TPTP Problem File: SLH0262^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00294_010942__14786342_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1436 ( 682 unt; 157 typ; 0 def)
% Number of atoms : 3329 (1484 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 9386 ( 459 ~; 100 |; 242 &;7331 @)
% ( 0 <=>;1254 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 1086 (1086 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 139 usr; 17 con; 0-5 aty)
% Number of variables : 3185 ( 149 ^;2784 !; 252 ?;3185 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:45:09.686
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__List__Olist_It__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J_J,type,
list_l7067579673986547845_nat_b: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J_J,type,
set_list_nat_b_nat_b: $tType ).
thf(ty_n_t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
list_nat_b_nat_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
set_nat_b_nat_b: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
set_list_real: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (139)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
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_OAbs__fps_001t__Real__Oreal,type,
formal798729627605919420s_real: ( nat > real ) > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Fun_Oid_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
id_nat_b_nat_b: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_Fun_Oid_001_062_It__Nat__Onat_Mtf__b_J,type,
id_nat_b: ( nat > b ) > nat > b ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_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_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_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_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__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_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
if_list_nat_b_nat_b: $o > list_nat_b_nat_b > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__List__Olist_It__Real__Oreal_J,type,
if_list_real: $o > list_real > list_real > list_real ).
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__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_List_Oappend_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
append_nat_b_nat_b: list_nat_b_nat_b > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
append3853175007740601498_nat_b: list_l7067579673986547845_nat_b > list_l7067579673986547845_nat_b > list_l7067579673986547845_nat_b ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Int__Oint_J,type,
append_list_int: list_list_int > list_list_int > list_list_int ).
thf(sy_c_List_Oappend_001t__Real__Oreal,type,
append_real: list_real > list_real > list_real ).
thf(sy_c_List_Obind_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
bind_n614534383549643347_nat_b: list_nat_b_nat_b > ( ( ( nat > b ) > nat > b ) > list_nat_b_nat_b ) > list_nat_b_nat_b ).
thf(sy_c_List_Obind_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Int__Oint,type,
bind_nat_b_nat_b_int: list_nat_b_nat_b > ( ( ( nat > b ) > nat > b ) > list_int ) > list_int ).
thf(sy_c_List_Obind_001t__Int__Oint_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
bind_int_nat_b_nat_b: list_int > ( int > list_nat_b_nat_b ) > list_nat_b_nat_b ).
thf(sy_c_List_Obind_001t__Int__Oint_001t__Int__Oint,type,
bind_int_int: list_int > ( int > list_int ) > list_int ).
thf(sy_c_List_Obutlast_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
butlast_nat_b_nat_b: list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Obutlast_001t__Int__Oint,type,
butlast_int: list_int > list_int ).
thf(sy_c_List_Obutlast_001t__Real__Oreal,type,
butlast_real: list_real > list_real ).
thf(sy_c_List_Ocan__select_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
can_se6628294081300001064_nat_b: ( ( ( nat > b ) > nat > b ) > $o ) > set_nat_b_nat_b > $o ).
thf(sy_c_List_Ocan__select_001t__Real__Oreal,type,
can_select_real: ( real > $o ) > set_real > $o ).
thf(sy_c_List_Oconcat_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
concat_nat_b_nat_b: list_l7067579673986547845_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Oconcat_001t__Int__Oint,type,
concat_int: list_list_int > list_int ).
thf(sy_c_List_Ofold_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
fold_n1883935874045088215_nat_b: ( ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_nat_b_nat_b > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__b_J,type,
fold_n5624292861071574516_nat_b: ( ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_nat_b_nat_b > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001t__Int__Oint_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
fold_int_nat_b_nat_b: ( int > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_int > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_List_Ofold_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
fold_r4529934814194841502_nat_b: ( real > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ) > list_real > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b ).
thf(sy_c_List_Ogen__length_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
gen_le7432612553264103954_nat_b: nat > list_nat_b_nat_b > nat ).
thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
gen_length_int: nat > list_int > nat ).
thf(sy_c_List_Oinsert_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
insert_nat_b_nat_b: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
insert_real: real > list_real > list_real ).
thf(sy_c_List_Olast_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
last_nat_b_nat_b: list_nat_b_nat_b > ( nat > b ) > nat > b ).
thf(sy_c_List_Olast_001t__Int__Oint,type,
last_int: list_int > int ).
thf(sy_c_List_Olast_001t__Real__Oreal,type,
last_real: list_real > real ).
thf(sy_c_List_Olist_OCons_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
cons_nat_b_nat_b: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
cons_l8694591469225861951_nat_b: list_nat_b_nat_b > list_l7067579673986547845_nat_b > list_l7067579673986547845_nat_b ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
thf(sy_c_List_Olist_ONil_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
nil_nat_b_nat_b: list_nat_b_nat_b ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
nil_list_nat_b_nat_b: list_l7067579673986547845_nat_b ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_Oset_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
set_nat_b_nat_b2: list_nat_b_nat_b > set_nat_b_nat_b ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
set_list_nat_b_nat_b2: list_l7067579673986547845_nat_b > set_list_nat_b_nat_b ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
set_list_int2: list_list_int > set_list_int ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist__ex1_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
list_ex1_nat_b_nat_b: ( ( ( nat > b ) > nat > b ) > $o ) > list_nat_b_nat_b > $o ).
thf(sy_c_List_Olist__ex1_001t__Int__Oint,type,
list_ex1_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Omaps_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
maps_n8879270819509224205_nat_b: ( ( ( nat > b ) > nat > b ) > list_nat_b_nat_b ) > list_nat_b_nat_b > list_nat_b_nat_b ).
thf(sy_c_List_Omaps_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Int__Oint,type,
maps_nat_b_nat_b_int: ( ( ( nat > b ) > nat > b ) > list_int ) > list_nat_b_nat_b > list_int ).
thf(sy_c_List_Omaps_001t__Int__Oint_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
maps_int_nat_b_nat_b: ( int > list_nat_b_nat_b ) > list_int > list_nat_b_nat_b ).
thf(sy_c_List_Omaps_001t__Int__Oint_001t__Int__Oint,type,
maps_int_int: ( int > list_int ) > list_int > list_int ).
thf(sy_c_List_On__lists_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
n_lists_nat_b_nat_b: nat > list_nat_b_nat_b > list_l7067579673986547845_nat_b ).
thf(sy_c_List_On__lists_001t__Int__Oint,type,
n_lists_int: nat > list_int > list_list_int ).
thf(sy_c_List_Onths_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
nths_nat_b_nat_b: list_nat_b_nat_b > set_nat > list_nat_b_nat_b ).
thf(sy_c_List_Onths_001t__Int__Oint,type,
nths_int: list_int > set_nat > list_int ).
thf(sy_c_List_Onths_001t__Real__Oreal,type,
nths_real: list_real > set_nat > list_real ).
thf(sy_c_List_Oproduct__lists_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
produc8493759205635542326_nat_b: list_l7067579673986547845_nat_b > list_l7067579673986547845_nat_b ).
thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
product_lists_int: list_list_int > list_list_int ).
thf(sy_c_List_Oset__Cons_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
set_Cons_nat_b_nat_b: set_nat_b_nat_b > set_list_nat_b_nat_b > set_list_nat_b_nat_b ).
thf(sy_c_List_Oset__Cons_001t__Int__Oint,type,
set_Cons_int: set_int > set_list_int > set_list_int ).
thf(sy_c_List_Oset__Cons_001t__Real__Oreal,type,
set_Cons_real: set_real > set_list_real > set_list_real ).
thf(sy_c_List_Osubseqs_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
subseqs_nat_b_nat_b: list_nat_b_nat_b > list_l7067579673986547845_nat_b ).
thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
subseqs_int: list_int > list_list_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Median_Osort__primitive_001t__Nat__Onat_001tf__b,type,
sort_primitive_nat_b: nat > nat > ( nat > b ) > nat > b ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_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__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
ord_le9047053354294502011_nat_b: set_nat_b_nat_b > set_nat_b_nat_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
collec3644297838412717002_nat_b: ( list_nat_b_nat_b > $o ) > set_list_nat_b_nat_b ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Real__Oreal_J,type,
collect_list_real: ( list_real > $o ) > set_list_real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Othe__elem_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
the_elem_nat_b_nat_b: set_nat_b_nat_b > ( nat > b ) > nat > b ).
thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
the_elem_int: set_int > int ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J,type,
member_nat_b_nat_b: ( ( nat > b ) > nat > b ) > set_nat_b_nat_b > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
member8456897207961558412_nat_b: list_nat_b_nat_b > set_list_nat_b_nat_b > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
member_list_real: list_real > set_list_real > $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_v_X,type,
x: nat > a > b ).
thf(sy_v_is__swap____,type,
is_swap: ( ( nat > b ) > nat > b ) > $o ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_meas__ptw____,type,
meas_ptw: ( nat > a > b ) > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x____,type,
x2: ( nat > b ) > nat > b ).
thf(sy_v_xs____,type,
xs: list_nat_b_nat_b ).
% Relevant facts (1265)
thf(fact_0_ind__step,axiom,
! [G: nat > a > b,X: ( nat > b ) > nat > b] :
( ( meas_ptw @ G )
=> ( ( is_swap @ X )
=> ( meas_ptw
@ ^ [K: nat,Omega: a] :
( X
@ ^ [I: nat] : ( G @ I @ Omega )
@ K ) ) ) ) ).
% ind_step
thf(fact_1_snoc_Oprems,axiom,
! [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X2 @ ( set_nat_b_nat_b2 @ ( append_nat_b_nat_b @ xs @ ( cons_nat_b_nat_b @ x2 @ nil_nat_b_nat_b ) ) ) )
=> ( is_swap @ X2 ) ) ).
% snoc.prems
thf(fact_2_is__swap__def,axiom,
( is_swap
= ( ^ [Ts: ( nat > b ) > nat > b] :
? [I: nat] :
( ( ord_less_nat @ I @ n )
& ? [J: nat] :
( ( ord_less_nat @ J @ n )
& ( Ts
= ( sort_primitive_nat_b @ I @ J ) ) ) ) ) ) ).
% is_swap_def
thf(fact_3_assms_I2_J,axiom,
ord_less_nat @ j @ n ).
% assms(2)
thf(fact_4_append1__eq__conv,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b,Ys: list_nat_b_nat_b,Y: ( nat > b ) > nat > b] :
( ( ( append_nat_b_nat_b @ Xs @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) )
= ( append_nat_b_nat_b @ Ys @ ( cons_nat_b_nat_b @ Y @ nil_nat_b_nat_b ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_5_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_6_append_Oright__neutral,axiom,
! [A: list_int] :
( ( append_int @ A @ nil_int )
= A ) ).
% append.right_neutral
thf(fact_7_append_Oright__neutral,axiom,
! [A: list_nat_b_nat_b] :
( ( append_nat_b_nat_b @ A @ nil_nat_b_nat_b )
= A ) ).
% append.right_neutral
thf(fact_8_append__Nil2,axiom,
! [Xs: list_int] :
( ( append_int @ Xs @ nil_int )
= Xs ) ).
% append_Nil2
thf(fact_9_append__Nil2,axiom,
! [Xs: list_nat_b_nat_b] :
( ( append_nat_b_nat_b @ Xs @ nil_nat_b_nat_b )
= Xs ) ).
% append_Nil2
thf(fact_10_append__self__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_int ) ) ).
% append_self_conv
thf(fact_11_append__self__conv,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat_b_nat_b ) ) ).
% append_self_conv
thf(fact_12_self__append__conv,axiom,
! [Y: list_int,Ys: list_int] :
( ( Y
= ( append_int @ Y @ Ys ) )
= ( Ys = nil_int ) ) ).
% self_append_conv
thf(fact_13_self__append__conv,axiom,
! [Y: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( Y
= ( append_nat_b_nat_b @ Y @ Ys ) )
= ( Ys = nil_nat_b_nat_b ) ) ).
% self_append_conv
thf(fact_14_append__self__conv2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_int ) ) ).
% append_self_conv2
thf(fact_15_append__self__conv2,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat_b_nat_b ) ) ).
% append_self_conv2
thf(fact_16_self__append__conv2,axiom,
! [Y: list_nat_b_nat_b,Xs: list_nat_b_nat_b] :
( ( Y
= ( append_nat_b_nat_b @ Xs @ Y ) )
= ( Xs = nil_nat_b_nat_b ) ) ).
% self_append_conv2
thf(fact_17_self__append__conv2,axiom,
! [Y: list_int,Xs: list_int] :
( ( Y
= ( append_int @ Xs @ Y ) )
= ( Xs = nil_int ) ) ).
% self_append_conv2
thf(fact_18_Nil__is__append__conv,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( nil_nat_b_nat_b
= ( append_nat_b_nat_b @ Xs @ Ys ) )
= ( ( Xs = nil_nat_b_nat_b )
& ( Ys = nil_nat_b_nat_b ) ) ) ).
% Nil_is_append_conv
thf(fact_19_Nil__is__append__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( nil_int
= ( append_int @ Xs @ Ys ) )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_20_append__is__Nil__conv,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Xs @ Ys )
= nil_nat_b_nat_b )
= ( ( Xs = nil_nat_b_nat_b )
& ( Ys = nil_nat_b_nat_b ) ) ) ).
% append_is_Nil_conv
thf(fact_21_append__is__Nil__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_22_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_23_split__list,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ? [Ys2: list_real,Zs: list_real] :
( Xs
= ( append_real @ Ys2 @ ( cons_real @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_24_split__list,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ? [Ys2: list_nat_b_nat_b,Zs: list_nat_b_nat_b] :
( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_25_split__list,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys2: list_int,Zs: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs ) ) ) ) ).
% split_list
thf(fact_26_split__list__last,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ? [Ys2: list_real,Zs: list_real] :
( ( Xs
= ( append_real @ Ys2 @ ( cons_real @ X @ Zs ) ) )
& ~ ( member_real @ X @ ( set_real2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_27_split__list__last,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ? [Ys2: list_nat_b_nat_b,Zs: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X @ Zs ) ) )
& ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_28_split__list__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys2: list_int,Zs: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_29_list_Oinject,axiom,
! [X21: ( nat > b ) > nat > b,X22: list_nat_b_nat_b,Y21: ( nat > b ) > nat > b,Y22: list_nat_b_nat_b] :
( ( ( cons_nat_b_nat_b @ X21 @ X22 )
= ( cons_nat_b_nat_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_30_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_31_same__append__eq,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,Zs2: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Xs @ Ys )
= ( append_nat_b_nat_b @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_32_same__append__eq,axiom,
! [Xs: list_int,Ys: list_int,Zs2: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_33_append__same__eq,axiom,
! [Ys: list_nat_b_nat_b,Xs: list_nat_b_nat_b,Zs2: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Ys @ Xs )
= ( append_nat_b_nat_b @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_34_append__same__eq,axiom,
! [Ys: list_int,Xs: list_int,Zs2: list_int] :
( ( ( append_int @ Ys @ Xs )
= ( append_int @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_35_append__assoc,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,Zs2: list_nat_b_nat_b] :
( ( append_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) @ Zs2 )
= ( append_nat_b_nat_b @ Xs @ ( append_nat_b_nat_b @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_36_append__assoc,axiom,
! [Xs: list_int,Ys: list_int,Zs2: list_int] :
( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs2 )
= ( append_int @ Xs @ ( append_int @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_37_append_Oassoc,axiom,
! [A: list_nat_b_nat_b,B: list_nat_b_nat_b,C: list_nat_b_nat_b] :
( ( append_nat_b_nat_b @ ( append_nat_b_nat_b @ A @ B ) @ C )
= ( append_nat_b_nat_b @ A @ ( append_nat_b_nat_b @ B @ C ) ) ) ).
% append.assoc
thf(fact_38_append_Oassoc,axiom,
! [A: list_int,B: list_int,C: list_int] :
( ( append_int @ ( append_int @ A @ B ) @ C )
= ( append_int @ A @ ( append_int @ B @ C ) ) ) ).
% append.assoc
thf(fact_39_not__Cons__self2,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( cons_nat_b_nat_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_40_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_41_append__eq__append__conv2,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,Zs2: list_nat_b_nat_b,Ts2: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Xs @ Ys )
= ( append_nat_b_nat_b @ Zs2 @ Ts2 ) )
= ( ? [Us: list_nat_b_nat_b] :
( ( ( Xs
= ( append_nat_b_nat_b @ Zs2 @ Us ) )
& ( ( append_nat_b_nat_b @ Us @ Ys )
= Ts2 ) )
| ( ( ( append_nat_b_nat_b @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append_nat_b_nat_b @ Us @ Ts2 ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_42_append__eq__append__conv2,axiom,
! [Xs: list_int,Ys: list_int,Zs2: list_int,Ts2: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs2 @ Ts2 ) )
= ( ? [Us: list_int] :
( ( ( Xs
= ( append_int @ Zs2 @ Us ) )
& ( ( append_int @ Us @ Ys )
= Ts2 ) )
| ( ( ( append_int @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append_int @ Us @ Ts2 ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_43_append__eq__appendI,axiom,
! [Xs: list_nat_b_nat_b,Xs1: list_nat_b_nat_b,Zs2: list_nat_b_nat_b,Ys: list_nat_b_nat_b,Us2: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_nat_b_nat_b @ Xs1 @ Us2 ) )
=> ( ( append_nat_b_nat_b @ Xs @ Ys )
= ( append_nat_b_nat_b @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_44_append__eq__appendI,axiom,
! [Xs: list_int,Xs1: list_int,Zs2: list_int,Ys: list_int,Us2: list_int] :
( ( ( append_int @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_int @ Xs1 @ Us2 ) )
=> ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_45_list__nonempty__induct,axiom,
! [Xs: list_nat_b_nat_b,P: list_nat_b_nat_b > $o] :
( ( Xs != nil_nat_b_nat_b )
=> ( ! [X3: ( nat > b ) > nat > b] : ( P @ ( cons_nat_b_nat_b @ X3 @ nil_nat_b_nat_b ) )
=> ( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] :
( ( Xs2 != nil_nat_b_nat_b )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat_b_nat_b @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_46_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X3: int] : ( P @ ( cons_int @ X3 @ nil_int ) )
=> ( ! [X3: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_47_list__induct2_H,axiom,
! [P: list_nat_b_nat_b > list_nat_b_nat_b > $o,Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( P @ nil_nat_b_nat_b @ nil_nat_b_nat_b )
=> ( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] : ( P @ ( cons_nat_b_nat_b @ X3 @ Xs2 ) @ nil_nat_b_nat_b )
=> ( ! [Y2: ( nat > b ) > nat > b,Ys2: list_nat_b_nat_b] : ( P @ nil_nat_b_nat_b @ ( cons_nat_b_nat_b @ Y2 @ Ys2 ) )
=> ( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b,Y2: ( nat > b ) > nat > b,Ys2: list_nat_b_nat_b] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat_b_nat_b @ X3 @ Xs2 ) @ ( cons_nat_b_nat_b @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_48_list__induct2_H,axiom,
! [P: list_nat_b_nat_b > list_int > $o,Xs: list_nat_b_nat_b,Ys: list_int] :
( ( P @ nil_nat_b_nat_b @ nil_int )
=> ( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] : ( P @ ( cons_nat_b_nat_b @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_nat_b_nat_b @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b,Y2: int,Ys2: list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat_b_nat_b @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_49_list__induct2_H,axiom,
! [P: list_int > list_nat_b_nat_b > $o,Xs: list_int,Ys: list_nat_b_nat_b] :
( ( P @ nil_int @ nil_nat_b_nat_b )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_nat_b_nat_b )
=> ( ! [Y2: ( nat > b ) > nat > b,Ys2: list_nat_b_nat_b] : ( P @ nil_int @ ( cons_nat_b_nat_b @ Y2 @ Ys2 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: ( nat > b ) > nat > b,Ys2: list_nat_b_nat_b] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat_b_nat_b @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_50_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_51_neq__Nil__conv,axiom,
! [Xs: list_nat_b_nat_b] :
( ( Xs != nil_nat_b_nat_b )
= ( ? [Y3: ( nat > b ) > nat > b,Ys3: list_nat_b_nat_b] :
( Xs
= ( cons_nat_b_nat_b @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_52_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y3: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_53_remdups__adj_Ocases,axiom,
! [X: list_nat_b_nat_b] :
( ( X != nil_nat_b_nat_b )
=> ( ! [X3: ( nat > b ) > nat > b] :
( X
!= ( cons_nat_b_nat_b @ X3 @ nil_nat_b_nat_b ) )
=> ~ ! [X3: ( nat > b ) > nat > b,Y2: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] :
( X
!= ( cons_nat_b_nat_b @ X3 @ ( cons_nat_b_nat_b @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_54_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X3: int] :
( X
!= ( cons_int @ X3 @ nil_int ) )
=> ~ ! [X3: int,Y2: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_55_transpose_Ocases,axiom,
! [X: list_l7067579673986547845_nat_b] :
( ( X != nil_list_nat_b_nat_b )
=> ( ! [Xss: list_l7067579673986547845_nat_b] :
( X
!= ( cons_l8694591469225861951_nat_b @ nil_nat_b_nat_b @ Xss ) )
=> ~ ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b,Xss: list_l7067579673986547845_nat_b] :
( X
!= ( cons_l8694591469225861951_nat_b @ ( cons_nat_b_nat_b @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_56_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X3: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_57_min__list_Ocases,axiom,
! [X: list_nat_b_nat_b] :
( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] :
( X
!= ( cons_nat_b_nat_b @ X3 @ Xs2 ) )
=> ( X = nil_nat_b_nat_b ) ) ).
% min_list.cases
thf(fact_58_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X3: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_59_list_Oexhaust,axiom,
! [Y: list_nat_b_nat_b] :
( ( Y != nil_nat_b_nat_b )
=> ~ ! [X212: ( nat > b ) > nat > b,X222: list_nat_b_nat_b] :
( Y
!= ( cons_nat_b_nat_b @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_60_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_61_list_OdiscI,axiom,
! [List: list_nat_b_nat_b,X21: ( nat > b ) > nat > b,X22: list_nat_b_nat_b] :
( ( List
= ( cons_nat_b_nat_b @ X21 @ X22 ) )
=> ( List != nil_nat_b_nat_b ) ) ).
% list.discI
thf(fact_62_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_63_list_Odistinct_I1_J,axiom,
! [X21: ( nat > b ) > nat > b,X22: list_nat_b_nat_b] :
( nil_nat_b_nat_b
!= ( cons_nat_b_nat_b @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_64_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_65_set__ConsD,axiom,
! [Y: real,X: real,Xs: list_real] :
( ( member_real @ Y @ ( set_real2 @ ( cons_real @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_real @ Y @ ( set_real2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_66_set__ConsD,axiom,
! [Y: ( nat > b ) > nat > b,X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ Y @ ( set_nat_b_nat_b2 @ ( cons_nat_b_nat_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat_b_nat_b @ Y @ ( set_nat_b_nat_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_67_set__ConsD,axiom,
! [Y: int,X: int,Xs: list_int] :
( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_68_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_list_Oset__cases,axiom,
! [E: real,A: list_real] :
( ( member_real @ E @ ( set_real2 @ A ) )
=> ( ! [Z2: list_real] :
( A
!= ( cons_real @ E @ Z2 ) )
=> ~ ! [Z1: real,Z2: list_real] :
( ( A
= ( cons_real @ Z1 @ Z2 ) )
=> ~ ( member_real @ E @ ( set_real2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_71_list_Oset__cases,axiom,
! [E: ( nat > b ) > nat > b,A: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ E @ ( set_nat_b_nat_b2 @ A ) )
=> ( ! [Z2: list_nat_b_nat_b] :
( A
!= ( cons_nat_b_nat_b @ E @ Z2 ) )
=> ~ ! [Z1: ( nat > b ) > nat > b,Z2: list_nat_b_nat_b] :
( ( A
= ( cons_nat_b_nat_b @ Z1 @ Z2 ) )
=> ~ ( member_nat_b_nat_b @ E @ ( set_nat_b_nat_b2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_72_list_Oset__cases,axiom,
! [E: int,A: list_int] :
( ( member_int @ E @ ( set_int2 @ A ) )
=> ( ! [Z2: list_int] :
( A
!= ( cons_int @ E @ Z2 ) )
=> ~ ! [Z1: int,Z2: list_int] :
( ( A
= ( cons_int @ Z1 @ Z2 ) )
=> ~ ( member_int @ E @ ( set_int2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_73_list_Oset__intros_I1_J,axiom,
! [X21: real,X22: list_real] : ( member_real @ X21 @ ( set_real2 @ ( cons_real @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_74_list_Oset__intros_I1_J,axiom,
! [X21: ( nat > b ) > nat > b,X22: list_nat_b_nat_b] : ( member_nat_b_nat_b @ X21 @ ( set_nat_b_nat_b2 @ ( cons_nat_b_nat_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_75_list_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_76_list_Oset__intros_I2_J,axiom,
! [Y: real,X22: list_real,X21: real] :
( ( member_real @ Y @ ( set_real2 @ X22 ) )
=> ( member_real @ Y @ ( set_real2 @ ( cons_real @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_77_list_Oset__intros_I2_J,axiom,
! [Y: ( nat > b ) > nat > b,X22: list_nat_b_nat_b,X21: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ Y @ ( set_nat_b_nat_b2 @ X22 ) )
=> ( member_nat_b_nat_b @ Y @ ( set_nat_b_nat_b2 @ ( cons_nat_b_nat_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_78_list_Oset__intros_I2_J,axiom,
! [Y: int,X22: list_int,X21: int] :
( ( member_int @ Y @ ( set_int2 @ X22 ) )
=> ( member_int @ Y @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_79_Cons__eq__appendI,axiom,
! [X: ( nat > b ) > nat > b,Xs1: list_nat_b_nat_b,Ys: list_nat_b_nat_b,Xs: list_nat_b_nat_b,Zs2: list_nat_b_nat_b] :
( ( ( cons_nat_b_nat_b @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat_b_nat_b @ Xs1 @ Zs2 ) )
=> ( ( cons_nat_b_nat_b @ X @ Xs )
= ( append_nat_b_nat_b @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_80_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs2: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs2 ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_81_append__Cons,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( append_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) @ Ys )
= ( cons_nat_b_nat_b @ X @ ( append_nat_b_nat_b @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_82_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_83_eq__Nil__appendI,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat_b_nat_b @ nil_nat_b_nat_b @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_84_eq__Nil__appendI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_int @ nil_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_85_append_Oleft__neutral,axiom,
! [A: list_nat_b_nat_b] :
( ( append_nat_b_nat_b @ nil_nat_b_nat_b @ A )
= A ) ).
% append.left_neutral
thf(fact_86_append_Oleft__neutral,axiom,
! [A: list_int] :
( ( append_int @ nil_int @ A )
= A ) ).
% append.left_neutral
thf(fact_87_append__Nil,axiom,
! [Ys: list_nat_b_nat_b] :
( ( append_nat_b_nat_b @ nil_nat_b_nat_b @ Ys )
= Ys ) ).
% append_Nil
thf(fact_88_append__Nil,axiom,
! [Ys: list_int] :
( ( append_int @ nil_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_89_rev__nonempty__induct,axiom,
! [Xs: list_nat_b_nat_b,P: list_nat_b_nat_b > $o] :
( ( Xs != nil_nat_b_nat_b )
=> ( ! [X3: ( nat > b ) > nat > b] : ( P @ ( cons_nat_b_nat_b @ X3 @ nil_nat_b_nat_b ) )
=> ( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] :
( ( Xs2 != nil_nat_b_nat_b )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat_b_nat_b @ Xs2 @ ( cons_nat_b_nat_b @ X3 @ nil_nat_b_nat_b ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_90_rev__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X3: int] : ( P @ ( cons_int @ X3 @ nil_int ) )
=> ( ! [X3: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X3 @ nil_int ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_91_append__eq__Cons__conv,axiom,
! [Ys: list_nat_b_nat_b,Zs2: list_nat_b_nat_b,X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Ys @ Zs2 )
= ( cons_nat_b_nat_b @ X @ Xs ) )
= ( ( ( Ys = nil_nat_b_nat_b )
& ( Zs2
= ( cons_nat_b_nat_b @ X @ Xs ) ) )
| ? [Ys4: list_nat_b_nat_b] :
( ( Ys
= ( cons_nat_b_nat_b @ X @ Ys4 ) )
& ( ( append_nat_b_nat_b @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_92_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs2: list_int,X: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs2 )
= ( cons_int @ X @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs2
= ( cons_int @ X @ Xs ) ) )
| ? [Ys4: list_int] :
( ( Ys
= ( cons_int @ X @ Ys4 ) )
& ( ( append_int @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_93_Cons__eq__append__conv,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,Zs2: list_nat_b_nat_b] :
( ( ( cons_nat_b_nat_b @ X @ Xs )
= ( append_nat_b_nat_b @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_nat_b_nat_b )
& ( ( cons_nat_b_nat_b @ X @ Xs )
= Zs2 ) )
| ? [Ys4: list_nat_b_nat_b] :
( ( ( cons_nat_b_nat_b @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat_b_nat_b @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_94_Cons__eq__append__conv,axiom,
! [X: int,Xs: list_int,Ys: list_int,Zs2: list_int] :
( ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X @ Xs )
= Zs2 ) )
| ? [Ys4: list_int] :
( ( ( cons_int @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_int @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_95_rev__exhaust,axiom,
! [Xs: list_nat_b_nat_b] :
( ( Xs != nil_nat_b_nat_b )
=> ~ ! [Ys2: list_nat_b_nat_b,Y2: ( nat > b ) > nat > b] :
( Xs
!= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ Y2 @ nil_nat_b_nat_b ) ) ) ) ).
% rev_exhaust
thf(fact_96_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys2: list_int,Y2: int] :
( Xs
!= ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_97_rev__induct,axiom,
! [P: list_nat_b_nat_b > $o,Xs: list_nat_b_nat_b] :
( ( P @ nil_nat_b_nat_b )
=> ( ! [X3: ( nat > b ) > nat > b,Xs2: list_nat_b_nat_b] :
( ( P @ Xs2 )
=> ( P @ ( append_nat_b_nat_b @ Xs2 @ ( cons_nat_b_nat_b @ X3 @ nil_nat_b_nat_b ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_98_rev__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ( P @ nil_int )
=> ( ! [X3: int,Xs2: list_int] :
( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X3 @ nil_int ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_99_split__list__first__prop__iff,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ( ? [X4: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X4 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_nat_b_nat_b,X4: ( nat > b ) > nat > b] :
( ? [Zs3: list_nat_b_nat_b] :
( Xs
= ( append_nat_b_nat_b @ Ys3 @ ( cons_nat_b_nat_b @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ Y3 @ ( set_nat_b_nat_b2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_100_split__list__first__prop__iff,axiom,
! [Xs: list_int,P: int > $o] :
( ( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_int,X4: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_101_split__list__last__prop__iff,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ( ? [X4: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X4 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_nat_b_nat_b,X4: ( nat > b ) > nat > b,Zs3: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Ys3 @ ( cons_nat_b_nat_b @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ Y3 @ ( set_nat_b_nat_b2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_102_split__list__last__prop__iff,axiom,
! [Xs: list_int,P: int > $o] :
( ( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_int,X4: int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_103_in__set__conv__decomp__first,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
= ( ? [Ys3: list_real,Zs3: list_real] :
( ( Xs
= ( append_real @ Ys3 @ ( cons_real @ X @ Zs3 ) ) )
& ~ ( member_real @ X @ ( set_real2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_104_in__set__conv__decomp__first,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
= ( ? [Ys3: list_nat_b_nat_b,Zs3: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Ys3 @ ( cons_nat_b_nat_b @ X @ Zs3 ) ) )
& ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_105_in__set__conv__decomp__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys3: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_106_in__set__conv__decomp__last,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
= ( ? [Ys3: list_real,Zs3: list_real] :
( ( Xs
= ( append_real @ Ys3 @ ( cons_real @ X @ Zs3 ) ) )
& ~ ( member_real @ X @ ( set_real2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_107_in__set__conv__decomp__last,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
= ( ? [Ys3: list_nat_b_nat_b,Zs3: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Ys3 @ ( cons_nat_b_nat_b @ X @ Zs3 ) ) )
& ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_108_in__set__conv__decomp__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys3: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_109_split__list__first__propE,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X2 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_nat_b_nat_b,X3: ( nat > b ) > nat > b] :
( ? [Zs: list_nat_b_nat_b] :
( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X3 @ Zs ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ Xa @ ( set_nat_b_nat_b2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_110_split__list__first__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_int,X3: int] :
( ? [Zs: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_111_split__list__last__propE,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X2 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_nat_b_nat_b,X3: ( nat > b ) > nat > b,Zs: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X3 @ Zs ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ Xa @ ( set_nat_b_nat_b2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_112_split__list__last__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_int,X3: int,Zs: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_113_split__list__first__prop,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X2 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_nat_b_nat_b,X3: ( nat > b ) > nat > b] :
( ? [Zs: list_nat_b_nat_b] :
( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X3 @ Zs ) ) )
& ( P @ X3 )
& ! [Xa: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ Xa @ ( set_nat_b_nat_b2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_114_split__list__first__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_int,X3: int] :
( ? [Zs: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs ) ) )
& ( P @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_115_split__list__last__prop,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X2 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_nat_b_nat_b,X3: ( nat > b ) > nat > b,Zs: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X3 @ Zs ) ) )
& ( P @ X3 )
& ! [Xa: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ Xa @ ( set_nat_b_nat_b2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_116_split__list__last__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_int,X3: int,Zs: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs ) ) )
& ( P @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_117_in__set__conv__decomp,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
= ( ? [Ys3: list_real,Zs3: list_real] :
( Xs
= ( append_real @ Ys3 @ ( cons_real @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_118_in__set__conv__decomp,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
= ( ? [Ys3: list_nat_b_nat_b,Zs3: list_nat_b_nat_b] :
( Xs
= ( append_nat_b_nat_b @ Ys3 @ ( cons_nat_b_nat_b @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_119_in__set__conv__decomp,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys3: list_int,Zs3: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_120_append__Cons__eq__iff,axiom,
! [X: real,Xs: list_real,Ys: list_real,Xs3: list_real,Ys5: list_real] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ~ ( member_real @ X @ ( set_real2 @ Ys ) )
=> ( ( ( append_real @ Xs @ ( cons_real @ X @ Ys ) )
= ( append_real @ Xs3 @ ( cons_real @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_121_append__Cons__eq__iff,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,Xs3: list_nat_b_nat_b,Ys5: list_nat_b_nat_b] :
( ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Ys ) )
=> ( ( ( append_nat_b_nat_b @ Xs @ ( cons_nat_b_nat_b @ X @ Ys ) )
= ( append_nat_b_nat_b @ Xs3 @ ( cons_nat_b_nat_b @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_122_append__Cons__eq__iff,axiom,
! [X: int,Xs: list_int,Ys: list_int,Xs3: list_int,Ys5: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int @ X @ ( set_int2 @ Ys ) )
=> ( ( ( append_int @ Xs @ ( cons_int @ X @ Ys ) )
= ( append_int @ Xs3 @ ( cons_int @ X @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_123_split__list__propE,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X2 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_nat_b_nat_b,X3: ( nat > b ) > nat > b] :
( ? [Zs: list_nat_b_nat_b] :
( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X3 @ Zs ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_124_split__list__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_int,X3: int] :
( ? [Zs: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_125_split__list__first,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ? [Ys2: list_real,Zs: list_real] :
( ( Xs
= ( append_real @ Ys2 @ ( cons_real @ X @ Zs ) ) )
& ~ ( member_real @ X @ ( set_real2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_126_split__list__first,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ? [Ys2: list_nat_b_nat_b,Zs: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X @ Zs ) ) )
& ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_127_split__list__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys2: list_int,Zs: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_128_split__list__prop,axiom,
! [Xs: list_nat_b_nat_b,P: ( ( nat > b ) > nat > b ) > $o] :
( ? [X2: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X2 @ ( set_nat_b_nat_b2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_nat_b_nat_b,X3: ( nat > b ) > nat > b] :
( ? [Zs: list_nat_b_nat_b] :
( Xs
= ( append_nat_b_nat_b @ Ys2 @ ( cons_nat_b_nat_b @ X3 @ Zs ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_129_split__list__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_int,X3: int] :
( ? [Zs: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X3 @ Zs ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_130_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_131_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_132_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_133_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_134_the__elem__set,axiom,
! [X: ( nat > b ) > nat > b] :
( ( the_elem_nat_b_nat_b @ ( set_nat_b_nat_b2 @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) ) )
= X ) ).
% the_elem_set
thf(fact_135_the__elem__set,axiom,
! [X: int] :
( ( the_elem_int @ ( set_int2 @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% the_elem_set
thf(fact_136_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_137_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_138_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_139_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_140_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_141_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_142_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_143_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_144_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_145_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_146_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_147_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_148_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_149_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_150_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_151_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_152_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_153_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_154_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_155_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_156_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_157_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_158_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat_b_nat_b @ N @ nil_nat_b_nat_b )
= ( cons_l8694591469225861951_nat_b @ nil_nat_b_nat_b @ nil_list_nat_b_nat_b ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat_b_nat_b @ N @ nil_nat_b_nat_b )
= nil_list_nat_b_nat_b ) ) ) ).
% n_lists_Nil
thf(fact_159_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= nil_list_int ) ) ) ).
% n_lists_Nil
thf(fact_160_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat_b_nat_b] :
( ( n_lists_nat_b_nat_b @ zero_zero_nat @ Xs )
= ( cons_l8694591469225861951_nat_b @ nil_nat_b_nat_b @ nil_list_nat_b_nat_b ) ) ).
% n_lists.simps(1)
thf(fact_161_n__lists_Osimps_I1_J,axiom,
! [Xs: list_int] :
( ( n_lists_int @ zero_zero_nat @ Xs )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% n_lists.simps(1)
thf(fact_162_snoc_OIH,axiom,
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X3 @ ( set_nat_b_nat_b2 @ xs ) )
=> ( is_swap @ X3 ) )
=> ( meas_ptw
@ ^ [K: nat,Omega: a] :
( fold_n5624292861071574516_nat_b @ id_nat_b_nat_b @ xs
@ ^ [L: nat] : ( x @ L @ Omega )
@ K ) ) ) ).
% snoc.IH
thf(fact_163_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_164_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_165_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_166_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_167_product__lists_Osimps_I1_J,axiom,
( ( produc8493759205635542326_nat_b @ nil_list_nat_b_nat_b )
= ( cons_l8694591469225861951_nat_b @ nil_nat_b_nat_b @ nil_list_nat_b_nat_b ) ) ).
% product_lists.simps(1)
thf(fact_168_product__lists_Osimps_I1_J,axiom,
( ( product_lists_int @ nil_list_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% product_lists.simps(1)
thf(fact_169_bind__simps_I2_J,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b] :
( ( bind_n614534383549643347_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) @ F )
= ( append_nat_b_nat_b @ ( F @ X ) @ ( bind_n614534383549643347_nat_b @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_170_bind__simps_I2_J,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > list_int] :
( ( bind_nat_b_nat_b_int @ ( cons_nat_b_nat_b @ X @ Xs ) @ F )
= ( append_int @ ( F @ X ) @ ( bind_nat_b_nat_b_int @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_171_bind__simps_I2_J,axiom,
! [X: int,Xs: list_int,F: int > list_nat_b_nat_b] :
( ( bind_int_nat_b_nat_b @ ( cons_int @ X @ Xs ) @ F )
= ( append_nat_b_nat_b @ ( F @ X ) @ ( bind_int_nat_b_nat_b @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_172_bind__simps_I2_J,axiom,
! [X: int,Xs: list_int,F: int > list_int] :
( ( bind_int_int @ ( cons_int @ X @ Xs ) @ F )
= ( append_int @ ( F @ X ) @ ( bind_int_int @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_173_a,axiom,
( meas_ptw
@ ^ [K: nat,Omega: a] :
( fold_n5624292861071574516_nat_b
@ ^ [A3: ( nat > b ) > nat > b] : A3
@ xs
@ ^ [L: nat] : ( x @ L @ Omega )
@ K ) ) ).
% a
thf(fact_174_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat_b_nat_b @ nil_nat_b_nat_b )
= ( cons_l8694591469225861951_nat_b @ nil_nat_b_nat_b @ nil_list_nat_b_nat_b ) ) ).
% subseqs.simps(1)
thf(fact_175_subseqs_Osimps_I1_J,axiom,
( ( subseqs_int @ nil_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% subseqs.simps(1)
thf(fact_176_set__Cons__def,axiom,
( set_Cons_real
= ( ^ [A4: set_real,XS: set_list_real] :
( collect_list_real
@ ^ [Z: list_real] :
? [X4: real,Xs4: list_real] :
( ( Z
= ( cons_real @ X4 @ Xs4 ) )
& ( member_real @ X4 @ A4 )
& ( member_list_real @ Xs4 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_177_set__Cons__def,axiom,
( set_Cons_nat_b_nat_b
= ( ^ [A4: set_nat_b_nat_b,XS: set_list_nat_b_nat_b] :
( collec3644297838412717002_nat_b
@ ^ [Z: list_nat_b_nat_b] :
? [X4: ( nat > b ) > nat > b,Xs4: list_nat_b_nat_b] :
( ( Z
= ( cons_nat_b_nat_b @ X4 @ Xs4 ) )
& ( member_nat_b_nat_b @ X4 @ A4 )
& ( member8456897207961558412_nat_b @ Xs4 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_178_set__Cons__def,axiom,
( set_Cons_int
= ( ^ [A4: set_int,XS: set_list_int] :
( collect_list_int
@ ^ [Z: list_int] :
? [X4: int,Xs4: list_int] :
( ( Z
= ( cons_int @ X4 @ Xs4 ) )
& ( member_int @ X4 @ A4 )
& ( member_list_int @ Xs4 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_179_bind__simps_I1_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b] :
( ( bind_n614534383549643347_nat_b @ nil_nat_b_nat_b @ F )
= nil_nat_b_nat_b ) ).
% bind_simps(1)
thf(fact_180_bind__simps_I1_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > list_int] :
( ( bind_nat_b_nat_b_int @ nil_nat_b_nat_b @ F )
= nil_int ) ).
% bind_simps(1)
thf(fact_181_bind__simps_I1_J,axiom,
! [F: int > list_nat_b_nat_b] :
( ( bind_int_nat_b_nat_b @ nil_int @ F )
= nil_nat_b_nat_b ) ).
% bind_simps(1)
thf(fact_182_bind__simps_I1_J,axiom,
! [F: int > list_int] :
( ( bind_int_int @ nil_int @ F )
= nil_int ) ).
% bind_simps(1)
thf(fact_183_fold__Nil,axiom,
! [F: ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( fold_n1883935874045088215_nat_b @ F @ nil_nat_b_nat_b )
= id_nat_b_nat_b ) ).
% fold_Nil
thf(fact_184_fold__Nil,axiom,
! [F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( fold_n5624292861071574516_nat_b @ F @ nil_nat_b_nat_b )
= id_nat_b ) ).
% fold_Nil
thf(fact_185_fold__Nil,axiom,
! [F: int > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( fold_int_nat_b_nat_b @ F @ nil_int )
= id_nat_b_nat_b ) ).
% fold_Nil
thf(fact_186_fold__id,axiom,
! [Xs: list_real,F: real > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b_nat_b ) )
=> ( ( fold_r4529934814194841502_nat_b @ F @ Xs )
= id_nat_b_nat_b ) ) ).
% fold_id
thf(fact_187_fold__id,axiom,
! [Xs: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b_nat_b ) )
=> ( ( fold_n1883935874045088215_nat_b @ F @ Xs )
= id_nat_b_nat_b ) ) ).
% fold_id
thf(fact_188_fold__id,axiom,
! [Xs: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( F @ X3 )
= id_nat_b ) )
=> ( ( fold_n5624292861071574516_nat_b @ F @ Xs )
= id_nat_b ) ) ).
% fold_id
thf(fact_189_fold__simps_I2_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b,X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,S: nat > b] :
( ( fold_n5624292861071574516_nat_b @ F @ ( cons_nat_b_nat_b @ X @ Xs ) @ S )
= ( fold_n5624292861071574516_nat_b @ F @ Xs @ ( F @ X @ S ) ) ) ).
% fold_simps(2)
thf(fact_190_fold__simps_I1_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b,S: nat > b] :
( ( fold_n5624292861071574516_nat_b @ F @ nil_nat_b_nat_b @ S )
= S ) ).
% fold_simps(1)
thf(fact_191_fold__invariant,axiom,
! [Xs: list_nat_b_nat_b,Q: ( ( nat > b ) > nat > b ) > $o,P: ( nat > b ) > $o,S: nat > b,F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( Q @ X3 ) )
=> ( ( P @ S )
=> ( ! [X3: ( nat > b ) > nat > b,S2: nat > b] :
( ( Q @ X3 )
=> ( ( P @ S2 )
=> ( P @ ( F @ X3 @ S2 ) ) ) )
=> ( P @ ( fold_n5624292861071574516_nat_b @ F @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_192_List_Ofold__cong,axiom,
! [A: nat > b,B: nat > b,Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b,F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b,G: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( A = B )
=> ( ( Xs = Ys )
=> ( ! [X3: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X3 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fold_n5624292861071574516_nat_b @ F @ Xs @ A )
= ( fold_n5624292861071574516_nat_b @ G @ Ys @ B ) ) ) ) ) ).
% List.fold_cong
thf(fact_193_Cons__in__subseqsD,axiom,
! [Y: ( nat > b ) > nat > b,Ys: list_nat_b_nat_b,Xs: list_nat_b_nat_b] :
( ( member8456897207961558412_nat_b @ ( cons_nat_b_nat_b @ Y @ Ys ) @ ( set_list_nat_b_nat_b2 @ ( subseqs_nat_b_nat_b @ Xs ) ) )
=> ( member8456897207961558412_nat_b @ Ys @ ( set_list_nat_b_nat_b2 @ ( subseqs_nat_b_nat_b @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_194_Cons__in__subseqsD,axiom,
! [Y: int,Ys: list_int,Xs: list_int] :
( ( member_list_int @ ( cons_int @ Y @ Ys ) @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) )
=> ( member_list_int @ Ys @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_195_id__apply,axiom,
( id_nat_b_nat_b
= ( ^ [X4: ( nat > b ) > nat > b] : X4 ) ) ).
% id_apply
thf(fact_196_maps__simps_I1_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b,X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( maps_n8879270819509224205_nat_b @ F @ ( cons_nat_b_nat_b @ X @ Xs ) )
= ( append_nat_b_nat_b @ ( F @ X ) @ ( maps_n8879270819509224205_nat_b @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_197_maps__simps_I1_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > list_int,X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( maps_nat_b_nat_b_int @ F @ ( cons_nat_b_nat_b @ X @ Xs ) )
= ( append_int @ ( F @ X ) @ ( maps_nat_b_nat_b_int @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_198_maps__simps_I1_J,axiom,
! [F: int > list_nat_b_nat_b,X: int,Xs: list_int] :
( ( maps_int_nat_b_nat_b @ F @ ( cons_int @ X @ Xs ) )
= ( append_nat_b_nat_b @ ( F @ X ) @ ( maps_int_nat_b_nat_b @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_199_maps__simps_I1_J,axiom,
! [F: int > list_int,X: int,Xs: list_int] :
( ( maps_int_int @ F @ ( cons_int @ X @ Xs ) )
= ( append_int @ ( F @ X ) @ ( maps_int_int @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_200_not__in__set__insert,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X @ Xs )
= ( cons_real @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_201_not__in__set__insert,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( insert_nat_b_nat_b @ X @ Xs )
= ( cons_nat_b_nat_b @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_202_not__in__set__insert,axiom,
! [X: int,Xs: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X @ Xs )
= ( cons_int @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_203_insert__Nil,axiom,
! [X: ( nat > b ) > nat > b] :
( ( insert_nat_b_nat_b @ X @ nil_nat_b_nat_b )
= ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) ) ).
% insert_Nil
thf(fact_204_insert__Nil,axiom,
! [X: int] :
( ( insert_int @ X @ nil_int )
= ( cons_int @ X @ nil_int ) ) ).
% insert_Nil
thf(fact_205_concat__eq__append__conv,axiom,
! [Xss2: list_l7067579673986547845_nat_b,Ys: list_nat_b_nat_b,Zs2: list_nat_b_nat_b] :
( ( ( concat_nat_b_nat_b @ Xss2 )
= ( append_nat_b_nat_b @ Ys @ Zs2 ) )
= ( ( ( Xss2 = nil_list_nat_b_nat_b )
=> ( ( Ys = nil_nat_b_nat_b )
& ( Zs2 = nil_nat_b_nat_b ) ) )
& ( ( Xss2 != nil_list_nat_b_nat_b )
=> ? [Xss1: list_l7067579673986547845_nat_b,Xs4: list_nat_b_nat_b,Xs5: list_nat_b_nat_b,Xss22: list_l7067579673986547845_nat_b] :
( ( Xss2
= ( append3853175007740601498_nat_b @ Xss1 @ ( cons_l8694591469225861951_nat_b @ ( append_nat_b_nat_b @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat_b_nat_b @ ( concat_nat_b_nat_b @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_nat_b_nat_b @ Xs5 @ ( concat_nat_b_nat_b @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_206_concat__eq__append__conv,axiom,
! [Xss2: list_list_int,Ys: list_int,Zs2: list_int] :
( ( ( concat_int @ Xss2 )
= ( append_int @ Ys @ Zs2 ) )
= ( ( ( Xss2 = nil_list_int )
=> ( ( Ys = nil_int )
& ( Zs2 = nil_int ) ) )
& ( ( Xss2 != nil_list_int )
=> ? [Xss1: list_list_int,Xs4: list_int,Xs5: list_int,Xss22: list_list_int] :
( ( Xss2
= ( append_list_int @ Xss1 @ ( cons_list_int @ ( append_int @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_int @ ( concat_int @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_int @ Xs5 @ ( concat_int @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_207_nths__singleton,axiom,
! [A2: set_nat,X: ( nat > b ) > nat > b] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) @ A2 )
= ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) @ A2 )
= nil_nat_b_nat_b ) ) ) ).
% nths_singleton
thf(fact_208_nths__singleton,axiom,
! [A2: set_nat,X: int] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A2 )
= ( cons_int @ X @ nil_int ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A2 )
= nil_int ) ) ) ).
% nths_singleton
thf(fact_209_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_210_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_211_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_212_butlast__snoc,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( butlast_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_213_butlast__snoc,axiom,
! [Xs: list_int,X: int] :
( ( butlast_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_214_list__ex1__simps_I1_J,axiom,
! [P: ( ( nat > b ) > nat > b ) > $o] :
~ ( list_ex1_nat_b_nat_b @ P @ nil_nat_b_nat_b ) ).
% list_ex1_simps(1)
thf(fact_215_list__ex1__simps_I1_J,axiom,
! [P: int > $o] :
~ ( list_ex1_int @ P @ nil_int ) ).
% list_ex1_simps(1)
thf(fact_216_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_217_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_218_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_nat_b_nat_b @ nil_nat_b_nat_b @ A2 )
= nil_nat_b_nat_b ) ).
% nths_nil
thf(fact_219_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_int @ nil_int @ A2 )
= nil_int ) ).
% nths_nil
thf(fact_220_in__set__insert,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_221_in__set__insert,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( ( insert_nat_b_nat_b @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_222_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_223_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_224_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_225_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_226_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_227_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_228_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_229_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_230_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_231_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_232_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_233_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_234_concat__eq__Nil__conv,axiom,
! [Xss2: list_l7067579673986547845_nat_b] :
( ( ( concat_nat_b_nat_b @ Xss2 )
= nil_nat_b_nat_b )
= ( ! [X4: list_nat_b_nat_b] :
( ( member8456897207961558412_nat_b @ X4 @ ( set_list_nat_b_nat_b2 @ Xss2 ) )
=> ( X4 = nil_nat_b_nat_b ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_235_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_int] :
( ( ( concat_int @ Xss2 )
= nil_int )
= ( ! [X4: list_int] :
( ( member_list_int @ X4 @ ( set_list_int2 @ Xss2 ) )
=> ( X4 = nil_int ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_236_Nil__eq__concat__conv,axiom,
! [Xss2: list_l7067579673986547845_nat_b] :
( ( nil_nat_b_nat_b
= ( concat_nat_b_nat_b @ Xss2 ) )
= ( ! [X4: list_nat_b_nat_b] :
( ( member8456897207961558412_nat_b @ X4 @ ( set_list_nat_b_nat_b2 @ Xss2 ) )
=> ( X4 = nil_nat_b_nat_b ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_237_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_int] :
( ( nil_int
= ( concat_int @ Xss2 ) )
= ( ! [X4: list_int] :
( ( member_list_int @ X4 @ ( set_list_int2 @ Xss2 ) )
=> ( X4 = nil_int ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_238_concat__append,axiom,
! [Xs: list_l7067579673986547845_nat_b,Ys: list_l7067579673986547845_nat_b] :
( ( concat_nat_b_nat_b @ ( append3853175007740601498_nat_b @ Xs @ Ys ) )
= ( append_nat_b_nat_b @ ( concat_nat_b_nat_b @ Xs ) @ ( concat_nat_b_nat_b @ Ys ) ) ) ).
% concat_append
thf(fact_239_concat__append,axiom,
! [Xs: list_list_int,Ys: list_list_int] :
( ( concat_int @ ( append_list_int @ Xs @ Ys ) )
= ( append_int @ ( concat_int @ Xs ) @ ( concat_int @ Ys ) ) ) ).
% concat_append
thf(fact_240_butlast_Osimps_I1_J,axiom,
( ( butlast_nat_b_nat_b @ nil_nat_b_nat_b )
= nil_nat_b_nat_b ) ).
% butlast.simps(1)
thf(fact_241_butlast_Osimps_I1_J,axiom,
( ( butlast_int @ nil_int )
= nil_int ) ).
% butlast.simps(1)
thf(fact_242_in__set__butlastD,axiom,
! [X: real,Xs: list_real] :
( ( member_real @ X @ ( set_real2 @ ( butlast_real @ Xs ) ) )
=> ( member_real @ X @ ( set_real2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_243_in__set__butlastD,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ ( butlast_nat_b_nat_b @ Xs ) ) )
=> ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_244_in__set__nthsD,axiom,
! [X: real,Xs: list_real,I2: set_nat] :
( ( member_real @ X @ ( set_real2 @ ( nths_real @ Xs @ I2 ) ) )
=> ( member_real @ X @ ( set_real2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_245_in__set__nthsD,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,I2: set_nat] :
( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ ( nths_nat_b_nat_b @ Xs @ I2 ) ) )
=> ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_246_notin__set__nthsI,axiom,
! [X: real,Xs: list_real,I2: set_nat] :
( ~ ( member_real @ X @ ( set_real2 @ Xs ) )
=> ~ ( member_real @ X @ ( set_real2 @ ( nths_real @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_247_notin__set__nthsI,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,I2: set_nat] :
( ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ Xs ) )
=> ~ ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ ( nths_nat_b_nat_b @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_248_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_249_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_250_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_251_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_252_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_253_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_254_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_255_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_256_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_257_concat_Osimps_I1_J,axiom,
( ( concat_nat_b_nat_b @ nil_list_nat_b_nat_b )
= nil_nat_b_nat_b ) ).
% concat.simps(1)
thf(fact_258_concat_Osimps_I1_J,axiom,
( ( concat_int @ nil_list_int )
= nil_int ) ).
% concat.simps(1)
thf(fact_259_concat_Osimps_I2_J,axiom,
! [X: list_nat_b_nat_b,Xs: list_l7067579673986547845_nat_b] :
( ( concat_nat_b_nat_b @ ( cons_l8694591469225861951_nat_b @ X @ Xs ) )
= ( append_nat_b_nat_b @ X @ ( concat_nat_b_nat_b @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_260_concat_Osimps_I2_J,axiom,
! [X: list_int,Xs: list_list_int] :
( ( concat_int @ ( cons_list_int @ X @ Xs ) )
= ( append_int @ X @ ( concat_int @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_261_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( ( Xs = nil_nat_b_nat_b )
=> ( ( butlast_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) )
= nil_nat_b_nat_b ) )
& ( ( Xs != nil_nat_b_nat_b )
=> ( ( butlast_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) )
= ( cons_nat_b_nat_b @ X @ ( butlast_nat_b_nat_b @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_262_butlast_Osimps_I2_J,axiom,
! [Xs: list_int,X: int] :
( ( ( Xs = nil_int )
=> ( ( butlast_int @ ( cons_int @ X @ Xs ) )
= nil_int ) )
& ( ( Xs != nil_int )
=> ( ( butlast_int @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( butlast_int @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_263_butlast__append,axiom,
! [Ys: list_nat_b_nat_b,Xs: list_nat_b_nat_b] :
( ( ( Ys = nil_nat_b_nat_b )
=> ( ( butlast_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) )
= ( butlast_nat_b_nat_b @ Xs ) ) )
& ( ( Ys != nil_nat_b_nat_b )
=> ( ( butlast_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) )
= ( append_nat_b_nat_b @ Xs @ ( butlast_nat_b_nat_b @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_264_butlast__append,axiom,
! [Ys: list_int,Xs: list_int] :
( ( ( Ys = nil_int )
=> ( ( butlast_int @ ( append_int @ Xs @ Ys ) )
= ( butlast_int @ Xs ) ) )
& ( ( Ys != nil_int )
=> ( ( butlast_int @ ( append_int @ Xs @ Ys ) )
= ( append_int @ Xs @ ( butlast_int @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_265_in__set__butlast__appendI,axiom,
! [X: real,Xs: list_real,Ys: list_real] :
( ( ( member_real @ X @ ( set_real2 @ ( butlast_real @ Xs ) ) )
| ( member_real @ X @ ( set_real2 @ ( butlast_real @ Ys ) ) ) )
=> ( member_real @ X @ ( set_real2 @ ( butlast_real @ ( append_real @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_266_in__set__butlast__appendI,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( ( member_int @ X @ ( set_int2 @ ( butlast_int @ Xs ) ) )
| ( member_int @ X @ ( set_int2 @ ( butlast_int @ Ys ) ) ) )
=> ( member_int @ X @ ( set_int2 @ ( butlast_int @ ( append_int @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_267_in__set__butlast__appendI,axiom,
! [X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
( ( ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ ( butlast_nat_b_nat_b @ Xs ) ) )
| ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ ( butlast_nat_b_nat_b @ Ys ) ) ) )
=> ( member_nat_b_nat_b @ X @ ( set_nat_b_nat_b2 @ ( butlast_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_268_maps__simps_I2_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > list_nat_b_nat_b] :
( ( maps_n8879270819509224205_nat_b @ F @ nil_nat_b_nat_b )
= nil_nat_b_nat_b ) ).
% maps_simps(2)
thf(fact_269_maps__simps_I2_J,axiom,
! [F: ( ( nat > b ) > nat > b ) > list_int] :
( ( maps_nat_b_nat_b_int @ F @ nil_nat_b_nat_b )
= nil_int ) ).
% maps_simps(2)
thf(fact_270_maps__simps_I2_J,axiom,
! [F: int > list_nat_b_nat_b] :
( ( maps_int_nat_b_nat_b @ F @ nil_int )
= nil_nat_b_nat_b ) ).
% maps_simps(2)
thf(fact_271_maps__simps_I2_J,axiom,
! [F: int > list_int] :
( ( maps_int_int @ F @ nil_int )
= nil_int ) ).
% maps_simps(2)
thf(fact_272_list__ex1__iff,axiom,
( list_ex1_real
= ( ^ [P2: real > $o,Xs4: list_real] :
? [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs4 ) )
& ( P2 @ X4 )
& ! [Y3: real] :
( ( ( member_real @ Y3 @ ( set_real2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_273_list__ex1__iff,axiom,
( list_ex1_nat_b_nat_b
= ( ^ [P2: ( ( nat > b ) > nat > b ) > $o,Xs4: list_nat_b_nat_b] :
? [X4: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X4 @ ( set_nat_b_nat_b2 @ Xs4 ) )
& ( P2 @ X4 )
& ! [Y3: ( nat > b ) > nat > b] :
( ( ( member_nat_b_nat_b @ Y3 @ ( set_nat_b_nat_b2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_274_id__def,axiom,
( id_nat_b_nat_b
= ( ^ [X4: ( nat > b ) > nat > b] : X4 ) ) ).
% id_def
thf(fact_275_eq__id__iff,axiom,
! [F: ( ( nat > b ) > nat > b ) > ( nat > b ) > nat > b] :
( ( ! [X4: ( nat > b ) > nat > b] :
( ( F @ X4 )
= X4 ) )
= ( F = id_nat_b_nat_b ) ) ).
% eq_id_iff
thf(fact_276_List_Oinsert__def,axiom,
( insert_real
= ( ^ [X4: real,Xs4: list_real] : ( if_list_real @ ( member_real @ X4 @ ( set_real2 @ Xs4 ) ) @ Xs4 @ ( cons_real @ X4 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_277_List_Oinsert__def,axiom,
( insert_nat_b_nat_b
= ( ^ [X4: ( nat > b ) > nat > b,Xs4: list_nat_b_nat_b] : ( if_list_nat_b_nat_b @ ( member_nat_b_nat_b @ X4 @ ( set_nat_b_nat_b2 @ Xs4 ) ) @ Xs4 @ ( cons_nat_b_nat_b @ X4 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_278_List_Oinsert__def,axiom,
( insert_int
= ( ^ [X4: int,Xs4: list_int] : ( if_list_int @ ( member_int @ X4 @ ( set_int2 @ Xs4 ) ) @ Xs4 @ ( cons_int @ X4 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_279_concat__eq__appendD,axiom,
! [Xss2: list_l7067579673986547845_nat_b,Ys: list_nat_b_nat_b,Zs2: list_nat_b_nat_b] :
( ( ( concat_nat_b_nat_b @ Xss2 )
= ( append_nat_b_nat_b @ Ys @ Zs2 ) )
=> ( ( Xss2 != nil_list_nat_b_nat_b )
=> ? [Xss12: list_l7067579673986547845_nat_b,Xs2: list_nat_b_nat_b,Xs6: list_nat_b_nat_b,Xss23: list_l7067579673986547845_nat_b] :
( ( Xss2
= ( append3853175007740601498_nat_b @ Xss12 @ ( cons_l8694591469225861951_nat_b @ ( append_nat_b_nat_b @ Xs2 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat_b_nat_b @ ( concat_nat_b_nat_b @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append_nat_b_nat_b @ Xs6 @ ( concat_nat_b_nat_b @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_280_concat__eq__appendD,axiom,
! [Xss2: list_list_int,Ys: list_int,Zs2: list_int] :
( ( ( concat_int @ Xss2 )
= ( append_int @ Ys @ Zs2 ) )
=> ( ( Xss2 != nil_list_int )
=> ? [Xss12: list_list_int,Xs2: list_int,Xs6: list_int,Xss23: list_list_int] :
( ( Xss2
= ( append_list_int @ Xss12 @ ( cons_list_int @ ( append_int @ Xs2 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_int @ ( concat_int @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append_int @ Xs6 @ ( concat_int @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_281_can__select__set__list__ex1,axiom,
! [P: ( ( nat > b ) > nat > b ) > $o,A2: list_nat_b_nat_b] :
( ( can_se6628294081300001064_nat_b @ P @ ( set_nat_b_nat_b2 @ A2 ) )
= ( list_ex1_nat_b_nat_b @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_282_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_283_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_284_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_285_append__butlast__last__id,axiom,
! [Xs: list_nat_b_nat_b] :
( ( Xs != nil_nat_b_nat_b )
=> ( ( append_nat_b_nat_b @ ( butlast_nat_b_nat_b @ Xs ) @ ( cons_nat_b_nat_b @ ( last_nat_b_nat_b @ Xs ) @ nil_nat_b_nat_b ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_286_append__butlast__last__id,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( append_int @ ( butlast_int @ Xs ) @ ( cons_int @ ( last_int @ Xs ) @ nil_int ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_287_nths__Cons,axiom,
! [X: ( nat > b ) > nat > b,L2: list_nat_b_nat_b,A2: set_nat] :
( ( nths_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ L2 ) @ A2 )
= ( append_nat_b_nat_b @ ( if_list_nat_b_nat_b @ ( member_nat @ zero_zero_nat @ A2 ) @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) @ nil_nat_b_nat_b )
@ ( nths_nat_b_nat_b @ L2
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_288_nths__Cons,axiom,
! [X: int,L2: list_int,A2: set_nat] :
( ( nths_int @ ( cons_int @ X @ L2 ) @ A2 )
= ( append_int @ ( if_list_int @ ( member_nat @ zero_zero_nat @ A2 ) @ ( cons_int @ X @ nil_int ) @ nil_int )
@ ( nths_int @ L2
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_289_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b,Ys: list_nat_b_nat_b] :
( ( ( append_nat_b_nat_b @ Xs @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) )
= Ys )
= ( ( Ys != nil_nat_b_nat_b )
& ( ( butlast_nat_b_nat_b @ Ys )
= Xs )
& ( ( last_nat_b_nat_b @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_290_snoc__eq__iff__butlast,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= Ys )
= ( ( Ys != nil_int )
& ( ( butlast_int @ Ys )
= Xs )
& ( ( last_int @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_291_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_292_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_293_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_294_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_295_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_296_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_297_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_298_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_299_last__appendR,axiom,
! [Ys: list_nat_b_nat_b,Xs: list_nat_b_nat_b] :
( ( Ys != nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) )
= ( last_nat_b_nat_b @ Ys ) ) ) ).
% last_appendR
thf(fact_300_last__appendR,axiom,
! [Ys: list_int,Xs: list_int] :
( ( Ys != nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Ys ) ) ) ).
% last_appendR
thf(fact_301_last__appendL,axiom,
! [Ys: list_nat_b_nat_b,Xs: list_nat_b_nat_b] :
( ( Ys = nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) )
= ( last_nat_b_nat_b @ Xs ) ) ) ).
% last_appendL
thf(fact_302_last__appendL,axiom,
! [Ys: list_int,Xs: list_int] :
( ( Ys = nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Xs ) ) ) ).
% last_appendL
thf(fact_303_last__snoc,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( last_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ ( cons_nat_b_nat_b @ X @ nil_nat_b_nat_b ) ) )
= X ) ).
% last_snoc
thf(fact_304_last__snoc,axiom,
! [Xs: list_int,X: int] :
( ( last_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% last_snoc
thf(fact_305_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_306_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_307_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_308_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_309_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_310_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_311_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_312_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_313_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_314_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_315_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_316_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_317_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_318_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_319_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_320_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_321_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_322_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_323_can__select__def,axiom,
( can_select_real
= ( ^ [P2: real > $o,A4: set_real] :
? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( P2 @ X4 )
& ! [Y3: real] :
( ( ( member_real @ Y3 @ A4 )
& ( P2 @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_324_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_325_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I4: nat] :
( ( J2
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_326_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I4 @ K3 ) ) ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_327_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_328_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_329_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_330_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_331_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_332_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_333_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_334_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_335_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_336_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_337_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_338_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_339_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_340_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_341_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_342_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_343_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_344_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_345_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_346_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_347_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_348_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_349_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_350_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_351_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_352_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_353_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_354_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_355_last__ConsR,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( Xs != nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) )
= ( last_nat_b_nat_b @ Xs ) ) ) ).
% last_ConsR
thf(fact_356_last__ConsR,axiom,
! [Xs: list_int,X: int] :
( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_357_last__ConsL,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( Xs = nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_358_last__ConsL,axiom,
! [Xs: list_int,X: int] :
( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_359_last_Osimps,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] :
( ( ( Xs = nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( cons_nat_b_nat_b @ X @ Xs ) )
= ( last_nat_b_nat_b @ Xs ) ) ) ) ).
% last.simps
thf(fact_360_last_Osimps,axiom,
! [Xs: list_int,X: int] :
( ( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_361_last__in__set,axiom,
! [As: list_real] :
( ( As != nil_real )
=> ( member_real @ ( last_real @ As ) @ ( set_real2 @ As ) ) ) ).
% last_in_set
thf(fact_362_last__in__set,axiom,
! [As: list_int] :
( ( As != nil_int )
=> ( member_int @ ( last_int @ As ) @ ( set_int2 @ As ) ) ) ).
% last_in_set
thf(fact_363_last__in__set,axiom,
! [As: list_nat_b_nat_b] :
( ( As != nil_nat_b_nat_b )
=> ( member_nat_b_nat_b @ ( last_nat_b_nat_b @ As ) @ ( set_nat_b_nat_b2 @ As ) ) ) ).
% last_in_set
thf(fact_364_longest__common__suffix,axiom,
! [Xs: list_nat_b_nat_b,Ys: list_nat_b_nat_b] :
? [Ss: list_nat_b_nat_b,Xs6: list_nat_b_nat_b,Ys6: list_nat_b_nat_b] :
( ( Xs
= ( append_nat_b_nat_b @ Xs6 @ Ss ) )
& ( Ys
= ( append_nat_b_nat_b @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_nat_b_nat_b )
| ( Ys6 = nil_nat_b_nat_b )
| ( ( last_nat_b_nat_b @ Xs6 )
!= ( last_nat_b_nat_b @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_365_longest__common__suffix,axiom,
! [Xs: list_int,Ys: list_int] :
? [Ss: list_int,Xs6: list_int,Ys6: list_int] :
( ( Xs
= ( append_int @ Xs6 @ Ss ) )
& ( Ys
= ( append_int @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_int )
| ( Ys6 = nil_int )
| ( ( last_int @ Xs6 )
!= ( last_int @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_366_last__append,axiom,
! [Ys: list_nat_b_nat_b,Xs: list_nat_b_nat_b] :
( ( ( Ys = nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) )
= ( last_nat_b_nat_b @ Xs ) ) )
& ( ( Ys != nil_nat_b_nat_b )
=> ( ( last_nat_b_nat_b @ ( append_nat_b_nat_b @ Xs @ Ys ) )
= ( last_nat_b_nat_b @ Ys ) ) ) ) ).
% last_append
thf(fact_367_last__append,axiom,
! [Ys: list_int,Xs: list_int] :
( ( ( Ys = nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Xs ) ) )
& ( ( Ys != nil_int )
=> ( ( last_int @ ( append_int @ Xs @ Ys ) )
= ( last_int @ Ys ) ) ) ) ).
% last_append
thf(fact_368_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_369_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_370_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_371_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_372_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_373_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_374_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_375_gen__length__code_I2_J,axiom,
! [N: nat,X: ( nat > b ) > nat > b,Xs: list_nat_b_nat_b] :
( ( gen_le7432612553264103954_nat_b @ N @ ( cons_nat_b_nat_b @ X @ Xs ) )
= ( gen_le7432612553264103954_nat_b @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_376_gen__length__code_I2_J,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_377_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_378_zero__less__nat__eq,axiom,
! [Z4: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z4 ) )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% zero_less_nat_eq
thf(fact_379_of__int__0__less__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z4 ) )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% of_int_0_less_iff
thf(fact_380_of__int__0__less__iff,axiom,
! [Z4: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z4 ) )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% of_int_0_less_iff
thf(fact_381_of__int__less__0__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z4 ) @ zero_zero_int )
= ( ord_less_int @ Z4 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_382_of__int__less__0__iff,axiom,
! [Z4: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ zero_zero_real )
= ( ord_less_int @ Z4 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_383_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_384_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_385_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_386_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_387_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_388_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_389_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_390_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_391_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_392_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_393_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_394_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_395_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_396_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_397_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_398_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_399_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_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_408_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_409_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_410_of__int__0__eq__iff,axiom,
! [Z4: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z4 ) )
= ( Z4 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_411_of__int__0__eq__iff,axiom,
! [Z4: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z4 ) )
= ( Z4 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_412_of__int__eq__0__iff,axiom,
! [Z4: int] :
( ( ( ring_1_of_int_int @ Z4 )
= zero_zero_int )
= ( Z4 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_413_of__int__eq__0__iff,axiom,
! [Z4: int] :
( ( ( ring_1_of_int_real @ Z4 )
= zero_zero_real )
= ( Z4 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_414_of__int__less__iff,axiom,
! [W: int,Z4: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z4 ) )
= ( ord_less_int @ W @ Z4 ) ) ).
% of_int_less_iff
thf(fact_415_of__int__less__iff,axiom,
! [W: int,Z4: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z4 ) )
= ( ord_less_int @ W @ Z4 ) ) ).
% of_int_less_iff
thf(fact_416_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_417_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_418_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_419_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_420_zless__nat__conj,axiom,
! [W: int,Z4: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z4 ) )
= ( ( ord_less_int @ zero_zero_int @ Z4 )
& ( ord_less_int @ W @ Z4 ) ) ) ).
% zless_nat_conj
thf(fact_421_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_422_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K: int] : ( if_int @ ( ord_less_int @ K @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K ) ) ) ) ) ).
% of_int_of_nat
thf(fact_423_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K: int] : ( if_real @ ( ord_less_int @ K @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K ) ) ) ) ) ).
% of_int_of_nat
thf(fact_424_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_425_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_426_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_427_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_428_ex__of__int__less,axiom,
! [X: real] :
? [Z5: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z5 ) @ X ) ).
% ex_of_int_less
thf(fact_429_ex__less__of__int,axiom,
! [X: real] :
? [Z5: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% ex_less_of_int
thf(fact_430_forall__pos__mono,axiom,
! [P: real > $o,E: real] :
( ! [D: real,E2: real] :
( ( ord_less_real @ D @ E2 )
=> ( ( P @ D )
=> ( P @ E2 ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_431_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
= ( ? [N4: nat] :
( ( N4 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_432_forall__pos__mono__1,axiom,
! [P: real > $o,E: real] :
( ! [D: real,E2: real] :
( ( ord_less_real @ D @ E2 )
=> ( ( P @ D )
=> ( P @ E2 ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_433_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_434_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_435_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_436_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_437_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_438_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_439_int__cases2,axiom,
! [Z4: int] :
( ! [N2: nat] :
( Z4
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z4
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_440_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_441_int__of__nat__induct,axiom,
! [P: int > $o,Z4: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z4 ) ) ) ).
% int_of_nat_induct
thf(fact_442_int__cases,axiom,
! [Z4: int] :
( ! [N2: nat] :
( Z4
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z4
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_443_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_444_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le7432612553264103954_nat_b @ N @ nil_nat_b_nat_b )
= N ) ).
% gen_length_code(1)
thf(fact_445_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_int @ N @ nil_int )
= N ) ).
% gen_length_code(1)
thf(fact_446_nat__mono__iff,axiom,
! [Z4: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ W @ Z4 ) ) ) ).
% nat_mono_iff
thf(fact_447_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z4: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z4 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z4 ) ) ).
% zless_nat_eq_int_zless
thf(fact_448_int__cases4,axiom,
! [M2: int] :
( ! [N2: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_449_reals__Archimedean,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_450_of__int__pos,axiom,
! [Z4: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z4 ) ) ) ).
% of_int_pos
thf(fact_451_of__int__pos,axiom,
! [Z4: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z4 ) ) ) ).
% of_int_pos
thf(fact_452_split__nat,axiom,
! [P: nat > $o,I3: int] :
( ( P @ ( nat2 @ I3 ) )
= ( ! [N4: nat] :
( ( I3
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I3 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_453_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_454_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_455_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_456_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_457_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_458_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_459_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_460_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_461_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_462_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_463_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_464_linordered__field__no__ub,axiom,
! [X2: real] :
? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_465_linordered__field__no__lb,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% linordered_field_no_lb
thf(fact_466_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_467_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_468_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_469_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_470_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_real__arch__invD,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [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 ) ) @ E ) ) ) ).
% real_arch_invD
thf(fact_481_one__less__nat__eq,axiom,
! [Z4: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ one_one_int @ Z4 ) ) ).
% one_less_nat_eq
thf(fact_482_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_483_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_484_of__nat__nat,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z4 ) )
= ( ring_1_of_int_int @ Z4 ) ) ) ).
% of_nat_nat
thf(fact_485_of__nat__nat,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z4 ) )
= ( ring_1_of_int_real @ Z4 ) ) ) ).
% of_nat_nat
thf(fact_486_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_487_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_488_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_489_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_490_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_491_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_492_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_493_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_494_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_495_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_496_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_497_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_498_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_499_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_500_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_501_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_502_of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
= X )
= ( ? [N4: int] :
( X
= ( ring_1_of_int_real @ N4 ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_514_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_515_ceiling__of__nat,axiom,
! [N: nat] :
( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% ceiling_of_nat
thf(fact_516_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_517_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_518_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_519_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_520_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_521_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_522_nat__le__0,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ Z4 @ zero_zero_int )
=> ( ( nat2 @ Z4 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_523_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_524_int__nat__eq,axiom,
! [Z4: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z4 ) )
= Z4 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z4 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_525_of__int__le__0__iff,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z4 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z4 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_526_of__int__le__0__iff,axiom,
! [Z4: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z4 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_527_of__int__0__le__iff,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z4 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z4 ) ) ).
% of_int_0_le_iff
thf(fact_528_of__int__0__le__iff,axiom,
! [Z4: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z4 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z4 ) ) ).
% of_int_0_le_iff
thf(fact_529_of__int__less__1__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z4 ) @ one_one_int )
= ( ord_less_int @ Z4 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_530_of__int__less__1__iff,axiom,
! [Z4: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ one_one_real )
= ( ord_less_int @ Z4 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_531_of__int__1__less__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z4 ) )
= ( ord_less_int @ one_one_int @ Z4 ) ) ).
% of_int_1_less_iff
thf(fact_532_of__int__1__less__iff,axiom,
! [Z4: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z4 ) )
= ( ord_less_int @ one_one_int @ Z4 ) ) ).
% of_int_1_less_iff
thf(fact_533_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_534_ceiling__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_less_one
thf(fact_535_one__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_le_ceiling
thf(fact_536_one__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ).
% one_less_ceiling
thf(fact_537_zero__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% zero_le_ceiling
thf(fact_538_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_539_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_540_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_541_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_542_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_543_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_544_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I3 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_545_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_546_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_547_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_548_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_549_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_550_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_551_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_552_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_553_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_554_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_555_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_556_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_557_ceiling__le,axiom,
! [X: real,A: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).
% ceiling_le
thf(fact_558_of__nat__ceiling,axiom,
! [R: real] : ( ord_less_eq_real @ R @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R ) ) ) ) ).
% of_nat_ceiling
thf(fact_559_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_560_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_561_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_562_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_563_inverse__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% inverse_le_1_iff
thf(fact_564_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_565_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_566_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_567_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_568_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A5 ) )
= ( ord_less_int @ A5 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_569_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A5 ) )
= ( ord_less_nat @ A5 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_570_verit__comp__simplify1_I3_J,axiom,
! [B3: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B3 @ A5 ) )
= ( ord_less_real @ A5 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_571_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_572_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_573_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_574_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_575_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_576_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_577_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_578_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_579_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_580_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_581_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_582_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_583_one__le__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_le_inverse
thf(fact_584_inverse__less__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% inverse_less_1_iff
thf(fact_585_one__le__inverse__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_586_of__int__nonneg,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z4 ) ) ) ).
% of_int_nonneg
thf(fact_587_of__int__nonneg,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z4 ) ) ) ).
% of_int_nonneg
thf(fact_588_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_589_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_590_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_591_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_592_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_593_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_594_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_595_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_596_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_597_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_598_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_599_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_600_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_601_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_602_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_603_less__ceiling__iff,axiom,
! [Z4: int,X: real] :
( ( ord_less_int @ Z4 @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X ) ) ).
% less_ceiling_iff
thf(fact_604_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_605_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_606_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_607_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_608_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_609_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_610_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_611_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_612_one__less__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_less_inverse
thf(fact_613_one__less__inverse__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_614_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_615_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N2: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_616_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_617_int__eq__iff,axiom,
! [M2: nat,Z4: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z4 )
= ( ( M2
= ( nat2 @ Z4 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z4 ) ) ) ).
% int_eq_iff
thf(fact_618_nat__0__le,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z4 ) )
= Z4 ) ) ).
% nat_0_le
thf(fact_619_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_620_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_621_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_622_nat__less__eq__zless,axiom,
! [W: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z4 ) )
= ( ord_less_int @ W @ Z4 ) ) ) ).
% nat_less_eq_zless
thf(fact_623_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_624_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_int @ one_one_int @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_625_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_626_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_627_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_628_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_629_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_630_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_631_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_632_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_633_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_634_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_635_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_636_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_637_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_638_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_639_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_640_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_641_assms_I1_J,axiom,
ord_less_eq_nat @ one_one_nat @ n ).
% assms(1)
thf(fact_642_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_643_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_644_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_645_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_646_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_647_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_648_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_649_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_650_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_651_le__trans,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_652_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_653_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_654_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_655_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_656_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_657_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_658_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_659_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_660_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_661_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_662_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_663_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_664_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_665_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_666_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z5: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z5 )
=> ( R2 @ X3 @ Z5 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_667_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_668_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_669_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_670_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_671_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_672_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_673_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_674_subset__code_I1_J,axiom,
! [Xs: list_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B4 )
= ( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs ) )
=> ( member_real @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_675_subset__code_I1_J,axiom,
! [Xs: list_nat_b_nat_b,B4: set_nat_b_nat_b] :
( ( ord_le9047053354294502011_nat_b @ ( set_nat_b_nat_b2 @ Xs ) @ B4 )
= ( ! [X4: ( nat > b ) > nat > b] :
( ( member_nat_b_nat_b @ X4 @ ( set_nat_b_nat_b2 @ Xs ) )
=> ( member_nat_b_nat_b @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_676_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_677_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_678_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_679_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_680_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_681_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_682_inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_683_dec__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_684_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_685_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_686_set__subset__Cons,axiom,
! [Xs: list_nat_b_nat_b,X: ( nat > b ) > nat > b] : ( ord_le9047053354294502011_nat_b @ ( set_nat_b_nat_b2 @ Xs ) @ ( set_nat_b_nat_b2 @ ( cons_nat_b_nat_b @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_687_set__subset__Cons,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_688_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_689_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_690_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_691_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_692_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_693_set__nths__subset,axiom,
! [Xs: list_nat_b_nat_b,I2: set_nat] : ( ord_le9047053354294502011_nat_b @ ( set_nat_b_nat_b2 @ ( nths_nat_b_nat_b @ Xs @ I2 ) ) @ ( set_nat_b_nat_b2 @ Xs ) ) ).
% set_nths_subset
thf(fact_694_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_695_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_696_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_697_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_698_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_699_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_700_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_701_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_702_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_703_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_704_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I4: nat] :
( ( Q @ I4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I4 ) )
& ( ord_less_eq_real @ ( X3 @ I4 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X2: nat > real,I5: nat] : ( ord_less_eq_nat @ ( L3 @ X2 @ I5 ) @ one_one_nat )
& ! [X2: nat > real,I5: nat] :
( ( ( P @ X2 )
& ( Q @ I5 )
& ( ( X2 @ I5 )
= zero_zero_real ) )
=> ( ( L3 @ X2 @ I5 )
= zero_zero_nat ) )
& ! [X2: nat > real,I5: nat] :
( ( ( P @ X2 )
& ( Q @ I5 )
& ( ( X2 @ I5 )
= one_one_real ) )
=> ( ( L3 @ X2 @ I5 )
= one_one_nat ) )
& ! [X2: nat > real,I5: nat] :
( ( ( P @ X2 )
& ( Q @ I5 )
& ( ( L3 @ X2 @ I5 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X2 @ I5 ) @ ( F @ X2 @ I5 ) ) )
& ! [X2: nat > real,I5: nat] :
( ( ( P @ X2 )
& ( Q @ I5 )
& ( ( L3 @ X2 @ I5 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X2 @ I5 ) @ ( X2 @ I5 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_705_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_706_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_707_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_708_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_709_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_710_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_711_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_712_complete__real,axiom,
! [S3: set_real] :
( ? [X2: real] : ( member_real @ X2 @ S3 )
=> ( ? [Z6: real] :
! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z6 ) )
=> ? [Y2: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S3 )
=> ( ord_less_eq_real @ X2 @ Y2 ) )
& ! [Z6: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ord_less_eq_real @ X3 @ Z6 ) )
=> ( ord_less_eq_real @ Y2 @ Z6 ) ) ) ) ) ).
% complete_real
thf(fact_713_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_714_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_715_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_716_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_717_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_718_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_719_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_720_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 )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_721_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 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_722_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_723_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 )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_724_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 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_725_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_726_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_727_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_728_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_729_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_730_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_731_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_732_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_733_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_734_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_735_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_736_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_737_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_738_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_739_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_740_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_741_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_742_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_743_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_744_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_745_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_746_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_747_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_748_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_749_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_750_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_751_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_752_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_753_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_754_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_755_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_756_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_757_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_758_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_759_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_760_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_761_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_762_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_763_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_764_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_765_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_766_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_767_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_768_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_769_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_770_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_771_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_772_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_773_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_774_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_775_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_776_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_777_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_778_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_779_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_780_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_781_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_782_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_783_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_784_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_785_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_786_order__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_787_order__less__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_788_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_789_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_790_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_791_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_792_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_793_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_794_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_795_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_796_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_797_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_798_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_799_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_800_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_801_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_802_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_803_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_804_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_805_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_806_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_807_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_808_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_809_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_810_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_811_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_812_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_813_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_814_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_815_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B5: nat] :
( ( ord_less_nat @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B5: nat] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_816_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A6: int,B5: int] :
( ( ord_less_int @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: int] : ( P @ A6 @ A6 )
=> ( ! [A6: int,B5: int] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_817_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A6: real,B5: real] :
( ( ord_less_real @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: real] : ( P @ A6 @ A6 )
=> ( ! [A6: real,B5: real] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_818_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P2: nat > $o] :
? [N4: nat] :
( ( P2 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P2 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_819_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_820_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_821_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_822_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_823_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_824_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_825_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_826_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_827_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_828_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_829_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_830_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_831_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_832_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_833_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_834_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_835_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_836_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_837_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_838_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_839_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_840_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_841_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_842_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_843_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_844_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z5: real] :
( ( ord_less_real @ X @ Z5 )
& ( ord_less_real @ Z5 @ Y ) ) ) ).
% dense
thf(fact_845_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_846_gt__ex,axiom,
! [X: int] :
? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).
% gt_ex
thf(fact_847_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_848_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_849_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_850_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_851_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_852_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_853_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_854_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_855_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_856_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_857_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_858_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_859_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_860_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_861_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_862_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_863_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_864_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_865_dense__ge,axiom,
! [Z4: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ).
% dense_ge
thf(fact_866_dense__le,axiom,
! [Y: real,Z4: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ).
% dense_le
thf(fact_867_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_868_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_869_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_870_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_871_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_872_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_dense__ge__bounded,axiom,
! [Z4: real,X: real,Y: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ! [W2: real] :
( ( ord_less_real @ Z4 @ W2 )
=> ( ( ord_less_real @ W2 @ X )
=> ( ord_less_eq_real @ Y @ W2 ) ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ).
% dense_ge_bounded
thf(fact_889_dense__le__bounded,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W2: real] :
( ( ord_less_real @ X @ W2 )
=> ( ( ord_less_real @ W2 @ Y )
=> ( ord_less_eq_real @ W2 @ Z4 ) ) )
=> ( ord_less_eq_real @ Y @ Z4 ) ) ) ).
% dense_le_bounded
thf(fact_890_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_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_912_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_913_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_914_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_915_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_916_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_917_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_918_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_919_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_920_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_921_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_922_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_923_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_924_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_925_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_926_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_927_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_928_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_929_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_930_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_931_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_932_order__le__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_933_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_934_order__le__less__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_935_order__less__le__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_936_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_937_order__less__le__trans,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z4 )
=> ( ord_less_real @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_938_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_939_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 )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_940_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 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_941_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_942_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 )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_943_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 )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_944_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_945_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_946_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_947_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_948_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 )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_949_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_950_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_951_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_952_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_953_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_954_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_955_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_956_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_957_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 )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_958_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_959_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_960_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_961_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_962_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_963_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_964_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_965_minf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_966_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_967_minf_I8_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_968_minf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_969_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_970_minf_I6_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ord_less_eq_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_971_pinf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_972_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_973_pinf_I8_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ord_less_eq_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_974_pinf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_975_pinf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_976_pinf_I6_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_977_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_978_minf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_979_minf_I7_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_980_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_981_minf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_982_minf_I5_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_983_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_984_minf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_985_minf_I4_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_986_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_987_minf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_988_minf_I3_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_989_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_990_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_991_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_992_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_993_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_994_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_995_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_996_pinf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_997_pinf_I7_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_998_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_999_pinf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1000_pinf_I5_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1001_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1002_pinf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1003_pinf_I4_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1004_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1005_pinf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1006_pinf_I3_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1007_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1008_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1009_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1010_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1011_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1012_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z6: real] :
! [X3: real] :
( ( ord_less_real @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1013_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 )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1014_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 )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1015_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 )
& ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1016_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_1017_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_1018_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_1019_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1020_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1021_ex__gt__or__lt,axiom,
! [A: real] :
? [B5: real] :
( ( ord_less_real @ A @ B5 )
| ( ord_less_real @ B5 @ A ) ) ).
% ex_gt_or_lt
thf(fact_1022_fps__inverse__gp,axiom,
( ( invers68952373231134600s_real
@ ( formal798729627605919420s_real
@ ^ [N4: nat] : one_one_real ) )
= ( formal798729627605919420s_real
@ ^ [N4: nat] : ( if_real @ ( N4 = zero_zero_nat ) @ one_one_real @ ( if_real @ ( N4 = one_one_nat ) @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ) ) ) ) ).
% fps_inverse_gp
thf(fact_1023_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X4 )
@ ^ [X4: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X4 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1024_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( 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 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1025_le__ceiling__iff,axiom,
! [Z4: int,X: real] :
( ( ord_less_eq_int @ Z4 @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z4 ) @ one_one_real ) @ X ) ) ).
% le_ceiling_iff
thf(fact_1026_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1027_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1028_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1029_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1030_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1031_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1032_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1033_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1034_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1035_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1036_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1037_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1038_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1039_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1040_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1041_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1042_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1043_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1044_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1045_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1046_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1047_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1048_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1049_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1050_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1051_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_1052_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1053_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1054_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_1055_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1056_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_1057_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1058_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1059_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1060_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1061_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1062_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1063_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1064_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1065_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1066_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1067_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_1068_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1069_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1070_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1071_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1072_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1073_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1074_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1075_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1076_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1077_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1078_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1079_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1080_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1081_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1082_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1083_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1084_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1085_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1086_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1087_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1088_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1089_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_1090_mult__minus1__right,axiom,
! [Z4: int] :
( ( times_times_int @ Z4 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z4 ) ) ).
% mult_minus1_right
thf(fact_1091_mult__minus1__right,axiom,
! [Z4: real] :
( ( times_times_real @ Z4 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z4 ) ) ).
% mult_minus1_right
thf(fact_1092_mult__minus1,axiom,
! [Z4: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z4 )
= ( uminus_uminus_int @ Z4 ) ) ).
% mult_minus1
thf(fact_1093_mult__minus1,axiom,
! [Z4: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z4 )
= ( uminus_uminus_real @ Z4 ) ) ).
% mult_minus1
thf(fact_1094_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1095_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1096_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_1097_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_1098_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_1099_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1100_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1101_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1102_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1103_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1104_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1105_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1106_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_1107_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1108_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1109_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_1110_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1111_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1112_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_1113_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1114_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1115_division__ring__inverse__diff,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_1116_int__minus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1117_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1118_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1119_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1120_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1121_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1122_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1123_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D4: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D4 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D4 ) ) ) ).
% diff_eq_diff_less
thf(fact_1124_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D4: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D4 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D4 ) ) ) ).
% diff_eq_diff_less
thf(fact_1125_diff__strict__mono,axiom,
! [A: int,B: int,D4: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D4 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D4 ) ) ) ) ).
% diff_strict_mono
thf(fact_1126_diff__strict__mono,axiom,
! [A: real,B: real,D4: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D4 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D4 ) ) ) ) ).
% diff_strict_mono
thf(fact_1127_mult__delta__left,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_1128_mult__delta__left,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_1129_mult__delta__left,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_1130_mult__delta__right,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_1131_mult__delta__right,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_1132_mult__delta__right,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_1133_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1134_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1135_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_1136_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_1137_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1138_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1139_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I3: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1140_less__imp__diff__less,axiom,
! [J2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1141_diff__less__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1142_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1143_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1144_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1145_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1146_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1147_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1148_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1149_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1150_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1151_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_1152_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_1153_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_1154_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1155_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1156_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1157_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D4: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D4 ) )
=> ( ( A = B )
= ( C = D4 ) ) ) ).
% diff_eq_diff_eq
thf(fact_1158_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D4: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D4 ) )
=> ( ( A = B )
= ( C = D4 ) ) ) ).
% diff_eq_diff_eq
thf(fact_1159_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1160_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1161_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1162_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_1163_diff__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_1164_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_1165_int__diff__cases,axiom,
! [Z4: int] :
~ ! [M3: nat,N2: nat] :
( Z4
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1166_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1167_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1168_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1169_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1170_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1171_le__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1172_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1173_diff__le__mono,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1174_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1175_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1176_diff__le__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1177_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1178_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1179_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1180_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1181_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1182_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1183_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1184_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1185_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1186_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1187_diff__nat__eq__if,axiom,
! [Z7: int,Z4: int] :
( ( ( ord_less_int @ Z7 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z4 ) @ ( nat2 @ Z7 ) )
= ( nat2 @ Z4 ) ) )
& ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z4 ) @ ( nat2 @ Z7 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z4 @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z4 @ Z7 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_1188_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1189_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1190_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1191_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1192_mult__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1193_mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1194_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1195_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1196_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1197_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1198_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1199_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1200_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1201_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1202_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1203_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1204_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1205_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_1206_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1207_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1208_diff__commute,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J2 ) ) ).
% diff_commute
thf(fact_1209_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_1210_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1211_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1212_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1213_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1214_mult__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1215_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1216_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1217_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1218_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1219_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1220_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1221_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1222_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1223_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1224_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1225_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1226_zmult__zless__mono2__lemma,axiom,
! [I3: int,J2: int,K2: nat] :
( ( ord_less_int @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1227_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1228_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1229_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1230_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1231_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1232_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1233_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I: int,J: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J @ I ) @ Js @ ( upto_aux @ I @ ( minus_minus_int @ J @ one_one_int ) @ ( cons_int @ J @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1234_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1235_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1236_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1237_nat__div__as__int,axiom,
( divide_divide_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_1238_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1239_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X4: real,Y3: real] : ( times_times_real @ X4 @ ( inverse_inverse_real @ Y3 ) ) ) ) ).
% divide_real_def
thf(fact_1240_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_1241_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1242_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1243_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_1244_nat__mult__div__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1245_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1246_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1247_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1248_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1249_zdiv__int,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_1250_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1251_less__mult__imp__div__less,axiom,
! [M2: nat,I3: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I3 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I3 ) ) ).
% less_mult_imp_div_less
thf(fact_1252_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1253_div__le__mono2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1254_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1255_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1256_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1257_div__if,axiom,
( divide_divide_nat
= ( ^ [M5: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M5 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_1258_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1259_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M2 )
=> ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M2 @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1260_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1261_split__div_H,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M2 )
& ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1262_int__power__div__base,axiom,
! [M2: nat,K2: int] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( divide_divide_int @ ( power_power_int @ K2 @ M2 ) @ K2 )
= ( power_power_int @ K2 @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1263_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( F @ I4 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1264_div__abs__eq__div__nat,axiom,
! [K2: int,L2: int] :
( ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L2 ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% div_abs_eq_div_nat
% Helper facts (13)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X: list_real,Y: list_real] :
( ( if_list_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X: list_real,Y: list_real] :
( ( if_list_real @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J_T,axiom,
! [X: list_nat_b_nat_b,Y: list_nat_b_nat_b] :
( ( if_list_nat_b_nat_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_I_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_J_T,axiom,
! [X: list_nat_b_nat_b,Y: list_nat_b_nat_b] :
( ( if_list_nat_b_nat_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
is_swap @ x2 ).
%------------------------------------------------------------------------------