TPTP Problem File: SLH0260^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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_00244_009063__14757682_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1474 ( 547 unt; 205 typ; 0 def)
% Number of atoms : 3824 (1261 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10046 ( 360 ~; 87 |; 155 &;7778 @)
% ( 0 <=>;1666 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 873 ( 873 >; 0 *; 0 +; 0 <<)
% Number of symbols : 189 ( 186 usr; 18 con; 0-4 aty)
% Number of variables : 3472 ( 314 ^;3020 !; 138 ?;3472 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:44:48.276
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
formal_Power_fps_nat: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
formal_Power_fps_int: $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__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__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $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 (186)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
invers68952373231134600s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Int__Oint,type,
formal3717847055265219294th_int: formal_Power_fps_int > nat > int ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Nat__Onat,type,
formal3720337525774269570th_nat: formal_Power_fps_nat > nat > nat ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Real__Oreal,type,
formal2580924720334399070h_real: formal3361831859752904756s_real > nat > real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Oradical_001t__Real__Oreal,type,
formal8005797870169972230l_real: ( nat > real > real ) > nat > formal3361831859752904756s_real > nat > real ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Real__Oreal,type,
inj_on_nat_real: ( nat > real ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__b,type,
inj_on_nat_b: ( nat > b ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001t__Int__Oint,type,
inj_on_b_int: ( b > int ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001t__Nat__Onat,type,
inj_on_b_nat: ( b > nat ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001t__Real__Oreal,type,
inj_on_b_real: ( b > real ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
inj_on_b_b: ( b > b ) > set_b > $o ).
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__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__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
times_3091854549176928185ps_int: formal_Power_fps_int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
times_7269705568686124893ps_nat: formal_Power_fps_nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
times_7561426564079326009s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
uminus_uminus_real_o: ( real > $o ) > real > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
zero_z4353722679246354365ps_int: formal_Power_fps_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
zero_z8531573698755551073ps_nat: formal_Power_fps_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
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__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_HOL_OUniq_001t__List__Olist_It__Int__Oint_J,type,
uniq_list_int: ( list_int > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__List__Olist_It__Nat__Onat_J,type,
uniq_list_nat: ( list_nat > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__List__Olist_It__Real__Oreal_J,type,
uniq_list_real: ( list_real > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__List__Olist_Itf__b_J,type,
uniq_list_b: ( list_b > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
uniq_nat: ( nat > $o ) > $o ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
ring_14195589558981938179ps_int: int > formal_Power_fps_int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
ring_13846316942275908227s_real: int > formal3361831859752904756s_real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__Nat__Onat_001t__Nat__Onat,type,
lattic8739620818006775868at_nat: ( nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_List_Ofilter_001t__Int__Oint,type,
filter_int: ( int > $o ) > list_int > list_int ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__Real__Oreal,type,
filter_real: ( real > $o ) > list_real > list_real ).
thf(sy_c_List_Ofilter_001tf__b,type,
filter_b: ( b > $o ) > list_b > list_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_001tf__b_001t__List__Olist_Itf__b_J,type,
fold_b_list_b: ( b > list_b > list_b ) > list_b > list_b > list_b ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Int__Oint_001t__Int__Oint,type,
linord2918399596068453212nt_int: ( int > int ) > int > list_int > list_int ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord1921536354676448932at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001tf__b,type,
linord5078064450519508267_nat_b: ( nat > b ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Real__Oreal_001t__Real__Oreal,type,
linord1891625487229344476l_real: ( real > real ) > real > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001tf__b_001tf__b,type,
linord7675407133644508226ey_b_b: ( b > b ) > b > list_b > list_b ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Int__Oint_001t__Int__Oint,type,
linord734827384618529109nt_int: ( int > int ) > int > list_int > list_int ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__Int__Oint,type,
linord8958845709572250361at_int: ( nat > int ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord8961336180081300637at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__Real__Oreal,type,
linord1530454444912894201t_real: ( nat > real ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001tf__b,type,
linord9222868044636945266_nat_b: ( nat > b ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Real__Oreal_001t__Int__Oint,type,
linord6095272547532742613al_int: ( real > int ) > real > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Real__Oreal_001t__Nat__Onat,type,
linord6097763018041792889al_nat: ( real > nat ) > real > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Real__Oreal_001t__Real__Oreal,type,
linord1674302359176591317l_real: ( real > real ) > real > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Real__Oreal_001tf__b,type,
linord3786284756137309590real_b: ( real > b ) > real > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001tf__b_001tf__b,type,
linord2228103134468323771ey_b_b: ( b > b ) > b > list_b > list_b ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001tf__b,type,
linord2522810402733747245_nat_b: ( nat > b ) > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Real__Oreal_001t__Int__Oint,type,
linord5469610843616646106al_int: ( real > int ) > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Real__Oreal_001t__Nat__Onat,type,
linord5472101314125696382al_nat: ( real > nat ) > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Real__Oreal_001t__Real__Oreal,type,
linord6132810859473402202l_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Real__Oreal_001tf__b,type,
linord2738525447772944593real_b: ( real > b ) > list_real > list_real ).
thf(sy_c_List_Olinorder__class_Osort__key_001tf__b_001t__Int__Oint,type,
linord8947204796857580843_b_int: ( b > int ) > list_b > list_b ).
thf(sy_c_List_Olinorder__class_Osort__key_001tf__b_001t__Nat__Onat,type,
linord8949695267366631119_b_nat: ( b > nat ) > list_b > list_b ).
thf(sy_c_List_Olinorder__class_Osort__key_001tf__b_001t__Real__Oreal,type,
linord7704725295488125483b_real: ( b > real ) > list_b > list_b ).
thf(sy_c_List_Olinorder__class_Osort__key_001tf__b_001tf__b,type,
linord5847811544569201088ey_b_b: ( b > b ) > list_b > list_b ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord227665693835759911at_nat: ( ( nat > nat ) > list_nat > list_nat ) > $o ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key_001tf__b_001tf__b,type,
linord1690884029915216197ey_b_b: ( ( b > b ) > list_b > list_b ) > $o ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_ONil_001tf__b,type,
nil_b: list_b ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Real__Oreal,type,
hd_real: list_real > real ).
thf(sy_c_List_Olist_Ohd_001tf__b,type,
hd_b: list_b > b ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
map_nat_int: ( nat > int ) > list_nat > list_int ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
map_nat_real: ( nat > real ) > list_nat > list_real ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__b,type,
map_nat_b: ( nat > b ) > list_nat > list_b ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Int__Oint,type,
map_real_int: ( real > int ) > list_real > list_int ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
map_real_nat: ( real > nat ) > list_real > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001tf__b,type,
map_real_b: ( real > b ) > list_real > list_b ).
thf(sy_c_List_Olist_Omap_001tf__b_001t__Int__Oint,type,
map_b_int: ( b > int ) > list_b > list_int ).
thf(sy_c_List_Olist_Omap_001tf__b_001t__Nat__Onat,type,
map_b_nat: ( b > nat ) > list_b > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__b_001t__Real__Oreal,type,
map_b_real: ( b > real ) > list_b > list_real ).
thf(sy_c_List_Olist_Omap_001tf__b_001tf__b,type,
map_b_b: ( b > b ) > list_b > list_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_Oset_001tf__b,type,
set_b2: list_b > set_b ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001tf__b,type,
map_tailrec_nat_b: ( nat > b ) > list_nat > list_b ).
thf(sy_c_List_Oremove1_001t__Int__Oint,type,
remove1_int: int > list_int > list_int ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001t__Real__Oreal,type,
remove1_real: real > list_real > list_real ).
thf(sy_c_List_Oremove1_001tf__b,type,
remove1_b: b > list_b > list_b ).
thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
replicate_int: nat > int > list_int ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
replicate_real: nat > real > list_real ).
thf(sy_c_List_Oreplicate_001tf__b,type,
replicate_b: nat > b > list_b ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
sorted_wrt_real: ( real > real > $o ) > list_real > $o ).
thf(sy_c_List_Osorted__wrt_001tf__b,type,
sorted_wrt_b: ( b > b > $o ) > list_b > $o ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Median_Odown__ray_001t__Int__Oint,type,
down_ray_int: set_int > $o ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001t__Real__Oreal,type,
down_ray_real: set_real > $o ).
thf(sy_c_Median_Odown__ray_001tf__b,type,
down_ray_b: set_b > $o ).
thf(sy_c_Median_Ointerval_001t__Int__Oint,type,
interval_int: set_int > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001t__Real__Oreal,type,
interval_real: set_real > $o ).
thf(sy_c_Median_Ointerval_001tf__b,type,
interval_b: set_b > $o ).
thf(sy_c_Median_Osort__map_001t__Int__Oint,type,
sort_map_int: ( nat > int ) > nat > nat > int ).
thf(sy_c_Median_Osort__map_001t__Nat__Onat,type,
sort_map_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Median_Osort__map_001t__Real__Oreal,type,
sort_map_real: ( nat > real ) > nat > nat > real ).
thf(sy_c_Median_Osort__map_001tf__b,type,
sort_map_b: ( nat > b ) > nat > nat > b ).
thf(sy_c_Median_Oup__ray_001t__Int__Oint,type,
up_ray_int: set_int > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001t__Real__Oreal,type,
up_ray_real: set_real > $o ).
thf(sy_c_Median_Oup__ray_001tf__b,type,
up_ray_b: set_b > $o ).
thf(sy_c_Multiset_Ofold__mset_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
fold_m3004868085856202685st_nat: ( nat > list_nat > list_nat ) > list_nat > multiset_nat > list_nat ).
thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001t__Nat__Onat,type,
image_mset_nat_nat: ( nat > nat ) > multiset_nat > multiset_nat ).
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_Oimage__mset_001tf__b_001tf__b,type,
image_mset_b_b: ( b > b ) > multiset_b > 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_Olinorder__class_Osorted__list__of__multiset_001tf__b,type,
linord814965612141868909iset_b: multiset_b > list_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_001tf__b,type,
mset_b: list_b > multiset_b ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
semiri6570152736363784213ps_int: nat > formal_Power_fps_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
semiri1524631719018205113ps_nat: nat > formal_Power_fps_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
semiri2475410149736220053s_real: nat > formal3361831859752904756s_real ).
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_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
neg_numeral_dbl_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $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_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__b,type,
ord_less_b: b > b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $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_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_001tf__b,type,
ord_less_eq_b: b > b > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__b,type,
order_Greatest_b: ( b > $o ) > b ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
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_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_f,type,
f: nat > b ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1261)
thf(fact_0__092_060open_062sorted_A_Imap_A_Isort__map_Af_An_J_A_0910_O_O_060n_093_J_092_060close_062,axiom,
sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ ( sort_map_b @ f @ n ) @ ( upt @ zero_zero_nat @ n ) ) ).
% \<open>sorted (map (sort_map f n) [0..<n])\<close>
thf(fact_1__092_060open_062mset_A_Imap_A_Isort__map_Af_An_J_A_0910_O_O_060n_093_J_A_061_Amset_A_Isort_A_Imap_Af_A_0910_O_O_060n_093_J_J_092_060close_062,axiom,
( ( mset_b @ ( map_nat_b @ ( sort_map_b @ f @ n ) @ ( upt @ zero_zero_nat @ n ) ) )
= ( mset_b
@ ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ ( map_nat_b @ f @ ( upt @ zero_zero_nat @ n ) ) ) ) ) ).
% \<open>mset (map (sort_map f n) [0..<n]) = mset (sort (map f [0..<n]))\<close>
thf(fact_2__092_060open_062sorted_A_Isort_A_Imap_Af_A_0910_O_O_060n_093_J_J_092_060close_062,axiom,
( sorted_wrt_b @ ord_less_eq_b
@ ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ ( map_nat_b @ f @ ( upt @ zero_zero_nat @ n ) ) ) ) ).
% \<open>sorted (sort (map f [0..<n]))\<close>
thf(fact_3_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs: list_nat] : Xs ) ) ).
% map_ident
thf(fact_4_sort__key__const,axiom,
! [C: b,Xs2: list_b] :
( ( linord5847811544569201088ey_b_b
@ ^ [X: b] : C
@ Xs2 )
= Xs2 ) ).
% sort_key_const
thf(fact_5_sort__key__const,axiom,
! [C: nat,Xs2: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : C
@ Xs2 )
= Xs2 ) ).
% sort_key_const
thf(fact_6_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_7_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_8_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_9_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_10_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_11_mset__sort,axiom,
! [K: b > b,Xs2: list_b] :
( ( mset_b @ ( linord5847811544569201088ey_b_b @ K @ Xs2 ) )
= ( mset_b @ Xs2 ) ) ).
% mset_sort
thf(fact_12_mset__sort,axiom,
! [K: nat > nat,Xs2: list_nat] :
( ( mset_nat @ ( linord738340561235409698at_nat @ K @ Xs2 ) )
= ( mset_nat @ Xs2 ) ) ).
% mset_sort
thf(fact_13_map__replicate__trivial,axiom,
! [X2: b,I: nat] :
( ( map_nat_b
@ ^ [I2: nat] : X2
@ ( upt @ zero_zero_nat @ I ) )
= ( replicate_b @ I @ X2 ) ) ).
% map_replicate_trivial
thf(fact_14_map__replicate__trivial,axiom,
! [X2: nat,I: nat] :
( ( map_nat_nat
@ ^ [I2: nat] : X2
@ ( upt @ zero_zero_nat @ I ) )
= ( replicate_nat @ I @ X2 ) ) ).
% map_replicate_trivial
thf(fact_15_sorted__sort__key,axiom,
! [F: nat > b,Xs2: list_nat] : ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ ( linord2522810402733747245_nat_b @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_16_sorted__sort__key,axiom,
! [F: b > b,Xs2: list_b] : ( sorted_wrt_b @ ord_less_eq_b @ ( map_b_b @ F @ ( linord5847811544569201088ey_b_b @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_17_sorted__sort__key,axiom,
! [F: nat > nat,Xs2: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord738340561235409698at_nat @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_18_sort__key__id__if__sorted,axiom,
! [F: nat > b,Xs2: list_nat] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) )
=> ( ( linord2522810402733747245_nat_b @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_19_sort__key__id__if__sorted,axiom,
! [F: b > b,Xs2: list_b] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_b_b @ F @ Xs2 ) )
=> ( ( linord5847811544569201088ey_b_b @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_20_sort__key__id__if__sorted,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_21_list__eq__iff,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ( mset_b @ Xs2 )
= ( mset_b @ Ys ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ Xs2 )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ Ys )
=> ( Xs2 = Ys ) ) ) ) ).
% list_eq_iff
thf(fact_22_list__eq__iff,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs2 )
= ( mset_nat @ Ys ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( Xs2 = Ys ) ) ) ) ).
% list_eq_iff
thf(fact_23_list__eq__iff,axiom,
! [Xs2: list_int,Ys: list_int] :
( ( ( mset_int @ Xs2 )
= ( mset_int @ Ys ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
=> ( Xs2 = Ys ) ) ) ) ).
% list_eq_iff
thf(fact_24_list__eq__iff,axiom,
! [Xs2: list_real,Ys: list_real] :
( ( ( mset_real @ Xs2 )
= ( mset_real @ Ys ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Ys )
=> ( Xs2 = Ys ) ) ) ) ).
% list_eq_iff
thf(fact_25_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_26_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_27_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_28_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_29_map__replicate,axiom,
! [F: nat > b,N: nat,X2: nat] :
( ( map_nat_b @ F @ ( replicate_nat @ N @ X2 ) )
= ( replicate_b @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_30_map__replicate,axiom,
! [F: nat > nat,N: nat,X2: nat] :
( ( map_nat_nat @ F @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_31_sorted__replicate,axiom,
! [N: nat,X2: b] : ( sorted_wrt_b @ ord_less_eq_b @ ( replicate_b @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_32_sorted__replicate,axiom,
! [N: nat,X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_33_sorted__replicate,axiom,
! [N: nat,X2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( replicate_int @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_34_sorted__replicate,axiom,
! [N: nat,X2: real] : ( sorted_wrt_real @ ord_less_eq_real @ ( replicate_real @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_35_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_36_ex__mset,axiom,
! [X3: multiset_b] :
? [Xs3: list_b] :
( ( mset_b @ Xs3 )
= X3 ) ).
% ex_mset
thf(fact_37_sorted__wrt__true,axiom,
! [Xs2: list_b] :
( sorted_wrt_b
@ ^ [Uu: b,Uv: b] : $true
@ Xs2 ) ).
% sorted_wrt_true
thf(fact_38_sorted__wrt__true,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs2 ) ).
% sorted_wrt_true
thf(fact_39_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_40_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_41_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_42_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_43_properties__for__sort,axiom,
! [Ys: list_int,Xs2: list_int] :
( ( ( mset_int @ Ys )
= ( mset_int @ Xs2 ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
=> ( ( linord1735203802627413978nt_int
@ ^ [X: int] : X
@ Xs2 )
= Ys ) ) ) ).
% properties_for_sort
thf(fact_44_properties__for__sort,axiom,
! [Ys: list_real,Xs2: list_real] :
( ( ( mset_real @ Ys )
= ( mset_real @ Xs2 ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Ys )
=> ( ( linord6132810859473402202l_real
@ ^ [X: real] : X
@ Xs2 )
= Ys ) ) ) ).
% properties_for_sort
thf(fact_45_properties__for__sort,axiom,
! [Ys: list_b,Xs2: list_b] :
( ( ( mset_b @ Ys )
= ( mset_b @ Xs2 ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ Ys )
=> ( ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ Xs2 )
= Ys ) ) ) ).
% properties_for_sort
thf(fact_46_properties__for__sort,axiom,
! [Ys: list_nat,Xs2: list_nat] :
( ( ( mset_nat @ Ys )
= ( mset_nat @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs2 )
= Ys ) ) ) ).
% properties_for_sort
thf(fact_47_sorted__map,axiom,
! [F: b > b,Xs2: list_b] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_b_b @ F @ Xs2 ) )
= ( sorted_wrt_b
@ ^ [X: b,Y: b] : ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_48_sorted__map,axiom,
! [F: nat > b,Xs2: list_nat] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_b @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_49_sorted__map,axiom,
! [F: b > nat,Xs2: list_b] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_b_nat @ F @ Xs2 ) )
= ( sorted_wrt_b
@ ^ [X: b,Y: b] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_50_sorted__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_51_sorted__map,axiom,
! [F: b > int,Xs2: list_b] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( map_b_int @ F @ Xs2 ) )
= ( sorted_wrt_b
@ ^ [X: b,Y: b] : ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_52_sorted__map,axiom,
! [F: nat > int,Xs2: list_nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( map_nat_int @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_53_sorted__map,axiom,
! [F: b > real,Xs2: list_b] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( map_b_real @ F @ Xs2 ) )
= ( sorted_wrt_b
@ ^ [X: b,Y: b] : ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_54_sorted__map,axiom,
! [F: nat > real,Xs2: list_nat] :
( ( sorted_wrt_real @ ord_less_eq_real @ ( map_nat_real @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_55_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_56_sorted__sort__id,axiom,
! [Xs2: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
=> ( ( linord1735203802627413978nt_int
@ ^ [X: int] : X
@ Xs2 )
= Xs2 ) ) ).
% sorted_sort_id
thf(fact_57_sorted__sort__id,axiom,
! [Xs2: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
=> ( ( linord6132810859473402202l_real
@ ^ [X: real] : X
@ Xs2 )
= Xs2 ) ) ).
% sorted_sort_id
thf(fact_58_sorted__sort__id,axiom,
! [Xs2: list_b] :
( ( sorted_wrt_b @ ord_less_eq_b @ Xs2 )
=> ( ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ Xs2 )
= Xs2 ) ) ).
% sorted_sort_id
thf(fact_59_sorted__sort__id,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs2 )
= Xs2 ) ) ).
% sorted_sort_id
thf(fact_60_sorted__sort,axiom,
! [Xs2: list_int] :
( sorted_wrt_int @ ord_less_eq_int
@ ( linord1735203802627413978nt_int
@ ^ [X: int] : X
@ Xs2 ) ) ).
% sorted_sort
thf(fact_61_sorted__sort,axiom,
! [Xs2: list_real] :
( sorted_wrt_real @ ord_less_eq_real
@ ( linord6132810859473402202l_real
@ ^ [X: real] : X
@ Xs2 ) ) ).
% sorted_sort
thf(fact_62_sorted__sort,axiom,
! [Xs2: list_b] :
( sorted_wrt_b @ ord_less_eq_b
@ ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ Xs2 ) ) ).
% sorted_sort
thf(fact_63_sorted__sort,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs2 ) ) ).
% sorted_sort
thf(fact_64_sorted__wrt__map,axiom,
! [R: b > b > $o,F: b > b,Xs2: list_b] :
( ( sorted_wrt_b @ R @ ( map_b_b @ F @ Xs2 ) )
= ( sorted_wrt_b
@ ^ [X: b,Y: b] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_65_sorted__wrt__map,axiom,
! [R: b > b > $o,F: nat > b,Xs2: list_nat] :
( ( sorted_wrt_b @ R @ ( map_nat_b @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_66_sorted__wrt__map,axiom,
! [R: nat > nat > $o,F: b > nat,Xs2: list_b] :
( ( sorted_wrt_nat @ R @ ( map_b_nat @ F @ Xs2 ) )
= ( sorted_wrt_b
@ ^ [X: b,Y: b] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_67_sorted__wrt__map,axiom,
! [R: nat > nat > $o,F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ R @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_68_order__refl,axiom,
! [X2: b] : ( ord_less_eq_b @ X2 @ X2 ) ).
% order_refl
thf(fact_69_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_70_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_71_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_72_dual__order_Orefl,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% dual_order.refl
thf(fact_73_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_74_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_75_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_76_sorted__list__of__multiset__mset,axiom,
! [Xs2: list_b] :
( ( linord814965612141868909iset_b @ ( mset_b @ Xs2 ) )
= ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ Xs2 ) ) ).
% sorted_list_of_multiset_mset
thf(fact_77_sorted__list__of__multiset__mset,axiom,
! [Xs2: list_nat] :
( ( linord3047872887403683810et_nat @ ( mset_nat @ Xs2 ) )
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs2 ) ) ).
% sorted_list_of_multiset_mset
thf(fact_78_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_79_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_80_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_81_sorted__insort__insert__key,axiom,
! [F: nat > b,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) )
=> ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ ( linord5078064450519508267_nat_b @ F @ X2 @ Xs2 ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_82_sorted__insort__insert__key,axiom,
! [F: nat > nat,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord1921536354676448932at_nat @ F @ X2 @ Xs2 ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_83_stable__sort__key__sort__key,axiom,
linord1690884029915216197ey_b_b @ linord5847811544569201088ey_b_b ).
% stable_sort_key_sort_key
thf(fact_84_stable__sort__key__sort__key,axiom,
linord227665693835759911at_nat @ linord738340561235409698at_nat ).
% stable_sort_key_sort_key
thf(fact_85_map__eq__map__tailrec,axiom,
map_nat_b = map_tailrec_nat_b ).
% map_eq_map_tailrec
thf(fact_86_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_87_interval__def,axiom,
( interval_b
= ( ^ [I3: set_b] :
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ I3 )
=> ( ( member_b @ Z @ I3 )
=> ( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ Z )
=> ( member_b @ Y @ I3 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_88_interval__def,axiom,
( interval_nat
= ( ^ [I3: set_nat] :
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ I3 )
=> ( ( member_nat @ Z @ I3 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( member_nat @ Y @ I3 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_89_interval__def,axiom,
( interval_int
= ( ^ [I3: set_int] :
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ I3 )
=> ( ( member_int @ Z @ I3 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( member_int @ Y @ I3 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_90_interval__def,axiom,
( interval_real
= ( ^ [I3: set_real] :
! [X: real,Y: real,Z: real] :
( ( member_real @ X @ I3 )
=> ( ( member_real @ Z @ I3 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( member_real @ Y @ I3 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_91_down__ray__def,axiom,
( down_ray_b
= ( ^ [I3: set_b] :
! [X: b,Y: b] :
( ( member_b @ Y @ I3 )
=> ( ( ord_less_eq_b @ X @ Y )
=> ( member_b @ X @ I3 ) ) ) ) ) ).
% down_ray_def
thf(fact_92_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I3: set_nat] :
! [X: nat,Y: nat] :
( ( member_nat @ Y @ I3 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( member_nat @ X @ I3 ) ) ) ) ) ).
% down_ray_def
thf(fact_93_down__ray__def,axiom,
( down_ray_int
= ( ^ [I3: set_int] :
! [X: int,Y: int] :
( ( member_int @ Y @ I3 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( member_int @ X @ I3 ) ) ) ) ) ).
% down_ray_def
thf(fact_94_down__ray__def,axiom,
( down_ray_real
= ( ^ [I3: set_real] :
! [X: real,Y: real] :
( ( member_real @ Y @ I3 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( member_real @ X @ I3 ) ) ) ) ) ).
% down_ray_def
thf(fact_95_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_97_up__ray__def,axiom,
( up_ray_b
= ( ^ [I3: set_b] :
! [X: b,Y: b] :
( ( member_b @ X @ I3 )
=> ( ( ord_less_eq_b @ X @ Y )
=> ( member_b @ Y @ I3 ) ) ) ) ) ).
% up_ray_def
thf(fact_98_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I3: set_nat] :
! [X: nat,Y: nat] :
( ( member_nat @ X @ I3 )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( member_nat @ Y @ I3 ) ) ) ) ) ).
% up_ray_def
thf(fact_99_up__ray__def,axiom,
( up_ray_int
= ( ^ [I3: set_int] :
! [X: int,Y: int] :
( ( member_int @ X @ I3 )
=> ( ( ord_less_eq_int @ X @ Y )
=> ( member_int @ Y @ I3 ) ) ) ) ) ).
% up_ray_def
thf(fact_100_up__ray__def,axiom,
( up_ray_real
= ( ^ [I3: set_real] :
! [X: real,Y: real] :
( ( member_real @ X @ I3 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( member_real @ Y @ I3 ) ) ) ) ) ).
% up_ray_def
thf(fact_101_sorted__insort__insert,axiom,
! [Xs2: list_b,X2: b] :
( ( sorted_wrt_b @ ord_less_eq_b @ Xs2 )
=> ( sorted_wrt_b @ ord_less_eq_b
@ ( linord7675407133644508226ey_b_b
@ ^ [X: b] : X
@ X2
@ Xs2 ) ) ) ).
% sorted_insort_insert
thf(fact_102_sorted__insort__insert,axiom,
! [Xs2: list_nat,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord1921536354676448932at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 ) ) ) ).
% sorted_insort_insert
thf(fact_103_sorted__insort__insert,axiom,
! [Xs2: list_int,X2: int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
=> ( sorted_wrt_int @ ord_less_eq_int
@ ( linord2918399596068453212nt_int
@ ^ [X: int] : X
@ X2
@ Xs2 ) ) ) ).
% sorted_insort_insert
thf(fact_104_sorted__insort__insert,axiom,
! [Xs2: list_real,X2: real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
=> ( sorted_wrt_real @ ord_less_eq_real
@ ( linord1891625487229344476l_real
@ ^ [X: real] : X
@ X2
@ Xs2 ) ) ) ).
% sorted_insort_insert
thf(fact_105_sorted__map__remove1,axiom,
! [F: nat > b,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) )
=> ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ ( remove1_nat @ X2 @ Xs2 ) ) ) ) ).
% sorted_map_remove1
thf(fact_106_sorted__map__remove1,axiom,
! [F: nat > nat,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( remove1_nat @ X2 @ Xs2 ) ) ) ) ).
% sorted_map_remove1
thf(fact_107_mset__sorted__list__of__multiset,axiom,
! [M2: multiset_b] :
( ( mset_b @ ( linord814965612141868909iset_b @ M2 ) )
= M2 ) ).
% mset_sorted_list_of_multiset
thf(fact_108_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_109_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_110_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_111_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_112_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_113_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_114_sorted__remove1,axiom,
! [Xs2: list_b,A: b] :
( ( sorted_wrt_b @ ord_less_eq_b @ Xs2 )
=> ( sorted_wrt_b @ ord_less_eq_b @ ( remove1_b @ A @ Xs2 ) ) ) ).
% sorted_remove1
thf(fact_115_sorted__remove1,axiom,
! [Xs2: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs2 ) ) ) ).
% sorted_remove1
thf(fact_116_sorted__remove1,axiom,
! [Xs2: list_int,A: int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
=> ( sorted_wrt_int @ ord_less_eq_int @ ( remove1_int @ A @ Xs2 ) ) ) ).
% sorted_remove1
thf(fact_117_sorted__remove1,axiom,
! [Xs2: list_real,A: real] :
( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
=> ( sorted_wrt_real @ ord_less_eq_real @ ( remove1_real @ A @ Xs2 ) ) ) ).
% sorted_remove1
thf(fact_118_order__antisym__conv,axiom,
! [Y4: b,X2: b] :
( ( ord_less_eq_b @ Y4 @ X2 )
=> ( ( ord_less_eq_b @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_119_order__antisym__conv,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_120_order__antisym__conv,axiom,
! [Y4: int,X2: int] :
( ( ord_less_eq_int @ Y4 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_121_order__antisym__conv,axiom,
! [Y4: real,X2: real] :
( ( ord_less_eq_real @ Y4 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_122_linorder__le__cases,axiom,
! [X2: b,Y4: b] :
( ~ ( ord_less_eq_b @ X2 @ Y4 )
=> ( ord_less_eq_b @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_123_linorder__le__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_124_linorder__le__cases,axiom,
! [X2: int,Y4: int] :
( ~ ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_125_linorder__le__cases,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_126_ord__le__eq__subst,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_127_ord__le__eq__subst,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_128_ord__le__eq__subst,axiom,
! [A: b,B: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_129_ord__le__eq__subst,axiom,
! [A: b,B: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_130_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_131_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_132_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_133_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_134_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > b,C: b] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_135_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_138_ord__eq__le__subst,axiom,
! [A: int,F: b > int,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_139_ord__eq__le__subst,axiom,
! [A: real,F: b > real,B: b,C: b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_140_ord__eq__le__subst,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_141_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_142_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_143_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_144_ord__eq__le__subst,axiom,
! [A: b,F: int > b,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_145_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_146_linorder__linear,axiom,
! [X2: b,Y4: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
| ( ord_less_eq_b @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_147_linorder__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_148_linorder__linear,axiom,
! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_149_linorder__linear,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
| ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_150_order__eq__refl,axiom,
! [X2: b,Y4: b] :
( ( X2 = Y4 )
=> ( ord_less_eq_b @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_151_order__eq__refl,axiom,
! [X2: nat,Y4: nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_152_order__eq__refl,axiom,
! [X2: int,Y4: int] :
( ( X2 = Y4 )
=> ( ord_less_eq_int @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_153_order__eq__refl,axiom,
! [X2: real,Y4: real] :
( ( X2 = Y4 )
=> ( ord_less_eq_real @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_154_order__subst2,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A: b,B: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A: b,B: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst2,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_159_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_160_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_161_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_162_order__subst2,axiom,
! [A: int,B: int,F: int > b,C: b] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_163_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_164_order__subst1,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_165_order__subst1,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_166_order__subst1,axiom,
! [A: b,F: int > b,B: int,C: int] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_167_order__subst1,axiom,
! [A: b,F: real > b,B: real,C: real] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_168_order__subst1,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_169_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_170_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_171_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_172_order__subst1,axiom,
! [A: int,F: b > int,B: b,C: b] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_173_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_174_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: b,Z2: b] : ( Y5 = Z2 ) )
= ( ^ [A3: b,B2: b] :
( ( ord_less_eq_b @ A3 @ B2 )
& ( ord_less_eq_b @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_175_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_176_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_177_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_178_antisym,axiom,
! [A: b,B: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_179_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_180_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_181_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_182_dual__order_Otrans,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_less_eq_b @ C @ B )
=> ( ord_less_eq_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_183_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_184_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_185_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_186_dual__order_Oantisym,axiom,
! [B: b,A: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_less_eq_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_187_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_188_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_189_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_190_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: b,Z2: b] : ( Y5 = Z2 ) )
= ( ^ [A3: b,B2: b] :
( ( ord_less_eq_b @ B2 @ A3 )
& ( ord_less_eq_b @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_191_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_192_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_193_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_194_linorder__wlog,axiom,
! [P: b > b > $o,A: b,B: b] :
( ! [A4: b,B3: b] :
( ( ord_less_eq_b @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: b,B3: b] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_195_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_196_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_197_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_198_order__trans,axiom,
! [X2: b,Y4: b,Z3: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
=> ( ( ord_less_eq_b @ Y4 @ Z3 )
=> ( ord_less_eq_b @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_199_order__trans,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_200_order__trans,axiom,
! [X2: int,Y4: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_201_order__trans,axiom,
! [X2: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_202_order_Otrans,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% order.trans
thf(fact_203_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_204_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_205_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_206_order__antisym,axiom,
! [X2: b,Y4: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
=> ( ( ord_less_eq_b @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_207_order__antisym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_208_order__antisym,axiom,
! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_209_order__antisym,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_210_ord__le__eq__trans,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_211_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_212_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_213_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_214_ord__eq__le__trans,axiom,
! [A: b,B: b,C: b] :
( ( A = B )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_215_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_216_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_217_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_218_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: b,Z2: b] : ( Y5 = Z2 ) )
= ( ^ [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
& ( ord_less_eq_b @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_219_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_220_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_221_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_222_le__cases3,axiom,
! [X2: b,Y4: b,Z3: b] :
( ( ( ord_less_eq_b @ X2 @ Y4 )
=> ~ ( ord_less_eq_b @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_b @ Y4 @ X2 )
=> ~ ( ord_less_eq_b @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_b @ X2 @ Z3 )
=> ~ ( ord_less_eq_b @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_b @ Z3 @ Y4 )
=> ~ ( ord_less_eq_b @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_b @ Y4 @ Z3 )
=> ~ ( ord_less_eq_b @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_b @ Z3 @ X2 )
=> ~ ( ord_less_eq_b @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_223_le__cases3,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_224_le__cases3,axiom,
! [X2: int,Y4: int,Z3: int] :
( ( ( ord_less_eq_int @ X2 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_225_le__cases3,axiom,
! [X2: real,Y4: real,Z3: real] :
( ( ( ord_less_eq_real @ X2 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_226_nle__le,axiom,
! [A: b,B: b] :
( ( ~ ( ord_less_eq_b @ A @ B ) )
= ( ( ord_less_eq_b @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_227_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_228_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_229_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_230_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_231_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > b] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_b @ ( sort_map_b @ F @ N @ I ) @ ( sort_map_b @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_232_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N @ I ) @ ( sort_map_nat @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_233_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > int] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( sort_map_int @ F @ N @ I ) @ ( sort_map_int @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_234_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > real] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N @ I ) @ ( sort_map_real @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_235_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_236_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_237_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_238_Greatest__equality,axiom,
! [P: b > $o,X2: b] :
( ( P @ X2 )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X2 ) )
=> ( ( order_Greatest_b @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_239_Greatest__equality,axiom,
! [P: int > $o,X2: int] :
( ( P @ X2 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( order_Greatest_int @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_240_Greatest__equality,axiom,
! [P: real > $o,X2: real] :
( ( P @ X2 )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( order_Greatest_real @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_241_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_242_GreatestI2__order,axiom,
! [P: b > $o,X2: b,Q: b > $o] :
( ( P @ X2 )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X2 ) )
=> ( ! [X4: b] :
( ( P @ X4 )
=> ( ! [Y3: b] :
( ( P @ Y3 )
=> ( ord_less_eq_b @ Y3 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_243_GreatestI2__order,axiom,
! [P: int > $o,X2: int,Q: int > $o] :
( ( P @ X2 )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_244_GreatestI2__order,axiom,
! [P: real > $o,X2: real,Q: real > $o] :
( ( P @ X2 )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ! [X4: real] :
( ( P @ X4 )
=> ( ! [Y3: real] :
( ( P @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_245_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_246_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_247_sorted__insort__key,axiom,
! [F: nat > b,X2: nat,Xs2: list_nat] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ ( linord9222868044636945266_nat_b @ F @ X2 @ Xs2 ) ) )
= ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) ) ) ).
% sorted_insort_key
thf(fact_248_sorted__insort__key,axiom,
! [F: nat > nat,X2: nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord8961336180081300637at_nat @ F @ X2 @ Xs2 ) ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% sorted_insort_key
thf(fact_249_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_250_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_251_sort__map__perm,axiom,
! [F: nat > b,N: nat] :
( ( image_mset_nat_b @ ( sort_map_b @ F @ N ) @ ( mset_nat @ ( upt @ zero_zero_nat @ N ) ) )
= ( image_mset_nat_b @ F @ ( mset_nat @ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% sort_map_perm
thf(fact_252_le__left__mono,axiom,
! [X2: b,Y4: b,A: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
=> ( ( ord_less_eq_b @ Y4 @ A )
=> ( ord_less_eq_b @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_253_le__left__mono,axiom,
! [X2: nat,Y4: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ A )
=> ( ord_less_eq_nat @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_254_le__left__mono,axiom,
! [X2: int,Y4: int,A: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ A )
=> ( ord_less_eq_int @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_255_le__left__mono,axiom,
! [X2: real,Y4: real,A: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ A )
=> ( ord_less_eq_real @ X2 @ A ) ) ) ).
% le_left_mono
thf(fact_256_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_257_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_258_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_259_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_260_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_261_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_262_map__is__Nil__conv,axiom,
! [F: nat > b,Xs2: list_nat] :
( ( ( map_nat_b @ F @ Xs2 )
= nil_b )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_263_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_264_Nil__is__map__conv,axiom,
! [F: nat > b,Xs2: list_nat] :
( ( nil_b
= ( map_nat_b @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_265_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_266_list_Omap__disc__iff,axiom,
! [F: nat > b,A: list_nat] :
( ( ( map_nat_b @ F @ A )
= nil_b )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_267_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_268_mset__zero__iff,axiom,
! [X2: list_nat] :
( ( ( mset_nat @ X2 )
= zero_z7348594199698428585et_nat )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff
thf(fact_269_mset__zero__iff,axiom,
! [X2: list_b] :
( ( ( mset_b @ X2 )
= zero_zero_multiset_b )
= ( X2 = nil_b ) ) ).
% mset_zero_iff
thf(fact_270_mset__zero__iff__right,axiom,
! [X2: list_nat] :
( ( zero_z7348594199698428585et_nat
= ( mset_nat @ X2 ) )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff_right
thf(fact_271_mset__zero__iff__right,axiom,
! [X2: list_b] :
( ( zero_zero_multiset_b
= ( mset_b @ X2 ) )
= ( X2 = nil_b ) ) ).
% mset_zero_iff_right
thf(fact_272_sort__key__simps_I1_J,axiom,
! [F: b > b] :
( ( linord5847811544569201088ey_b_b @ F @ nil_b )
= nil_b ) ).
% sort_key_simps(1)
thf(fact_273_sort__key__simps_I1_J,axiom,
! [F: nat > nat] :
( ( linord738340561235409698at_nat @ F @ nil_nat )
= nil_nat ) ).
% sort_key_simps(1)
thf(fact_274_sorted__list__of__multiset__empty,axiom,
( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
= nil_nat ) ).
% sorted_list_of_multiset_empty
thf(fact_275_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_276_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_277_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_278_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_279_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_280_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_281_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_282_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_283_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_284_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_285_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_286_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_287_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_288_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_289_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_290_empty__replicate,axiom,
! [N: nat,X2: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_291_replicate__empty,axiom,
! [N: nat,X2: nat] :
( ( ( replicate_nat @ N @ X2 )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_292_mset__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( mset_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( image_mset_nat_nat @ F @ ( mset_nat @ Xs2 ) ) ) ).
% mset_map
thf(fact_293_mset__map,axiom,
! [F: nat > b,Xs2: list_nat] :
( ( mset_b @ ( map_nat_b @ F @ Xs2 ) )
= ( image_mset_nat_b @ F @ ( mset_nat @ Xs2 ) ) ) ).
% mset_map
thf(fact_294_mset__map,axiom,
! [F: b > b,Xs2: list_b] :
( ( mset_b @ ( map_b_b @ F @ Xs2 ) )
= ( image_mset_b_b @ F @ ( mset_b @ Xs2 ) ) ) ).
% mset_map
thf(fact_295_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_296_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_297_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_298_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_299_order__less__imp__not__less,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_300_order__less__imp__not__less,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_301_order__less__imp__not__less,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_302_order__less__imp__not__eq2,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( Y4 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_303_order__less__imp__not__eq2,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( Y4 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_304_order__less__imp__not__eq2,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( Y4 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_305_order__less__imp__not__eq,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_306_order__less__imp__not__eq,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_307_order__less__imp__not__eq,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_308_linorder__less__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 )
| ( ord_less_nat @ Y4 @ X2 ) ) ).
% linorder_less_linear
thf(fact_309_linorder__less__linear,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
| ( X2 = Y4 )
| ( ord_less_int @ Y4 @ X2 ) ) ).
% linorder_less_linear
thf(fact_310_linorder__less__linear,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 )
| ( ord_less_real @ Y4 @ X2 ) ) ).
% linorder_less_linear
thf(fact_311_order__less__imp__triv,axiom,
! [X2: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_312_order__less__imp__triv,axiom,
! [X2: int,Y4: int,P: $o] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ( ord_less_int @ Y4 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_313_order__less__imp__triv,axiom,
! [X2: real,Y4: real,P: $o] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_real @ Y4 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_314_order__less__not__sym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X2 ) ) ).
% order_less_not_sym
thf(fact_315_order__less__not__sym,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X2 ) ) ).
% order_less_not_sym
thf(fact_316_order__less__not__sym,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X2 ) ) ).
% order_less_not_sym
thf(fact_317_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_318_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_319_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_320_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_321_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_322_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_323_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_324_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_325_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_326_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_327_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_328_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_329_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_330_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_331_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_332_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_333_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_334_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_335_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_336_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_337_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_338_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_339_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_340_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_341_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_342_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_343_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_344_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_345_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_346_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_347_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_348_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_349_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_350_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_351_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_352_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_353_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_354_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_355_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_356_order__less__trans,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_357_order__less__trans,axiom,
! [X2: int,Y4: int,Z3: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_358_order__less__trans,axiom,
! [X2: real,Y4: real,Z3: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_359_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_360_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_361_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_362_linorder__neq__iff,axiom,
! [X2: nat,Y4: nat] :
( ( X2 != Y4 )
= ( ( ord_less_nat @ X2 @ Y4 )
| ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_363_linorder__neq__iff,axiom,
! [X2: int,Y4: int] :
( ( X2 != Y4 )
= ( ( ord_less_int @ X2 @ Y4 )
| ( ord_less_int @ Y4 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_364_linorder__neq__iff,axiom,
! [X2: real,Y4: real] :
( ( X2 != Y4 )
= ( ( ord_less_real @ X2 @ Y4 )
| ( ord_less_real @ Y4 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_365_order__less__asym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X2 ) ) ).
% order_less_asym
thf(fact_366_order__less__asym,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X2 ) ) ).
% order_less_asym
thf(fact_367_order__less__asym,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X2 ) ) ).
% order_less_asym
thf(fact_368_linorder__neqE,axiom,
! [X2: nat,Y4: nat] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_369_linorder__neqE,axiom,
! [X2: int,Y4: int] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_int @ Y4 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_370_linorder__neqE,axiom,
! [X2: real,Y4: real] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_real @ Y4 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_371_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_372_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_373_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_374_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_375_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_376_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_377_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_378_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_379_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_380_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X2 )
| ( X2 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_381_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y4: int] :
( ( ~ ( ord_less_int @ X2 @ Y4 ) )
= ( ( ord_less_int @ Y4 @ X2 )
| ( X2 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_382_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y4: real] :
( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( ( ord_less_real @ Y4 @ X2 )
| ( X2 = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_383_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_384_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_385_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_386_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_387_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_388_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_389_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_390_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_391_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_392_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_393_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_394_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_395_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_396_linorder__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ( X2 != Y4 )
=> ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_cases
thf(fact_397_linorder__cases,axiom,
! [X2: int,Y4: int] :
( ~ ( ord_less_int @ X2 @ Y4 )
=> ( ( X2 != Y4 )
=> ( ord_less_int @ Y4 @ X2 ) ) ) ).
% linorder_cases
thf(fact_398_linorder__cases,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ( X2 != Y4 )
=> ( ord_less_real @ Y4 @ X2 ) ) ) ).
% linorder_cases
thf(fact_399_antisym__conv3,axiom,
! [Y4: nat,X2: nat] :
( ~ ( ord_less_nat @ Y4 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_400_antisym__conv3,axiom,
! [Y4: int,X2: int] :
( ~ ( ord_less_int @ Y4 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_401_antisym__conv3,axiom,
! [Y4: real,X2: real] :
( ~ ( ord_less_real @ Y4 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv3
thf(fact_402_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X4 )
=> ( P @ Y3 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_403_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_404_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_405_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_406_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_407_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_408_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_409_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_410_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_411_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_412_less__imp__neq,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% less_imp_neq
thf(fact_413_less__imp__neq,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% less_imp_neq
thf(fact_414_less__imp__neq,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( X2 != Y4 ) ) ).
% less_imp_neq
thf(fact_415_dense,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ? [Z4: real] :
( ( ord_less_real @ X2 @ Z4 )
& ( ord_less_real @ Z4 @ Y4 ) ) ) ).
% dense
thf(fact_416_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_417_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_418_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_419_lt__ex,axiom,
! [X2: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X2 ) ).
% lt_ex
thf(fact_420_lt__ex,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% lt_ex
thf(fact_421_linorder__neqE__nat,axiom,
! [X2: nat,Y4: nat] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_422_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_423_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_424_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_425_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_426_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_427_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_428_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_429_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_430_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_431_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_432_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_433_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_434_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_435_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_436_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_437_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_438_strict__sorted__simps_I1_J,axiom,
sorted_wrt_b @ ord_less_b @ nil_b ).
% strict_sorted_simps(1)
thf(fact_439_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_440_strict__sorted__simps_I1_J,axiom,
sorted_wrt_int @ ord_less_int @ nil_int ).
% strict_sorted_simps(1)
thf(fact_441_strict__sorted__simps_I1_J,axiom,
sorted_wrt_real @ ord_less_real @ nil_real ).
% strict_sorted_simps(1)
thf(fact_442_leD,axiom,
! [Y4: b,X2: b] :
( ( ord_less_eq_b @ Y4 @ X2 )
=> ~ ( ord_less_b @ X2 @ Y4 ) ) ).
% leD
thf(fact_443_leD,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y4 ) ) ).
% leD
thf(fact_444_leD,axiom,
! [Y4: int,X2: int] :
( ( ord_less_eq_int @ Y4 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y4 ) ) ).
% leD
thf(fact_445_leD,axiom,
! [Y4: real,X2: real] :
( ( ord_less_eq_real @ Y4 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y4 ) ) ).
% leD
thf(fact_446_leI,axiom,
! [X2: b,Y4: b] :
( ~ ( ord_less_b @ X2 @ Y4 )
=> ( ord_less_eq_b @ Y4 @ X2 ) ) ).
% leI
thf(fact_447_leI,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% leI
thf(fact_448_leI,axiom,
! [X2: int,Y4: int] :
( ~ ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X2 ) ) ).
% leI
thf(fact_449_leI,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% leI
thf(fact_450_nless__le,axiom,
! [A: b,B: b] :
( ( ~ ( ord_less_b @ A @ B ) )
= ( ~ ( ord_less_eq_b @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_451_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_452_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_453_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_454_antisym__conv1,axiom,
! [X2: b,Y4: b] :
( ~ ( ord_less_b @ X2 @ Y4 )
=> ( ( ord_less_eq_b @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_455_antisym__conv1,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_456_antisym__conv1,axiom,
! [X2: int,Y4: int] :
( ~ ( ord_less_int @ X2 @ Y4 )
=> ( ( ord_less_eq_int @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_457_antisym__conv1,axiom,
! [X2: real,Y4: real] :
( ~ ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% antisym_conv1
thf(fact_458_antisym__conv2,axiom,
! [X2: b,Y4: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
=> ( ( ~ ( ord_less_b @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_459_antisym__conv2,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_460_antisym__conv2,axiom,
! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ( ~ ( ord_less_int @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_461_antisym__conv2,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( X2 = Y4 ) ) ) ).
% antisym_conv2
thf(fact_462_dense__ge,axiom,
! [Z3: real,Y4: real] :
( ! [X4: real] :
( ( ord_less_real @ Z3 @ X4 )
=> ( ord_less_eq_real @ Y4 @ X4 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ).
% dense_ge
thf(fact_463_dense__le,axiom,
! [Y4: real,Z3: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ X4 @ Z3 ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ).
% dense_le
thf(fact_464_less__le__not__le,axiom,
( ord_less_b
= ( ^ [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
& ~ ( ord_less_eq_b @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_465_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_466_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_467_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_468_not__le__imp__less,axiom,
! [Y4: b,X2: b] :
( ~ ( ord_less_eq_b @ Y4 @ X2 )
=> ( ord_less_b @ X2 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_469_not__le__imp__less,axiom,
! [Y4: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ord_less_nat @ X2 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_470_not__le__imp__less,axiom,
! [Y4: int,X2: int] :
( ~ ( ord_less_eq_int @ Y4 @ X2 )
=> ( ord_less_int @ X2 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_471_not__le__imp__less,axiom,
! [Y4: real,X2: real] :
( ~ ( ord_less_eq_real @ Y4 @ X2 )
=> ( ord_less_real @ X2 @ Y4 ) ) ).
% not_le_imp_less
thf(fact_472_order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [A3: b,B2: b] :
( ( ord_less_b @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_473_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_474_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_475_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_476_order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [A3: b,B2: b] :
( ( ord_less_eq_b @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_477_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_478_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_479_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_480_order_Ostrict__trans1,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_b @ B @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_481_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_482_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_483_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_484_order_Ostrict__trans2,axiom,
! [A: b,B: b,C: b] :
( ( ord_less_b @ A @ B )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_485_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_486_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_487_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_488_order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [A3: b,B2: b] :
( ( ord_less_eq_b @ A3 @ B2 )
& ~ ( ord_less_eq_b @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_489_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_490_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_491_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_492_dense__ge__bounded,axiom,
! [Z3: real,X2: real,Y4: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y4 @ W ) ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_493_dense__le__bounded,axiom,
! [X2: real,Y4: real,Z3: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y4 )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_494_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [B2: b,A3: b] :
( ( ord_less_b @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_495_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_496_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_497_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_real @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_498_dual__order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [B2: b,A3: b] :
( ( ord_less_eq_b @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_499_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_500_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_501_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_502_dual__order_Ostrict__trans1,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_eq_b @ B @ A )
=> ( ( ord_less_b @ C @ B )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_503_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_504_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_505_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_506_dual__order_Ostrict__trans2,axiom,
! [B: b,A: b,C: b] :
( ( ord_less_b @ B @ A )
=> ( ( ord_less_eq_b @ C @ B )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_507_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_508_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_509_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_510_dual__order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [B2: b,A3: b] :
( ( ord_less_eq_b @ B2 @ A3 )
& ~ ( ord_less_eq_b @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_511_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_512_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_513_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_514_order_Ostrict__implies__order,axiom,
! [A: b,B: b] :
( ( ord_less_b @ A @ B )
=> ( ord_less_eq_b @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_515_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_516_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_517_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_518_dual__order_Ostrict__implies__order,axiom,
! [B: b,A: b] :
( ( ord_less_b @ B @ A )
=> ( ord_less_eq_b @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_519_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_520_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_521_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_522_order__le__less,axiom,
( ord_less_eq_b
= ( ^ [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_523_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_524_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_525_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_526_order__less__le,axiom,
( ord_less_b
= ( ^ [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_527_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_528_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_529_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_530_linorder__not__le,axiom,
! [X2: b,Y4: b] :
( ( ~ ( ord_less_eq_b @ X2 @ Y4 ) )
= ( ord_less_b @ Y4 @ X2 ) ) ).
% linorder_not_le
thf(fact_531_linorder__not__le,axiom,
! [X2: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y4 ) )
= ( ord_less_nat @ Y4 @ X2 ) ) ).
% linorder_not_le
thf(fact_532_linorder__not__le,axiom,
! [X2: int,Y4: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y4 ) )
= ( ord_less_int @ Y4 @ X2 ) ) ).
% linorder_not_le
thf(fact_533_linorder__not__le,axiom,
! [X2: real,Y4: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y4 ) )
= ( ord_less_real @ Y4 @ X2 ) ) ).
% linorder_not_le
thf(fact_534_linorder__not__less,axiom,
! [X2: b,Y4: b] :
( ( ~ ( ord_less_b @ X2 @ Y4 ) )
= ( ord_less_eq_b @ Y4 @ X2 ) ) ).
% linorder_not_less
thf(fact_535_linorder__not__less,axiom,
! [X2: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_not_less
thf(fact_536_linorder__not__less,axiom,
! [X2: int,Y4: int] :
( ( ~ ( ord_less_int @ X2 @ Y4 ) )
= ( ord_less_eq_int @ Y4 @ X2 ) ) ).
% linorder_not_less
thf(fact_537_linorder__not__less,axiom,
! [X2: real,Y4: real] :
( ( ~ ( ord_less_real @ X2 @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X2 ) ) ).
% linorder_not_less
thf(fact_538_order__less__imp__le,axiom,
! [X2: b,Y4: b] :
( ( ord_less_b @ X2 @ Y4 )
=> ( ord_less_eq_b @ X2 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_539_order__less__imp__le,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_540_order__less__imp__le,axiom,
! [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ord_less_eq_int @ X2 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_541_order__less__imp__le,axiom,
! [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ X2 @ Y4 ) ) ).
% order_less_imp_le
thf(fact_542_order__le__neq__trans,axiom,
! [A: b,B: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( A != B )
=> ( ord_less_b @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_543_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_544_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_545_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_546_order__neq__le__trans,axiom,
! [A: b,B: b] :
( ( A != B )
=> ( ( ord_less_eq_b @ A @ B )
=> ( ord_less_b @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_547_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_548_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_549_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_550_order__le__less__trans,axiom,
! [X2: b,Y4: b,Z3: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
=> ( ( ord_less_b @ Y4 @ Z3 )
=> ( ord_less_b @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_551_order__le__less__trans,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_552_order__le__less__trans,axiom,
! [X2: int,Y4: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_553_order__le__less__trans,axiom,
! [X2: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_554_order__less__le__trans,axiom,
! [X2: b,Y4: b,Z3: b] :
( ( ord_less_b @ X2 @ Y4 )
=> ( ( ord_less_eq_b @ Y4 @ Z3 )
=> ( ord_less_b @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_555_order__less__le__trans,axiom,
! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_556_order__less__le__trans,axiom,
! [X2: int,Y4: int,Z3: int] :
( ( ord_less_int @ X2 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_557_order__less__le__trans,axiom,
! [X2: real,Y4: real,Z3: real] :
( ( ord_less_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z3 )
=> ( ord_less_real @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_558_order__le__less__subst1,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_559_order__le__less__subst1,axiom,
! [A: b,F: int > b,B: int,C: int] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_560_order__le__less__subst1,axiom,
! [A: b,F: real > b,B: real,C: real] :
( ( ord_less_eq_b @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_561_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_562_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_563_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_564_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_565_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_566_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_567_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_568_order__le__less__subst2,axiom,
! [A: b,B: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_569_order__le__less__subst2,axiom,
! [A: b,B: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_570_order__le__less__subst2,axiom,
! [A: b,B: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_571_order__le__less__subst2,axiom,
! [A: b,B: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_572_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_573_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_574_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_575_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_576_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > b,C: b] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_b @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_577_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_578_order__less__le__subst1,axiom,
! [A: b,F: b > b,B: b,C: b] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_579_order__less__le__subst1,axiom,
! [A: nat,F: b > nat,B: b,C: b] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_580_order__less__le__subst1,axiom,
! [A: int,F: b > int,B: b,C: b] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_581_order__less__le__subst1,axiom,
! [A: real,F: b > real,B: b,C: b] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_b @ B @ C )
=> ( ! [X4: b,Y2: b] :
( ( ord_less_eq_b @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_582_order__less__le__subst1,axiom,
! [A: b,F: nat > b,B: nat,C: nat] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_583_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_584_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_585_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_586_order__less__le__subst1,axiom,
! [A: b,F: int > b,B: int,C: int] :
( ( ord_less_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_587_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_588_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_589_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > b,C: b] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_590_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > b,C: b] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_b @ ( F @ B ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_b @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_591_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_592_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_593_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_594_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_595_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_596_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: real,Y2: real] :
( ( ord_less_real @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_597_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_598_linorder__le__less__linear,axiom,
! [X2: b,Y4: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
| ( ord_less_b @ Y4 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_599_linorder__le__less__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_nat @ Y4 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_600_linorder__le__less__linear,axiom,
! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
| ( ord_less_int @ Y4 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_601_linorder__le__less__linear,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
| ( ord_less_real @ Y4 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_602_order__le__imp__less__or__eq,axiom,
! [X2: b,Y4: b] :
( ( ord_less_eq_b @ X2 @ Y4 )
=> ( ( ord_less_b @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_603_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_604_order__le__imp__less__or__eq,axiom,
! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ( ord_less_int @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_605_order__le__imp__less__or__eq,axiom,
! [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_606_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_607_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_608_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_609_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_610_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_611_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_612_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_613_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_614_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_615_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_616_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_617_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_618_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_619_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_620_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_621_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_622_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_623_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_624_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_625_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_626_list_Osimps_I8_J,axiom,
! [F: nat > b] :
( ( map_nat_b @ F @ nil_nat )
= nil_b ) ).
% list.simps(8)
thf(fact_627_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_628_sorted__wrt_Osimps_I1_J,axiom,
! [P: b > b > $o] : ( sorted_wrt_b @ P @ nil_b ) ).
% sorted_wrt.simps(1)
thf(fact_629_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_630_mset_Osimps_I1_J,axiom,
( ( mset_nat @ nil_nat )
= zero_z7348594199698428585et_nat ) ).
% mset.simps(1)
thf(fact_631_mset_Osimps_I1_J,axiom,
( ( mset_b @ nil_b )
= zero_zero_multiset_b ) ).
% mset.simps(1)
thf(fact_632_remove1_Osimps_I1_J,axiom,
! [X2: nat] :
( ( remove1_nat @ X2 @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_633_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_634_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_635_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_636_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_637_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_638_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_639_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_640_strict__sorted__imp__sorted,axiom,
! [Xs2: list_b] :
( ( sorted_wrt_b @ ord_less_b @ Xs2 )
=> ( sorted_wrt_b @ ord_less_eq_b @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_641_strict__sorted__imp__sorted,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_642_strict__sorted__imp__sorted,axiom,
! [Xs2: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs2 )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_643_strict__sorted__imp__sorted,axiom,
! [Xs2: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs2 )
=> ( sorted_wrt_real @ ord_less_eq_real @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_644_sorted0,axiom,
sorted_wrt_b @ ord_less_eq_b @ nil_b ).
% sorted0
thf(fact_645_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_646_sorted0,axiom,
sorted_wrt_int @ ord_less_eq_int @ nil_int ).
% sorted0
thf(fact_647_sorted0,axiom,
sorted_wrt_real @ ord_less_eq_real @ nil_real ).
% sorted0
thf(fact_648_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_649_replicate__0,axiom,
! [X2: nat] :
( ( replicate_nat @ zero_zero_nat @ X2 )
= nil_nat ) ).
% replicate_0
thf(fact_650_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > b] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_b @ ( sort_map_b @ F @ N @ I ) @ ( sort_map_b @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_651_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N @ I ) @ ( sort_map_nat @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_652_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > int] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_int @ ( sort_map_int @ F @ N @ I ) @ ( sort_map_int @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_653_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > real] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N @ I ) @ ( sort_map_real @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_654_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_655_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_656_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_657_sorted__insort,axiom,
! [X2: b,Xs2: list_b] :
( ( sorted_wrt_b @ ord_less_eq_b
@ ( linord2228103134468323771ey_b_b
@ ^ [X: b] : X
@ X2
@ Xs2 ) )
= ( sorted_wrt_b @ ord_less_eq_b @ Xs2 ) ) ).
% sorted_insort
thf(fact_658_sorted__insort,axiom,
! [X2: nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% sorted_insort
thf(fact_659_sorted__insort,axiom,
! [X2: int,Xs2: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int
@ ( linord734827384618529109nt_int
@ ^ [X: int] : X
@ X2
@ Xs2 ) )
= ( sorted_wrt_int @ ord_less_eq_int @ Xs2 ) ) ).
% sorted_insort
thf(fact_660_sorted__insort,axiom,
! [X2: real,Xs2: list_real] :
( ( sorted_wrt_real @ ord_less_eq_real
@ ( linord1674302359176591317l_real
@ ^ [X: real] : X
@ X2
@ Xs2 ) )
= ( sorted_wrt_real @ ord_less_eq_real @ Xs2 ) ) ).
% sorted_insort
thf(fact_661_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_662_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_663_reals__Archimedean2,axiom,
! [X2: real] :
? [N3: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_664_real__arch__simple,axiom,
! [X2: real] :
? [N3: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_665_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_666_minf_I8_J,axiom,
! [T: b] :
? [Z4: b] :
! [X5: b] :
( ( ord_less_b @ X5 @ Z4 )
=> ~ ( ord_less_eq_b @ T @ X5 ) ) ).
% minf(8)
thf(fact_667_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_668_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_669_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_670_minf_I6_J,axiom,
! [T: b] :
? [Z4: b] :
! [X5: b] :
( ( ord_less_b @ X5 @ Z4 )
=> ( ord_less_eq_b @ X5 @ T ) ) ).
% minf(6)
thf(fact_671_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_672_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_673_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_674_pinf_I8_J,axiom,
! [T: b] :
? [Z4: b] :
! [X5: b] :
( ( ord_less_b @ Z4 @ X5 )
=> ( ord_less_eq_b @ T @ X5 ) ) ).
% pinf(8)
thf(fact_675_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_676_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_677_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_678_pinf_I6_J,axiom,
! [T: b] :
? [Z4: b] :
! [X5: b] :
( ( ord_less_b @ Z4 @ X5 )
=> ~ ( ord_less_eq_b @ X5 @ T ) ) ).
% pinf(6)
thf(fact_679_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_680_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_681_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_682_verit__comp__simplify1_I3_J,axiom,
! [B4: b,A5: b] :
( ( ~ ( ord_less_eq_b @ B4 @ A5 ) )
= ( ord_less_b @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_683_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_684_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_685_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_686_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_687_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_688_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_689_verit__la__disequality,axiom,
! [A: b,B: b] :
( ( A = B )
| ~ ( ord_less_eq_b @ A @ B )
| ~ ( ord_less_eq_b @ B @ A ) ) ).
% verit_la_disequality
thf(fact_690_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_691_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_692_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_693_verit__comp__simplify1_I2_J,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_694_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_695_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_696_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_697_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_698_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_699_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_700_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_701_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_702_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_703_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_704_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_705_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ Z5 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_706_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_707_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_708_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_709_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_710_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_711_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_712_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_713_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_714_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_715_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_716_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ Z4 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_717_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ Z4 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_718_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_719_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_720_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_721_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_722_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_723_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z5 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_724_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_725_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_726_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_727_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_728_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_729_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_730_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_731_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_732_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_733_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_734_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z4 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_735_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z4 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_736_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_737_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_738_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_739_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_740_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_741_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_742_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_743_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A @ X5 )
& ( ord_less_real @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_744_sorted__list__of__multiset__def,axiom,
( linord3047872887403683810et_nat
= ( fold_m3004868085856202685st_nat
@ ( linord8961336180081300637at_nat
@ ^ [X: nat] : X )
@ nil_nat ) ) ).
% sorted_list_of_multiset_def
thf(fact_745_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_746_insort__remove1,axiom,
! [A: b,Xs2: list_b] :
( ( member_b @ A @ ( set_b2 @ Xs2 ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ Xs2 )
=> ( ( linord2228103134468323771ey_b_b
@ ^ [X: b] : X
@ A
@ ( remove1_b @ A @ Xs2 ) )
= Xs2 ) ) ) ).
% insort_remove1
thf(fact_747_insort__remove1,axiom,
! [A: nat,Xs2: list_nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ A
@ ( remove1_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ).
% insort_remove1
thf(fact_748_insort__remove1,axiom,
! [A: int,Xs2: list_int] :
( ( member_int @ A @ ( set_int2 @ Xs2 ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Xs2 )
=> ( ( linord734827384618529109nt_int
@ ^ [X: int] : X
@ A
@ ( remove1_int @ A @ Xs2 ) )
= Xs2 ) ) ) ).
% insort_remove1
thf(fact_749_insort__remove1,axiom,
! [A: real,Xs2: list_real] :
( ( member_real @ A @ ( set_real2 @ Xs2 ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ Xs2 )
=> ( ( linord1674302359176591317l_real
@ ^ [X: real] : X
@ A
@ ( remove1_real @ A @ Xs2 ) )
= Xs2 ) ) ) ).
% insort_remove1
thf(fact_750_map__eq__conv,axiom,
! [F: nat > b,Xs2: list_nat,G: nat > b] :
( ( ( map_nat_b @ F @ Xs2 )
= ( map_nat_b @ G @ Xs2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_751_map__eq__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_752_set__sort,axiom,
! [F: b > b,Xs2: list_b] :
( ( set_b2 @ ( linord5847811544569201088ey_b_b @ F @ Xs2 ) )
= ( set_b2 @ Xs2 ) ) ).
% set_sort
thf(fact_753_set__sort,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( set_nat2 @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_sort
thf(fact_754_in__set__remove1,axiom,
! [A: real,B: real,Xs2: list_real] :
( ( A != B )
=> ( ( member_real @ A @ ( set_real2 @ ( remove1_real @ B @ Xs2 ) ) )
= ( member_real @ A @ ( set_real2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_755_in__set__replicate,axiom,
! [X2: real,N: nat,Y4: real] :
( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N @ Y4 ) ) )
= ( ( X2 = Y4 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_756_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_757_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_758_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_759_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_760_imp__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_761_conj__le__cong,axiom,
! [X2: int,X7: int,P: $o,P4: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_762_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_763_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_764_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_765_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_766_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_767_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_768_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_769_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_770_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_771_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_772_subset__code_I1_J,axiom,
! [Xs2: list_real,B5: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
= ( ! [X: real] :
( ( member_real @ X @ ( set_real2 @ Xs2 ) )
=> ( member_real @ X @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_773_ex__map__conv,axiom,
! [Ys: list_b,F: nat > b] :
( ( ? [Xs: list_nat] :
( Ys
= ( map_nat_b @ F @ Xs ) ) )
= ( ! [X: b] :
( ( member_b @ X @ ( set_b2 @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_774_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_775_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > b,G: nat > b] :
( ( Xs2 = Ys )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_b @ F @ Xs2 )
= ( map_nat_b @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_776_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 = Ys )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_777_map__idI,axiom,
! [Xs2: list_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_real_real @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_778_map__idI,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_779_map__ext,axiom,
! [Xs2: list_nat,F: nat > b,G: nat > b] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_b @ F @ Xs2 )
= ( map_nat_b @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_780_map__ext,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_781_list_Omap__ident__strong,axiom,
! [T: list_real,F: real > real] :
( ! [Z4: real] :
( ( member_real @ Z4 @ ( set_real2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_real_real @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_782_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_783_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > b,Fa: nat > b] :
( ! [Z4: nat,Za: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_nat_b @ F @ X2 )
= ( map_nat_b @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_784_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z4: nat,Za: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_785_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > b,G: nat > b] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_nat_b @ F @ X2 )
= ( map_nat_b @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_786_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_787_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > b,G: nat > b] :
( ( X2 = Ya )
=> ( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_nat_b @ F @ X2 )
= ( map_nat_b @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_788_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z4: nat] :
( ( member_nat @ Z4 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_789_sorted__wrt__mono__rel,axiom,
! [Xs2: list_real,P: real > real > $o,Q: real > real > $o] :
( ! [X4: real,Y2: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
=> ( ( member_real @ Y2 @ ( set_real2 @ Xs2 ) )
=> ( ( P @ X4 @ Y2 )
=> ( Q @ X4 @ Y2 ) ) ) )
=> ( ( sorted_wrt_real @ P @ Xs2 )
=> ( sorted_wrt_real @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_790_sorted__wrt__mono__rel,axiom,
! [Xs2: list_b,P: b > b > $o,Q: b > b > $o] :
( ! [X4: b,Y2: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
=> ( ( member_b @ Y2 @ ( set_b2 @ Xs2 ) )
=> ( ( P @ X4 @ Y2 )
=> ( Q @ X4 @ Y2 ) ) ) )
=> ( ( sorted_wrt_b @ P @ Xs2 )
=> ( sorted_wrt_b @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_791_sorted__wrt__mono__rel,axiom,
! [Xs2: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X4: nat,Y2: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ Xs2 ) )
=> ( ( P @ X4 @ Y2 )
=> ( Q @ X4 @ Y2 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs2 )
=> ( sorted_wrt_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_792_mset__eq__setD,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ( mset_b @ Xs2 )
= ( mset_b @ Ys ) )
=> ( ( set_b2 @ Xs2 )
= ( set_b2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_793_notin__set__remove1,axiom,
! [X2: real,Xs2: list_real,Y4: real] :
( ~ ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
=> ~ ( member_real @ X2 @ ( set_real2 @ ( remove1_real @ Y4 @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_794_remove1__idem,axiom,
! [X2: real,Xs2: list_real] :
( ~ ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
=> ( ( remove1_real @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_795_insort__insert__triv,axiom,
! [X2: real,Xs2: list_real] :
( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
=> ( ( linord1891625487229344476l_real
@ ^ [X: real] : X
@ X2
@ Xs2 )
= Xs2 ) ) ).
% insort_insert_triv
thf(fact_796_strict__sorted__equal,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( sorted_wrt_b @ ord_less_b @ Xs2 )
=> ( ( sorted_wrt_b @ ord_less_b @ Ys )
=> ( ( ( set_b2 @ Ys )
= ( set_b2 @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ).
% strict_sorted_equal
thf(fact_797_strict__sorted__equal,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ).
% strict_sorted_equal
thf(fact_798_strict__sorted__equal,axiom,
! [Xs2: list_int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs2 )
=> ( ( sorted_wrt_int @ ord_less_int @ Ys )
=> ( ( ( set_int2 @ Ys )
= ( set_int2 @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ).
% strict_sorted_equal
thf(fact_799_strict__sorted__equal,axiom,
! [Xs2: list_real,Ys: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs2 )
=> ( ( sorted_wrt_real @ ord_less_real @ Ys )
=> ( ( ( set_real2 @ Ys )
= ( set_real2 @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ).
% strict_sorted_equal
thf(fact_800_insort__insert__insort,axiom,
! [X2: real,Xs2: list_real] :
( ~ ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
=> ( ( linord1891625487229344476l_real
@ ^ [X: real] : X
@ X2
@ Xs2 )
= ( linord1674302359176591317l_real
@ ^ [X: real] : X
@ X2
@ Xs2 ) ) ) ).
% insort_insert_insort
thf(fact_801_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_802_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_803_sort__key__inj__key__eq,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > b] :
( ( ( mset_nat @ Xs2 )
= ( mset_nat @ Ys ) )
=> ( ( inj_on_nat_b @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Ys ) )
=> ( ( linord2522810402733747245_nat_b @ F @ Xs2 )
= Ys ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_804_sort__key__inj__key__eq,axiom,
! [Xs2: list_b,Ys: list_b,F: b > nat] :
( ( ( mset_b @ Xs2 )
= ( mset_b @ Ys ) )
=> ( ( inj_on_b_nat @ F @ ( set_b2 @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_b_nat @ F @ Ys ) )
=> ( ( linord8949695267366631119_b_nat @ F @ Xs2 )
= Ys ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_805_sort__key__inj__key__eq,axiom,
! [Xs2: list_b,Ys: list_b,F: b > int] :
( ( ( mset_b @ Xs2 )
= ( mset_b @ Ys ) )
=> ( ( inj_on_b_int @ F @ ( set_b2 @ Xs2 ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_b_int @ F @ Ys ) )
=> ( ( linord8947204796857580843_b_int @ F @ Xs2 )
= Ys ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_806_sort__key__inj__key__eq,axiom,
! [Xs2: list_b,Ys: list_b,F: b > real] :
( ( ( mset_b @ Xs2 )
= ( mset_b @ Ys ) )
=> ( ( inj_on_b_real @ F @ ( set_b2 @ Xs2 ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_b_real @ F @ Ys ) )
=> ( ( linord7704725295488125483b_real @ F @ Xs2 )
= Ys ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_807_sort__key__inj__key__eq,axiom,
! [Xs2: list_b,Ys: list_b,F: b > b] :
( ( ( mset_b @ Xs2 )
= ( mset_b @ Ys ) )
=> ( ( inj_on_b_b @ F @ ( set_b2 @ Xs2 ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ ( map_b_b @ F @ Ys ) )
=> ( ( linord5847811544569201088ey_b_b @ F @ Xs2 )
= Ys ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_808_sort__key__inj__key__eq,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat] :
( ( ( mset_nat @ Xs2 )
= ( mset_nat @ Ys ) )
=> ( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Ys ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs2 )
= Ys ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_809_properties__for__sort__key,axiom,
! [Ys: list_real,Xs2: list_real,F: real > b] :
( ( ( mset_real @ Ys )
= ( mset_real @ Xs2 ) )
=> ( ! [K2: real] :
( ( member_real @ K2 @ ( set_real2 @ Ys ) )
=> ( ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ ( map_real_b @ F @ Ys ) )
=> ( ( linord2738525447772944593real_b @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_810_properties__for__sort__key,axiom,
! [Ys: list_nat,Xs2: list_nat,F: nat > b] :
( ( ( mset_nat @ Ys )
= ( mset_nat @ Xs2 ) )
=> ( ! [K2: nat] :
( ( member_nat @ K2 @ ( set_nat2 @ Ys ) )
=> ( ( filter_nat
@ ^ [X: nat] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_nat
@ ^ [X: nat] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Ys ) )
=> ( ( linord2522810402733747245_nat_b @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_811_properties__for__sort__key,axiom,
! [Ys: list_real,Xs2: list_real,F: real > nat] :
( ( ( mset_real @ Ys )
= ( mset_real @ Xs2 ) )
=> ( ! [K2: real] :
( ( member_real @ K2 @ ( set_real2 @ Ys ) )
=> ( ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_real_nat @ F @ Ys ) )
=> ( ( linord5472101314125696382al_nat @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_812_properties__for__sort__key,axiom,
! [Ys: list_b,Xs2: list_b,F: b > nat] :
( ( ( mset_b @ Ys )
= ( mset_b @ Xs2 ) )
=> ( ! [K2: b] :
( ( member_b @ K2 @ ( set_b2 @ Ys ) )
=> ( ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_b_nat @ F @ Ys ) )
=> ( ( linord8949695267366631119_b_nat @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_813_properties__for__sort__key,axiom,
! [Ys: list_real,Xs2: list_real,F: real > int] :
( ( ( mset_real @ Ys )
= ( mset_real @ Xs2 ) )
=> ( ! [K2: real] :
( ( member_real @ K2 @ ( set_real2 @ Ys ) )
=> ( ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_real_int @ F @ Ys ) )
=> ( ( linord5469610843616646106al_int @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_814_properties__for__sort__key,axiom,
! [Ys: list_b,Xs2: list_b,F: b > int] :
( ( ( mset_b @ Ys )
= ( mset_b @ Xs2 ) )
=> ( ! [K2: b] :
( ( member_b @ K2 @ ( set_b2 @ Ys ) )
=> ( ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_b_int @ F @ Ys ) )
=> ( ( linord8947204796857580843_b_int @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_815_properties__for__sort__key,axiom,
! [Ys: list_real,Xs2: list_real,F: real > real] :
( ( ( mset_real @ Ys )
= ( mset_real @ Xs2 ) )
=> ( ! [K2: real] :
( ( member_real @ K2 @ ( set_real2 @ Ys ) )
=> ( ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_real
@ ^ [X: real] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_real_real @ F @ Ys ) )
=> ( ( linord6132810859473402202l_real @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_816_properties__for__sort__key,axiom,
! [Ys: list_b,Xs2: list_b,F: b > real] :
( ( ( mset_b @ Ys )
= ( mset_b @ Xs2 ) )
=> ( ! [K2: b] :
( ( member_b @ K2 @ ( set_b2 @ Ys ) )
=> ( ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_b_real @ F @ Ys ) )
=> ( ( linord7704725295488125483b_real @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_817_properties__for__sort__key,axiom,
! [Ys: list_b,Xs2: list_b,F: b > b] :
( ( ( mset_b @ Ys )
= ( mset_b @ Xs2 ) )
=> ( ! [K2: b] :
( ( member_b @ K2 @ ( set_b2 @ Ys ) )
=> ( ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_b
@ ^ [X: b] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ ( map_b_b @ F @ Ys ) )
=> ( ( linord5847811544569201088ey_b_b @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_818_properties__for__sort__key,axiom,
! [Ys: list_nat,Xs2: list_nat,F: nat > nat] :
( ( ( mset_nat @ Ys )
= ( mset_nat @ Xs2 ) )
=> ( ! [K2: nat] :
( ( member_nat @ K2 @ ( set_nat2 @ Ys ) )
=> ( ( filter_nat
@ ^ [X: nat] :
( ( F @ K2 )
= ( F @ X ) )
@ Ys )
= ( filter_nat
@ ^ [X: nat] :
( ( F @ K2 )
= ( F @ X ) )
@ Xs2 ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Ys ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs2 )
= Ys ) ) ) ) ).
% properties_for_sort_key
thf(fact_819_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_820_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_821_strict__sorted__equal__Uniq,axiom,
! [A2: set_b] :
( uniq_list_b
@ ^ [Xs: list_b] :
( ( sorted_wrt_b @ ord_less_b @ Xs )
& ( ( set_b2 @ Xs )
= A2 ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_822_strict__sorted__equal__Uniq,axiom,
! [A2: set_nat] :
( uniq_list_nat
@ ^ [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
& ( ( set_nat2 @ Xs )
= A2 ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_823_strict__sorted__equal__Uniq,axiom,
! [A2: set_int] :
( uniq_list_int
@ ^ [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
& ( ( set_int2 @ Xs )
= A2 ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_824_strict__sorted__equal__Uniq,axiom,
! [A2: set_real] :
( uniq_list_real
@ ^ [Xs: list_real] :
( ( sorted_wrt_real @ ord_less_real @ Xs )
& ( ( set_real2 @ Xs )
= A2 ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_825_ex__inverse__of__nat__less,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X2 ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_826_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_827_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_828_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_829_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_830_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_831_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_832_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_833_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_834_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_835_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_836_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_837_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_838_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_839_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_840_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_841_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_842_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_843_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_844_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_845_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_846_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_847_set__filter,axiom,
! [P: real > $o,Xs2: list_real] :
( ( set_real2 @ ( filter_real @ P @ Xs2 ) )
= ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ ( set_real2 @ Xs2 ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_848_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_849_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_850_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_851_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_852_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_853_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_854_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_855_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_856_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_857_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_858_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_859_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_860_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_861_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_862_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_863_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_864_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_865_filter__False,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X4 ) )
=> ( ( filter_nat @ P @ Xs2 )
= nil_nat ) ) ).
% filter_False
thf(fact_866_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_867_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_868_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_869_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_870_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_871_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_872_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_873_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_874_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_875_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_876_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_877_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_878_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_879_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_880_filter__cong,axiom,
! [Xs2: list_real,Ys: list_real,P: real > $o,Q: real > $o] :
( ( Xs2 = Ys )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Ys ) )
=> ( ( P @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_real @ P @ Xs2 )
= ( filter_real @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_881_inj__on__of__nat,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ semiri1316708129612266289at_nat @ N4 ) ).
% inj_on_of_nat
thf(fact_882_inj__on__of__nat,axiom,
! [N4: set_nat] : ( inj_on_nat_int @ semiri1314217659103216013at_int @ N4 ) ).
% inj_on_of_nat
thf(fact_883_inj__on__of__nat,axiom,
! [N4: set_nat] : ( inj_on_nat_real @ semiri5074537144036343181t_real @ N4 ) ).
% inj_on_of_nat
thf(fact_884_sorted__wrt__filter,axiom,
! [F: b > b > $o,Xs2: list_b,P: b > $o] :
( ( sorted_wrt_b @ F @ Xs2 )
=> ( sorted_wrt_b @ F @ ( filter_b @ P @ Xs2 ) ) ) ).
% sorted_wrt_filter
thf(fact_885_sorted__wrt__filter,axiom,
! [F: nat > nat > $o,Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ F @ Xs2 )
=> ( sorted_wrt_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ).
% sorted_wrt_filter
thf(fact_886_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N2: nat] :
( ( N2 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_887_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D3: real,E: real] :
( ( ord_less_real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_888_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_889_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_890_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_891_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_892_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_893_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_894_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_895_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_896_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_897_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_898_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_899_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_900_filter__sort,axiom,
! [P: b > $o,F: b > b,Xs2: list_b] :
( ( filter_b @ P @ ( linord5847811544569201088ey_b_b @ F @ Xs2 ) )
= ( linord5847811544569201088ey_b_b @ F @ ( filter_b @ P @ Xs2 ) ) ) ).
% filter_sort
thf(fact_901_filter__sort,axiom,
! [P: nat > $o,F: nat > nat,Xs2: list_nat] :
( ( filter_nat @ P @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( linord738340561235409698at_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ).
% filter_sort
thf(fact_902_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_903_int__cases2,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_904_sort__key__stable,axiom,
! [F: b > b,K: b,Xs2: list_b] :
( ( filter_b
@ ^ [Y: b] :
( ( F @ Y )
= K )
@ ( linord5847811544569201088ey_b_b @ F @ Xs2 ) )
= ( filter_b
@ ^ [Y: b] :
( ( F @ Y )
= K )
@ Xs2 ) ) ).
% sort_key_stable
thf(fact_905_sort__key__stable,axiom,
! [F: nat > nat,K: nat,Xs2: list_nat] :
( ( filter_nat
@ ^ [Y: nat] :
( ( F @ Y )
= K )
@ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( filter_nat
@ ^ [Y: nat] :
( ( F @ Y )
= K )
@ Xs2 ) ) ).
% sort_key_stable
thf(fact_906_empty__filter__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( nil_nat
= ( filter_nat @ P @ Xs2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_907_filter__empty__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ( filter_nat @ P @ Xs2 )
= nil_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_908_filter__replicate,axiom,
! [P: nat > $o,X2: nat,N: nat] :
( ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) ) ) ).
% filter_replicate
thf(fact_909_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_910_nat__mono,axiom,
! [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y4 ) ) ) ).
% nat_mono
thf(fact_911_eq__nat__nat__iff,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z6 ) )
= ( Z3 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_912_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% all_nat
thf(fact_913_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
& ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% ex_nat
thf(fact_914_sorted__same,axiom,
! [G: list_b > b,Xs2: list_b] :
( sorted_wrt_b @ ord_less_eq_b
@ ( filter_b
@ ^ [X: b] :
( X
= ( G @ Xs2 ) )
@ Xs2 ) ) ).
% sorted_same
thf(fact_915_sorted__same,axiom,
! [G: list_nat > nat,Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( filter_nat
@ ^ [X: nat] :
( X
= ( G @ Xs2 ) )
@ Xs2 ) ) ).
% sorted_same
thf(fact_916_sorted__same,axiom,
! [G: list_int > int,Xs2: list_int] :
( sorted_wrt_int @ ord_less_eq_int
@ ( filter_int
@ ^ [X: int] :
( X
= ( G @ Xs2 ) )
@ Xs2 ) ) ).
% sorted_same
thf(fact_917_sorted__same,axiom,
! [G: list_real > real,Xs2: list_real] :
( sorted_wrt_real @ ord_less_eq_real
@ ( filter_real
@ ^ [X: real] :
( X
= ( G @ Xs2 ) )
@ Xs2 ) ) ).
% sorted_same
thf(fact_918_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_919_sorted__filter,axiom,
! [F: nat > b,Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) )
=> ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ ( filter_nat @ P @ Xs2 ) ) ) ) ).
% sorted_filter
thf(fact_920_sorted__filter,axiom,
! [F: nat > nat,Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ) ).
% sorted_filter
thf(fact_921_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_922_nat__le__iff,axiom,
! [X2: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
= ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_923_zless__nat__eq__int__zless,axiom,
! [M: nat,Z3: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_924_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_925_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_926_sorted__map__same,axiom,
! [F: nat > b,G: list_nat > b,Xs2: list_nat] :
( sorted_wrt_b @ ord_less_eq_b
@ ( map_nat_b @ F
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ X )
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_map_same
thf(fact_927_sorted__map__same,axiom,
! [F: nat > nat,G: list_nat > nat,Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( map_nat_nat @ F
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ X )
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_map_same
thf(fact_928_sort__key__eq__sort__key,axiom,
! [Xs2: list_b,Ys: list_b,F: b > b] :
( ( ( mset_b @ Xs2 )
= ( mset_b @ Ys ) )
=> ( ( inj_on_b_b @ F @ ( set_b2 @ Xs2 ) )
=> ( ( linord5847811544569201088ey_b_b @ F @ Xs2 )
= ( linord5847811544569201088ey_b_b @ F @ Ys ) ) ) ) ).
% sort_key_eq_sort_key
thf(fact_929_sort__key__eq__sort__key,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat] :
( ( ( mset_nat @ Xs2 )
= ( mset_nat @ Ys ) )
=> ( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs2 )
= ( linord738340561235409698at_nat @ F @ Ys ) ) ) ) ).
% sort_key_eq_sort_key
thf(fact_930_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_931_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_932_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_933_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_934_filter__insort,axiom,
! [F: nat > b,Xs2: list_nat,P: nat > $o,X2: nat] :
( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) )
=> ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( linord9222868044636945266_nat_b @ F @ X2 @ Xs2 ) )
= ( linord9222868044636945266_nat_b @ F @ X2 @ ( filter_nat @ P @ Xs2 ) ) ) ) ) ).
% filter_insort
thf(fact_935_filter__insort,axiom,
! [F: nat > nat,Xs2: list_nat,P: nat > $o,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( linord8961336180081300637at_nat @ F @ X2 @ Xs2 ) )
= ( linord8961336180081300637at_nat @ F @ X2 @ ( filter_nat @ P @ Xs2 ) ) ) ) ) ).
% filter_insort
thf(fact_936_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_937_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_938_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_939_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_940_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_941_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_942_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_943_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_944_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_945_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_946_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_947_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_948_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_949_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_950_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_951_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_952_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_953_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_954_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X5 ) ).
% linordered_field_no_lb
thf(fact_955_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_956_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_957_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_958_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_959_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_960_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_961_positive__imp__inverse__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_962_negative__imp__inverse__negative,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_963_inverse__positive__imp__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_964_inverse__negative__imp__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_965_less__imp__inverse__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_966_inverse__less__imp__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_967_less__imp__inverse__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_968_inverse__less__imp__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_969_nonzero__inverse__minus__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_970_inverse__le__imp__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_971_le__imp__inverse__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_972_inverse__le__imp__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_973_le__imp__inverse__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_974_real__arch__invD,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ).
% real_arch_invD
thf(fact_975_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ( F @ X4 )
!= ( F @ Y2 ) ) ) ) )
=> ( ! [X4: nat,Y2: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X4 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_976_complete__real,axiom,
! [S2: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S2 )
=> ( ? [Z5: real] :
! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z5 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Y2 ) )
& ! [Z5: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z5 ) )
=> ( ord_less_eq_real @ Y2 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_977_inj__on__id2,axiom,
! [A2: set_nat] :
( inj_on_nat_nat
@ ^ [X: nat] : X
@ A2 ) ).
% inj_on_id2
thf(fact_978_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [I4: nat,J2: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ( member_nat @ J2 @ A2 )
=> ( ( ord_less_nat @ I4 @ J2 )
=> ( ( F @ I4 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_979_subset__inj__on,axiom,
! [F: nat > nat,B5: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_980_inj__on__subset,axiom,
! [F: nat > nat,A2: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( inj_on_nat_nat @ F @ B5 ) ) ) ).
% inj_on_subset
thf(fact_981_inj__on__iff__Uniq,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A6: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A6 )
=> ( uniq_nat
@ ^ [Y: nat] :
( ( member_nat @ Y @ A6 )
& ( ( F2 @ X )
= ( F2 @ Y ) ) ) ) ) ) ) ).
% inj_on_iff_Uniq
thf(fact_982_seq__mono__lemma,axiom,
! [M: nat,D4: nat > real,E2: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_real @ ( D4 @ N3 ) @ ( E2 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( E2 @ N3 ) @ ( E2 @ M ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ord_less_real @ ( D4 @ N5 ) @ ( E2 @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_983_subsetI,axiom,
! [A2: set_real,B5: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B5 ) )
=> ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% subsetI
thf(fact_984_psubsetD,axiom,
! [A2: set_real,B5: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B5 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_985_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_real_o
@ ^ [X: real] : ( member_real @ X @ A6 )
@ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_986_in__mono,axiom,
! [A2: set_real,B5: set_real,X2: real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_987_subsetD,axiom,
! [A2: set_real,B5: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B5 ) ) ) ).
% subsetD
thf(fact_988_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X: real] :
( ( member_real @ X @ A6 )
=> ( member_real @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_989_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A6 )
=> ( member_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_990_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_991_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ A6 )
@ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_992_insort__key__remove1,axiom,
! [A: real,Xs2: list_real,F: real > b] :
( ( member_real @ A @ ( set_real2 @ Xs2 ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ ( map_real_b @ F @ Xs2 ) )
=> ( ( ( hd_real
@ ( filter_real
@ ^ [X: real] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord3786284756137309590real_b @ F @ A @ ( remove1_real @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_993_insort__key__remove1,axiom,
! [A: nat,Xs2: list_nat,F: nat > b] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_b @ ord_less_eq_b @ ( map_nat_b @ F @ Xs2 ) )
=> ( ( ( hd_nat
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord9222868044636945266_nat_b @ F @ A @ ( remove1_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_994_insort__key__remove1,axiom,
! [A: real,Xs2: list_real,F: real > nat] :
( ( member_real @ A @ ( set_real2 @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_real_nat @ F @ Xs2 ) )
=> ( ( ( hd_real
@ ( filter_real
@ ^ [X: real] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord6097763018041792889al_nat @ F @ A @ ( remove1_real @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_995_insort__key__remove1,axiom,
! [A: nat,Xs2: list_nat,F: nat > nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( ( ( hd_nat
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord8961336180081300637at_nat @ F @ A @ ( remove1_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_996_insort__key__remove1,axiom,
! [A: real,Xs2: list_real,F: real > int] :
( ( member_real @ A @ ( set_real2 @ Xs2 ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_real_int @ F @ Xs2 ) )
=> ( ( ( hd_real
@ ( filter_real
@ ^ [X: real] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord6095272547532742613al_int @ F @ A @ ( remove1_real @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_997_insort__key__remove1,axiom,
! [A: nat,Xs2: list_nat,F: nat > int] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_nat_int @ F @ Xs2 ) )
=> ( ( ( hd_nat
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord8958845709572250361at_int @ F @ A @ ( remove1_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_998_insort__key__remove1,axiom,
! [A: real,Xs2: list_real,F: real > real] :
( ( member_real @ A @ ( set_real2 @ Xs2 ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_real_real @ F @ Xs2 ) )
=> ( ( ( hd_real
@ ( filter_real
@ ^ [X: real] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord1674302359176591317l_real @ F @ A @ ( remove1_real @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_999_insort__key__remove1,axiom,
! [A: nat,Xs2: list_nat,F: nat > real] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_real @ ord_less_eq_real @ ( map_nat_real @ F @ Xs2 ) )
=> ( ( ( hd_nat
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord1530454444912894201t_real @ F @ A @ ( remove1_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_1000_fps__inverse__zero_H,axiom,
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_zero'
thf(fact_1001_bgauge__existence__lemma,axiom,
! [S: set_real,Q3: real > real > $o] :
( ( ! [X: real] :
( ( member_real @ X @ S )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ( Q3 @ D5 @ X ) ) ) )
= ( ! [X: real] :
? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ( ( member_real @ X @ S )
=> ( Q3 @ D5 @ X ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_1002_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_1003_hd__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( hd_nat @ ( replicate_nat @ N @ X2 ) )
= X2 ) ) ).
% hd_replicate
thf(fact_1004_Compl__eq,axiom,
( uminus612125837232591019t_real
= ( ^ [A6: set_real] :
( collect_real
@ ^ [X: real] :
~ ( member_real @ X @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_1005_uminus__set__def,axiom,
( uminus612125837232591019t_real
= ( ^ [A6: set_real] :
( collect_real
@ ( uminus_uminus_real_o
@ ^ [X: real] : ( member_real @ X @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1006_list_Oset__sel_I1_J,axiom,
! [A: list_real] :
( ( A != nil_real )
=> ( member_real @ ( hd_real @ A ) @ ( set_real2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1007_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1008_hd__in__set,axiom,
! [Xs2: list_real] :
( ( Xs2 != nil_real )
=> ( member_real @ ( hd_real @ Xs2 ) @ ( set_real2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_1009_hd__in__set,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs2 ) @ ( set_nat2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_1010_hd__map,axiom,
! [Xs2: list_nat,F: nat > b] :
( ( Xs2 != nil_nat )
=> ( ( hd_b @ ( map_nat_b @ F @ Xs2 ) )
= ( F @ ( hd_nat @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_1011_hd__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( hd_nat @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_1012_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > b] :
( ( A != nil_nat )
=> ( ( hd_b @ ( map_nat_b @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1013_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1014_pred__subset__eq,axiom,
! [R: set_real,S2: set_real] :
( ( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ R )
@ ^ [X: real] : ( member_real @ X @ S2 ) )
= ( ord_less_eq_set_real @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_1015_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_1016_real__eq__0__iff__le__ge__0,axiom,
! [X2: real] :
( ( X2 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1017_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1018_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1019_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_1020_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_1021_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1022_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1023_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_int @ Z3 )
= zero_zero_int )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1024_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_real @ Z3 )
= zero_zero_real )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1025_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_1026_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_1027_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_1028_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_1029_of__int__minus,axiom,
! [Z3: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z3 ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_minus
thf(fact_1030_of__int__minus,axiom,
! [Z3: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z3 ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_int_minus
thf(fact_1031_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_1032_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_1033_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_1034_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_1035_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1036_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1037_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1038_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1039_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_1040_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_1041_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_1042_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_1043_ex__less__of__int,axiom,
! [X2: real] :
? [Z4: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).
% ex_less_of_int
thf(fact_1044_ex__of__int__less,axiom,
! [X2: real] :
? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X2 ) ).
% ex_of_int_less
thf(fact_1045_ex__le__of__int,axiom,
! [X2: real] :
? [Z4: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).
% ex_le_of_int
thf(fact_1046_of__int__nonneg,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_nonneg
thf(fact_1047_of__int__nonneg,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_int_nonneg
thf(fact_1048_of__int__pos,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_pos
thf(fact_1049_of__int__pos,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_int_pos
thf(fact_1050_of__nat__less__of__int__iff,axiom,
! [N: nat,X2: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% of_nat_less_of_int_iff
thf(fact_1051_of__nat__less__of__int__iff,axiom,
! [N: nat,X2: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% of_nat_less_of_int_iff
thf(fact_1052_arg__min__inj__eq,axiom,
! [F: nat > nat,P: nat > $o,A: nat] :
( ( inj_on_nat_nat @ F @ ( collect_nat @ P ) )
=> ( ( P @ A )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ ( F @ A ) @ ( F @ Y2 ) ) )
=> ( ( lattic8739620818006775868at_nat @ F @ P )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_1053_sort__key__conv__fold,axiom,
! [F: b > b,Xs2: list_b] :
( ( inj_on_b_b @ F @ ( set_b2 @ Xs2 ) )
=> ( ( linord5847811544569201088ey_b_b @ F @ Xs2 )
= ( fold_b_list_b @ ( linord2228103134468323771ey_b_b @ F ) @ Xs2 @ nil_b ) ) ) ).
% sort_key_conv_fold
thf(fact_1054_sort__key__conv__fold,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs2 )
= ( fold_nat_list_nat @ ( linord8961336180081300637at_nat @ F ) @ Xs2 @ nil_nat ) ) ) ).
% sort_key_conv_fold
thf(fact_1055_sort__conv__fold,axiom,
! [Xs2: list_b] :
( ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ Xs2 )
= ( fold_b_list_b
@ ( linord2228103134468323771ey_b_b
@ ^ [X: b] : X )
@ Xs2
@ nil_b ) ) ).
% sort_conv_fold
thf(fact_1056_sort__conv__fold,axiom,
! [Xs2: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs2 )
= ( fold_nat_list_nat
@ ( linord8961336180081300637at_nat
@ ^ [X: nat] : X )
@ Xs2
@ nil_nat ) ) ).
% sort_conv_fold
thf(fact_1057_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X2 ) @ C ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1058_radical__0,axiom,
! [N: nat,R2: nat > real > real,A: formal3361831859752904756s_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( formal8005797870169972230l_real @ R2 @ zero_zero_nat @ A @ N )
= zero_zero_real ) ) ).
% radical_0
thf(fact_1059_fps__inverse__eq__0_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_eq_0'
thf(fact_1060_fps__inverse__eq__0__iff_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real )
= ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real ) ) ).
% fps_inverse_eq_0_iff'
thf(fact_1061_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1062_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1063_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_1064_of__int__mult,axiom,
! [W2: int,Z3: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z3 ) )
= ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_int_mult
thf(fact_1065_of__int__mult,axiom,
! [W2: int,Z3: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z3 ) )
= ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_mult
thf(fact_1066_fps__mult__nth__0,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ zero_zero_nat )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_1067_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ zero_zero_nat )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_1068_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ zero_zero_nat )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_1069_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_nat,K: nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ ( semiri1524631719018205113ps_nat @ K ) ) @ N )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( semiri1316708129612266289at_nat @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_1070_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_int,K: nat,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ ( semiri6570152736363784213ps_int @ K ) ) @ N )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ N ) @ ( semiri1314217659103216013at_int @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_1071_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal3361831859752904756s_real,K: nat,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ ( semiri2475410149736220053s_real @ K ) ) @ N )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ N ) @ ( semiri5074537144036343181t_real @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_1072_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( semiri1524631719018205113ps_nat @ K ) @ F ) @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ K ) @ ( formal3720337525774269570th_nat @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_1073_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( semiri6570152736363784213ps_int @ K ) @ F ) @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ ( formal3717847055265219294th_int @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_1074_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( semiri2475410149736220053s_real @ K ) @ F ) @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( formal2580924720334399070h_real @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_1075_fps__zero__nth,axiom,
! [N: nat] :
( ( formal3720337525774269570th_nat @ zero_z8531573698755551073ps_nat @ N )
= zero_zero_nat ) ).
% fps_zero_nth
thf(fact_1076_fps__zero__nth,axiom,
! [N: nat] :
( ( formal3717847055265219294th_int @ zero_z4353722679246354365ps_int @ N )
= zero_zero_int ) ).
% fps_zero_nth
thf(fact_1077_fps__zero__nth,axiom,
! [N: nat] :
( ( formal2580924720334399070h_real @ zero_z7760665558314615101s_real @ N )
= zero_zero_real ) ).
% fps_zero_nth
thf(fact_1078_fps__nth__of__nat,axiom,
! [N: nat,C: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N )
= ( semiri1316708129612266289at_nat @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N )
= zero_zero_nat ) ) ) ).
% fps_nth_of_nat
thf(fact_1079_fps__nth__of__nat,axiom,
! [N: nat,C: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N )
= ( semiri1314217659103216013at_int @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N )
= zero_zero_int ) ) ) ).
% fps_nth_of_nat
thf(fact_1080_fps__nth__of__nat,axiom,
! [N: nat,C: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N )
= ( semiri5074537144036343181t_real @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N )
= zero_zero_real ) ) ) ).
% fps_nth_of_nat
thf(fact_1081_fps__nth__of__int,axiom,
! [N: nat,C: int] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( ring_14195589558981938179ps_int @ C ) @ N )
= ( ring_1_of_int_int @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( ring_14195589558981938179ps_int @ C ) @ N )
= zero_zero_int ) ) ) ).
% fps_nth_of_int
thf(fact_1082_fps__nth__of__int,axiom,
! [N: nat,C: int] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( ring_13846316942275908227s_real @ C ) @ N )
= ( ring_1_of_int_real @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( ring_13846316942275908227s_real @ C ) @ N )
= zero_zero_real ) ) ) ).
% fps_nth_of_int
thf(fact_1083_fps__inverse__idempotent,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( invers68952373231134600s_real @ ( invers68952373231134600s_real @ F ) )
= F ) ) ).
% fps_inverse_idempotent
thf(fact_1084_fps__inverse__0__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= zero_zero_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real ) ) ).
% fps_inverse_0_iff
thf(fact_1085_fps__inverse__nth__0,axiom,
! [F: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ).
% fps_inverse_nth_0
thf(fact_1086_real__minus__mult__self__le,axiom,
! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_1087_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y3: real] :
? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_1088_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1089_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1090_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1091_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1092_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1093_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1094_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1095_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1096_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1097_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1098_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1099_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1100_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1101_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1102_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1103_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1104_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_1105_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1106_nat__mult__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1107_nat__mult__distrib__neg,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1108_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1109_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1110_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1111_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1112_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1113_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1114_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1115_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1116_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1117_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1118_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1119_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1120_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1121_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1122_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1123_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1124_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1125_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1126_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1127_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1128_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1129_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1130_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1131_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1132_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I4: nat] :
( ( Q @ I4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I4 ) )
& ( ord_less_eq_real @ ( X4 @ I4 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I5: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I5 ) @ one_one_nat )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q @ I5 )
& ( ( X5 @ I5 )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I5 )
= zero_zero_nat ) )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q @ I5 )
& ( ( X5 @ I5 )
= one_one_real ) )
=> ( ( L2 @ X5 @ I5 )
= one_one_nat ) )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q @ I5 )
& ( ( L2 @ X5 @ I5 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I5 ) @ ( F @ X5 @ I5 ) ) )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q @ I5 )
& ( ( L2 @ X5 @ I5 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I5 ) @ ( X5 @ I5 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1133_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_1134_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1135_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1136_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1137_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1138_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1139_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1140_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1141_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1142_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1143_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1144_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1145_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1146_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1147_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1148_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1149_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1150_inj__Suc,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_1151_Suc__inject,axiom,
! [X2: nat,Y4: nat] :
( ( ( suc @ X2 )
= ( suc @ Y4 ) )
=> ( X2 = Y4 ) ) ).
% Suc_inject
thf(fact_1152_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1153_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1154_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y2: nat,Z4: nat] :
( ( R @ X4 @ Y2 )
=> ( ( R @ Y2 @ Z4 )
=> ( R @ X4 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1155_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1156_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1157_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1158_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1159_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1160_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1161_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1162_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1163_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1164_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1165_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1166_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1167_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1168_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1169_old_Onat_Oexhaust,axiom,
! [Y4: nat] :
( ( Y4 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y4
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1170_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1171_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] :
( ( P @ X4 @ Y2 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1172_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1173_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1174_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1175_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1176_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1177_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1178_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1179_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1180_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1181_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1182_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1183_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1184_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1185_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1186_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1187_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1188_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1189_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1190_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1191_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1192_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1193_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1194_int__cases,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1195_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_1196_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1197_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1198_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1199_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1200_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1201_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1202_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1203_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1204_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1205_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1206_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1207_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1208_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1209_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1210_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1211_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1212_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1213_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1214_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1215_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1216_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1217_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D3: real,E: real] :
( ( ord_less_real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1218_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1219_negD,axiom,
! [X2: int] :
( ( ord_less_int @ X2 @ zero_zero_int )
=> ? [N3: nat] :
( X2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1220_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1221_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_1222_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1223_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1224_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1225_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1226_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1227_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1228_real__add__minus__iff,axiom,
! [X2: real,A: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X2 = A ) ) ).
% real_add_minus_iff
thf(fact_1229_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1230_zle__add1__eq__le,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1231_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1232_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1233_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1234_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1235_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_1236_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1237_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1238_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1239_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1240_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1241_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1242_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1243_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1244_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1245_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1246_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1247_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z: int] :
? [N2: nat] :
( Z
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1248_zadd__int__left,axiom,
! [M: nat,N: nat,Z3: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_1249_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1250_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_1251_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_1252_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1253_nat__add__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( nat2 @ ( plus_plus_int @ Z3 @ Z6 ) )
= ( plus_plus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1254_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1255_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1256_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1257_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1258_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1259_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1260_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y4: int] :
( ( if_int @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y4: int] :
( ( if_int @ $true @ X2 @ Y4 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y4: nat] :
( ( if_nat @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y4: nat] :
( ( if_nat @ $true @ X2 @ Y4 )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y4: real] :
( ( if_real @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y4: real] :
( ( if_real @ $true @ X2 @ Y4 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( map_nat_b @ ( sort_map_b @ f @ n ) @ ( upt @ zero_zero_nat @ n ) )
= ( linord5847811544569201088ey_b_b
@ ^ [X: b] : X
@ ( map_nat_b @ f @ ( upt @ zero_zero_nat @ n ) ) ) ) ).
%------------------------------------------------------------------------------