TPTP Problem File: SLH0884^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00184_006575__14706118_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1402 ( 586 unt; 129 typ; 0 def)
% Number of atoms : 3780 (1227 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10416 ( 371 ~; 118 |; 217 &;8051 @)
% ( 0 <=>;1659 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 430 ( 430 >; 0 *; 0 +; 0 <<)
% Number of symbols : 117 ( 114 usr; 18 con; 0-8 aty)
% Number of variables : 3221 ( 128 ^;2956 !; 137 ?;3221 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:44:08.999
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
formal_Power_fps_nat: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
formal_Power_fps_int: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__b_J,type,
multiset_b: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (114)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Cancellation_Oiterate__add_001t__Int__Oint,type,
iterate_add_int: nat > int > int ).
thf(sy_c_Cancellation_Oiterate__add_001t__Nat__Onat,type,
iterate_add_nat: nat > nat > nat ).
thf(sy_c_Cancellation_Oiterate__add_001t__Real__Oreal,type,
iterate_add_real: nat > real > real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
invers68952373231134600s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Int__Oint,type,
formal3717847055265219294th_int: formal_Power_fps_int > nat > int ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Nat__Onat,type,
formal3720337525774269570th_nat: formal_Power_fps_nat > nat > nat ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Real__Oreal,type,
formal2580924720334399070h_real: formal3361831859752904756s_real > nat > real ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Int__Oint,type,
formal812433016830480481XD_int: formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Nat__Onat,type,
formal814923487339530757XD_nat: formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Real__Oreal,type,
formal4292469010823635553D_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__cutoff_001t__Int__Oint,type,
formal4815718713524518466ff_int: nat > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__cutoff_001t__Nat__Onat,type,
formal4818209184033568742ff_nat: nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__cutoff_001t__Real__Oreal,type,
formal1487479903726251970f_real: nat > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__ln_001t__Real__Oreal,type,
formal8688746759596762231n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Oradical_001t__Real__Oreal,type,
formal8005797870169972230l_real: ( nat > real > real ) > nat > formal3361831859752904756s_real > nat > real ).
thf(sy_c_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__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
one_on8395608022581818233ps_int: formal_Power_fps_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
one_on3350087005236239133ps_nat: formal_Power_fps_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
one_on8598947968683843321s_real: formal3361831859752904756s_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__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
times_3091854549176928185ps_int: formal_Power_fps_int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
times_7269705568686124893ps_nat: formal_Power_fps_nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
times_7561426564079326009s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_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__Int__Oint_J,type,
zero_z4353722679246354365ps_int: formal_Power_fps_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
zero_z8531573698755551073ps_nat: formal_Power_fps_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__b_J,type,
zero_zero_multiset_b: multiset_b ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Harmonic__Numbers_Oharm_001t__Real__Oreal,type,
harmonic_harm_real: nat > real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__b,type,
map_nat_b: ( nat > b ) > list_nat > list_b ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Median_Odown__ray_001t__Int__Oint,type,
down_ray_int: set_int > $o ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001t__Real__Oreal,type,
down_ray_real: set_real > $o ).
thf(sy_c_Median_Ointerval_001t__Int__Oint,type,
interval_int: set_int > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001t__Real__Oreal,type,
interval_real: set_real > $o ).
thf(sy_c_Median_Osort__map_001t__Int__Oint,type,
sort_map_int: ( nat > int ) > nat > nat > int ).
thf(sy_c_Median_Osort__map_001t__Nat__Onat,type,
sort_map_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Median_Osort__map_001t__Real__Oreal,type,
sort_map_real: ( nat > real ) > nat > nat > real ).
thf(sy_c_Median_Osort__map_001tf__b,type,
sort_map_b: ( nat > b ) > nat > nat > b ).
thf(sy_c_Median_Oup__ray_001t__Int__Oint,type,
up_ray_int: set_int > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001t__Real__Oreal,type,
up_ray_real: set_real > $o ).
thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001t__Nat__Onat,type,
image_mset_nat_nat: ( nat > nat ) > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001tf__b,type,
image_mset_nat_b: ( nat > b ) > multiset_nat > multiset_b ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
linord3047872887403683810et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001tf__b,type,
mset_b: list_b > multiset_b ).
thf(sy_c_Multiset_Oreplicate__mset_001t__Nat__Onat,type,
replicate_mset_nat: nat > nat > multiset_nat ).
thf(sy_c_Multiset_Oreplicate__mset_001tf__b,type,
replicate_mset_b: nat > b > multiset_b ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__b,type,
subseteq_mset_b: multiset_b > multiset_b > $o ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
semiri6570152736363784213ps_int: nat > formal_Power_fps_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
semiri1524631719018205113ps_nat: nat > formal_Power_fps_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
semiri2475410149736220053s_real: nat > formal3361831859752904756s_real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
size_s5917832649809541300et_nat: multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__b_J,type,
size_size_multiset_b: multiset_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
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_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_001tf__b,type,
ord_less_eq_b: b > b > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
exp_real: real > real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_f,type,
f: nat > b ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1265)
thf(fact_0_ex__mset,axiom,
! [X: multiset_nat] :
? [Xs: list_nat] :
( ( mset_nat @ Xs )
= X ) ).
% ex_mset
thf(fact_1_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_2_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_3_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_4_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_5_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_6_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_7_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_8_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_9_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_10_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_11_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_12_iterate__add__simps_I1_J,axiom,
! [A: nat] :
( ( iterate_add_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% iterate_add_simps(1)
thf(fact_13_iterate__add__simps_I1_J,axiom,
! [A: int] :
( ( iterate_add_int @ zero_zero_nat @ A )
= zero_zero_int ) ).
% iterate_add_simps(1)
thf(fact_14_iterate__add__simps_I1_J,axiom,
! [A: real] :
( ( iterate_add_real @ zero_zero_nat @ A )
= zero_zero_real ) ).
% iterate_add_simps(1)
thf(fact_15_harm__expand_I1_J,axiom,
( ( harmonic_harm_real @ zero_zero_nat )
= zero_zero_real ) ).
% harm_expand(1)
thf(fact_16_image__mset__is__empty__iff,axiom,
! [F: nat > b,M: multiset_nat] :
( ( ( image_mset_nat_b @ F @ M )
= zero_zero_multiset_b )
= ( M = zero_z7348594199698428585et_nat ) ) ).
% image_mset_is_empty_iff
thf(fact_17_image__mset__empty,axiom,
! [F: nat > b] :
( ( image_mset_nat_b @ F @ zero_z7348594199698428585et_nat )
= zero_zero_multiset_b ) ).
% image_mset_empty
thf(fact_18_iterate__add__empty,axiom,
! [N: nat] :
( ( iterate_add_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% iterate_add_empty
thf(fact_19_iterate__add__empty,axiom,
! [N: nat] :
( ( iterate_add_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% iterate_add_empty
thf(fact_20_iterate__add__empty,axiom,
! [N: nat] :
( ( iterate_add_real @ N @ zero_zero_real )
= zero_zero_real ) ).
% iterate_add_empty
thf(fact_21_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_22_char_Osize__gen,axiom,
! [X1: $o,X22: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X22 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_23_mset__sorted__list__of__multiset,axiom,
! [M: multiset_nat] :
( ( mset_nat @ ( linord3047872887403683810et_nat @ M ) )
= M ) ).
% mset_sorted_list_of_multiset
thf(fact_24_mset__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( mset_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( image_mset_nat_nat @ F @ ( mset_nat @ Xs2 ) ) ) ).
% mset_map
thf(fact_25_mset__map,axiom,
! [F: nat > b,Xs2: list_nat] :
( ( mset_b @ ( map_nat_b @ F @ Xs2 ) )
= ( image_mset_nat_b @ F @ ( mset_nat @ Xs2 ) ) ) ).
% mset_map
thf(fact_26_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_27_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_28_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_29_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_30_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_31_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_32_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_33_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_34_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_35_harm__nonneg,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( harmonic_harm_real @ N ) ) ).
% harm_nonneg
thf(fact_36_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_37_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_38_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_39_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_40_char_Oinject,axiom,
! [X1: $o,X22: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y2: $o,Y3: $o,Y4: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
( ( ( char2 @ X1 @ X22 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 )
= ( char2 @ Y1 @ Y2 @ Y3 @ Y4 @ Y5 @ Y6 @ Y7 @ Y8 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 )
& ( X3 = Y3 )
& ( X4 = Y4 )
& ( X5 = Y5 )
& ( X6 = Y6 )
& ( X7 = Y7 )
& ( X8 = Y8 ) ) ) ).
% char.inject
thf(fact_41_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_42_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_43_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_44_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_45_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_46_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_47_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_48_char_Oexhaust,axiom,
! [Y: char] :
~ ! [X12: $o,X23: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( Y
!= ( char2 @ X12 @ X23 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) ) ).
% char.exhaust
thf(fact_49_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_50_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_51_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_52_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_53_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_54_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_55_harm__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_real @ ( harmonic_harm_real @ M2 ) @ ( harmonic_harm_real @ N ) ) ) ).
% harm_mono
thf(fact_56_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X3: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_57_size__neq__size__imp__neq,axiom,
! [X2: char,Y: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_58_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_59_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_60_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_61_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_62_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_63_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X9: real] : ( member_real @ X9 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_66_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% of_nat_mask_eq
thf(fact_67_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_nat_mask_eq
thf(fact_68_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_71_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_72_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_73_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_74_real__arch__simple,axiom,
! [X2: real] :
? [N2: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_75_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_76_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_77_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_78_interval__def,axiom,
( interval_nat
= ( ^ [I2: set_nat] :
! [X9: nat,Y9: nat,Z: nat] :
( ( member_nat @ X9 @ I2 )
=> ( ( member_nat @ Z @ I2 )
=> ( ( ord_less_eq_nat @ X9 @ Y9 )
=> ( ( ord_less_eq_nat @ Y9 @ Z )
=> ( member_nat @ Y9 @ I2 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_79_interval__def,axiom,
( interval_int
= ( ^ [I2: set_int] :
! [X9: int,Y9: int,Z: int] :
( ( member_int @ X9 @ I2 )
=> ( ( member_int @ Z @ I2 )
=> ( ( ord_less_eq_int @ X9 @ Y9 )
=> ( ( ord_less_eq_int @ Y9 @ Z )
=> ( member_int @ Y9 @ I2 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_80_interval__def,axiom,
( interval_real
= ( ^ [I2: set_real] :
! [X9: real,Y9: real,Z: real] :
( ( member_real @ X9 @ I2 )
=> ( ( member_real @ Z @ I2 )
=> ( ( ord_less_eq_real @ X9 @ Y9 )
=> ( ( ord_less_eq_real @ Y9 @ Z )
=> ( member_real @ Y9 @ I2 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_81_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I2: set_nat] :
! [X9: nat,Y9: nat] :
( ( member_nat @ Y9 @ I2 )
=> ( ( ord_less_eq_nat @ X9 @ Y9 )
=> ( member_nat @ X9 @ I2 ) ) ) ) ) ).
% down_ray_def
thf(fact_82_down__ray__def,axiom,
( down_ray_int
= ( ^ [I2: set_int] :
! [X9: int,Y9: int] :
( ( member_int @ Y9 @ I2 )
=> ( ( ord_less_eq_int @ X9 @ Y9 )
=> ( member_int @ X9 @ I2 ) ) ) ) ) ).
% down_ray_def
thf(fact_83_down__ray__def,axiom,
( down_ray_real
= ( ^ [I2: set_real] :
! [X9: real,Y9: real] :
( ( member_real @ Y9 @ I2 )
=> ( ( ord_less_eq_real @ X9 @ Y9 )
=> ( member_real @ X9 @ I2 ) ) ) ) ) ).
% down_ray_def
thf(fact_84_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I2: set_nat] :
! [X9: nat,Y9: nat] :
( ( member_nat @ X9 @ I2 )
=> ( ( ord_less_eq_nat @ X9 @ Y9 )
=> ( member_nat @ Y9 @ I2 ) ) ) ) ) ).
% up_ray_def
thf(fact_85_up__ray__def,axiom,
( up_ray_int
= ( ^ [I2: set_int] :
! [X9: int,Y9: int] :
( ( member_int @ X9 @ I2 )
=> ( ( ord_less_eq_int @ X9 @ Y9 )
=> ( member_int @ Y9 @ I2 ) ) ) ) ) ).
% up_ray_def
thf(fact_86_up__ray__def,axiom,
( up_ray_real
= ( ^ [I2: set_real] :
! [X9: real,Y9: real] :
( ( member_real @ X9 @ I2 )
=> ( ( ord_less_eq_real @ X9 @ Y9 )
=> ( member_real @ Y9 @ I2 ) ) ) ) ) ).
% up_ray_def
thf(fact_87_size__image__mset,axiom,
! [F: nat > b,M: multiset_nat] :
( ( size_size_multiset_b @ ( image_mset_nat_b @ F @ M ) )
= ( size_s5917832649809541300et_nat @ M ) ) ).
% size_image_mset
thf(fact_88_size__mset,axiom,
! [Xs2: list_nat] :
( ( size_s5917832649809541300et_nat @ ( mset_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% size_mset
thf(fact_89_image__replicate__mset,axiom,
! [F: nat > b,N: nat,A: nat] :
( ( image_mset_nat_b @ F @ ( replicate_mset_nat @ N @ A ) )
= ( replicate_mset_b @ N @ ( F @ A ) ) ) ).
% image_replicate_mset
thf(fact_90_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_91_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_92_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_93_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_94_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_95_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ B ) )
=> ? [X10: nat] :
( ( P @ X10 )
& ! [Y11: nat] :
( ( P @ Y11 )
=> ( ord_less_eq_nat @ Y11 @ X10 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_96_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_97_mset__eq__length,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs2 )
= ( mset_nat @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% mset_eq_length
thf(fact_98_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_99_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_100_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_101_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_102_le__cases3,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_103_le__cases3,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_104_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y12: nat,Z3: nat] : ( Y12 = Z3 ) )
= ( ^ [X9: nat,Y9: nat] :
( ( ord_less_eq_nat @ X9 @ Y9 )
& ( ord_less_eq_nat @ Y9 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_105_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y12: int,Z3: int] : ( Y12 = Z3 ) )
= ( ^ [X9: int,Y9: int] :
( ( ord_less_eq_int @ X9 @ Y9 )
& ( ord_less_eq_int @ Y9 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_106_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y12: real,Z3: real] : ( Y12 = Z3 ) )
= ( ^ [X9: real,Y9: real] :
( ( ord_less_eq_real @ X9 @ Y9 )
& ( ord_less_eq_real @ Y9 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_107_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_108_ord__eq__le__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_109_ord__eq__le__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_110_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_111_ord__le__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_112_ord__le__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_113_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_114_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_115_order__antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_116_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_117_order_Otrans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_118_order_Otrans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_119_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_120_order__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_121_order__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_122_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_123_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_124_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_125_dual__order_Oeq__iff,axiom,
( ( ^ [Y12: nat,Z3: nat] : ( Y12 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_126_dual__order_Oeq__iff,axiom,
( ( ^ [Y12: int,Z3: int] : ( Y12 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_127_dual__order_Oeq__iff,axiom,
( ( ^ [Y12: real,Z3: real] : ( Y12 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_128_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_129_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_130_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_131_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_132_dual__order_Otrans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_133_dual__order_Otrans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_134_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_135_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_136_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_137_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y12: nat,Z3: nat] : ( Y12 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_138_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y12: int,Z3: int] : ( Y12 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_139_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y12: real,Z3: real] : ( Y12 = Z3 ) )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_140_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_150_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_151_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_152_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_158_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_159_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_160_order__eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_161_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_162_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_163_linorder__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_164_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_165_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_166_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_167_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_168_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_169_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_170_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_171_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_172_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_173_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_174_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_175_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_176_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_177_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_178_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_179_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_180_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_181_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_182_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_183_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_184_linorder__le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_185_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_186_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_187_order__antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_188_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_189_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > b] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_b @ ( sort_map_b @ F @ N @ I ) @ ( sort_map_b @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_190_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N @ I ) @ ( sort_map_nat @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_191_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > int] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( sort_map_int @ F @ N @ I ) @ ( sort_map_int @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_192_sort__map__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > real] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N @ I ) @ ( sort_map_real @ F @ N @ J ) ) ) ) ).
% sort_map_mono
thf(fact_193_GreatestI2__order,axiom,
! [P: int > $o,X2: int,Q: int > $o] :
( ( P @ X2 )
=> ( ! [Y10: int] :
( ( P @ Y10 )
=> ( ord_less_eq_int @ Y10 @ X2 ) )
=> ( ! [X10: int] :
( ( P @ X10 )
=> ( ! [Y11: int] :
( ( P @ Y11 )
=> ( ord_less_eq_int @ Y11 @ X10 ) )
=> ( Q @ X10 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_194_GreatestI2__order,axiom,
! [P: real > $o,X2: real,Q: real > $o] :
( ( P @ X2 )
=> ( ! [Y10: real] :
( ( P @ Y10 )
=> ( ord_less_eq_real @ Y10 @ X2 ) )
=> ( ! [X10: real] :
( ( P @ X10 )
=> ( ! [Y11: real] :
( ( P @ Y11 )
=> ( ord_less_eq_real @ Y11 @ X10 ) )
=> ( Q @ X10 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_195_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ X2 ) )
=> ( ! [X10: nat] :
( ( P @ X10 )
=> ( ! [Y11: nat] :
( ( P @ Y11 )
=> ( ord_less_eq_nat @ Y11 @ X10 ) )
=> ( Q @ X10 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_196_Greatest__equality,axiom,
! [P: int > $o,X2: int] :
( ( P @ X2 )
=> ( ! [Y10: int] :
( ( P @ Y10 )
=> ( ord_less_eq_int @ Y10 @ X2 ) )
=> ( ( order_Greatest_int @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_197_Greatest__equality,axiom,
! [P: real > $o,X2: real] :
( ( P @ X2 )
=> ( ! [Y10: real] :
( ( P @ Y10 )
=> ( ord_less_eq_real @ Y10 @ X2 ) )
=> ( ( order_Greatest_real @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_198_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_199_sorted__list__of__multiset__empty,axiom,
( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
= nil_nat ) ).
% sorted_list_of_multiset_empty
thf(fact_200_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X10: nat] :
( ( P @ X10 )
=> ( ord_less_eq_nat @ X10 @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X11: nat] :
( ( P @ X11 )
=> ( ord_less_eq_nat @ X11 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_201_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_202_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_203_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_204_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_205_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_206_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_207_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_208_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_209_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_210_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_211_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_212_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_213_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_214_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_215_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_216_mset__zero__iff__right,axiom,
! [X2: list_nat] :
( ( zero_z7348594199698428585et_nat
= ( mset_nat @ X2 ) )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff_right
thf(fact_217_mset__zero__iff,axiom,
! [X2: list_nat] :
( ( ( mset_nat @ X2 )
= zero_z7348594199698428585et_nat )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff
thf(fact_218_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_219_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_220_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_221_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_222_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_223_harm__pos__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% harm_pos_iff
thf(fact_224_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_225_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_226_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_227_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_228_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_229_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_230_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_231_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_232_lt__ex,axiom,
! [X2: int] :
? [Y10: int] : ( ord_less_int @ Y10 @ X2 ) ).
% lt_ex
thf(fact_233_lt__ex,axiom,
! [X2: real] :
? [Y10: real] : ( ord_less_real @ Y10 @ X2 ) ).
% lt_ex
thf(fact_234_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_235_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_236_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_237_dense,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ? [Z4: real] :
( ( ord_less_real @ X2 @ Z4 )
& ( ord_less_real @ Z4 @ Y ) ) ) ).
% dense
thf(fact_238_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_239_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_240_less__imp__neq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_241_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_242_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_243_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_244_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_245_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_246_ord__eq__less__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_247_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_248_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_249_ord__less__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_250_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X10: nat] :
( ! [Y11: nat] :
( ( ord_less_nat @ Y11 @ X10 )
=> ( P @ Y11 ) )
=> ( P @ X10 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_251_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_252_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_253_antisym__conv3,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_real @ Y @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_254_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_255_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_256_linorder__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_257_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_258_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_259_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_260_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_261_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_262_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_263_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X13: nat] : ( P2 @ X13 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_264_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_265_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_266_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_267_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_268_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_269_order_Ostrict__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_270_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_271_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_272_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ( ord_less_real @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_273_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_274_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_275_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_276_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_277_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_278_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_279_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_280_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_281_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_282_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_283_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_284_linorder__neqE,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_285_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_286_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_287_order__less__asym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_288_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_289_linorder__neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_290_linorder__neq__iff,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
= ( ( ord_less_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_291_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_292_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_293_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_294_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_295_order__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_296_order__less__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_297_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_298_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_299_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_300_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_301_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_302_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_303_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_304_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_305_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_306_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_307_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_308_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_309_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_310_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_311_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_312_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_313_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_314_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_315_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_316_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_317_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_318_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_319_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_320_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_321_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_322_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_323_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_324_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_325_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_326_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_327_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_328_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_329_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_330_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_331_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_332_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_333_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_334_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_335_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_336_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_337_order__less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_338_order__less__not__sym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_339_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_340_order__less__imp__triv,axiom,
! [X2: int,Y: int,P: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_341_order__less__imp__triv,axiom,
! [X2: real,Y: real,P: $o] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_342_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_343_linorder__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_344_linorder__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_345_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_346_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_347_order__less__imp__not__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_348_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_349_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_350_order__less__imp__not__eq2,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_351_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_352_order__less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_353_order__less__imp__not__less,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_354_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_355_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_356_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_357_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_358_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_359_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_360_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_361_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_362_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_363_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_364_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_365_nat__int__comparison_I1_J,axiom,
( ( ^ [Y12: nat,Z3: nat] : ( Y12 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_366_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_367_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_368_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_369_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_370_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_371_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_372_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_373_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_374_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_375_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_376_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_377_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_378_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_379_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_380_order__le__imp__less__or__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_381_order__le__imp__less__or__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_382_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_383_linorder__le__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_384_linorder__le__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_385_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_386_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_387_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_388_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_389_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_390_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_391_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_392_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_393_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_394_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_395_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_396_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_397_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_398_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_399_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_400_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_401_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_402_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_403_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_404_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_405_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_eq_nat @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_406_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_407_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_408_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_eq_int @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_409_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_410_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_411_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_eq_real @ X10 @ Y10 )
=> ( ord_less_eq_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_412_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_413_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_414_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_nat @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_415_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_416_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_417_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_int @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_418_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X10: nat,Y10: nat] :
( ( ord_less_nat @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_419_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X10: int,Y10: int] :
( ( ord_less_int @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_420_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X10: real,Y10: real] :
( ( ord_less_real @ X10 @ Y10 )
=> ( ord_less_real @ ( F @ X10 ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_421_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_422_order__less__le__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_423_order__less__le__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_424_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_425_order__le__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_426_order__le__less__trans,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_427_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_428_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_429_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_430_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_431_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_432_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_433_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_434_order__less__imp__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_435_order__less__imp__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_436_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_437_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_438_linorder__not__less,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_439_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_440_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_441_linorder__not__le,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
= ( ord_less_real @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_442_order__less__le,axiom,
( ord_less_nat
= ( ^ [X9: nat,Y9: nat] :
( ( ord_less_eq_nat @ X9 @ Y9 )
& ( X9 != Y9 ) ) ) ) ).
% order_less_le
thf(fact_443_order__less__le,axiom,
( ord_less_int
= ( ^ [X9: int,Y9: int] :
( ( ord_less_eq_int @ X9 @ Y9 )
& ( X9 != Y9 ) ) ) ) ).
% order_less_le
thf(fact_444_order__less__le,axiom,
( ord_less_real
= ( ^ [X9: real,Y9: real] :
( ( ord_less_eq_real @ X9 @ Y9 )
& ( X9 != Y9 ) ) ) ) ).
% order_less_le
thf(fact_445_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X9: nat,Y9: nat] :
( ( ord_less_nat @ X9 @ Y9 )
| ( X9 = Y9 ) ) ) ) ).
% order_le_less
thf(fact_446_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X9: int,Y9: int] :
( ( ord_less_int @ X9 @ Y9 )
| ( X9 = Y9 ) ) ) ) ).
% order_le_less
thf(fact_447_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X9: real,Y9: real] :
( ( ord_less_real @ X9 @ Y9 )
| ( X9 = Y9 ) ) ) ) ).
% order_le_less
thf(fact_448_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_449_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_450_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_451_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_452_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_453_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_454_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_455_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_456_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_457_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_458_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_459_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_460_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_461_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_462_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_463_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_464_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_465_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_466_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_467_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_468_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_469_dense__le__bounded,axiom,
! [X2: real,Y: real,Z2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X2 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_470_dense__ge__bounded,axiom,
! [Z2: real,X2: real,Y: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X2 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_471_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_472_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_473_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_474_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_475_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_476_order_Ostrict__trans2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_477_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_478_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_479_order_Ostrict__trans1,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_480_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_481_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_482_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_483_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_484_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_485_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_486_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_487_not__le__imp__less,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_eq_int @ Y @ X2 )
=> ( ord_less_int @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_488_not__le__imp__less,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq_real @ Y @ X2 )
=> ( ord_less_real @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_489_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X9: nat,Y9: nat] :
( ( ord_less_eq_nat @ X9 @ Y9 )
& ~ ( ord_less_eq_nat @ Y9 @ X9 ) ) ) ) ).
% less_le_not_le
thf(fact_490_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X9: int,Y9: int] :
( ( ord_less_eq_int @ X9 @ Y9 )
& ~ ( ord_less_eq_int @ Y9 @ X9 ) ) ) ) ).
% less_le_not_le
thf(fact_491_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X9: real,Y9: real] :
( ( ord_less_eq_real @ X9 @ Y9 )
& ~ ( ord_less_eq_real @ Y9 @ X9 ) ) ) ) ).
% less_le_not_le
thf(fact_492_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X10: real] :
( ( ord_less_real @ X10 @ Y )
=> ( ord_less_eq_real @ X10 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_493_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X10: real] :
( ( ord_less_real @ Z2 @ X10 )
=> ( ord_less_eq_real @ Y @ X10 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_494_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_495_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_496_antisym__conv2,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_497_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_498_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_499_antisym__conv1,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_500_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_501_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_502_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_503_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_504_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_505_leI,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% leI
thf(fact_506_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_507_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_508_leD,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_real @ X2 @ Y ) ) ).
% leD
thf(fact_509_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_510_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_511_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_512_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_513_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_514_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_515_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_516_reals__Archimedean2,axiom,
! [X2: real] :
? [N2: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_517_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_518_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_519_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_520_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_521_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_522_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_523_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_524_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_525_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_526_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_527_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_528_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_529_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_530_image__mset__subseteq__mono,axiom,
! [A2: multiset_nat,B5: multiset_nat,F: nat > b] :
( ( subseteq_mset_nat @ A2 @ B5 )
=> ( subseteq_mset_b @ ( image_mset_nat_b @ F @ A2 ) @ ( image_mset_nat_b @ F @ B5 ) ) ) ).
% image_mset_subseteq_mono
thf(fact_531_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_532_harm__pos,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N ) ) ) ).
% harm_pos
thf(fact_533_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_534_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_535_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_536_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_537_mset_Osimps_I1_J,axiom,
( ( mset_nat @ nil_nat )
= zero_z7348594199698428585et_nat ) ).
% mset.simps(1)
thf(fact_538_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_539_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_540_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_541_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > b] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_b @ ( sort_map_b @ F @ N @ I ) @ ( sort_map_b @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_542_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > nat] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_nat @ ( sort_map_nat @ F @ N @ I ) @ ( sort_map_nat @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_543_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > int] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_int @ ( sort_map_int @ F @ N @ I ) @ ( sort_map_int @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_544_sort__map__strict__mono,axiom,
! [J: nat,N: nat,I: nat,F: nat > real] :
( ( ord_less_nat @ J @ N )
=> ( ( ord_less_nat @ I @ J )
=> ( ord_less_eq_real @ ( sort_map_real @ F @ N @ I ) @ ( sort_map_real @ F @ N @ J ) ) ) ) ).
% sort_map_strict_mono
thf(fact_545_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_546_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_547_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_548_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_549_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_550_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_551_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_552_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_553_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_554_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X11: nat] :
( ( ( ord_less_eq_nat @ A @ X11 )
& ( ord_less_nat @ X11 @ C3 ) )
=> ( P @ X11 ) )
& ! [D: nat] :
( ! [X10: nat] :
( ( ( ord_less_eq_nat @ A @ X10 )
& ( ord_less_nat @ X10 @ D ) )
=> ( P @ X10 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_555_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X11: int] :
( ( ( ord_less_eq_int @ A @ X11 )
& ( ord_less_int @ X11 @ C3 ) )
=> ( P @ X11 ) )
& ! [D: int] :
( ! [X10: int] :
( ( ( ord_less_eq_int @ A @ X10 )
& ( ord_less_int @ X10 @ D ) )
=> ( P @ X10 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_556_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: real] :
( ( ord_less_eq_real @ A @ C3 )
& ( ord_less_eq_real @ C3 @ B )
& ! [X11: real] :
( ( ( ord_less_eq_real @ A @ X11 )
& ( ord_less_real @ X11 @ C3 ) )
=> ( P @ X11 ) )
& ! [D: real] :
( ! [X10: real] :
( ( ( ord_less_eq_real @ A @ X10 )
& ( ord_less_real @ X10 @ D ) )
=> ( P @ X10 ) )
=> ( ord_less_eq_real @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_557_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X9: real,Y9: real] :
( ( ord_less_eq_real @ X9 @ Y9 )
& ~ ( ord_less_eq_real @ Y9 @ X9 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_558_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ~ ( ord_less_eq_nat @ X11 @ T ) ) ).
% pinf(6)
thf(fact_559_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ~ ( ord_less_eq_int @ X11 @ T ) ) ).
% pinf(6)
thf(fact_560_pinf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ~ ( ord_less_eq_real @ X11 @ T ) ) ).
% pinf(6)
thf(fact_561_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ( ord_less_eq_nat @ T @ X11 ) ) ).
% pinf(8)
thf(fact_562_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ( ord_less_eq_int @ T @ X11 ) ) ).
% pinf(8)
thf(fact_563_pinf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ( ord_less_eq_real @ T @ X11 ) ) ).
% pinf(8)
thf(fact_564_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ( ord_less_eq_nat @ X11 @ T ) ) ).
% minf(6)
thf(fact_565_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ( ord_less_eq_int @ X11 @ T ) ) ).
% minf(6)
thf(fact_566_minf_I6_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ( ord_less_eq_real @ X11 @ T ) ) ).
% minf(6)
thf(fact_567_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X11 ) ) ).
% minf(8)
thf(fact_568_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X11 ) ) ).
% minf(8)
thf(fact_569_minf_I8_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ~ ( ord_less_eq_real @ T @ X11 ) ) ).
% minf(8)
thf(fact_570_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_571_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_572_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_573_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_574_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_575_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_576_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ~ ( ord_less_nat @ T @ X11 ) ) ).
% minf(7)
thf(fact_577_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ~ ( ord_less_int @ T @ X11 ) ) ).
% minf(7)
thf(fact_578_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ~ ( ord_less_real @ T @ X11 ) ) ).
% minf(7)
thf(fact_579_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ( ord_less_nat @ X11 @ T ) ) ).
% minf(5)
thf(fact_580_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ( ord_less_int @ X11 @ T ) ) ).
% minf(5)
thf(fact_581_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ( ord_less_real @ X11 @ T ) ) ).
% minf(5)
thf(fact_582_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ( X11 != T ) ) ).
% minf(4)
thf(fact_583_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ( X11 != T ) ) ).
% minf(4)
thf(fact_584_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ( X11 != T ) ) ).
% minf(4)
thf(fact_585_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ( X11 != T ) ) ).
% minf(3)
thf(fact_586_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ( X11 != T ) ) ).
% minf(3)
thf(fact_587_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ( X11 != T ) ) ).
% minf(3)
thf(fact_588_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ X10 @ Z5 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ X10 @ Z5 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ( ( ( P @ X11 )
| ( Q @ X11 ) )
= ( ( P4 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(2)
thf(fact_589_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ X10 @ Z5 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ X10 @ Z5 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ( ( ( P @ X11 )
| ( Q @ X11 ) )
= ( ( P4 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(2)
thf(fact_590_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ X10 @ Z5 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ X10 @ Z5 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ( ( ( P @ X11 )
| ( Q @ X11 ) )
= ( ( P4 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(2)
thf(fact_591_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ X10 @ Z5 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ X10 @ Z5 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ X11 @ Z4 )
=> ( ( ( P @ X11 )
& ( Q @ X11 ) )
= ( ( P4 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(1)
thf(fact_592_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ X10 @ Z5 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ X10 @ Z5 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ X11 @ Z4 )
=> ( ( ( P @ X11 )
& ( Q @ X11 ) )
= ( ( P4 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(1)
thf(fact_593_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ X10 @ Z5 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ X10 @ Z5 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ X11 @ Z4 )
=> ( ( ( P @ X11 )
& ( Q @ X11 ) )
= ( ( P4 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% minf(1)
thf(fact_594_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ( ord_less_nat @ T @ X11 ) ) ).
% pinf(7)
thf(fact_595_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ( ord_less_int @ T @ X11 ) ) ).
% pinf(7)
thf(fact_596_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ( ord_less_real @ T @ X11 ) ) ).
% pinf(7)
thf(fact_597_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ~ ( ord_less_nat @ X11 @ T ) ) ).
% pinf(5)
thf(fact_598_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ~ ( ord_less_int @ X11 @ T ) ) ).
% pinf(5)
thf(fact_599_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ~ ( ord_less_real @ X11 @ T ) ) ).
% pinf(5)
thf(fact_600_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ( X11 != T ) ) ).
% pinf(4)
thf(fact_601_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ( X11 != T ) ) ).
% pinf(4)
thf(fact_602_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ( X11 != T ) ) ).
% pinf(4)
thf(fact_603_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ( X11 != T ) ) ).
% pinf(3)
thf(fact_604_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ( X11 != T ) ) ).
% pinf(3)
thf(fact_605_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ( X11 != T ) ) ).
% pinf(3)
thf(fact_606_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ Z5 @ X10 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ Z5 @ X10 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ( ( ( P @ X11 )
| ( Q @ X11 ) )
= ( ( P4 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_607_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ Z5 @ X10 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ Z5 @ X10 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ( ( ( P @ X11 )
| ( Q @ X11 ) )
= ( ( P4 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_608_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ Z5 @ X10 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ Z5 @ X10 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ( ( ( P @ X11 )
| ( Q @ X11 ) )
= ( ( P4 @ X11 )
| ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_609_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ Z5 @ X10 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: nat] :
! [X10: nat] :
( ( ord_less_nat @ Z5 @ X10 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: nat] :
! [X11: nat] :
( ( ord_less_nat @ Z4 @ X11 )
=> ( ( ( P @ X11 )
& ( Q @ X11 ) )
= ( ( P4 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_610_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ Z5 @ X10 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ Z5 @ X10 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: int] :
! [X11: int] :
( ( ord_less_int @ Z4 @ X11 )
=> ( ( ( P @ X11 )
& ( Q @ X11 ) )
= ( ( P4 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_611_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ Z5 @ X10 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [Z5: real] :
! [X10: real] :
( ( ord_less_real @ Z5 @ X10 )
=> ( ( Q @ X10 )
= ( Q2 @ X10 ) ) )
=> ? [Z4: real] :
! [X11: real] :
( ( ord_less_real @ Z4 @ X11 )
=> ( ( ( P @ X11 )
& ( Q @ X11 ) )
= ( ( P4 @ X11 )
& ( Q2 @ X11 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_612_ex__gt__or__lt,axiom,
! [A: real] :
? [B2: real] :
( ( ord_less_real @ A @ B2 )
| ( ord_less_real @ B2 @ A ) ) ).
% ex_gt_or_lt
thf(fact_613_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_614_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_615_conj__le__cong,axiom,
! [X2: int,X14: int,P: $o,P4: $o] :
( ( X2 = X14 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X14 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X14 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_616_imp__le__cong,axiom,
! [X2: int,X14: int,P: $o,P4: $o] :
( ( X2 = X14 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X14 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X14 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_617_ex__inverse__of__nat__less,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X2 ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_618_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D3: real,E: real] :
( ( ord_less_real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_619_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_620_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_621_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_622_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_623_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_624_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_625_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_626_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_627_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_628_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_629_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_630_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X9: real,Y9: real] :
( ( ord_less_real @ X9 @ Y9 )
| ( X9 = Y9 ) ) ) ) ).
% less_eq_real_def
thf(fact_631_linordered__field__no__lb,axiom,
! [X11: real] :
? [Y10: real] : ( ord_less_real @ Y10 @ X11 ) ).
% linordered_field_no_lb
thf(fact_632_linordered__field__no__ub,axiom,
! [X11: real] :
? [X_1: real] : ( ord_less_real @ X11 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_633_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_634_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_635_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_636_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_637_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_638_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_639_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_640_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_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_real__arch__invD,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N2: nat] :
( ( N2 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ).
% real_arch_invD
thf(fact_651_seq__mono__lemma,axiom,
! [M2: nat,D4: nat > real,E2: nat > real] :
( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_real @ ( D4 @ N2 ) @ ( E2 @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( E2 @ N2 ) @ ( E2 @ M2 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D4 @ N4 ) @ ( E2 @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_652_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_653_bgauge__existence__lemma,axiom,
! [S: set_real,Q3: real > real > $o] :
( ( ! [X9: real] :
( ( member_real @ X9 @ S )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ( Q3 @ D5 @ X9 ) ) ) )
= ( ! [X9: real] :
? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ( ( member_real @ X9 @ S )
=> ( Q3 @ D5 @ X9 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_654_complete__real,axiom,
! [S2: set_real] :
( ? [X11: real] : ( member_real @ X11 @ S2 )
=> ( ? [Z5: real] :
! [X10: real] :
( ( member_real @ X10 @ S2 )
=> ( ord_less_eq_real @ X10 @ Z5 ) )
=> ? [Y10: real] :
( ! [X11: real] :
( ( member_real @ X11 @ S2 )
=> ( ord_less_eq_real @ X11 @ Y10 ) )
& ! [Z5: real] :
( ! [X10: real] :
( ( member_real @ X10 @ S2 )
=> ( ord_less_eq_real @ X10 @ Z5 ) )
=> ( ord_less_eq_real @ Y10 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_655_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_656_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X2 ) @ C2 ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_657_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_658_radical__0,axiom,
! [N: nat,R: nat > real > real,A: formal3361831859752904756s_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( formal8005797870169972230l_real @ R @ zero_zero_nat @ A @ N )
= zero_zero_real ) ) ).
% radical_0
thf(fact_659_fps__inverse__eq__0_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_eq_0'
thf(fact_660_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_661_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_662_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_663_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_664_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_665_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_666_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_667_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_668_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_669_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_670_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_671_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_672_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_673_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_674_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_675_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_676_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_677_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_fps__mult__nth__0,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ zero_zero_nat )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_700_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ zero_zero_nat )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_701_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ zero_zero_nat )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_702_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_nat,K: nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ ( semiri1524631719018205113ps_nat @ K ) ) @ N )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( semiri1316708129612266289at_nat @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_703_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_int,K: nat,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ ( semiri6570152736363784213ps_int @ K ) ) @ N )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ N ) @ ( semiri1314217659103216013at_int @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_704_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal3361831859752904756s_real,K: nat,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ ( semiri2475410149736220053s_real @ K ) ) @ N )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ N ) @ ( semiri5074537144036343181t_real @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_705_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( semiri1524631719018205113ps_nat @ K ) @ F ) @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ K ) @ ( formal3720337525774269570th_nat @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_706_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( semiri6570152736363784213ps_int @ K ) @ F ) @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ ( formal3717847055265219294th_int @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_707_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( semiri2475410149736220053s_real @ K ) @ F ) @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( formal2580924720334399070h_real @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_708_fps__zero__nth,axiom,
! [N: nat] :
( ( formal3720337525774269570th_nat @ zero_z8531573698755551073ps_nat @ N )
= zero_zero_nat ) ).
% fps_zero_nth
thf(fact_709_fps__zero__nth,axiom,
! [N: nat] :
( ( formal3717847055265219294th_int @ zero_z4353722679246354365ps_int @ N )
= zero_zero_int ) ).
% fps_zero_nth
thf(fact_710_fps__zero__nth,axiom,
! [N: nat] :
( ( formal2580924720334399070h_real @ zero_z7760665558314615101s_real @ N )
= zero_zero_real ) ).
% fps_zero_nth
thf(fact_711_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_712_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_713_fps__nth__of__nat,axiom,
! [N: nat,C2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C2 ) @ N )
= ( semiri1316708129612266289at_nat @ C2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C2 ) @ N )
= zero_zero_nat ) ) ) ).
% fps_nth_of_nat
thf(fact_714_fps__nth__of__nat,axiom,
! [N: nat,C2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C2 ) @ N )
= ( semiri1314217659103216013at_int @ C2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C2 ) @ N )
= zero_zero_int ) ) ) ).
% fps_nth_of_nat
thf(fact_715_fps__nth__of__nat,axiom,
! [N: nat,C2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C2 ) @ N )
= ( semiri5074537144036343181t_real @ C2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C2 ) @ N )
= zero_zero_real ) ) ) ).
% fps_nth_of_nat
thf(fact_716_fps__inverse__0__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= zero_zero_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real ) ) ).
% fps_inverse_0_iff
thf(fact_717_fps__inverse__idempotent,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( invers68952373231134600s_real @ ( invers68952373231134600s_real @ F ) )
= F ) ) ).
% fps_inverse_idempotent
thf(fact_718_fps__inverse__nth__0,axiom,
! [F: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ).
% fps_inverse_nth_0
thf(fact_719_fps__inverse__eq__0__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real ) ) ).
% fps_inverse_eq_0_iff
thf(fact_720_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_721_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_722_mult_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_723_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_724_mult_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_725_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_726_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_727_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_728_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_729_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_730_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_731_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_732_mult_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_733_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_734_mult_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_735_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_736_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_737_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_mult__delta__left,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_751_mult__delta__left,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_752_mult__delta__left,axiom,
! [B: $o,X2: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X2 @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_753_mult__delta__right,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_754_mult__delta__right,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_755_mult__delta__right,axiom,
! [B: $o,X2: int,Y: int] :
( ( B
=> ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X2 @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_756_mult__of__nat__commute,axiom,
! [X2: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_757_mult__of__nat__commute,axiom,
! [X2: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_758_mult__of__nat__commute,axiom,
! [X2: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_759_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_760_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_761_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_762_fps__nonzeroI,axiom,
! [F: formal_Power_fps_nat,N: nat] :
( ( ( formal3720337525774269570th_nat @ F @ N )
!= zero_zero_nat )
=> ( F != zero_z8531573698755551073ps_nat ) ) ).
% fps_nonzeroI
thf(fact_763_fps__nonzeroI,axiom,
! [F: formal_Power_fps_int,N: nat] :
( ( ( formal3717847055265219294th_int @ F @ N )
!= zero_zero_int )
=> ( F != zero_z4353722679246354365ps_int ) ) ).
% fps_nonzeroI
thf(fact_764_fps__nonzeroI,axiom,
! [F: formal3361831859752904756s_real,N: nat] :
( ( ( formal2580924720334399070h_real @ F @ N )
!= zero_zero_real )
=> ( F != zero_z7760665558314615101s_real ) ) ).
% fps_nonzeroI
thf(fact_765_fps__nonzero__nth,axiom,
! [F: formal_Power_fps_nat] :
( ( F != zero_z8531573698755551073ps_nat )
= ( ? [N3: nat] :
( ( formal3720337525774269570th_nat @ F @ N3 )
!= zero_zero_nat ) ) ) ).
% fps_nonzero_nth
thf(fact_766_fps__nonzero__nth,axiom,
! [F: formal_Power_fps_int] :
( ( F != zero_z4353722679246354365ps_int )
= ( ? [N3: nat] :
( ( formal3717847055265219294th_int @ F @ N3 )
!= zero_zero_int ) ) ) ).
% fps_nonzero_nth
thf(fact_767_fps__nonzero__nth,axiom,
! [F: formal3361831859752904756s_real] :
( ( F != zero_z7760665558314615101s_real )
= ( ? [N3: nat] :
( ( formal2580924720334399070h_real @ F @ N3 )
!= zero_zero_real ) ) ) ).
% fps_nonzero_nth
thf(fact_768_nonzero__inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_769_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X2: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X2 )
= ( times_times_real @ X2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_770_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_771_fps__nonzero__nth__minimal,axiom,
! [F: formal_Power_fps_nat] :
( ( F != zero_z8531573698755551073ps_nat )
= ( ? [N3: nat] :
( ( ( formal3720337525774269570th_nat @ F @ N3 )
!= zero_zero_nat )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ( formal3720337525774269570th_nat @ F @ M4 )
= zero_zero_nat ) ) ) ) ) ).
% fps_nonzero_nth_minimal
thf(fact_772_fps__nonzero__nth__minimal,axiom,
! [F: formal_Power_fps_int] :
( ( F != zero_z4353722679246354365ps_int )
= ( ? [N3: nat] :
( ( ( formal3717847055265219294th_int @ F @ N3 )
!= zero_zero_int )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ( formal3717847055265219294th_int @ F @ M4 )
= zero_zero_int ) ) ) ) ) ).
% fps_nonzero_nth_minimal
thf(fact_773_fps__nonzero__nth__minimal,axiom,
! [F: formal3361831859752904756s_real] :
( ( F != zero_z7760665558314615101s_real )
= ( ? [N3: nat] :
( ( ( formal2580924720334399070h_real @ F @ N3 )
!= zero_zero_real )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ( formal2580924720334399070h_real @ F @ M4 )
= zero_zero_real ) ) ) ) ) ).
% fps_nonzero_nth_minimal
thf(fact_774_ex__less__of__nat__mult,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) ) ).
% ex_less_of_nat_mult
thf(fact_775_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_776_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_777_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y11: real] :
? [N2: nat] : ( ord_less_real @ Y11 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_778_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_779_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_780_inverse__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_real @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_781_inverse__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_eq_real @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_782_fps__inverse__0__iff_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= zero_zero_real )
= ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real ) ) ).
% fps_inverse_0_iff'
thf(fact_783_fps__inverse__eq__0,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_eq_0
thf(fact_784_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_785_fps__inverse__eq__0__iff_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real )
= ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real ) ) ).
% fps_inverse_eq_0_iff'
thf(fact_786_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_787_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_788_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_789_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_790_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_791_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_792_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_793_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_794_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_795_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_796_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_797_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_798_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_799_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_800_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_801_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_802_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_803_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_804_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_805_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_806_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_807_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_808_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_809_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_810_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_811_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_812_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_813_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_814_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_815_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_816_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_817_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_818_real__minus__mult__self__le,axiom,
! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_819_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_820_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_821_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_822_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_823_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_824_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_825_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_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_mult__left__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_835_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_836_mult__left__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_837_mult__right__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_838_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_839_mult__right__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_840_mult__mono,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D4 ) ) ) ) ) ) ).
% mult_mono
thf(fact_841_mult__mono,axiom,
! [A: int,B: int,C2: int,D4: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D4 ) ) ) ) ) ) ).
% mult_mono
thf(fact_842_mult__mono,axiom,
! [A: real,B: real,C2: real,D4: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D4 ) ) ) ) ) ) ).
% mult_mono
thf(fact_843_mult__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D4 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_844_mult__mono_H,axiom,
! [A: int,B: int,C2: int,D4: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D4 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_845_mult__mono_H,axiom,
! [A: real,B: real,C2: real,D4: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D4 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_846_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_847_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_848_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_849_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_850_mult__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_851_mult__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_852_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_853_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_854_mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_855_mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_856_mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_857_mult__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_858_mult__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_859_mult__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_860_mult__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_861_mult__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_862_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_863_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_864_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_865_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_866_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_867_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_868_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_869_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_870_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_871_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_872_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_873_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_874_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_875_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_876_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_877_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_878_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_879_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_880_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_881_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_882_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_883_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_884_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_885_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_886_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_887_mult__less__cancel__right__disj,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_888_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_889_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_890_mult__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_891_mult__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_892_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_893_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_894_mult__less__cancel__left__disj,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_895_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_896_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_897_mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_898_mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_899_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_900_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_901_mult__less__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_902_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_903_mult__less__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_904_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_905_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_906_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_907_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_908_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_909_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_910_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_911_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_912_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_913_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_914_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_915_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_916_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_917_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_918_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_919_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_920_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_921_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_922_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_923_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_924_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_925_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_926_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_927_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_928_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_929_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_930_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_931_mult__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_932_mult__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_933_mult__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_934_mult__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_935_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_936_mult__left__less__imp__less,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_937_mult__left__less__imp__less,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_938_mult__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D4 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D4 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_939_mult__strict__mono,axiom,
! [A: int,B: int,C2: int,D4: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D4 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_940_mult__strict__mono,axiom,
! [A: real,B: real,C2: real,D4: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D4 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D4 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_941_mult__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_942_mult__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_943_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_944_mult__right__less__imp__less,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_945_mult__right__less__imp__less,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_946_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D4 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_947_mult__strict__mono_H,axiom,
! [A: int,B: int,C2: int,D4: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D4 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_948_mult__strict__mono_H,axiom,
! [A: real,B: real,C2: real,D4: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D4 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_949_mult__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_950_mult__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_951_mult__le__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_952_mult__le__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_953_mult__le__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_954_mult__le__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_955_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_956_mult__left__le__imp__le,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_957_mult__left__le__imp__le,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_958_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_959_mult__right__le__imp__le,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_960_mult__right__le__imp__le,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_961_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D4 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_962_mult__le__less__imp__less,axiom,
! [A: int,B: int,C2: int,D4: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D4 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_963_mult__le__less__imp__less,axiom,
! [A: real,B: real,C2: real,D4: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D4 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D4 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_964_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D4: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D4 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_965_mult__less__le__imp__less,axiom,
! [A: int,B: int,C2: int,D4: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D4 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_966_mult__less__le__imp__less,axiom,
! [A: real,B: real,C2: real,D4: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D4 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_967_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_968_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_969_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X2: real,B: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_970_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X2: real,Y: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_971_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_972_arsinh__minus__real,axiom,
! [X2: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( arsinh_real @ X2 ) ) ) ).
% arsinh_minus_real
thf(fact_973_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X2: real] :
( ( ( times_times_real @ A @ X2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_974_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_975_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_976_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X2: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_977_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_978_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_979_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_980_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_981_mult__le__cancel__iff2,axiom,
! [Z2: int,X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X2 ) @ ( times_times_int @ Z2 @ Y ) )
= ( ord_less_eq_int @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_982_mult__le__cancel__iff2,axiom,
! [Z2: real,X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X2 ) @ ( times_times_real @ Z2 @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_983_mult__le__cancel__iff1,axiom,
! [Z2: int,X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_eq_int @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_984_mult__le__cancel__iff1,axiom,
! [Z2: real,X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_985_real__eq__0__iff__le__ge__0,axiom,
! [X2: real] :
( ( X2 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_986_mult__less__iff1,axiom,
! [Z2: int,X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X2 @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X2 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_987_mult__less__iff1,axiom,
! [Z2: real,X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X2 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_988_fps__ln__0,axiom,
! [C2: real] :
( ( formal2580924720334399070h_real @ ( formal8688746759596762231n_real @ C2 ) @ zero_zero_nat )
= zero_zero_real ) ).
% fps_ln_0
thf(fact_989_inverse__mult__eq__1,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( times_7561426564079326009s_real @ ( invers68952373231134600s_real @ F ) @ F )
= one_on8598947968683843321s_real ) ) ).
% inverse_mult_eq_1
thf(fact_990_fps__cutoff__nth,axiom,
! [I: nat,N: nat,F: formal_Power_fps_nat] :
( ( ( ord_less_nat @ I @ N )
=> ( ( formal3720337525774269570th_nat @ ( formal4818209184033568742ff_nat @ N @ F ) @ I )
= ( formal3720337525774269570th_nat @ F @ I ) ) )
& ( ~ ( ord_less_nat @ I @ N )
=> ( ( formal3720337525774269570th_nat @ ( formal4818209184033568742ff_nat @ N @ F ) @ I )
= zero_zero_nat ) ) ) ).
% fps_cutoff_nth
thf(fact_991_fps__cutoff__nth,axiom,
! [I: nat,N: nat,F: formal_Power_fps_int] :
( ( ( ord_less_nat @ I @ N )
=> ( ( formal3717847055265219294th_int @ ( formal4815718713524518466ff_int @ N @ F ) @ I )
= ( formal3717847055265219294th_int @ F @ I ) ) )
& ( ~ ( ord_less_nat @ I @ N )
=> ( ( formal3717847055265219294th_int @ ( formal4815718713524518466ff_int @ N @ F ) @ I )
= zero_zero_int ) ) ) ).
% fps_cutoff_nth
thf(fact_992_fps__cutoff__nth,axiom,
! [I: nat,N: nat,F: formal3361831859752904756s_real] :
( ( ( ord_less_nat @ I @ N )
=> ( ( formal2580924720334399070h_real @ ( formal1487479903726251970f_real @ N @ F ) @ I )
= ( formal2580924720334399070h_real @ F @ I ) ) )
& ( ~ ( ord_less_nat @ I @ N )
=> ( ( formal2580924720334399070h_real @ ( formal1487479903726251970f_real @ N @ F ) @ I )
= zero_zero_real ) ) ) ).
% fps_cutoff_nth
thf(fact_993_fps__XD__nth,axiom,
! [A: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( formal3720337525774269570th_nat @ A @ N ) ) ) ).
% fps_XD_nth
thf(fact_994_fps__XD__nth,axiom,
! [A: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( formal3717847055265219294th_int @ A @ N ) ) ) ).
% fps_XD_nth
thf(fact_995_fps__XD__nth,axiom,
! [A: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( formal4292469010823635553D_real @ A ) @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( formal2580924720334399070h_real @ A @ N ) ) ) ).
% fps_XD_nth
thf(fact_996_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_997_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_998_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_999_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1000_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1001_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1002_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1003_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1004_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_1005_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_1006_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_1007_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_1008_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_1009_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_1010_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_1011_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1012_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1013_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1014_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1015_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1016_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1017_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1018_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1019_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1020_mult__minus1__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1021_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_1022_mult__minus1,axiom,
! [Z2: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1
thf(fact_1023_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_1024_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_1025_fps__XD__0th,axiom,
! [A: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ zero_zero_nat )
= zero_zero_nat ) ).
% fps_XD_0th
thf(fact_1026_fps__XD__0th,axiom,
! [A: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ zero_zero_nat )
= zero_zero_int ) ).
% fps_XD_0th
thf(fact_1027_fps__XD__0th,axiom,
! [A: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( formal4292469010823635553D_real @ A ) @ zero_zero_nat )
= zero_zero_real ) ).
% fps_XD_0th
thf(fact_1028_fps__one__nth,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N )
= zero_zero_nat ) ) ) ).
% fps_one_nth
thf(fact_1029_fps__one__nth,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N )
= zero_zero_int ) ) ) ).
% fps_one_nth
thf(fact_1030_fps__one__nth,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N )
= zero_zero_real ) ) ) ).
% fps_one_nth
thf(fact_1031_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1032_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1033_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_1034_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1035_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1036_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_1037_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1038_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_1039_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_1040_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_1041_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_1042_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_1043_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1044_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_1045_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1046_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1047_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1048_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1049_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1050_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_1051_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G2: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ G2 @ F ) ) )
= ( ? [K3: real] :
( one_one_real
= ( times_times_real @ K3 @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_1052_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal_Power_fps_int] :
( ( ? [G2: formal_Power_fps_int] :
( one_on8395608022581818233ps_int
= ( times_3091854549176928185ps_int @ G2 @ F ) ) )
= ( ? [K3: int] :
( one_one_int
= ( times_times_int @ K3 @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_1053_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G2: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ F @ G2 ) ) )
= ( ? [K3: real] :
( one_one_real
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ K3 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_1054_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal_Power_fps_int] :
( ( ? [G2: formal_Power_fps_int] :
( one_on8395608022581818233ps_int
= ( times_3091854549176928185ps_int @ F @ G2 ) ) )
= ( ? [K3: int] :
( one_one_int
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ K3 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_1055_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1056_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1057_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1065_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1066_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_1067_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1068_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1069_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1070_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1071_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1072_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1073_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_1074_less__1__mult,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ one_one_int @ M2 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_1075_less__1__mult,axiom,
! [M2: real,N: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_1076_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_1077_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_1078_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_1079_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_1080_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1081_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_1082_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_1083_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_1084_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_1085_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_1086_iterate__add__1,axiom,
! [N: nat] :
( ( iterate_add_nat @ N @ one_one_nat )
= ( semiri1316708129612266289at_nat @ N ) ) ).
% iterate_add_1
thf(fact_1087_iterate__add__1,axiom,
! [N: nat] :
( ( iterate_add_int @ N @ one_one_int )
= ( semiri1314217659103216013at_int @ N ) ) ).
% iterate_add_1
thf(fact_1088_iterate__add__1,axiom,
! [N: nat] :
( ( iterate_add_real @ N @ one_one_real )
= ( semiri5074537144036343181t_real @ N ) ) ).
% iterate_add_1
thf(fact_1089_radical_Osimps_I1_J,axiom,
! [R: nat > real > real,A: formal3361831859752904756s_real] :
( ( formal8005797870169972230l_real @ R @ zero_zero_nat @ A @ zero_zero_nat )
= one_one_real ) ).
% radical.simps(1)
thf(fact_1090_mult__left__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1091_mult__left__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1092_mult__right__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1093_mult__right__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1094_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1095_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1096_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1097_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_1098_mult__left__le,axiom,
! [C2: int,A: int] :
( ( ord_less_eq_int @ C2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_1099_mult__left__le,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_inverse__le__1__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X2 ) @ one_one_real )
= ( ( ord_less_eq_real @ X2 @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% inverse_le_1_iff
thf(fact_1109_one__less__inverse__iff,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X2 ) )
= ( ( ord_less_real @ zero_zero_real @ X2 )
& ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_1110_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_1111_field__class_Ofield__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_1112_field__le__mult__one__interval,axiom,
! [X2: real,Y: real] :
( ! [Z4: real] :
( ( ord_less_real @ zero_zero_real @ Z4 )
=> ( ( ord_less_real @ Z4 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z4 @ X2 ) @ Y ) ) )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_1113_mult__le__cancel__left1,axiom,
! [C2: int,B: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1114_mult__le__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1115_mult__le__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1116_mult__le__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1117_mult__le__cancel__right1,axiom,
! [C2: int,B: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1118_mult__le__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1119_mult__le__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1120_mult__le__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1121_mult__less__cancel__left1,axiom,
! [C2: int,B: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1122_mult__less__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1123_mult__less__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1124_mult__less__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1125_mult__less__cancel__right1,axiom,
! [C2: int,B: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1126_mult__less__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1127_mult__less__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1128_mult__less__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1129_one__le__inverse__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X2 ) )
= ( ( ord_less_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_1130_inverse__less__1__iff,axiom,
! [X2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X2 ) @ one_one_real )
= ( ( ord_less_eq_real @ X2 @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% inverse_less_1_iff
thf(fact_1131_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_1132_fps__unit__dvd__right,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ? [G3: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ G3 @ F ) ) ) ).
% fps_unit_dvd_right
thf(fact_1133_fps__unit__dvd__left,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ? [G3: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ F @ G3 ) ) ) ).
% fps_unit_dvd_left
thf(fact_1134_inverse__mult__eq__1_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( times_7561426564079326009s_real @ F @ ( invers68952373231134600s_real @ F ) )
= one_on8598947968683843321s_real ) ) ).
% inverse_mult_eq_1'
thf(fact_1135_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_1136_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_1137_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_1138_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_1139_mult__eq__1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= one_one_nat )
= ( ( A = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_1140_mult__eq__1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= one_one_int )
= ( ( A = one_one_int )
& ( B = one_one_int ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_1141_mult__eq__1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= one_one_real )
= ( ( A = one_one_real )
& ( B = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_1142_mult__if__delta,axiom,
! [P: $o,Q3: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_1143_mult__if__delta,axiom,
! [P: $o,Q3: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_1144_mult__if__delta,axiom,
! [P: $o,Q3: int] :
( ( P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q3 )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_1145_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_1146_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_1147_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1148_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1149_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1150_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1151_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1152_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_1153_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
=> ( ( M2 = one_one_int )
| ( M2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1154_zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( ( M2 = one_one_int )
& ( N = one_one_int ) )
| ( ( M2
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1155_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1156_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1157_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X10: nat > real] :
( ( P @ X10 )
=> ( P @ ( F @ X10 ) ) )
=> ( ! [X10: nat > real] :
( ( P @ X10 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X10 @ I3 ) )
& ( ord_less_eq_real @ ( X10 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X11: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X11 @ I4 ) @ one_one_nat )
& ! [X11: nat > real,I4: nat] :
( ( ( P @ X11 )
& ( Q @ I4 )
& ( ( X11 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X11 @ I4 )
= zero_zero_nat ) )
& ! [X11: nat > real,I4: nat] :
( ( ( P @ X11 )
& ( Q @ I4 )
& ( ( X11 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X11 @ I4 )
= one_one_nat ) )
& ! [X11: nat > real,I4: nat] :
( ( ( P @ X11 )
& ( Q @ I4 )
& ( ( L2 @ X11 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X11 @ I4 ) @ ( F @ X11 @ I4 ) ) )
& ! [X11: nat > real,I4: nat] :
( ( ( P @ X11 )
& ( Q @ I4 )
& ( ( L2 @ X11 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X11 @ I4 ) @ ( X11 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1158_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_1159_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1160_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_1161_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_1162_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_1163_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1164_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1165_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1166_real__arch__pow,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X2 @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1167_real__arch__pow__inv,axiom,
! [Y: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1168_power__le__one__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real )
= ( ( N = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1169_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X10: real] :
( ( ord_less_real @ zero_zero_real @ X10 )
& ( ( power_power_real @ X10 @ N )
= A )
& ! [Y11: real] :
( ( ( ord_less_real @ zero_zero_real @ Y11 )
& ( ( power_power_real @ Y11 @ N )
= A ) )
=> ( Y11 = X10 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1170_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1171_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1172_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1173_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1174_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1175_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_1176_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_1177_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_1178_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1179_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_1180_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1181_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1182_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1183_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1184_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1185_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1186_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1187_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1188_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1189_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1190_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1191_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1192_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_1193_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1194_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1195_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1196_real__of__nat__div2,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).
% real_of_nat_div2
thf(fact_1197_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1198_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1199_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1200_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1201_real__of__nat__div3,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1202_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1203_real__of__nat__div4,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% real_of_nat_div4
thf(fact_1204_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_1205_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1206_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1207_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1208_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X9: real,Y9: real] : ( times_times_real @ X9 @ ( inverse_inverse_real @ Y9 ) ) ) ) ).
% divide_real_def
thf(fact_1209_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1210_minusinfinity,axiom,
! [D4: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ! [X10: int,K2: int] :
( ( P1 @ X10 )
= ( P1 @ ( minus_minus_int @ X10 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ X10 @ Z5 )
=> ( ( P @ X10 )
= ( P1 @ X10 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1211_plusinfinity,axiom,
! [D4: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ! [X10: int,K2: int] :
( ( P4 @ X10 )
= ( P4 @ ( minus_minus_int @ X10 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ? [Z5: int] :
! [X10: int] :
( ( ord_less_int @ Z5 @ X10 )
=> ( ( P @ X10 )
= ( P4 @ X10 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1212_verit__less__mono__div__int2,axiom,
! [A2: int,B5: int,N: int] :
( ( ord_less_eq_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1213_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_1214_decr__mult__lemma,axiom,
! [D4: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ! [X10: int] :
( ( P @ X10 )
=> ( P @ ( minus_minus_int @ X10 @ D4 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X11: int] :
( ( P @ X11 )
=> ( P @ ( minus_minus_int @ X11 @ ( times_times_int @ K @ D4 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1215_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X2 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1216_reals__power__lt__ex,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K2 ) @ X2 ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1217_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1218_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1219_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1220_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_1221_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_1222_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1223_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1224_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_1225_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_1226_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1227_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1228_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1229_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1230_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1231_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1232_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_1233_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1234_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_1235_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_1236_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_1237_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X2 )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1238_zdiv__mono1,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1239_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1240_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1241_zdiv__mono1__neg,axiom,
! [A: int,A5: int,B: int] :
( ( ord_less_eq_int @ A @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1242_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1243_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1244_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1245_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1246_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1247_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1248_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1249_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1250_zdiv__zmult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1251_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_1252_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1253_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B2: real,C3: real] :
( ( P @ A3 @ B2 )
=> ( ( P @ B2 @ C3 )
=> ( ( ord_less_eq_real @ A3 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C3 )
=> ( P @ A3 @ C3 ) ) ) ) )
=> ( ! [X10: real] :
( ( ord_less_eq_real @ A @ X10 )
=> ( ( ord_less_eq_real @ X10 @ B )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [A3: real,B2: real] :
( ( ( ord_less_eq_real @ A3 @ X10 )
& ( ord_less_eq_real @ X10 @ B2 )
& ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D ) )
=> ( P @ A3 @ B2 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1254_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_1255_exp__inj__iff,axiom,
! [X2: real,Y: real] :
( ( ( exp_real @ X2 )
= ( exp_real @ Y ) )
= ( X2 = Y ) ) ).
% exp_inj_iff
thf(fact_1256_exp__le__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_1257_exp__less__mono,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y ) ) ) ).
% exp_less_mono
thf(fact_1258_exp__less__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y ) )
= ( ord_less_real @ X2 @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_1259_exp__eq__one__iff,axiom,
! [X2: real] :
( ( ( exp_real @ X2 )
= one_one_real )
= ( X2 = zero_zero_real ) ) ).
% exp_eq_one_iff
thf(fact_1260_one__le__exp__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
= ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% one_le_exp_iff
thf(fact_1261_exp__le__one__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
= ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% exp_le_one_iff
thf(fact_1262_exp__less__one__iff,axiom,
! [X2: real] :
( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
= ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% exp_less_one_iff
thf(fact_1263_one__less__exp__iff,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
= ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% one_less_exp_iff
thf(fact_1264_exp__gt__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).
% exp_gt_one
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( image_mset_nat_b @ ( sort_map_b @ f @ n ) @ ( mset_nat @ ( upt @ zero_zero_nat @ n ) ) )
= ( image_mset_nat_b @ f @ ( mset_nat @ ( upt @ zero_zero_nat @ n ) ) ) ) ).
%------------------------------------------------------------------------------