,S3: set_int,H: int > real,G: int > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X23: real,Y23: real] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int2 @ X3 @ S3 )
=> ( R @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups8778361861064173332t_real @ H @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_782_mset__set__set,axiom,
! [Xs: list_nat_b_nat_b] :
( ( distinct_nat_b_nat_b @ Xs )
=> ( ( mset_set_nat_b_nat_b @ ( set_nat_b_nat_b2 @ Xs ) )
= ( mset_nat_b_nat_b @ Xs ) ) ) ).
% mset_set_set
thf(fact_783_mset__set__set,axiom,
! [Xs: list_int] :
( ( distinct_int @ Xs )
=> ( ( mset_set_int @ ( set_int2 @ Xs ) )
= ( mset_int @ Xs ) ) ) ).
% mset_set_set
thf(fact_784_mset__set__set,axiom,
! [Xs: list_real] :
( ( distinct_real @ Xs )
=> ( ( mset_set_real @ ( set_real2 @ Xs ) )
= ( mset_real @ Xs ) ) ) ).
% mset_set_set
thf(fact_785_mset__set__set,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( mset_set_nat @ ( set_nat2 @ Xs ) )
= ( mset_nat @ Xs ) ) ) ).
% mset_set_set
thf(fact_786_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_787_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_788_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_789_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_790_sum_OatLeast0__lessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_791_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_792_sum_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_793_sum_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_794_sum_OatLeast__Suc__lessThan,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_795_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_796_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_797_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_798_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_799_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_800_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_801_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_802_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_803_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_804_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_805_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_806_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_807_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_808_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_809_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_810_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_811_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_812_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_813_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_814_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_815_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_12: real] : ( ord_less_real @ X5 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_816_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X5 ) ).
% linordered_field_no_lb
thf(fact_817_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_818_divide__neg__neg,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y3 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_neg_neg
thf(fact_819_divide__neg__pos,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_820_divide__pos__neg,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y3 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_821_divide__pos__pos,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_pos_pos
thf(fact_822_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_823_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_824_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_825_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_826_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_827_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_828_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_829_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_830_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_831_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_832_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_833_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_834_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_835_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_836_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_837_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_838_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_839_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_840_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_841_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_842_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_843_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_844_zdiv__int,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_845_int__if,axiom,
! [P2: $o,A: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_846_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_847_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_848_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_849_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_850_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_851_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_852_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_853_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_854_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_855_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_856_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_857_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_858_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_859_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_860_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_861_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_862_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_863_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_864_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_865_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_866_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_867_div__if,axiom,
( divide_divide_nat
= ( ^ [M5: nat,N3: nat] :
( if_nat
@ ( ( ord_less_nat @ M5 @ N3 )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_868_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_869_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_870_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_871_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_872_int__power__div__base,axiom,
! [M2: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M2 ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_873_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B5: nat] :
( ( P2 @ A6 @ B5 )
= ( P2 @ B5 @ A6 ) )
=> ( ! [A6: nat] : ( P2 @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B5: nat] :
( ( P2 @ A6 @ B5 )
=> ( P2 @ A6 @ ( plus_plus_nat @ A6 @ B5 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_874_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_875_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_876_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_877_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_878_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_879_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_880_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_881_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_882_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_883_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_884_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_885_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_886_lessThan__subset__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y3 ) )
= ( ord_less_eq_int @ X @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_887_lessThan__subset__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y3 ) )
= ( ord_less_eq_real @ X @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_888_lessThan__subset__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_889_ivl__subset,axiom,
! [I2: real,J2: real,M2: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I2 @ J2 ) @ ( set_or66887138388493659n_real @ M2 @ N ) )
= ( ( ord_less_eq_real @ J2 @ I2 )
| ( ( ord_less_eq_real @ M2 @ I2 )
& ( ord_less_eq_real @ J2 @ N ) ) ) ) ).
% ivl_subset
thf(fact_890_ivl__subset,axiom,
! [I2: nat,J2: nat,M2: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ J2 @ I2 )
| ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ J2 @ N ) ) ) ) ).
% ivl_subset
thf(fact_891_ivl__subset,axiom,
! [I2: int,J2: int,M2: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I2 @ J2 ) @ ( set_or4662586982721622107an_int @ M2 @ N ) )
= ( ( ord_less_eq_int @ J2 @ I2 )
| ( ( ord_less_eq_int @ M2 @ I2 )
& ( ord_less_eq_int @ J2 @ N ) ) ) ) ).
% ivl_subset
thf(fact_892_atMost__iff,axiom,
! [I2: ( nat > b ) > nat > b,K: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ I2 @ ( set_or5633471081860273246_nat_b @ K ) )
= ( ord_le535081054522138693_nat_b @ I2 @ K ) ) ).
% atMost_iff
thf(fact_893_atMost__iff,axiom,
! [I2: int,K: int] :
( ( member_int2 @ I2 @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I2 @ K ) ) ).
% atMost_iff
thf(fact_894_atMost__iff,axiom,
! [I2: real,K: real] :
( ( member_real2 @ I2 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_895_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_896_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat2 @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_897_atMost__subset__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y3 ) )
= ( ord_less_eq_int @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_898_atMost__subset__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y3 ) )
= ( ord_less_eq_real @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_899_atMost__subset__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y3 ) )
= ( ord_less_eq_set_nat @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_900_atMost__subset__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y3 ) )
= ( ord_less_eq_nat @ X @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_901_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_902_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_903_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_904_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_905_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_906_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_907_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_908_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_909_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_910_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_911_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_912_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_913_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_914_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_915_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_916_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_917_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_918_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_919_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_920_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_921_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_922_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_923_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_924_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_925_atLeastLessThan__iff,axiom,
! [I2: ( nat > b ) > nat > b,L: ( nat > b ) > nat > b,U: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ I2 @ ( set_or4963087151523820308_nat_b @ L @ U ) )
= ( ( ord_le535081054522138693_nat_b @ L @ I2 )
& ( ord_less_nat_b_nat_b @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_926_atLeastLessThan__iff,axiom,
! [I2: real,L: real,U: real] :
( ( member_real2 @ I2 @ ( set_or66887138388493659n_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_real @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_927_atLeastLessThan__iff,axiom,
! [I2: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I2 @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I2 )
& ( ord_less_set_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_928_atLeastLessThan__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat2 @ I2 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_929_atLeastLessThan__iff,axiom,
! [I2: int,L: int,U: int] :
( ( member_int2 @ I2 @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I2 )
& ( ord_less_int @ I2 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_930_ivl__diff,axiom,
! [I2: real,N: real,M2: real] :
( ( ord_less_eq_real @ I2 @ N )
=> ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I2 @ M2 ) @ ( set_or66887138388493659n_real @ I2 @ N ) )
= ( set_or66887138388493659n_real @ N @ M2 ) ) ) ).
% ivl_diff
thf(fact_931_ivl__diff,axiom,
! [I2: nat,N: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M2 ) @ ( set_or4665077453230672383an_nat @ I2 @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M2 ) ) ) ).
% ivl_diff
thf(fact_932_ivl__diff,axiom,
! [I2: int,N: int,M2: int] :
( ( ord_less_eq_int @ I2 @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I2 @ M2 ) @ ( set_or4662586982721622107an_int @ I2 @ N ) )
= ( set_or4662586982721622107an_int @ N @ M2 ) ) ) ).
% ivl_diff
thf(fact_933_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_934_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_935_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_936_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_937_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_938_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_939_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_940_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_941_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_942_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_943_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_944_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_945_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_946_atLeastLessThan__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_947_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_948_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_949_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_950_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_951_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_952_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_953_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_954_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_955_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_956_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_957_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_958_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_959_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( I2 = J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_960_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_961_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_962_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_963_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_964_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_965_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_966_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_967_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_968_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_969_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_970_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_971_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_972_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_973_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_974_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_975_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A5 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_976_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_977_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_978_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_979_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_980_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_981_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_982_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_983_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_984_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_985_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_986_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_987_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_988_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_989_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_990_finite__has__minimal2,axiom,
! [A2: set_nat_b_nat_b,A: ( nat > b ) > nat > b] :
( ( finite1660923644538309244_nat_b @ A2 )
=> ( ( member_nat_b_nat_b2 @ A @ A2 )
=> ? [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ A2 )
& ( ord_le535081054522138693_nat_b @ X3 @ A )
& ! [Xa: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ Xa @ A2 )
=> ( ( ord_le535081054522138693_nat_b @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_991_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int2 @ A @ A2 )
=> ? [X3: int] :
( ( member_int2 @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int2 @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_992_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_993_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real2 @ A @ A2 )
=> ? [X3: real] :
( ( member_real2 @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real2 @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_994_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_995_finite__has__maximal2,axiom,
! [A2: set_nat_b_nat_b,A: ( nat > b ) > nat > b] :
( ( finite1660923644538309244_nat_b @ A2 )
=> ( ( member_nat_b_nat_b2 @ A @ A2 )
=> ? [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ A2 )
& ( ord_le535081054522138693_nat_b @ A @ X3 )
& ! [Xa: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ Xa @ A2 )
=> ( ( ord_le535081054522138693_nat_b @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_996_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int2 @ A @ A2 )
=> ? [X3: int] :
( ( member_int2 @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int2 @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_997_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_998_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real2 @ A @ A2 )
=> ? [X3: real] :
( ( member_real2 @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real2 @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_999_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1000_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_1001_ge__iff__diff__ge__0,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] : ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A5 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_1002_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A5 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_1003_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_int @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1004_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1005_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_real @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1006_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1007_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1008_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1009_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1010_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1011_harm__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_real @ ( harmonic_harm_real @ M2 ) @ ( harmonic_harm_real @ N ) ) ) ).
% harm_mono
thf(fact_1012_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_1013_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_1014_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_1015_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
= ( ord_less_int @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1016_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1017_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A7: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
= ( ord_less_real @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1018_inverse__of__nat__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_1019_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: real > int] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1020_sum__mono2,axiom,
! [B2: set_int,A2: set_int,F: int > int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1021_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: real > nat] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1022_sum__mono2,axiom,
! [B2: set_int,A2: set_int,F: int > nat] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1023_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: real > real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [B5: real] :
( ( member_real2 @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1024_sum__mono2,axiom,
! [B2: set_int,A2: set_int,F: int > real] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [B5: int] :
( ( member_int2 @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1025_sum__mono2,axiom,
! [B2: set_nat,A2: set_nat,F: nat > int] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1026_sum__mono2,axiom,
! [B2: set_nat,A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1027_sum__mono2,axiom,
! [B2: set_nat,A2: set_nat,F: nat > real] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B5: nat] :
( ( member_nat2 @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1028_sum__mono2,axiom,
! [B2: set_nat_b_nat_b,A2: set_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > int] :
( ( finite1660923644538309244_nat_b @ B2 )
=> ( ( ord_le9047053354294502011_nat_b @ A2 @ B2 )
=> ( ! [B5: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ B5 @ ( minus_2952184192894347252_nat_b @ B2 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
=> ( ord_less_eq_int @ ( groups3087159172733796685_b_int @ F @ A2 ) @ ( groups3087159172733796685_b_int @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_1029_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1030_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1031_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1032_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1033_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1034_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1035_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1036_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1037_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1038_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1039_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1040_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1041_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1042_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1043_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1044_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1045_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1046_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1047_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1048_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1049_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1050_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1051_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1052_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1053_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1054_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1055_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1056_add__nonneg__eq__0__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
=> ( ( ( plus_plus_int @ X @ Y3 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1057_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1058_add__nonneg__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1059_add__nonpos__eq__0__iff,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y3 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1060_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1061_add__nonpos__eq__0__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y3 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1062_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1063_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1064_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1065_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1066_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1067_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1068_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1069_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1070_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1071_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1072_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J2 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1073_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1074_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1075_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1076_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1077_divide__nonpos__nonpos,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1078_divide__nonpos__nonneg,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1079_divide__nonneg__nonpos,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1080_divide__nonneg__nonneg,axiom,
! [X: real,Y3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1081_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1082_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_1083_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_1084_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1085_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1086_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1087_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1088_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1089_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1090_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1091_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1092_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1093_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1094_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1095_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1096_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1097_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1098_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1099_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1100_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1101_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1102_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1103_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N: real,J2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1104_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real2 @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_1105_sum__nonpos,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int2 @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_1106_sum__nonpos,axiom,
! [A2: set_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > real] :
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b2 @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8666044050912730445b_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_1107_sum__nonpos,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_1108_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real2 @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_1109_sum__nonpos,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int2 @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_1110_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1111_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1112_int__ge__induct,axiom,
! [K: int,I2: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1113_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1114_int__le__induct,axiom,
! [I2: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1115_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1116_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1117_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1118_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1119_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1120_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1121_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
= ( ord_less_eq_int @ K @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1122_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1123_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1124_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1125_zdiv__mono2__neg,axiom,
! [A: int,B7: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B7 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1126_zdiv__mono1__neg,axiom,
! [A: int,A7: int,B: int] :
( ( ord_less_eq_int @ A @ A7 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1127_zdiv__eq__0__iff,axiom,
! [I2: int,K: int] :
( ( ( divide_divide_int @ I2 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I2 )
& ( ord_less_int @ I2 @ K ) )
| ( ( ord_less_eq_int @ I2 @ zero_zero_int )
& ( ord_less_int @ K @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1128_zdiv__mono2,axiom,
! [A: int,B7: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B7 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1129_zdiv__mono1,axiom,
! [A: int,A7: int,B: int] :
( ( ord_less_eq_int @ A @ A7 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A7 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1130_int__induct,axiom,
! [P2: int > $o,K: int,I2: int] :
( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_induct
thf(fact_1131_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1132_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power_nat @ X @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1133_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1134_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1135_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1136_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 ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1137_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1138_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1139_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1140_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1141_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1142_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1143_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1144_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1145_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1146_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1147_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1148_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1149_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1150_power__le__one__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real )
= ( ( N = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1151_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1152_trans__le__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1153_trans__le__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1154_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1155_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1156_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1157_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1158_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1159_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1160_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1161_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1162_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1163_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1164_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1165_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1166_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z2: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1167_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1168_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P2 @ M4 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1169_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1170_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1171_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1172_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1173_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1174_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1175_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1176_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1177_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1178_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1179_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1180_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1181_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1182_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1183_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M3: nat] :
( ( P2 @ M3 )
=> ~ ! [X5: nat] :
( ( P2 @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1184_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1185_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1186_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1187_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1188_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1189_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1190_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_1191_infinite__nat__iff__unbounded__le,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M5: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( member_nat2 @ N3 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1192_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M5: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1193_nat__power__less__imp__less,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1194_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_1195_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1196_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1197_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1198_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1199_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1200_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1201_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1202_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1203_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1204_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1205_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1206_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1207_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1208_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1209_inc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P2 @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1210_dec__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P2 @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) ) )
=> ( P2 @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1211_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1212_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1213_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1214_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1215_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1216_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1217_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1218_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1219_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1220_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1221_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1222_Suc__div__le__mono,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1223_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1224_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_1225_finite__nat__bounded,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_1226_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S5: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_1227_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S5: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_1228_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1229_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1230_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1231_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1232_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1233_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1234_zdiff__int__split,axiom,
! [P2: int > $o,X: nat,Y3: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y3 ) ) )
= ( ( ( ord_less_eq_nat @ Y3 @ X )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
& ( ( ord_less_nat @ X @ Y3 )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1235_kuhn__lemma,axiom,
! [P3: nat,N: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P3 )
=> ( ! [X3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_nat @ ( X3 @ I4 ) @ P3 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( ( Label @ X3 @ I3 )
= zero_zero_nat )
| ( ( Label @ X3 @ I3 )
= one_one_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_nat @ ( X3 @ I4 ) @ P3 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( ( X3 @ I3 )
= zero_zero_nat )
=> ( ( Label @ X3 @ I3 )
= zero_zero_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_nat @ ( X3 @ I4 ) @ P3 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( ( X3 @ I3 )
= P3 )
=> ( ( Label @ X3 @ I3 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_nat @ ( Q2 @ I4 ) @ P3 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ? [R2: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( R2 @ J4 ) )
& ( ord_less_eq_nat @ ( R2 @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ? [S2: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( S2 @ J4 ) )
& ( ord_less_eq_nat @ ( S2 @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I4 )
!= ( Label @ S2 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1236_complete__real,axiom,
! [S3: set_real] :
( ? [X5: real] : ( member_real2 @ X5 @ S3 )
=> ( ? [Z5: real] :
! [X3: real] :
( ( member_real2 @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real2 @ X5 @ S3 )
=> ( ord_less_eq_real @ X5 @ Y2 ) )
& ! [Z5: real] :
( ! [X3: real] :
( ( member_real2 @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ( ord_less_eq_real @ Y2 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_1237_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_1238_real__arch__pow,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1239_real__arch__pow__inv,axiom,
! [Y3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y3 ) ) ) ).
% real_arch_pow_inv
thf(fact_1240_kuhn__labelling__lemma_H,axiom,
! [P2: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P2 @ X3 )
=> ( P2 @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P2 @ X3 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) )
& ( ord_less_eq_real @ ( X3 @ I3 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L3 @ X5 @ I4 ) @ one_one_nat )
& ! [X5: nat > real,I4: nat] :
( ( ( P2 @ X5 )
& ( Q @ I4 )
& ( ( X5 @ I4 )
= zero_zero_real ) )
=> ( ( L3 @ X5 @ I4 )
= zero_zero_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P2 @ X5 )
& ( Q @ I4 )
& ( ( X5 @ I4 )
= one_one_real ) )
=> ( ( L3 @ X5 @ I4 )
= one_one_nat ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P2 @ X5 )
& ( Q @ I4 )
& ( ( L3 @ X5 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I4 ) @ ( F @ X5 @ I4 ) ) )
& ! [X5: nat > real,I4: nat] :
( ( ( P2 @ X5 )
& ( Q @ I4 )
& ( ( L3 @ X5 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I4 ) @ ( X5 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1241_reals__power__lt__ex,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y3 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y3 ) @ K2 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1242_real__of__nat__ge__one__iff,axiom,
! [N: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ one_one_nat @ N ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1243_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1244_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1245_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1246_Bolzano,axiom,
! [A: real,B: real,P2: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A6: real,B5: real,C4: real] :
( ( P2 @ A6 @ B5 )
=> ( ( P2 @ B5 @ C4 )
=> ( ( ord_less_eq_real @ A6 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C4 )
=> ( P2 @ A6 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A6: real,B5: real] :
( ( ( ord_less_eq_real @ A6 @ X3 )
& ( ord_less_eq_real @ X3 @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A6 ) @ D4 ) )
=> ( P2 @ A6 @ B5 ) ) ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1247_seq__mono__lemma,axiom,
! [M2: nat,D: nat > real,E2: nat > real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_real @ ( D @ N2 ) @ ( E2 @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( E2 @ N2 ) @ ( E2 @ M2 ) ) )
=> ! [N7: nat] :
( ( ord_less_eq_nat @ M2 @ N7 )
=> ( ord_less_real @ ( D @ N7 ) @ ( E2 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1248_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1249_eq__diff__eq_H,axiom,
! [X: real,Y3: real,Z: real] :
( ( X
= ( minus_minus_real @ Y3 @ Z ) )
= ( Y3
= ( plus_plus_real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_1250_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1251_infinite__int__iff__unbounded__le,axiom,
! [S3: set_int] :
( ( ~ ( finite_finite_int @ S3 ) )
= ( ! [M5: int] :
? [N3: int] :
( ( ord_less_eq_int @ M5 @ ( abs_abs_int @ N3 ) )
& ( member_int2 @ N3 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_1252_infinite__int__iff__unbounded,axiom,
! [S3: set_int] :
( ( ~ ( finite_finite_int @ S3 ) )
= ( ! [M5: int] :
? [N3: int] :
( ( ord_less_int @ M5 @ ( abs_abs_int @ N3 ) )
& ( member_int2 @ N3 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_1253_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1254_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1255_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1256_sin__bound__lemma,axiom,
! [X: real,Y3: real,U: real,V: real] :
( ( X = Y3 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y3 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_1257_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D5 )
=> ( ( ord_less_eq_real @ A @ Y5 )
& ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1258_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D5 )
=> ( ( ord_less_real @ A @ Y5 )
& ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1259_abs__harm,axiom,
! [N: nat] :
( ( abs_abs_real @ ( harmonic_harm_real @ N ) )
= ( harmonic_harm_real @ N ) ) ).
% abs_harm
thf(fact_1260_decr__lemma,axiom,
! [D: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1261_abs__zmult__eq__1,axiom,
! [M2: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M2 )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1262_zmult__zless__mono2,axiom,
! [I2: int,J2: int,K: int] :
( ( ord_less_int @ I2 @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
member_nat_b_nat_b2 @ x @ ( set_nat_b_nat_b2 @ t ) ).
thf(conj_1,conjecture,
is_swap @ x ).
%------------------------------------------------------------------------------