TPTP Problem File: SLH0741^1.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 : CRYSTALS-Kyber/0015_Kyber_spec/prob_00090_003590__25615580_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1457 ( 677 unt; 166 typ; 0 def)
% Number of atoms : 3177 (1983 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 7760 ( 308 ~; 97 |; 197 &;6291 @)
% ( 0 <=>; 867 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 344 ( 344 >; 0 *; 0 +; 0 <<)
% Number of symbols : 150 ( 147 usr; 36 con; 0-3 aty)
% Number of variables : 2435 ( 45 ^;2334 !; 56 ?;2435 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:29:50.145
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_J,type,
poly_p2743341848350813180ring_n: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
poly_p6692042823160534382ring_n: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J_J,type,
poly_poly_poly_int: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
poly_F4222894760850802144ring_n: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
poly_poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
poly_poly_nat: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
poly_poly_int: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Finite____Field__Omod____ring_Itf__n_J,type,
finite_mod_ring_n: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_nat: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_int: $tType ).
thf(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
set_literal: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__n_J,type,
set_n: $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 ).
% Explicit typings (147)
thf(sy_c_Field__as__Ring_Ofield__class_Oeuclidean__size__field_001t__Real__Oreal,type,
field_5283244131969691238d_real: real > nat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ocard_001t__String__Oliteral,type,
finite_card_literal: set_literal > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__n,type,
finite_card_n: set_n > nat ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Fundamental__Theorem__Algebra_Opsize_001t__Int__Oint,type,
fundam7803151596185808025ze_int: poly_int > nat ).
thf(sy_c_Fundamental__Theorem__Algebra_Opsize_001t__Nat__Onat,type,
fundam7805642066694858301ze_nat: poly_nat > nat ).
thf(sy_c_Fundamental__Theorem__Algebra_Opsize_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
fundam881691483918006195ring_n: poly_p6692042823160534382ring_n > nat ).
thf(sy_c_Fundamental__Theorem__Algebra_Opsize_001t__Polynomial__Opoly_It__Int__Oint_J,type,
fundam3750135516382849185ly_int: poly_poly_int > nat ).
thf(sy_c_Fundamental__Theorem__Algebra_Opsize_001t__Real__Oreal,type,
fundam22707326917796505e_real: poly_real > nat ).
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__Finite____Field__Omod____ring_Itf__n_J,type,
one_on2109788483843180749ring_n: finite_mod_ring_n ).
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__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
one_on4318287115420659547ring_n: poly_F4222894760850802144ring_n ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Int__Oint_J,type,
one_one_poly_int: poly_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
one_one_poly_nat: poly_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
one_on5457780782968151273ring_n: poly_p6692042823160534382ring_n ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
one_on1166514126663969025ly_int: poly_poly_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
one_on3656597271595695781ly_nat: poly_poly_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_J,type,
one_on281575345490252151ring_n: poly_p2743341848350813180ring_n ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J_J,type,
one_on7423179019345326345ly_int: poly_poly_poly_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
one_on1191988272081909249y_real: poly_poly_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
one_one_poly_real: poly_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Finite____Field__Omod____ring_Itf__n_J,type,
times_5121417632533718157ring_n: finite_mod_ring_n > finite_mod_ring_n > finite_mod_ring_n ).
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__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
times_4166049284782705435ring_n: poly_F4222894760850802144ring_n > poly_F4222894760850802144ring_n > poly_F4222894760850802144ring_n ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Int__Oint_J,type,
times_times_poly_int: poly_int > poly_int > poly_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
times_times_poly_nat: poly_nat > poly_nat > poly_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
times_2573333606529333417ring_n: poly_p6692042823160534382ring_n > poly_p6692042823160534382ring_n > poly_p6692042823160534382ring_n ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
times_4739760418287672641ly_int: poly_poly_int > poly_poly_int > poly_poly_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
times_7229843563219399397ly_nat: poly_poly_nat > poly_poly_nat > poly_poly_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_J,type,
times_4617534433836805431ring_n: poly_p2743341848350813180ring_n > poly_p2743341848350813180ring_n > poly_p2743341848350813180ring_n ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J_J,type,
times_4100521150541653321ly_int: poly_poly_poly_int > poly_poly_poly_int > poly_poly_poly_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
times_4423207553272384065y_real: poly_poly_real > poly_poly_real > poly_poly_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
times_7914811829580426937y_real: poly_real > poly_real > poly_real ).
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_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__n_J,type,
zero_z7902377597758090121ring_n: finite_mod_ring_n ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
zero_z2753989067526334999ring_n: poly_F4222894760850802144ring_n ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Int__Oint_J,type,
zero_zero_poly_int: poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
zero_zero_poly_nat: poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
zero_z5482829069124612005ring_n: poly_p6692042823160534382ring_n ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
zero_z799223564134138693ly_int: poly_poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
zero_z3289306709065865449ly_nat: poly_poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_J,type,
zero_z3442457038203223091ring_n: poly_p2743341848350813180ring_n ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J_J,type,
zero_z240508265545053005ly_int: poly_poly_poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
zero_z5583686468110200389y_real: poly_poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
zero_zero_poly_real: poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Finite____Field__Omod____ring_Itf__n_J,type,
if_Finite_mod_ring_n: $o > finite_mod_ring_n > finite_mod_ring_n > finite_mod_ring_n ).
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__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
if_pol9129390727684501670ring_n: $o > poly_F4222894760850802144ring_n > poly_F4222894760850802144ring_n > poly_F4222894760850802144ring_n ).
thf(sy_c_If_001t__Polynomial__Opoly_It__Int__Oint_J,type,
if_poly_int: $o > poly_int > poly_int > poly_int ).
thf(sy_c_If_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
if_poly_nat: $o > poly_nat > poly_nat > poly_nat ).
thf(sy_c_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
if_pol5337421645260315700ring_n: $o > poly_p6692042823160534382ring_n > poly_p6692042823160534382ring_n > poly_p6692042823160534382ring_n ).
thf(sy_c_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
if_poly_poly_int: $o > poly_poly_int > poly_poly_int > poly_poly_int ).
thf(sy_c_If_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
if_poly_real: $o > poly_real > poly_real > poly_real ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Finite____Field__Omod____ring_Itf__n_J,type,
ring_18169885536585341379ring_n: int > finite_mod_ring_n ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
ring_18712857867054464081ring_n: int > poly_F4222894760850802144ring_n ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Int__Oint_J,type,
ring_17892525584911698563ly_int: int > poly_int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
ring_14208964510912816607ring_n: int > poly_p6692042823160534382ring_n ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
ring_14695796289142966411ly_int: int > poly_poly_int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
ring_12936506555246842115y_real: int > poly_real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Kyber__spec_Omod__poly__irreducible,type,
kyber_5366887534115960522ucible: nat > poly_int > $o ).
thf(sy_c_Kyber__spec_Omod__poly__is__unit,type,
kyber_6337687584560828108s_unit: nat > poly_int > $o ).
thf(sy_c_Kyber__spec_Omod__poly__rel,type,
kyber_mod_poly_rel: nat > poly_int > poly_int > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > 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_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
top_top_set_literal: set_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__n_J,type,
top_top_set_n: set_n ).
thf(sy_c_Permutations_Osign_001t__Nat__Onat,type,
sign_nat: ( nat > nat ) > int ).
thf(sy_c_Polynomial_Omap__poly_001t__Finite____Field__Omod____ring_Itf__n_J_001t__Int__Oint,type,
map_po7622579134325003606_n_int: ( finite_mod_ring_n > int ) > poly_F4222894760850802144ring_n > poly_int ).
thf(sy_c_Polynomial_Omap__poly_001t__Finite____Field__Omod____ring_Itf__n_J_001t__Nat__Onat,type,
map_po7625069604834053882_n_nat: ( finite_mod_ring_n > nat ) > poly_F4222894760850802144ring_n > poly_nat ).
thf(sy_c_Polynomial_Omap__poly_001t__Int__Oint_001t__Finite____Field__Omod____ring_Itf__n_J,type,
map_po1011533443592629756ring_n: ( int > finite_mod_ring_n ) > poly_int > poly_F4222894760850802144ring_n ).
thf(sy_c_Polynomial_Omap__poly_001t__Int__Oint_001t__Int__Oint,type,
map_poly_int_int: ( int > int ) > poly_int > poly_int ).
thf(sy_c_Polynomial_Omap__poly_001t__Int__Oint_001t__Nat__Onat,type,
map_poly_int_nat: ( int > nat ) > poly_int > poly_nat ).
thf(sy_c_Polynomial_Omap__poly_001t__Int__Oint_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
map_po7854272679927642762ring_n: ( int > poly_F4222894760850802144ring_n ) > poly_int > poly_p6692042823160534382ring_n ).
thf(sy_c_Polynomial_Omap__poly_001t__Int__Oint_001t__Polynomial__Opoly_It__Int__Oint_J,type,
map_po8616709625927008010ly_int: ( int > poly_int ) > poly_int > poly_poly_int ).
thf(sy_c_Polynomial_Omap__poly_001t__Int__Oint_001t__Real__Oreal,type,
map_poly_int_real: ( int > real ) > poly_int > poly_real ).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Int__Oint,type,
map_poly_nat_int: ( nat > int ) > poly_nat > poly_int ).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Nat__Onat,type,
map_poly_nat_nat: ( nat > nat ) > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Real__Oreal,type,
map_poly_nat_real: ( nat > real ) > poly_nat > poly_real ).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Int__Oint_J_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
map_po1386818991929012354ring_n: ( poly_int > poly_F4222894760850802144ring_n ) > poly_poly_int > poly_p6692042823160534382ring_n ).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Int__Oint_J_001t__Polynomial__Opoly_It__Int__Oint_J,type,
map_po136720853902459666ly_int: ( poly_int > poly_int ) > poly_poly_int > poly_poly_int ).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Int__Oint_J_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
map_po2544297806126724112ring_n: ( poly_int > poly_p6692042823160534382ring_n ) > poly_poly_int > poly_p2743341848350813180ring_n ).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Int__Oint_J_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
map_po7381751157747918618ly_int: ( poly_int > poly_poly_int ) > poly_poly_int > poly_poly_poly_int ).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Int__Oint_J_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
map_po9185419567230169618y_real: ( poly_int > poly_real ) > poly_poly_int > poly_poly_real ).
thf(sy_c_Polynomial_Omap__poly_001t__Real__Oreal_001t__Int__Oint,type,
map_poly_real_int: ( real > int ) > poly_real > poly_int ).
thf(sy_c_Polynomial_Omap__poly_001t__Real__Oreal_001t__Nat__Onat,type,
map_poly_real_nat: ( real > nat ) > poly_real > poly_nat ).
thf(sy_c_Polynomial_Omap__poly_001t__Real__Oreal_001t__Real__Oreal,type,
map_poly_real_real: ( real > real ) > poly_real > poly_real ).
thf(sy_c_Polynomial_OpCons_001t__Finite____Field__Omod____ring_Itf__n_J,type,
pCons_8126420873123957872ring_n: finite_mod_ring_n > poly_F4222894760850802144ring_n > poly_F4222894760850802144ring_n ).
thf(sy_c_Polynomial_OpCons_001t__Int__Oint,type,
pCons_int: int > poly_int > poly_int ).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat,type,
pCons_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
pCons_6246009715029582078ring_n: poly_F4222894760850802144ring_n > poly_p6692042823160534382ring_n > poly_p6692042823160534382ring_n ).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Int__Oint_J,type,
pCons_poly_int: poly_int > poly_poly_int > poly_poly_int ).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
pCons_poly_nat: poly_nat > poly_poly_nat > poly_poly_nat ).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
pCons_2385395009258896524ring_n: poly_p6692042823160534382ring_n > poly_p2743341848350813180ring_n > poly_p2743341848350813180ring_n ).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
pCons_poly_poly_int: poly_poly_int > poly_poly_poly_int > poly_poly_poly_int ).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
pCons_poly_real: poly_real > poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_OpCons_001t__Real__Oreal,type,
pCons_real: real > poly_real > poly_real ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Finite____Field__Omod____ring_Itf__n_J,type,
poly_c8149583629457385976ring_n: nat > poly_F4222894760850802144ring_n > poly_F4222894760850802144ring_n ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Int__Oint,type,
poly_cutoff_int: nat > poly_int > poly_int ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
poly_cutoff_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
poly_c5946732737480674950ring_n: nat > poly_p6692042823160534382ring_n > poly_p6692042823160534382ring_n ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Int__Oint_J,type,
poly_cutoff_poly_int: nat > poly_poly_int > poly_poly_int ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
poly_cutoff_real: nat > poly_real > poly_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_Rings_Odvd__class_Odvd_001t__Finite____Field__Omod____ring_Itf__n_J,type,
dvd_dv7258769396337835820ring_n: finite_mod_ring_n > finite_mod_ring_n > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J,type,
dvd_dv8138414522854976442ring_n: poly_F4222894760850802144ring_n > poly_F4222894760850802144ring_n > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Int__Oint_J,type,
dvd_dvd_poly_int: poly_int > poly_int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
dvd_dvd_poly_nat: poly_nat > poly_nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J,type,
dvd_dv3135175980337127240ring_n: poly_p6692042823160534382ring_n > poly_p6692042823160534382ring_n > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J,type,
dvd_dv6998304861263046114ly_int: poly_poly_int > poly_poly_int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
dvd_dv265015969339997062ly_nat: poly_poly_nat > poly_poly_nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_J,type,
dvd_dv3919477662729673174ring_n: poly_p2743341848350813180ring_n > poly_p2743341848350813180ring_n > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J_J,type,
dvd_dv7705178354154678250ly_int: poly_poly_poly_int > poly_poly_poly_int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
dvd_dv4532039564868358754y_real: poly_poly_real > poly_poly_real > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
dvd_dvd_poly_real: poly_real > poly_real > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_v_p,type,
p: poly_int ).
% Relevant facts (1269)
thf(fact_0_of__int__poly__hom_Ohom__one,axiom,
( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ one_one_poly_int )
= one_on1166514126663969025ly_int ) ).
% of_int_poly_hom.hom_one
thf(fact_1_of__int__poly__hom_Ohom__one,axiom,
( ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ one_one_poly_int )
= one_on5457780782968151273ring_n ) ).
% of_int_poly_hom.hom_one
thf(fact_2_of__int__poly__hom_Ohom__one,axiom,
( ( map_poly_int_int @ ring_1_of_int_int @ one_one_poly_int )
= one_one_poly_int ) ).
% of_int_poly_hom.hom_one
thf(fact_3_of__int__poly__hom_Ohom__one,axiom,
( ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ one_one_poly_int )
= one_on4318287115420659547ring_n ) ).
% of_int_poly_hom.hom_one
thf(fact_4_of__int__poly__hom_Ohom__one,axiom,
( ( map_poly_int_real @ ring_1_of_int_real @ one_one_poly_int )
= one_one_poly_real ) ).
% of_int_poly_hom.hom_one
thf(fact_5_of__int__poly__hom_Ohom__1__iff,axiom,
! [X: poly_int] :
( ( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X )
= one_on1166514126663969025ly_int )
= ( X = one_one_poly_int ) ) ).
% of_int_poly_hom.hom_1_iff
thf(fact_6_of__int__poly__hom_Ohom__1__iff,axiom,
! [X: poly_int] :
( ( ( map_poly_int_int @ ring_1_of_int_int @ X )
= one_one_poly_int )
= ( X = one_one_poly_int ) ) ).
% of_int_poly_hom.hom_1_iff
thf(fact_7_of__int__poly__hom_Ohom__1__iff,axiom,
! [X: poly_int] :
( ( ( map_poly_int_real @ ring_1_of_int_real @ X )
= one_one_poly_real )
= ( X = one_one_poly_int ) ) ).
% of_int_poly_hom.hom_1_iff
thf(fact_8_of__int__poly__hom_Ohom__dvd,axiom,
! [P: poly_int,Q: poly_int] :
( ( dvd_dvd_poly_int @ P @ Q )
=> ( dvd_dv3135175980337127240ring_n @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ P ) @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ Q ) ) ) ).
% of_int_poly_hom.hom_dvd
thf(fact_9_of__int__poly__hom_Ohom__dvd,axiom,
! [P: poly_int,Q: poly_int] :
( ( dvd_dvd_poly_int @ P @ Q )
=> ( dvd_dv6998304861263046114ly_int @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ P ) @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ Q ) ) ) ).
% of_int_poly_hom.hom_dvd
thf(fact_10_of__int__poly__hom_Ohom__dvd,axiom,
! [P: poly_int,Q: poly_int] :
( ( dvd_dvd_poly_int @ P @ Q )
=> ( dvd_dvd_poly_int @ ( map_poly_int_int @ ring_1_of_int_int @ P ) @ ( map_poly_int_int @ ring_1_of_int_int @ Q ) ) ) ).
% of_int_poly_hom.hom_dvd
thf(fact_11_of__int__poly__hom_Ohom__dvd,axiom,
! [P: poly_int,Q: poly_int] :
( ( dvd_dvd_poly_int @ P @ Q )
=> ( dvd_dv8138414522854976442ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ P ) @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ Q ) ) ) ).
% of_int_poly_hom.hom_dvd
thf(fact_12_of__int__poly__hom_Ohom__dvd,axiom,
! [P: poly_int,Q: poly_int] :
( ( dvd_dvd_poly_int @ P @ Q )
=> ( dvd_dvd_poly_real @ ( map_poly_int_real @ ring_1_of_int_real @ P ) @ ( map_poly_int_real @ ring_1_of_int_real @ Q ) ) ) ).
% of_int_poly_hom.hom_dvd
thf(fact_13_of__int__poly__hom_Ohom__zero,axiom,
( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ zero_zero_poly_int )
= zero_z799223564134138693ly_int ) ).
% of_int_poly_hom.hom_zero
thf(fact_14_of__int__poly__hom_Ohom__zero,axiom,
( ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ zero_zero_poly_int )
= zero_z5482829069124612005ring_n ) ).
% of_int_poly_hom.hom_zero
thf(fact_15_of__int__poly__hom_Ohom__zero,axiom,
( ( map_poly_int_int @ ring_1_of_int_int @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% of_int_poly_hom.hom_zero
thf(fact_16_of__int__poly__hom_Ohom__zero,axiom,
( ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ zero_zero_poly_int )
= zero_z2753989067526334999ring_n ) ).
% of_int_poly_hom.hom_zero
thf(fact_17_of__int__poly__hom_Ohom__zero,axiom,
( ( map_poly_int_real @ ring_1_of_int_real @ zero_zero_poly_int )
= zero_zero_poly_real ) ).
% of_int_poly_hom.hom_zero
thf(fact_18_of__int__poly__hom_Ohom__0__iff,axiom,
! [X: poly_int] :
( ( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X )
= zero_z799223564134138693ly_int )
= ( X = zero_zero_poly_int ) ) ).
% of_int_poly_hom.hom_0_iff
thf(fact_19_of__int__poly__hom_Ohom__0__iff,axiom,
! [X: poly_int] :
( ( ( map_poly_int_int @ ring_1_of_int_int @ X )
= zero_zero_poly_int )
= ( X = zero_zero_poly_int ) ) ).
% of_int_poly_hom.hom_0_iff
thf(fact_20_of__int__poly__hom_Ohom__0__iff,axiom,
! [X: poly_int] :
( ( ( map_poly_int_real @ ring_1_of_int_real @ X )
= zero_zero_poly_real )
= ( X = zero_zero_poly_int ) ) ).
% of_int_poly_hom.hom_0_iff
thf(fact_21_map__poly__1_H,axiom,
! [F: nat > nat] :
( ( ( F @ one_one_nat )
= one_one_nat )
=> ( ( map_poly_nat_nat @ F @ one_one_poly_nat )
= one_one_poly_nat ) ) ).
% map_poly_1'
thf(fact_22_map__poly__1_H,axiom,
! [F: nat > int] :
( ( ( F @ one_one_nat )
= one_one_int )
=> ( ( map_poly_nat_int @ F @ one_one_poly_nat )
= one_one_poly_int ) ) ).
% map_poly_1'
thf(fact_23_map__poly__1_H,axiom,
! [F: nat > real] :
( ( ( F @ one_one_nat )
= one_one_real )
=> ( ( map_poly_nat_real @ F @ one_one_poly_nat )
= one_one_poly_real ) ) ).
% map_poly_1'
thf(fact_24_map__poly__1_H,axiom,
! [F: int > nat] :
( ( ( F @ one_one_int )
= one_one_nat )
=> ( ( map_poly_int_nat @ F @ one_one_poly_int )
= one_one_poly_nat ) ) ).
% map_poly_1'
thf(fact_25_map__poly__1_H,axiom,
! [F: int > int] :
( ( ( F @ one_one_int )
= one_one_int )
=> ( ( map_poly_int_int @ F @ one_one_poly_int )
= one_one_poly_int ) ) ).
% map_poly_1'
thf(fact_26_map__poly__1_H,axiom,
! [F: int > real] :
( ( ( F @ one_one_int )
= one_one_real )
=> ( ( map_poly_int_real @ F @ one_one_poly_int )
= one_one_poly_real ) ) ).
% map_poly_1'
thf(fact_27_map__poly__1_H,axiom,
! [F: real > nat] :
( ( ( F @ one_one_real )
= one_one_nat )
=> ( ( map_poly_real_nat @ F @ one_one_poly_real )
= one_one_poly_nat ) ) ).
% map_poly_1'
thf(fact_28_map__poly__1_H,axiom,
! [F: real > int] :
( ( ( F @ one_one_real )
= one_one_int )
=> ( ( map_poly_real_int @ F @ one_one_poly_real )
= one_one_poly_int ) ) ).
% map_poly_1'
thf(fact_29_map__poly__1_H,axiom,
! [F: real > real] :
( ( ( F @ one_one_real )
= one_one_real )
=> ( ( map_poly_real_real @ F @ one_one_poly_real )
= one_one_poly_real ) ) ).
% map_poly_1'
thf(fact_30_map__poly__1_H,axiom,
! [F: finite_mod_ring_n > nat] :
( ( ( F @ one_on2109788483843180749ring_n )
= one_one_nat )
=> ( ( map_po7625069604834053882_n_nat @ F @ one_on4318287115420659547ring_n )
= one_one_poly_nat ) ) ).
% map_poly_1'
thf(fact_31_map__poly__zero,axiom,
! [F: int > int,P: poly_int] :
( ! [C: int] :
( ( ( F @ C )
= zero_zero_int )
=> ( C = zero_zero_int ) )
=> ( ( ( map_poly_int_int @ F @ P )
= zero_zero_poly_int )
= ( P = zero_zero_poly_int ) ) ) ).
% map_poly_zero
thf(fact_32_map__poly__zero,axiom,
! [F: nat > int,P: poly_nat] :
( ! [C: nat] :
( ( ( F @ C )
= zero_zero_int )
=> ( C = zero_zero_nat ) )
=> ( ( ( map_poly_nat_int @ F @ P )
= zero_zero_poly_int )
= ( P = zero_zero_poly_nat ) ) ) ).
% map_poly_zero
thf(fact_33_map__poly__zero,axiom,
! [F: real > int,P: poly_real] :
( ! [C: real] :
( ( ( F @ C )
= zero_zero_int )
=> ( C = zero_zero_real ) )
=> ( ( ( map_poly_real_int @ F @ P )
= zero_zero_poly_int )
= ( P = zero_zero_poly_real ) ) ) ).
% map_poly_zero
thf(fact_34_map__poly__zero,axiom,
! [F: int > nat,P: poly_int] :
( ! [C: int] :
( ( ( F @ C )
= zero_zero_nat )
=> ( C = zero_zero_int ) )
=> ( ( ( map_poly_int_nat @ F @ P )
= zero_zero_poly_nat )
= ( P = zero_zero_poly_int ) ) ) ).
% map_poly_zero
thf(fact_35_map__poly__zero,axiom,
! [F: nat > nat,P: poly_nat] :
( ! [C: nat] :
( ( ( F @ C )
= zero_zero_nat )
=> ( C = zero_zero_nat ) )
=> ( ( ( map_poly_nat_nat @ F @ P )
= zero_zero_poly_nat )
= ( P = zero_zero_poly_nat ) ) ) ).
% map_poly_zero
thf(fact_36_map__poly__zero,axiom,
! [F: real > nat,P: poly_real] :
( ! [C: real] :
( ( ( F @ C )
= zero_zero_nat )
=> ( C = zero_zero_real ) )
=> ( ( ( map_poly_real_nat @ F @ P )
= zero_zero_poly_nat )
= ( P = zero_zero_poly_real ) ) ) ).
% map_poly_zero
thf(fact_37_map__poly__zero,axiom,
! [F: int > real,P: poly_int] :
( ! [C: int] :
( ( ( F @ C )
= zero_zero_real )
=> ( C = zero_zero_int ) )
=> ( ( ( map_poly_int_real @ F @ P )
= zero_zero_poly_real )
= ( P = zero_zero_poly_int ) ) ) ).
% map_poly_zero
thf(fact_38_map__poly__zero,axiom,
! [F: nat > real,P: poly_nat] :
( ! [C: nat] :
( ( ( F @ C )
= zero_zero_real )
=> ( C = zero_zero_nat ) )
=> ( ( ( map_poly_nat_real @ F @ P )
= zero_zero_poly_real )
= ( P = zero_zero_poly_nat ) ) ) ).
% map_poly_zero
thf(fact_39_map__poly__zero,axiom,
! [F: real > real,P: poly_real] :
( ! [C: real] :
( ( ( F @ C )
= zero_zero_real )
=> ( C = zero_zero_real ) )
=> ( ( ( map_poly_real_real @ F @ P )
= zero_zero_poly_real )
= ( P = zero_zero_poly_real ) ) ) ).
% map_poly_zero
thf(fact_40_map__poly__zero,axiom,
! [F: int > finite_mod_ring_n,P: poly_int] :
( ! [C: int] :
( ( ( F @ C )
= zero_z7902377597758090121ring_n )
=> ( C = zero_zero_int ) )
=> ( ( ( map_po1011533443592629756ring_n @ F @ P )
= zero_z2753989067526334999ring_n )
= ( P = zero_zero_poly_int ) ) ) ).
% map_poly_zero
thf(fact_41_algebraic__semidom__class_Ounit__prod,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ one_one_poly_real ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_42_algebraic__semidom__class_Ounit__prod,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ one_on1166514126663969025ly_int ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_43_algebraic__semidom__class_Ounit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_44_algebraic__semidom__class_Ounit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_45_algebraic__semidom__class_Ounit__prod,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ one_one_poly_int ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_46_algebraic__semidom__class_Ounit__prod,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_47_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ one_one_poly_real ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_48_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ one_on1166514126663969025ly_int ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_49_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: poly_nat,B: poly_nat] :
( ( dvd_dvd_poly_nat @ A @ one_one_poly_nat )
=> ( ( dvd_dvd_poly_nat @ B @ one_one_poly_nat )
=> ( dvd_dvd_poly_nat @ ( times_times_poly_nat @ A @ B ) @ one_one_poly_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_50_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: poly_p6692042823160534382ring_n,B: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ A @ one_on5457780782968151273ring_n )
=> ( ( dvd_dv3135175980337127240ring_n @ B @ one_on5457780782968151273ring_n )
=> ( dvd_dv3135175980337127240ring_n @ ( times_2573333606529333417ring_n @ A @ B ) @ one_on5457780782968151273ring_n ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_51_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: finite_mod_ring_n,B: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ A @ one_on2109788483843180749ring_n )
=> ( ( dvd_dv7258769396337835820ring_n @ B @ one_on2109788483843180749ring_n )
=> ( dvd_dv7258769396337835820ring_n @ ( times_5121417632533718157ring_n @ A @ B ) @ one_on2109788483843180749ring_n ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_52_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ A @ one_on4318287115420659547ring_n )
=> ( ( dvd_dv8138414522854976442ring_n @ B @ one_on4318287115420659547ring_n )
=> ( dvd_dv8138414522854976442ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ one_on4318287115420659547ring_n ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_53_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_54_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_55_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ one_one_poly_int ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_56_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_57_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ one_one_poly_real )
= ( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
& ( dvd_dvd_poly_real @ B @ one_one_poly_real ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_58_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ one_on1166514126663969025ly_int )
= ( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
& ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_59_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: poly_nat,B: poly_nat] :
( ( dvd_dvd_poly_nat @ ( times_times_poly_nat @ A @ B ) @ one_one_poly_nat )
= ( ( dvd_dvd_poly_nat @ A @ one_one_poly_nat )
& ( dvd_dvd_poly_nat @ B @ one_one_poly_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_60_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: poly_p6692042823160534382ring_n,B: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ ( times_2573333606529333417ring_n @ A @ B ) @ one_on5457780782968151273ring_n )
= ( ( dvd_dv3135175980337127240ring_n @ A @ one_on5457780782968151273ring_n )
& ( dvd_dv3135175980337127240ring_n @ B @ one_on5457780782968151273ring_n ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_61_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: finite_mod_ring_n,B: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ ( times_5121417632533718157ring_n @ A @ B ) @ one_on2109788483843180749ring_n )
= ( ( dvd_dv7258769396337835820ring_n @ A @ one_on2109788483843180749ring_n )
& ( dvd_dv7258769396337835820ring_n @ B @ one_on2109788483843180749ring_n ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_62_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ one_on4318287115420659547ring_n )
= ( ( dvd_dv8138414522854976442ring_n @ A @ one_on4318287115420659547ring_n )
& ( dvd_dv8138414522854976442ring_n @ B @ one_on4318287115420659547ring_n ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_63_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_64_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_65_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ one_one_poly_int )
= ( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
& ( dvd_dvd_poly_int @ B @ one_one_poly_int ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_66_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ one_one_real )
= ( ( dvd_dvd_real @ A @ one_one_real )
& ( dvd_dvd_real @ B @ one_one_real ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_67_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: poly_real,A: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ C2 )
= ( dvd_dvd_poly_real @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_68_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: poly_poly_int,A: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ C2 )
= ( dvd_dv6998304861263046114ly_int @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_69_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: poly_nat,A: poly_nat,C2: poly_nat] :
( ( dvd_dvd_poly_nat @ B @ one_one_poly_nat )
=> ( ( dvd_dvd_poly_nat @ ( times_times_poly_nat @ A @ B ) @ C2 )
= ( dvd_dvd_poly_nat @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_70_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: poly_p6692042823160534382ring_n,A: poly_p6692042823160534382ring_n,C2: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ B @ one_on5457780782968151273ring_n )
=> ( ( dvd_dv3135175980337127240ring_n @ ( times_2573333606529333417ring_n @ A @ B ) @ C2 )
= ( dvd_dv3135175980337127240ring_n @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_71_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: finite_mod_ring_n,A: finite_mod_ring_n,C2: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ B @ one_on2109788483843180749ring_n )
=> ( ( dvd_dv7258769396337835820ring_n @ ( times_5121417632533718157ring_n @ A @ B ) @ C2 )
= ( dvd_dv7258769396337835820ring_n @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_72_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: poly_F4222894760850802144ring_n,A: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ B @ one_on4318287115420659547ring_n )
=> ( ( dvd_dv8138414522854976442ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ C2 )
= ( dvd_dv8138414522854976442ring_n @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_73_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_74_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_75_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: poly_int,A: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
= ( dvd_dvd_poly_int @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_76_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: real,A: real,C2: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_77_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ C2 )
= ( dvd_dvd_poly_real @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_78_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ C2 )
= ( dvd_dv6998304861263046114ly_int @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_79_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_nat,B: poly_nat,C2: poly_nat] :
( ( dvd_dvd_poly_nat @ A @ one_one_poly_nat )
=> ( ( dvd_dvd_poly_nat @ ( times_times_poly_nat @ A @ B ) @ C2 )
= ( dvd_dvd_poly_nat @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_80_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_p6692042823160534382ring_n,B: poly_p6692042823160534382ring_n,C2: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ A @ one_on5457780782968151273ring_n )
=> ( ( dvd_dv3135175980337127240ring_n @ ( times_2573333606529333417ring_n @ A @ B ) @ C2 )
= ( dvd_dv3135175980337127240ring_n @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_81_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: finite_mod_ring_n,B: finite_mod_ring_n,C2: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ A @ one_on2109788483843180749ring_n )
=> ( ( dvd_dv7258769396337835820ring_n @ ( times_5121417632533718157ring_n @ A @ B ) @ C2 )
= ( dvd_dv7258769396337835820ring_n @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_82_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ A @ one_on4318287115420659547ring_n )
=> ( ( dvd_dv8138414522854976442ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ C2 )
= ( dvd_dv8138414522854976442ring_n @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_83_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_84_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_85_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
= ( dvd_dvd_poly_int @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_86_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
= ( dvd_dvd_real @ B @ C2 ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_87_mult__cancel__right,axiom,
! [A: poly_real,C2: poly_real,B: poly_real] :
( ( ( times_7914811829580426937y_real @ A @ C2 )
= ( times_7914811829580426937y_real @ B @ C2 ) )
= ( ( C2 = zero_zero_poly_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_88_mult__cancel__right,axiom,
! [A: poly_poly_int,C2: poly_poly_int,B: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ A @ C2 )
= ( times_4739760418287672641ly_int @ B @ C2 ) )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_89_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_90_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_91_mult__cancel__right,axiom,
! [A: poly_int,C2: poly_int,B: poly_int] :
( ( ( times_times_poly_int @ A @ C2 )
= ( times_times_poly_int @ B @ C2 ) )
= ( ( C2 = zero_zero_poly_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_92_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_93_mult__cancel__left,axiom,
! [C2: poly_real,A: poly_real,B: poly_real] :
( ( ( times_7914811829580426937y_real @ C2 @ A )
= ( times_7914811829580426937y_real @ C2 @ B ) )
= ( ( C2 = zero_zero_poly_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_94_mult__cancel__left,axiom,
! [C2: poly_poly_int,A: poly_poly_int,B: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ C2 @ A )
= ( times_4739760418287672641ly_int @ C2 @ B ) )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_95_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_96_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_97_mult__cancel__left,axiom,
! [C2: poly_int,A: poly_int,B: poly_int] :
( ( ( times_times_poly_int @ C2 @ A )
= ( times_times_poly_int @ C2 @ B ) )
= ( ( C2 = zero_zero_poly_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_98_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_99_mult__eq__0__iff,axiom,
! [A: poly_real,B: poly_real] :
( ( ( times_7914811829580426937y_real @ A @ B )
= zero_zero_poly_real )
= ( ( A = zero_zero_poly_real )
| ( B = zero_zero_poly_real ) ) ) ).
% mult_eq_0_iff
thf(fact_100_mult__eq__0__iff,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ A @ B )
= zero_z799223564134138693ly_int )
= ( ( A = zero_z799223564134138693ly_int )
| ( B = zero_z799223564134138693ly_int ) ) ) ).
% mult_eq_0_iff
thf(fact_101_mult__eq__0__iff,axiom,
! [A: poly_nat,B: poly_nat] :
( ( ( times_times_poly_nat @ A @ B )
= zero_zero_poly_nat )
= ( ( A = zero_zero_poly_nat )
| ( B = zero_zero_poly_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_102_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_103_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_104_mult__eq__0__iff,axiom,
! [A: poly_int,B: poly_int] :
( ( ( times_times_poly_int @ A @ B )
= zero_zero_poly_int )
= ( ( A = zero_zero_poly_int )
| ( B = zero_zero_poly_int ) ) ) ).
% mult_eq_0_iff
thf(fact_105_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_106_mult__zero__right,axiom,
! [A: poly_real] :
( ( times_7914811829580426937y_real @ A @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% mult_zero_right
thf(fact_107_mult__zero__right,axiom,
! [A: poly_poly_int] :
( ( times_4739760418287672641ly_int @ A @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int ) ).
% mult_zero_right
thf(fact_108_mult__zero__right,axiom,
! [A: poly_nat] :
( ( times_times_poly_nat @ A @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% mult_zero_right
thf(fact_109_mult__zero__right,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ A @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n ) ).
% mult_zero_right
thf(fact_110_mult__zero__right,axiom,
! [A: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ A @ zero_z7902377597758090121ring_n )
= zero_z7902377597758090121ring_n ) ).
% mult_zero_right
thf(fact_111_mult__zero__right,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ A @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n ) ).
% mult_zero_right
thf(fact_112_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_113_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_114_mult__zero__right,axiom,
! [A: poly_int] :
( ( times_times_poly_int @ A @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% mult_zero_right
thf(fact_115_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_116_mult__zero__left,axiom,
! [A: poly_real] :
( ( times_7914811829580426937y_real @ zero_zero_poly_real @ A )
= zero_zero_poly_real ) ).
% mult_zero_left
thf(fact_117_mult__zero__left,axiom,
! [A: poly_poly_int] :
( ( times_4739760418287672641ly_int @ zero_z799223564134138693ly_int @ A )
= zero_z799223564134138693ly_int ) ).
% mult_zero_left
thf(fact_118_mult__zero__left,axiom,
! [A: poly_nat] :
( ( times_times_poly_nat @ zero_zero_poly_nat @ A )
= zero_zero_poly_nat ) ).
% mult_zero_left
thf(fact_119_mult__zero__left,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ zero_z5482829069124612005ring_n @ A )
= zero_z5482829069124612005ring_n ) ).
% mult_zero_left
thf(fact_120_mult__zero__left,axiom,
! [A: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ zero_z7902377597758090121ring_n @ A )
= zero_z7902377597758090121ring_n ) ).
% mult_zero_left
thf(fact_121_mult__zero__left,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ zero_z2753989067526334999ring_n @ A )
= zero_z2753989067526334999ring_n ) ).
% mult_zero_left
thf(fact_122_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_123_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_124_mult__zero__left,axiom,
! [A: poly_int] :
( ( times_times_poly_int @ zero_zero_poly_int @ A )
= zero_zero_poly_int ) ).
% mult_zero_left
thf(fact_125_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_126_dvd__0__left__iff,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ zero_z2753989067526334999ring_n @ A )
= ( A = zero_z2753989067526334999ring_n ) ) ).
% dvd_0_left_iff
thf(fact_127_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_128_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_129_dvd__0__left__iff,axiom,
! [A: poly_int] :
( ( dvd_dvd_poly_int @ zero_zero_poly_int @ A )
= ( A = zero_zero_poly_int ) ) ).
% dvd_0_left_iff
thf(fact_130_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_131_dvd__0__left__iff,axiom,
! [A: poly_real] :
( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A )
= ( A = zero_zero_poly_real ) ) ).
% dvd_0_left_iff
thf(fact_132_dvd__0__left__iff,axiom,
! [A: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ zero_z799223564134138693ly_int @ A )
= ( A = zero_z799223564134138693ly_int ) ) ).
% dvd_0_left_iff
thf(fact_133_dvd__0__left__iff,axiom,
! [A: poly_nat] :
( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A )
= ( A = zero_zero_poly_nat ) ) ).
% dvd_0_left_iff
thf(fact_134_dvd__0__left__iff,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ zero_z5482829069124612005ring_n @ A )
= ( A = zero_z5482829069124612005ring_n ) ) ).
% dvd_0_left_iff
thf(fact_135_dvd__0__left__iff,axiom,
! [A: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ zero_z7902377597758090121ring_n @ A )
= ( A = zero_z7902377597758090121ring_n ) ) ).
% dvd_0_left_iff
thf(fact_136_dvd__0__right,axiom,
! [A: poly_F4222894760850802144ring_n] : ( dvd_dv8138414522854976442ring_n @ A @ zero_z2753989067526334999ring_n ) ).
% dvd_0_right
thf(fact_137_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_138_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_139_dvd__0__right,axiom,
! [A: poly_int] : ( dvd_dvd_poly_int @ A @ zero_zero_poly_int ) ).
% dvd_0_right
thf(fact_140_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_141_dvd__0__right,axiom,
! [A: poly_real] : ( dvd_dvd_poly_real @ A @ zero_zero_poly_real ) ).
% dvd_0_right
thf(fact_142_dvd__0__right,axiom,
! [A: poly_poly_int] : ( dvd_dv6998304861263046114ly_int @ A @ zero_z799223564134138693ly_int ) ).
% dvd_0_right
thf(fact_143_dvd__0__right,axiom,
! [A: poly_nat] : ( dvd_dvd_poly_nat @ A @ zero_zero_poly_nat ) ).
% dvd_0_right
thf(fact_144_dvd__0__right,axiom,
! [A: poly_p6692042823160534382ring_n] : ( dvd_dv3135175980337127240ring_n @ A @ zero_z5482829069124612005ring_n ) ).
% dvd_0_right
thf(fact_145_dvd__0__right,axiom,
! [A: finite_mod_ring_n] : ( dvd_dv7258769396337835820ring_n @ A @ zero_z7902377597758090121ring_n ) ).
% dvd_0_right
thf(fact_146_of__int__poly__hom_Obase_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dv8138414522854976442ring_n @ ( ring_18712857867054464081ring_n @ P ) @ ( ring_18712857867054464081ring_n @ Q ) ) ) ).
% of_int_poly_hom.base.hom_dvd
thf(fact_147_of__int__poly__hom_Obase_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_poly_int @ ( ring_17892525584911698563ly_int @ P ) @ ( ring_17892525584911698563ly_int @ Q ) ) ) ).
% of_int_poly_hom.base.hom_dvd
thf(fact_148_of__int__poly__hom_Obase_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_poly_real @ ( ring_12936506555246842115y_real @ P ) @ ( ring_12936506555246842115y_real @ Q ) ) ) ).
% of_int_poly_hom.base.hom_dvd
thf(fact_149_of__int__poly__hom_Obase_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dv7258769396337835820ring_n @ ( ring_18169885536585341379ring_n @ P ) @ ( ring_18169885536585341379ring_n @ Q ) ) ) ).
% of_int_poly_hom.base.hom_dvd
thf(fact_150_of__int__poly__hom_Obase_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_real @ ( ring_1_of_int_real @ P ) @ ( ring_1_of_int_real @ Q ) ) ) ).
% of_int_poly_hom.base.hom_dvd
thf(fact_151_of__int__poly__hom_Obase_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_int @ ( ring_1_of_int_int @ P ) @ ( ring_1_of_int_int @ Q ) ) ) ).
% of_int_poly_hom.base.hom_dvd
thf(fact_152_map__poly__0,axiom,
! [F: int > int] :
( ( map_poly_int_int @ F @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% map_poly_0
thf(fact_153_map__poly__0,axiom,
! [F: int > real] :
( ( map_poly_int_real @ F @ zero_zero_poly_int )
= zero_zero_poly_real ) ).
% map_poly_0
thf(fact_154_map__poly__0,axiom,
! [F: int > nat] :
( ( map_poly_int_nat @ F @ zero_zero_poly_int )
= zero_zero_poly_nat ) ).
% map_poly_0
thf(fact_155_map__poly__0,axiom,
! [F: real > int] :
( ( map_poly_real_int @ F @ zero_zero_poly_real )
= zero_zero_poly_int ) ).
% map_poly_0
thf(fact_156_map__poly__0,axiom,
! [F: real > real] :
( ( map_poly_real_real @ F @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% map_poly_0
thf(fact_157_map__poly__0,axiom,
! [F: real > nat] :
( ( map_poly_real_nat @ F @ zero_zero_poly_real )
= zero_zero_poly_nat ) ).
% map_poly_0
thf(fact_158_map__poly__0,axiom,
! [F: nat > int] :
( ( map_poly_nat_int @ F @ zero_zero_poly_nat )
= zero_zero_poly_int ) ).
% map_poly_0
thf(fact_159_map__poly__0,axiom,
! [F: nat > real] :
( ( map_poly_nat_real @ F @ zero_zero_poly_nat )
= zero_zero_poly_real ) ).
% map_poly_0
thf(fact_160_map__poly__0,axiom,
! [F: nat > nat] :
( ( map_poly_nat_nat @ F @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% map_poly_0
thf(fact_161_map__poly__0,axiom,
! [F: finite_mod_ring_n > int] :
( ( map_po7622579134325003606_n_int @ F @ zero_z2753989067526334999ring_n )
= zero_zero_poly_int ) ).
% map_poly_0
thf(fact_162_of__int__poly__hom_Oeq__iff,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X )
= ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ Y ) )
= ( X = Y ) ) ).
% of_int_poly_hom.eq_iff
thf(fact_163_of__int__poly__hom_Oeq__iff,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( map_poly_int_real @ ring_1_of_int_real @ X )
= ( map_poly_int_real @ ring_1_of_int_real @ Y ) )
= ( X = Y ) ) ).
% of_int_poly_hom.eq_iff
thf(fact_164_of__int__poly__hom_Oeq__iff,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( map_poly_int_int @ ring_1_of_int_int @ X )
= ( map_poly_int_int @ ring_1_of_int_int @ Y ) )
= ( X = Y ) ) ).
% of_int_poly_hom.eq_iff
thf(fact_165_mult__cancel__right2,axiom,
! [A: poly_real,C2: poly_real] :
( ( ( times_7914811829580426937y_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_poly_real )
| ( A = one_one_poly_real ) ) ) ).
% mult_cancel_right2
thf(fact_166_mult__cancel__right2,axiom,
! [A: poly_poly_int,C2: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( A = one_on1166514126663969025ly_int ) ) ) ).
% mult_cancel_right2
thf(fact_167_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_168_mult__cancel__right2,axiom,
! [A: poly_int,C2: poly_int] :
( ( ( times_times_poly_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_poly_int )
| ( A = one_one_poly_int ) ) ) ).
% mult_cancel_right2
thf(fact_169_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_170_mult__cancel__right1,axiom,
! [C2: poly_real,B: poly_real] :
( ( C2
= ( times_7914811829580426937y_real @ B @ C2 ) )
= ( ( C2 = zero_zero_poly_real )
| ( B = one_one_poly_real ) ) ) ).
% mult_cancel_right1
thf(fact_171_mult__cancel__right1,axiom,
! [C2: poly_poly_int,B: poly_poly_int] :
( ( C2
= ( times_4739760418287672641ly_int @ B @ C2 ) )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( B = one_on1166514126663969025ly_int ) ) ) ).
% mult_cancel_right1
thf(fact_172_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_173_mult__cancel__right1,axiom,
! [C2: poly_int,B: poly_int] :
( ( C2
= ( times_times_poly_int @ B @ C2 ) )
= ( ( C2 = zero_zero_poly_int )
| ( B = one_one_poly_int ) ) ) ).
% mult_cancel_right1
thf(fact_174_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_175_mult__cancel__left2,axiom,
! [C2: poly_real,A: poly_real] :
( ( ( times_7914811829580426937y_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_poly_real )
| ( A = one_one_poly_real ) ) ) ).
% mult_cancel_left2
thf(fact_176_mult__cancel__left2,axiom,
! [C2: poly_poly_int,A: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( A = one_on1166514126663969025ly_int ) ) ) ).
% mult_cancel_left2
thf(fact_177_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_178_mult__cancel__left2,axiom,
! [C2: poly_int,A: poly_int] :
( ( ( times_times_poly_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_poly_int )
| ( A = one_one_poly_int ) ) ) ).
% mult_cancel_left2
thf(fact_179_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_180_mult__cancel__left1,axiom,
! [C2: poly_real,B: poly_real] :
( ( C2
= ( times_7914811829580426937y_real @ C2 @ B ) )
= ( ( C2 = zero_zero_poly_real )
| ( B = one_one_poly_real ) ) ) ).
% mult_cancel_left1
thf(fact_181_mult__cancel__left1,axiom,
! [C2: poly_poly_int,B: poly_poly_int] :
( ( C2
= ( times_4739760418287672641ly_int @ C2 @ B ) )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( B = one_on1166514126663969025ly_int ) ) ) ).
% mult_cancel_left1
thf(fact_182_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_183_mult__cancel__left1,axiom,
! [C2: poly_int,B: poly_int] :
( ( C2
= ( times_times_poly_int @ C2 @ B ) )
= ( ( C2 = zero_zero_poly_int )
| ( B = one_one_poly_int ) ) ) ).
% mult_cancel_left1
thf(fact_184_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_185_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( A != zero_zero_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ B @ A ) @ ( times_7914811829580426937y_real @ C2 @ A ) )
= ( dvd_dvd_poly_real @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_186_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( A != zero_z799223564134138693ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ B @ A ) @ ( times_4739760418287672641ly_int @ C2 @ A ) )
= ( dvd_dv6998304861263046114ly_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_187_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_188_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_189_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( A != zero_zero_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ B @ A ) @ ( times_times_poly_int @ C2 @ A ) )
= ( dvd_dvd_poly_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_190_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: real,B: real,C2: real] :
( ( A != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C2 @ A ) )
= ( dvd_dvd_real @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_191_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( A != zero_zero_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ ( times_7914811829580426937y_real @ A @ C2 ) )
= ( dvd_dvd_poly_real @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_192_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( A != zero_z799223564134138693ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ ( times_4739760418287672641ly_int @ A @ C2 ) )
= ( dvd_dv6998304861263046114ly_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_193_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_194_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_195_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( A != zero_zero_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ ( times_times_poly_int @ A @ C2 ) )
= ( dvd_dvd_poly_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_196_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: real,B: real,C2: real] :
( ( A != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) )
= ( dvd_dvd_real @ B @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_197_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( A != zero_zero_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ B @ A ) @ ( times_7914811829580426937y_real @ C2 @ A ) )
= ( dvd_dvd_poly_real @ B @ C2 ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_198_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( A != zero_z799223564134138693ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ B @ A ) @ ( times_4739760418287672641ly_int @ C2 @ A ) )
= ( dvd_dv6998304861263046114ly_int @ B @ C2 ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_199_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_200_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( A != zero_zero_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ B @ A ) @ ( times_times_poly_int @ C2 @ A ) )
= ( dvd_dvd_poly_int @ B @ C2 ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_201_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: real,B: real,C2: real] :
( ( A != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C2 @ A ) )
= ( dvd_dvd_real @ B @ C2 ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_202_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( A != zero_zero_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ ( times_7914811829580426937y_real @ A @ C2 ) )
= ( dvd_dvd_poly_real @ B @ C2 ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_203_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( A != zero_z799223564134138693ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ ( times_4739760418287672641ly_int @ A @ C2 ) )
= ( dvd_dv6998304861263046114ly_int @ B @ C2 ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_204_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_205_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( A != zero_zero_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ ( times_times_poly_int @ A @ C2 ) )
= ( dvd_dvd_poly_int @ B @ C2 ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_206_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: real,B: real,C2: real] :
( ( A != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) )
= ( dvd_dvd_real @ B @ C2 ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_207_dvd__mult__cancel__right,axiom,
! [A: poly_real,C2: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ C2 ) @ ( times_7914811829580426937y_real @ B @ C2 ) )
= ( ( C2 = zero_zero_poly_real )
| ( dvd_dvd_poly_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_208_dvd__mult__cancel__right,axiom,
! [A: poly_poly_int,C2: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ C2 ) @ ( times_4739760418287672641ly_int @ B @ C2 ) )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( dvd_dv6998304861263046114ly_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_209_dvd__mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_210_dvd__mult__cancel__right,axiom,
! [A: poly_int,C2: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ C2 ) @ ( times_times_poly_int @ B @ C2 ) )
= ( ( C2 = zero_zero_poly_int )
| ( dvd_dvd_poly_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_211_dvd__mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_212_dvd__mult__cancel__left,axiom,
! [C2: poly_real,A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ C2 @ A ) @ ( times_7914811829580426937y_real @ C2 @ B ) )
= ( ( C2 = zero_zero_poly_real )
| ( dvd_dvd_poly_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_213_dvd__mult__cancel__left,axiom,
! [C2: poly_poly_int,A: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ C2 @ A ) @ ( times_4739760418287672641ly_int @ C2 @ B ) )
= ( ( C2 = zero_z799223564134138693ly_int )
| ( dvd_dv6998304861263046114ly_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_214_dvd__mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_215_dvd__mult__cancel__left,axiom,
! [C2: poly_int,A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ ( times_times_poly_int @ C2 @ A ) @ ( times_times_poly_int @ C2 @ B ) )
= ( ( C2 = zero_zero_poly_int )
| ( dvd_dvd_poly_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_216_dvd__mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_217_dvd__trans,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ A @ B )
=> ( ( dvd_dv8138414522854976442ring_n @ B @ C2 )
=> ( dvd_dv8138414522854976442ring_n @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_218_dvd__trans,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C2 )
=> ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_219_dvd__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C2 )
=> ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_220_dvd__trans,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ B )
=> ( ( dvd_dvd_poly_int @ B @ C2 )
=> ( dvd_dvd_poly_int @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_221_dvd__trans,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ B )
=> ( ( dvd_dvd_poly_real @ B @ C2 )
=> ( dvd_dvd_poly_real @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_222_dvd__trans,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ B @ C2 )
=> ( dvd_dvd_real @ A @ C2 ) ) ) ).
% dvd_trans
thf(fact_223_dvd__refl,axiom,
! [A: poly_F4222894760850802144ring_n] : ( dvd_dv8138414522854976442ring_n @ A @ A ) ).
% dvd_refl
thf(fact_224_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_225_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_226_dvd__refl,axiom,
! [A: poly_int] : ( dvd_dvd_poly_int @ A @ A ) ).
% dvd_refl
thf(fact_227_dvd__refl,axiom,
! [A: poly_real] : ( dvd_dvd_poly_real @ A @ A ) ).
% dvd_refl
thf(fact_228_dvd__refl,axiom,
! [A: real] : ( dvd_dvd_real @ A @ A ) ).
% dvd_refl
thf(fact_229_mult__right__cancel,axiom,
! [C2: poly_real,A: poly_real,B: poly_real] :
( ( C2 != zero_zero_poly_real )
=> ( ( ( times_7914811829580426937y_real @ A @ C2 )
= ( times_7914811829580426937y_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_230_mult__right__cancel,axiom,
! [C2: poly_poly_int,A: poly_poly_int,B: poly_poly_int] :
( ( C2 != zero_z799223564134138693ly_int )
=> ( ( ( times_4739760418287672641ly_int @ A @ C2 )
= ( times_4739760418287672641ly_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_231_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_232_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_233_mult__right__cancel,axiom,
! [C2: poly_int,A: poly_int,B: poly_int] :
( ( C2 != zero_zero_poly_int )
=> ( ( ( times_times_poly_int @ A @ C2 )
= ( times_times_poly_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_234_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_235_mult__left__cancel,axiom,
! [C2: poly_real,A: poly_real,B: poly_real] :
( ( C2 != zero_zero_poly_real )
=> ( ( ( times_7914811829580426937y_real @ C2 @ A )
= ( times_7914811829580426937y_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_236_mult__left__cancel,axiom,
! [C2: poly_poly_int,A: poly_poly_int,B: poly_poly_int] :
( ( C2 != zero_z799223564134138693ly_int )
=> ( ( ( times_4739760418287672641ly_int @ C2 @ A )
= ( times_4739760418287672641ly_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_237_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_238_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_239_mult__left__cancel,axiom,
! [C2: poly_int,A: poly_int,B: poly_int] :
( ( C2 != zero_zero_poly_int )
=> ( ( ( times_times_poly_int @ C2 @ A )
= ( times_times_poly_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_240_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_241_no__zero__divisors,axiom,
! [A: poly_real,B: poly_real] :
( ( A != zero_zero_poly_real )
=> ( ( B != zero_zero_poly_real )
=> ( ( times_7914811829580426937y_real @ A @ B )
!= zero_zero_poly_real ) ) ) ).
% no_zero_divisors
thf(fact_242_no__zero__divisors,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( A != zero_z799223564134138693ly_int )
=> ( ( B != zero_z799223564134138693ly_int )
=> ( ( times_4739760418287672641ly_int @ A @ B )
!= zero_z799223564134138693ly_int ) ) ) ).
% no_zero_divisors
thf(fact_243_no__zero__divisors,axiom,
! [A: poly_nat,B: poly_nat] :
( ( A != zero_zero_poly_nat )
=> ( ( B != zero_zero_poly_nat )
=> ( ( times_times_poly_nat @ A @ B )
!= zero_zero_poly_nat ) ) ) ).
% no_zero_divisors
thf(fact_244_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_245_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_246_no__zero__divisors,axiom,
! [A: poly_int,B: poly_int] :
( ( A != zero_zero_poly_int )
=> ( ( B != zero_zero_poly_int )
=> ( ( times_times_poly_int @ A @ B )
!= zero_zero_poly_int ) ) ) ).
% no_zero_divisors
thf(fact_247_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_248_divisors__zero,axiom,
! [A: poly_real,B: poly_real] :
( ( ( times_7914811829580426937y_real @ A @ B )
= zero_zero_poly_real )
=> ( ( A = zero_zero_poly_real )
| ( B = zero_zero_poly_real ) ) ) ).
% divisors_zero
thf(fact_249_divisors__zero,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ A @ B )
= zero_z799223564134138693ly_int )
=> ( ( A = zero_z799223564134138693ly_int )
| ( B = zero_z799223564134138693ly_int ) ) ) ).
% divisors_zero
thf(fact_250_divisors__zero,axiom,
! [A: poly_nat,B: poly_nat] :
( ( ( times_times_poly_nat @ A @ B )
= zero_zero_poly_nat )
=> ( ( A = zero_zero_poly_nat )
| ( B = zero_zero_poly_nat ) ) ) ).
% divisors_zero
thf(fact_251_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_252_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_253_divisors__zero,axiom,
! [A: poly_int,B: poly_int] :
( ( ( times_times_poly_int @ A @ B )
= zero_zero_poly_int )
=> ( ( A = zero_zero_poly_int )
| ( B = zero_zero_poly_int ) ) ) ).
% divisors_zero
thf(fact_254_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_255_mult__not__zero,axiom,
! [A: poly_real,B: poly_real] :
( ( ( times_7914811829580426937y_real @ A @ B )
!= zero_zero_poly_real )
=> ( ( A != zero_zero_poly_real )
& ( B != zero_zero_poly_real ) ) ) ).
% mult_not_zero
thf(fact_256_mult__not__zero,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ A @ B )
!= zero_z799223564134138693ly_int )
=> ( ( A != zero_z799223564134138693ly_int )
& ( B != zero_z799223564134138693ly_int ) ) ) ).
% mult_not_zero
thf(fact_257_mult__not__zero,axiom,
! [A: poly_nat,B: poly_nat] :
( ( ( times_times_poly_nat @ A @ B )
!= zero_zero_poly_nat )
=> ( ( A != zero_zero_poly_nat )
& ( B != zero_zero_poly_nat ) ) ) ).
% mult_not_zero
thf(fact_258_mult__not__zero,axiom,
! [A: poly_p6692042823160534382ring_n,B: poly_p6692042823160534382ring_n] :
( ( ( times_2573333606529333417ring_n @ A @ B )
!= zero_z5482829069124612005ring_n )
=> ( ( A != zero_z5482829069124612005ring_n )
& ( B != zero_z5482829069124612005ring_n ) ) ) ).
% mult_not_zero
thf(fact_259_mult__not__zero,axiom,
! [A: finite_mod_ring_n,B: finite_mod_ring_n] :
( ( ( times_5121417632533718157ring_n @ A @ B )
!= zero_z7902377597758090121ring_n )
=> ( ( A != zero_z7902377597758090121ring_n )
& ( B != zero_z7902377597758090121ring_n ) ) ) ).
% mult_not_zero
thf(fact_260_mult__not__zero,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n] :
( ( ( times_4166049284782705435ring_n @ A @ B )
!= zero_z2753989067526334999ring_n )
=> ( ( A != zero_z2753989067526334999ring_n )
& ( B != zero_z2753989067526334999ring_n ) ) ) ).
% mult_not_zero
thf(fact_261_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_262_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_263_mult__not__zero,axiom,
! [A: poly_int,B: poly_int] :
( ( ( times_times_poly_int @ A @ B )
!= zero_zero_poly_int )
=> ( ( A != zero_zero_poly_int )
& ( B != zero_zero_poly_int ) ) ) ).
% mult_not_zero
thf(fact_264_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_265_zero__neq__one,axiom,
zero_z2753989067526334999ring_n != one_on4318287115420659547ring_n ).
% zero_neq_one
thf(fact_266_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_267_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_268_zero__neq__one,axiom,
zero_zero_poly_int != one_one_poly_int ).
% zero_neq_one
thf(fact_269_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_270_zero__neq__one,axiom,
zero_zero_poly_real != one_one_poly_real ).
% zero_neq_one
thf(fact_271_zero__neq__one,axiom,
zero_z799223564134138693ly_int != one_on1166514126663969025ly_int ).
% zero_neq_one
thf(fact_272_zero__neq__one,axiom,
zero_zero_poly_nat != one_one_poly_nat ).
% zero_neq_one
thf(fact_273_zero__neq__one,axiom,
zero_z5482829069124612005ring_n != one_on5457780782968151273ring_n ).
% zero_neq_one
thf(fact_274_zero__neq__one,axiom,
zero_z7902377597758090121ring_n != one_on2109788483843180749ring_n ).
% zero_neq_one
thf(fact_275_dvd__0__left,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ zero_z2753989067526334999ring_n @ A )
=> ( A = zero_z2753989067526334999ring_n ) ) ).
% dvd_0_left
thf(fact_276_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_277_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_278_dvd__0__left,axiom,
! [A: poly_int] :
( ( dvd_dvd_poly_int @ zero_zero_poly_int @ A )
=> ( A = zero_zero_poly_int ) ) ).
% dvd_0_left
thf(fact_279_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_280_dvd__0__left,axiom,
! [A: poly_real] :
( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A )
=> ( A = zero_zero_poly_real ) ) ).
% dvd_0_left
thf(fact_281_dvd__0__left,axiom,
! [A: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ zero_z799223564134138693ly_int @ A )
=> ( A = zero_z799223564134138693ly_int ) ) ).
% dvd_0_left
thf(fact_282_dvd__0__left,axiom,
! [A: poly_nat] :
( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A )
=> ( A = zero_zero_poly_nat ) ) ).
% dvd_0_left
thf(fact_283_dvd__0__left,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ zero_z5482829069124612005ring_n @ A )
=> ( A = zero_z5482829069124612005ring_n ) ) ).
% dvd_0_left
thf(fact_284_dvd__0__left,axiom,
! [A: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ zero_z7902377597758090121ring_n @ A )
=> ( A = zero_z7902377597758090121ring_n ) ) ).
% dvd_0_left
thf(fact_285_dvd__triv__right,axiom,
! [A: poly_real,B: poly_real] : ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_286_dvd__triv__right,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n] : ( dvd_dv8138414522854976442ring_n @ A @ ( times_4166049284782705435ring_n @ B @ A ) ) ).
% dvd_triv_right
thf(fact_287_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_288_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_289_dvd__triv__right,axiom,
! [A: poly_int,B: poly_int] : ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_290_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_291_dvd__mult__right,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ C2 )
=> ( dvd_dvd_poly_real @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_292_dvd__mult__right,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ C2 )
=> ( dvd_dv8138414522854976442ring_n @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_293_dvd__mult__right,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
=> ( dvd_dvd_int @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_294_dvd__mult__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
=> ( dvd_dvd_nat @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_295_dvd__mult__right,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
=> ( dvd_dvd_poly_int @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_296_dvd__mult__right,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
=> ( dvd_dvd_real @ B @ C2 ) ) ).
% dvd_mult_right
thf(fact_297_mult__dvd__mono,axiom,
! [A: poly_real,B: poly_real,C2: poly_real,D: poly_real] :
( ( dvd_dvd_poly_real @ A @ B )
=> ( ( dvd_dvd_poly_real @ C2 @ D )
=> ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ C2 ) @ ( times_7914811829580426937y_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_298_mult__dvd__mono,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n,D: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ A @ B )
=> ( ( dvd_dv8138414522854976442ring_n @ C2 @ D )
=> ( dvd_dv8138414522854976442ring_n @ ( times_4166049284782705435ring_n @ A @ C2 ) @ ( times_4166049284782705435ring_n @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_299_mult__dvd__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C2 @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_300_mult__dvd__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C2 @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_301_mult__dvd__mono,axiom,
! [A: poly_int,B: poly_int,C2: poly_int,D: poly_int] :
( ( dvd_dvd_poly_int @ A @ B )
=> ( ( dvd_dvd_poly_int @ C2 @ D )
=> ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ C2 ) @ ( times_times_poly_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_302_mult__dvd__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ C2 @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_303_dvd__triv__left,axiom,
! [A: poly_real,B: poly_real] : ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_304_dvd__triv__left,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n] : ( dvd_dv8138414522854976442ring_n @ A @ ( times_4166049284782705435ring_n @ A @ B ) ) ).
% dvd_triv_left
thf(fact_305_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_306_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_307_dvd__triv__left,axiom,
! [A: poly_int,B: poly_int] : ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_308_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_309_dvd__mult__left,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ C2 )
=> ( dvd_dvd_poly_real @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_310_dvd__mult__left,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ C2 )
=> ( dvd_dv8138414522854976442ring_n @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_311_dvd__mult__left,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
=> ( dvd_dvd_int @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_312_dvd__mult__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
=> ( dvd_dvd_nat @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_313_dvd__mult__left,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
=> ( dvd_dvd_poly_int @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_314_dvd__mult__left,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
=> ( dvd_dvd_real @ A @ C2 ) ) ).
% dvd_mult_left
thf(fact_315_dvd__mult2,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ B )
=> ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_316_dvd__mult2,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ A @ B )
=> ( dvd_dv8138414522854976442ring_n @ A @ ( times_4166049284782705435ring_n @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_317_dvd__mult2,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_318_dvd__mult2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_319_dvd__mult2,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ B )
=> ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_320_dvd__mult2,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ B )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% dvd_mult2
thf(fact_321_dvd__mult,axiom,
! [A: poly_real,C2: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ A @ C2 )
=> ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_322_dvd__mult,axiom,
! [A: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ A @ C2 )
=> ( dvd_dv8138414522854976442ring_n @ A @ ( times_4166049284782705435ring_n @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_323_dvd__mult,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ C2 )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_324_dvd__mult,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C2 )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_325_dvd__mult,axiom,
! [A: poly_int,C2: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ A @ C2 )
=> ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_326_dvd__mult,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ A @ C2 )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% dvd_mult
thf(fact_327_dvd__def,axiom,
( dvd_dvd_poly_real
= ( ^ [B2: poly_real,A2: poly_real] :
? [K: poly_real] :
( A2
= ( times_7914811829580426937y_real @ B2 @ K ) ) ) ) ).
% dvd_def
thf(fact_328_dvd__def,axiom,
( dvd_dv8138414522854976442ring_n
= ( ^ [B2: poly_F4222894760850802144ring_n,A2: poly_F4222894760850802144ring_n] :
? [K: poly_F4222894760850802144ring_n] :
( A2
= ( times_4166049284782705435ring_n @ B2 @ K ) ) ) ) ).
% dvd_def
thf(fact_329_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A2: int] :
? [K: int] :
( A2
= ( times_times_int @ B2 @ K ) ) ) ) ).
% dvd_def
thf(fact_330_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A2: nat] :
? [K: nat] :
( A2
= ( times_times_nat @ B2 @ K ) ) ) ) ).
% dvd_def
thf(fact_331_dvd__def,axiom,
( dvd_dvd_poly_int
= ( ^ [B2: poly_int,A2: poly_int] :
? [K: poly_int] :
( A2
= ( times_times_poly_int @ B2 @ K ) ) ) ) ).
% dvd_def
thf(fact_332_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B2: real,A2: real] :
? [K: real] :
( A2
= ( times_times_real @ B2 @ K ) ) ) ) ).
% dvd_def
thf(fact_333_dvdI,axiom,
! [A: poly_real,B: poly_real,K2: poly_real] :
( ( A
= ( times_7914811829580426937y_real @ B @ K2 ) )
=> ( dvd_dvd_poly_real @ B @ A ) ) ).
% dvdI
thf(fact_334_dvdI,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,K2: poly_F4222894760850802144ring_n] :
( ( A
= ( times_4166049284782705435ring_n @ B @ K2 ) )
=> ( dvd_dv8138414522854976442ring_n @ B @ A ) ) ).
% dvdI
thf(fact_335_dvdI,axiom,
! [A: int,B: int,K2: int] :
( ( A
= ( times_times_int @ B @ K2 ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_336_dvdI,axiom,
! [A: nat,B: nat,K2: nat] :
( ( A
= ( times_times_nat @ B @ K2 ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_337_dvdI,axiom,
! [A: poly_int,B: poly_int,K2: poly_int] :
( ( A
= ( times_times_poly_int @ B @ K2 ) )
=> ( dvd_dvd_poly_int @ B @ A ) ) ).
% dvdI
thf(fact_338_dvdI,axiom,
! [A: real,B: real,K2: real] :
( ( A
= ( times_times_real @ B @ K2 ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_339_dvdE,axiom,
! [B: poly_real,A: poly_real] :
( ( dvd_dvd_poly_real @ B @ A )
=> ~ ! [K3: poly_real] :
( A
!= ( times_7914811829580426937y_real @ B @ K3 ) ) ) ).
% dvdE
thf(fact_340_dvdE,axiom,
! [B: poly_F4222894760850802144ring_n,A: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ B @ A )
=> ~ ! [K3: poly_F4222894760850802144ring_n] :
( A
!= ( times_4166049284782705435ring_n @ B @ K3 ) ) ) ).
% dvdE
thf(fact_341_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K3: int] :
( A
!= ( times_times_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_342_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K3: nat] :
( A
!= ( times_times_nat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_343_dvdE,axiom,
! [B: poly_int,A: poly_int] :
( ( dvd_dvd_poly_int @ B @ A )
=> ~ ! [K3: poly_int] :
( A
!= ( times_times_poly_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_344_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K3: real] :
( A
!= ( times_times_real @ B @ K3 ) ) ) ).
% dvdE
thf(fact_345_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_346_dvd__unit__imp__unit,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ A @ B )
=> ( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( dvd_dvd_poly_int @ A @ one_one_poly_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_347_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_348_dvd__unit__imp__unit,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ A @ one_one_real ) ) ) ).
% dvd_unit_imp_unit
thf(fact_349_dvd__unit__imp__unit,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ A @ B )
=> ( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( dvd_dvd_poly_real @ A @ one_one_poly_real ) ) ) ).
% dvd_unit_imp_unit
thf(fact_350_dvd__unit__imp__unit,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ B )
=> ( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_351_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_352_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: poly_int,A: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( dvd_dvd_poly_int @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_353_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_354_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_355_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: poly_real,A: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( dvd_dvd_poly_real @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_356_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: poly_poly_int,A: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( dvd_dv6998304861263046114ly_int @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_357_idom__class_Ounit__imp__dvd,axiom,
! [B: poly_int,A: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( dvd_dvd_poly_int @ B @ A ) ) ).
% idom_class.unit_imp_dvd
thf(fact_358_idom__class_Ounit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% idom_class.unit_imp_dvd
thf(fact_359_idom__class_Ounit__imp__dvd,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ B @ A ) ) ).
% idom_class.unit_imp_dvd
thf(fact_360_idom__class_Ounit__imp__dvd,axiom,
! [B: poly_real,A: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( dvd_dvd_poly_real @ B @ A ) ) ).
% idom_class.unit_imp_dvd
thf(fact_361_idom__class_Ounit__imp__dvd,axiom,
! [B: poly_poly_int,A: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( dvd_dv6998304861263046114ly_int @ B @ A ) ) ).
% idom_class.unit_imp_dvd
thf(fact_362_one__dvd,axiom,
! [A: poly_F4222894760850802144ring_n] : ( dvd_dv8138414522854976442ring_n @ one_on4318287115420659547ring_n @ A ) ).
% one_dvd
thf(fact_363_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_364_one__dvd,axiom,
! [A: poly_int] : ( dvd_dvd_poly_int @ one_one_poly_int @ A ) ).
% one_dvd
thf(fact_365_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_366_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_367_one__dvd,axiom,
! [A: poly_real] : ( dvd_dvd_poly_real @ one_one_poly_real @ A ) ).
% one_dvd
thf(fact_368_one__dvd,axiom,
! [A: poly_poly_int] : ( dvd_dv6998304861263046114ly_int @ one_on1166514126663969025ly_int @ A ) ).
% one_dvd
thf(fact_369_one__dvd,axiom,
! [A: poly_nat] : ( dvd_dvd_poly_nat @ one_one_poly_nat @ A ) ).
% one_dvd
thf(fact_370_one__dvd,axiom,
! [A: poly_p6692042823160534382ring_n] : ( dvd_dv3135175980337127240ring_n @ one_on5457780782968151273ring_n @ A ) ).
% one_dvd
thf(fact_371_one__dvd,axiom,
! [A: finite_mod_ring_n] : ( dvd_dv7258769396337835820ring_n @ one_on2109788483843180749ring_n @ A ) ).
% one_dvd
thf(fact_372_of__int__poly__hom_Oinjectivity,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X )
= ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ Y ) )
=> ( X = Y ) ) ).
% of_int_poly_hom.injectivity
thf(fact_373_of__int__poly__hom_Oinjectivity,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( map_poly_int_real @ ring_1_of_int_real @ X )
= ( map_poly_int_real @ ring_1_of_int_real @ Y ) )
=> ( X = Y ) ) ).
% of_int_poly_hom.injectivity
thf(fact_374_of__int__poly__hom_Oinjectivity,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( map_poly_int_int @ ring_1_of_int_int @ X )
= ( map_poly_int_int @ ring_1_of_int_int @ Y ) )
=> ( X = Y ) ) ).
% of_int_poly_hom.injectivity
thf(fact_375_mult__poly__0__right,axiom,
! [P: poly_real] :
( ( times_7914811829580426937y_real @ P @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% mult_poly_0_right
thf(fact_376_mult__poly__0__right,axiom,
! [P: poly_poly_int] :
( ( times_4739760418287672641ly_int @ P @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int ) ).
% mult_poly_0_right
thf(fact_377_mult__poly__0__right,axiom,
! [P: poly_nat] :
( ( times_times_poly_nat @ P @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% mult_poly_0_right
thf(fact_378_mult__poly__0__right,axiom,
! [P: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ P @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n ) ).
% mult_poly_0_right
thf(fact_379_mult__poly__0__right,axiom,
! [P: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ P @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n ) ).
% mult_poly_0_right
thf(fact_380_mult__poly__0__right,axiom,
! [P: poly_int] :
( ( times_times_poly_int @ P @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% mult_poly_0_right
thf(fact_381_mult__poly__0__left,axiom,
! [Q: poly_real] :
( ( times_7914811829580426937y_real @ zero_zero_poly_real @ Q )
= zero_zero_poly_real ) ).
% mult_poly_0_left
thf(fact_382_mult__poly__0__left,axiom,
! [Q: poly_poly_int] :
( ( times_4739760418287672641ly_int @ zero_z799223564134138693ly_int @ Q )
= zero_z799223564134138693ly_int ) ).
% mult_poly_0_left
thf(fact_383_mult__poly__0__left,axiom,
! [Q: poly_nat] :
( ( times_times_poly_nat @ zero_zero_poly_nat @ Q )
= zero_zero_poly_nat ) ).
% mult_poly_0_left
thf(fact_384_mult__poly__0__left,axiom,
! [Q: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ zero_z5482829069124612005ring_n @ Q )
= zero_z5482829069124612005ring_n ) ).
% mult_poly_0_left
thf(fact_385_mult__poly__0__left,axiom,
! [Q: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ zero_z2753989067526334999ring_n @ Q )
= zero_z2753989067526334999ring_n ) ).
% mult_poly_0_left
thf(fact_386_mult__poly__0__left,axiom,
! [Q: poly_int] :
( ( times_times_poly_int @ zero_zero_poly_int @ Q )
= zero_zero_poly_int ) ).
% mult_poly_0_left
thf(fact_387_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_388_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_389_not__is__unit__0,axiom,
~ ( dvd_dvd_poly_int @ zero_zero_poly_int @ one_one_poly_int ) ).
% not_is_unit_0
thf(fact_390_not__is__unit__0,axiom,
~ ( dvd_dvd_real @ zero_zero_real @ one_one_real ) ).
% not_is_unit_0
thf(fact_391_not__is__unit__0,axiom,
~ ( dvd_dvd_poly_real @ zero_zero_poly_real @ one_one_poly_real ) ).
% not_is_unit_0
thf(fact_392_not__is__unit__0,axiom,
~ ( dvd_dv6998304861263046114ly_int @ zero_z799223564134138693ly_int @ one_on1166514126663969025ly_int ) ).
% not_is_unit_0
thf(fact_393_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( ( times_7914811829580426937y_real @ B @ A )
= ( times_7914811829580426937y_real @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_394_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( ( times_4739760418287672641ly_int @ B @ A )
= ( times_4739760418287672641ly_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_395_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_396_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_397_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( ( times_times_poly_int @ B @ A )
= ( times_times_poly_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_398_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( ( times_times_real @ B @ A )
= ( times_times_real @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_399_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( ( times_7914811829580426937y_real @ A @ B )
= ( times_7914811829580426937y_real @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_400_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( ( times_4739760418287672641ly_int @ A @ B )
= ( times_4739760418287672641ly_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_401_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_402_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_403_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( ( times_times_poly_int @ A @ B )
= ( times_times_poly_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_404_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= ( times_times_real @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_405_idom__class_Ounit__mult__right__cancel,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( ( times_7914811829580426937y_real @ B @ A )
= ( times_7914811829580426937y_real @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_406_idom__class_Ounit__mult__right__cancel,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( ( times_4739760418287672641ly_int @ B @ A )
= ( times_4739760418287672641ly_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_407_idom__class_Ounit__mult__right__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_408_idom__class_Ounit__mult__right__cancel,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( ( times_times_poly_int @ B @ A )
= ( times_times_poly_int @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_409_idom__class_Ounit__mult__right__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( ( times_times_real @ B @ A )
= ( times_times_real @ C2 @ A ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_410_idom__class_Ounit__mult__left__cancel,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( ( times_7914811829580426937y_real @ A @ B )
= ( times_7914811829580426937y_real @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_411_idom__class_Ounit__mult__left__cancel,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( ( times_4739760418287672641ly_int @ A @ B )
= ( times_4739760418287672641ly_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_412_idom__class_Ounit__mult__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_413_idom__class_Ounit__mult__left__cancel,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( ( times_times_poly_int @ A @ B )
= ( times_times_poly_int @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_414_idom__class_Ounit__mult__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= ( times_times_real @ A @ C2 ) )
= ( B = C2 ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_415_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ C2 )
= ( dvd_dvd_poly_real @ B @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_416_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_poly_int,B: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ C2 )
= ( dvd_dv6998304861263046114ly_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_417_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_418_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_419_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
= ( dvd_dvd_poly_int @ B @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_420_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
= ( dvd_dvd_real @ B @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_421_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: poly_real,A: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ B @ C2 ) )
= ( dvd_dvd_poly_real @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_422_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: poly_poly_int,A: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ A @ ( times_4739760418287672641ly_int @ B @ C2 ) )
= ( dvd_dv6998304861263046114ly_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_423_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_424_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_425_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: poly_int,A: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) )
= ( dvd_dvd_poly_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_426_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: real,A: real,C2: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_427_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: poly_real,A: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ C2 )
= ( dvd_dvd_poly_real @ A @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_428_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: poly_poly_int,A: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ C2 )
= ( dvd_dv6998304861263046114ly_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_429_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_430_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_431_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: poly_int,A: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
= ( dvd_dvd_poly_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_432_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: real,A: real,C2: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_433_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: poly_real,A: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ C2 @ B ) )
= ( dvd_dvd_poly_real @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_434_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: poly_poly_int,A: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ A @ ( times_4739760418287672641ly_int @ C2 @ B ) )
= ( dvd_dv6998304861263046114ly_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_435_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_436_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C2 @ B ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_437_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: poly_int,A: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ C2 @ B ) )
= ( dvd_dvd_poly_int @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_438_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: real,A: real,C2: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A @ ( times_times_real @ C2 @ B ) )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_439_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ ( times_7914811829580426937y_real @ A @ B ) @ one_one_poly_real )
= ( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
& ( dvd_dvd_poly_real @ B @ one_one_poly_real ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_440_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( times_4739760418287672641ly_int @ A @ B ) @ one_on1166514126663969025ly_int )
= ( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
& ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_441_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_442_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_443_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ ( times_times_poly_int @ A @ B ) @ one_one_poly_int )
= ( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
& ( dvd_dvd_poly_int @ B @ one_one_poly_int ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_444_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ one_one_real )
= ( ( dvd_dvd_real @ A @ one_one_real )
& ( dvd_dvd_real @ B @ one_one_real ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_445_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: poly_real,A: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ B @ C2 ) )
= ( dvd_dvd_poly_real @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_446_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: poly_poly_int,A: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ A @ ( times_4739760418287672641ly_int @ B @ C2 ) )
= ( dvd_dv6998304861263046114ly_int @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_447_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_448_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: poly_int,A: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) )
= ( dvd_dvd_poly_int @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_449_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: real,A: real,C2: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_450_idom__class_Odvd__mult__unit__iff,axiom,
! [B: poly_real,A: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ B @ one_one_poly_real )
=> ( ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ C2 @ B ) )
= ( dvd_dvd_poly_real @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_451_idom__class_Odvd__mult__unit__iff,axiom,
! [B: poly_poly_int,A: poly_poly_int,C2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B @ one_on1166514126663969025ly_int )
=> ( ( dvd_dv6998304861263046114ly_int @ A @ ( times_4739760418287672641ly_int @ C2 @ B ) )
= ( dvd_dv6998304861263046114ly_int @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_452_idom__class_Odvd__mult__unit__iff,axiom,
! [B: int,A: int,C2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_453_idom__class_Odvd__mult__unit__iff,axiom,
! [B: poly_int,A: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ B @ one_one_poly_int )
=> ( ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ C2 @ B ) )
= ( dvd_dvd_poly_int @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_454_idom__class_Odvd__mult__unit__iff,axiom,
! [B: real,A: real,C2: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A @ ( times_times_real @ C2 @ B ) )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_455_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv8138414522854976442ring_n @ ( ring_18712857867054464081ring_n @ X ) @ one_on4318287115420659547ring_n ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_456_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_poly_int @ ( ring_17892525584911698563ly_int @ X ) @ one_one_poly_int ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_457_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_poly_real @ ( ring_12936506555246842115y_real @ X ) @ one_one_poly_real ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_458_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv6998304861263046114ly_int @ ( ring_14695796289142966411ly_int @ X ) @ one_on1166514126663969025ly_int ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_459_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv3135175980337127240ring_n @ ( ring_14208964510912816607ring_n @ X ) @ one_on5457780782968151273ring_n ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_460_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv7258769396337835820ring_n @ ( ring_18169885536585341379ring_n @ X ) @ one_on2109788483843180749ring_n ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_461_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_real @ ( ring_1_of_int_real @ X ) @ one_one_real ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_462_of__int__poly__hom_Obase_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_int @ ( ring_1_of_int_int @ X ) @ one_one_int ) ) ).
% of_int_poly_hom.base.hom_dvd_1
thf(fact_463_of__int__poly__hom_Ohom__0,axiom,
! [X: poly_int] :
( ( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X )
= zero_z799223564134138693ly_int )
=> ( X = zero_zero_poly_int ) ) ).
% of_int_poly_hom.hom_0
thf(fact_464_of__int__poly__hom_Ohom__0,axiom,
! [X: poly_int] :
( ( ( map_poly_int_real @ ring_1_of_int_real @ X )
= zero_zero_poly_real )
=> ( X = zero_zero_poly_int ) ) ).
% of_int_poly_hom.hom_0
thf(fact_465_of__int__poly__hom_Ohom__0,axiom,
! [X: poly_int] :
( ( ( map_poly_int_int @ ring_1_of_int_int @ X )
= zero_zero_poly_int )
=> ( X = zero_zero_poly_int ) ) ).
% of_int_poly_hom.hom_0
thf(fact_466_of__int__poly__hom_Ohom__mult,axiom,
! [X: poly_int,Y: poly_int] :
( ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ ( times_times_poly_int @ X @ Y ) )
= ( times_2573333606529333417ring_n @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ X ) @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ Y ) ) ) ).
% of_int_poly_hom.hom_mult
thf(fact_467_of__int__poly__hom_Ohom__mult,axiom,
! [X: poly_int,Y: poly_int] :
( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ ( times_times_poly_int @ X @ Y ) )
= ( times_4739760418287672641ly_int @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X ) @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ Y ) ) ) ).
% of_int_poly_hom.hom_mult
thf(fact_468_of__int__poly__hom_Ohom__mult,axiom,
! [X: poly_int,Y: poly_int] :
( ( map_poly_int_real @ ring_1_of_int_real @ ( times_times_poly_int @ X @ Y ) )
= ( times_7914811829580426937y_real @ ( map_poly_int_real @ ring_1_of_int_real @ X ) @ ( map_poly_int_real @ ring_1_of_int_real @ Y ) ) ) ).
% of_int_poly_hom.hom_mult
thf(fact_469_of__int__poly__hom_Ohom__mult,axiom,
! [X: poly_int,Y: poly_int] :
( ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ ( times_times_poly_int @ X @ Y ) )
= ( times_4166049284782705435ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ X ) @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ Y ) ) ) ).
% of_int_poly_hom.hom_mult
thf(fact_470_of__int__poly__hom_Ohom__mult,axiom,
! [X: poly_int,Y: poly_int] :
( ( map_poly_int_int @ ring_1_of_int_int @ ( times_times_poly_int @ X @ Y ) )
= ( times_times_poly_int @ ( map_poly_int_int @ ring_1_of_int_int @ X ) @ ( map_poly_int_int @ ring_1_of_int_int @ Y ) ) ) ).
% of_int_poly_hom.hom_mult
thf(fact_471_of__int__poly__hom_Ohom__1,axiom,
! [X: poly_int] :
( ( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X )
= one_on1166514126663969025ly_int )
=> ( X = one_one_poly_int ) ) ).
% of_int_poly_hom.hom_1
thf(fact_472_of__int__poly__hom_Ohom__1,axiom,
! [X: poly_int] :
( ( ( map_poly_int_real @ ring_1_of_int_real @ X )
= one_one_poly_real )
=> ( X = one_one_poly_int ) ) ).
% of_int_poly_hom.hom_1
thf(fact_473_of__int__poly__hom_Ohom__1,axiom,
! [X: poly_int] :
( ( ( map_poly_int_int @ ring_1_of_int_int @ X )
= one_one_poly_int )
=> ( X = one_one_poly_int ) ) ).
% of_int_poly_hom.hom_1
thf(fact_474_unit__dvdE,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ A @ one_one_poly_real )
=> ~ ( ( A != zero_zero_poly_real )
=> ! [C: poly_real] :
( B
!= ( times_7914811829580426937y_real @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_475_unit__dvdE,axiom,
! [A: poly_poly_int,B: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int )
=> ~ ( ( A != zero_z799223564134138693ly_int )
=> ! [C: poly_poly_int] :
( B
!= ( times_4739760418287672641ly_int @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_476_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C: int] :
( B
!= ( times_times_int @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_477_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C: nat] :
( B
!= ( times_times_nat @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_478_unit__dvdE,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ A @ one_one_poly_int )
=> ~ ( ( A != zero_zero_poly_int )
=> ! [C: poly_int] :
( B
!= ( times_times_poly_int @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_479_unit__dvdE,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ~ ( ( A != zero_zero_real )
=> ! [C: real] :
( B
!= ( times_times_real @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_480_of__int__poly__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( times_times_poly_int @ X @ Y )
= zero_zero_poly_int )
=> ( ( times_4739760418287672641ly_int @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X ) @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ Y ) )
= zero_z799223564134138693ly_int ) ) ).
% of_int_poly_hom.hom_mult_eq_zero
thf(fact_481_of__int__poly__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( times_times_poly_int @ X @ Y )
= zero_zero_poly_int )
=> ( ( times_2573333606529333417ring_n @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ X ) @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ Y ) )
= zero_z5482829069124612005ring_n ) ) ).
% of_int_poly_hom.hom_mult_eq_zero
thf(fact_482_of__int__poly__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( times_times_poly_int @ X @ Y )
= zero_zero_poly_int )
=> ( ( times_7914811829580426937y_real @ ( map_poly_int_real @ ring_1_of_int_real @ X ) @ ( map_poly_int_real @ ring_1_of_int_real @ Y ) )
= zero_zero_poly_real ) ) ).
% of_int_poly_hom.hom_mult_eq_zero
thf(fact_483_of__int__poly__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( times_times_poly_int @ X @ Y )
= zero_zero_poly_int )
=> ( ( times_4166049284782705435ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ X ) @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ Y ) )
= zero_z2753989067526334999ring_n ) ) ).
% of_int_poly_hom.hom_mult_eq_zero
thf(fact_484_of__int__poly__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( times_times_poly_int @ X @ Y )
= zero_zero_poly_int )
=> ( ( times_times_poly_int @ ( map_poly_int_int @ ring_1_of_int_int @ X ) @ ( map_poly_int_int @ ring_1_of_int_int @ Y ) )
= zero_zero_poly_int ) ) ).
% of_int_poly_hom.hom_mult_eq_zero
thf(fact_485_of__int__poly__hom_Ohom__dvd__1,axiom,
! [X: poly_int] :
( ( dvd_dvd_poly_int @ X @ one_one_poly_int )
=> ( dvd_dv6998304861263046114ly_int @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ X ) @ one_on1166514126663969025ly_int ) ) ).
% of_int_poly_hom.hom_dvd_1
thf(fact_486_of__int__poly__hom_Ohom__dvd__1,axiom,
! [X: poly_int] :
( ( dvd_dvd_poly_int @ X @ one_one_poly_int )
=> ( dvd_dv3135175980337127240ring_n @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ X ) @ one_on5457780782968151273ring_n ) ) ).
% of_int_poly_hom.hom_dvd_1
thf(fact_487_of__int__poly__hom_Ohom__dvd__1,axiom,
! [X: poly_int] :
( ( dvd_dvd_poly_int @ X @ one_one_poly_int )
=> ( dvd_dv8138414522854976442ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ X ) @ one_on4318287115420659547ring_n ) ) ).
% of_int_poly_hom.hom_dvd_1
thf(fact_488_of__int__poly__hom_Ohom__dvd__1,axiom,
! [X: poly_int] :
( ( dvd_dvd_poly_int @ X @ one_one_poly_int )
=> ( dvd_dvd_poly_real @ ( map_poly_int_real @ ring_1_of_int_real @ X ) @ one_one_poly_real ) ) ).
% of_int_poly_hom.hom_dvd_1
thf(fact_489_of__int__poly__hom_Ohom__dvd__1,axiom,
! [X: poly_int] :
( ( dvd_dvd_poly_int @ X @ one_one_poly_int )
=> ( dvd_dvd_poly_int @ ( map_poly_int_int @ ring_1_of_int_int @ X ) @ one_one_poly_int ) ) ).
% of_int_poly_hom.hom_dvd_1
thf(fact_490_of__int__hom_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dv8138414522854976442ring_n @ ( ring_18712857867054464081ring_n @ P ) @ ( ring_18712857867054464081ring_n @ Q ) ) ) ).
% of_int_hom.hom_dvd
thf(fact_491_of__int__hom_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_poly_int @ ( ring_17892525584911698563ly_int @ P ) @ ( ring_17892525584911698563ly_int @ Q ) ) ) ).
% of_int_hom.hom_dvd
thf(fact_492_of__int__hom_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_poly_real @ ( ring_12936506555246842115y_real @ P ) @ ( ring_12936506555246842115y_real @ Q ) ) ) ).
% of_int_hom.hom_dvd
thf(fact_493_of__int__hom_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dv7258769396337835820ring_n @ ( ring_18169885536585341379ring_n @ P ) @ ( ring_18169885536585341379ring_n @ Q ) ) ) ).
% of_int_hom.hom_dvd
thf(fact_494_of__int__hom_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_real @ ( ring_1_of_int_real @ P ) @ ( ring_1_of_int_real @ Q ) ) ) ).
% of_int_hom.hom_dvd
thf(fact_495_of__int__hom_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_int @ ( ring_1_of_int_int @ P ) @ ( ring_1_of_int_int @ Q ) ) ) ).
% of_int_hom.hom_dvd
thf(fact_496_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_17892525584911698563ly_int @ Z )
= one_one_poly_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_497_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_12936506555246842115y_real @ Z )
= one_one_poly_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_498_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_14695796289142966411ly_int @ Z )
= one_on1166514126663969025ly_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_499_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_500_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_501_of__int__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( ring_17892525584911698563ly_int @ X )
= one_one_poly_int )
= ( X = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_502_of__int__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( ring_12936506555246842115y_real @ X )
= one_one_poly_real )
= ( X = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_503_of__int__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( ring_14695796289142966411ly_int @ X )
= one_on1166514126663969025ly_int )
= ( X = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_504_of__int__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= one_one_real )
= ( X = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_505_of__int__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= one_one_int )
= ( X = one_one_int ) ) ).
% of_int_hom.hom_1_iff
thf(fact_506_of__int__hom_Ohom__one,axiom,
( ( ring_18712857867054464081ring_n @ one_one_int )
= one_on4318287115420659547ring_n ) ).
% of_int_hom.hom_one
thf(fact_507_of__int__hom_Ohom__one,axiom,
( ( ring_17892525584911698563ly_int @ one_one_int )
= one_one_poly_int ) ).
% of_int_hom.hom_one
thf(fact_508_of__int__hom_Ohom__one,axiom,
( ( ring_12936506555246842115y_real @ one_one_int )
= one_one_poly_real ) ).
% of_int_hom.hom_one
thf(fact_509_of__int__hom_Ohom__one,axiom,
( ( ring_14695796289142966411ly_int @ one_one_int )
= one_on1166514126663969025ly_int ) ).
% of_int_hom.hom_one
thf(fact_510_of__int__hom_Ohom__one,axiom,
( ( ring_14208964510912816607ring_n @ one_one_int )
= one_on5457780782968151273ring_n ) ).
% of_int_hom.hom_one
thf(fact_511_of__int__hom_Ohom__one,axiom,
( ( ring_18169885536585341379ring_n @ one_one_int )
= one_on2109788483843180749ring_n ) ).
% of_int_hom.hom_one
thf(fact_512_of__int__hom_Ohom__one,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_hom.hom_one
thf(fact_513_of__int__hom_Ohom__one,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_hom.hom_one
thf(fact_514_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_18169885536585341379ring_n @ ( times_times_int @ X @ Y ) )
= ( times_5121417632533718157ring_n @ ( ring_18169885536585341379ring_n @ X ) @ ( ring_18169885536585341379ring_n @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_515_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_18712857867054464081ring_n @ ( times_times_int @ X @ Y ) )
= ( times_4166049284782705435ring_n @ ( ring_18712857867054464081ring_n @ X ) @ ( ring_18712857867054464081ring_n @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_516_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_int @ ( times_times_int @ X @ Y ) )
= ( times_times_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_517_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_17892525584911698563ly_int @ ( times_times_int @ X @ Y ) )
= ( times_times_poly_int @ ( ring_17892525584911698563ly_int @ X ) @ ( ring_17892525584911698563ly_int @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_518_of__int__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( ring_1_of_int_real @ ( times_times_int @ X @ Y ) )
= ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_hom.hom_mult
thf(fact_519_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_18169885536585341379ring_n @ ( times_times_int @ W @ Z ) )
= ( times_5121417632533718157ring_n @ ( ring_18169885536585341379ring_n @ W ) @ ( ring_18169885536585341379ring_n @ Z ) ) ) ).
% of_int_mult
thf(fact_520_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_18712857867054464081ring_n @ ( times_times_int @ W @ Z ) )
= ( times_4166049284782705435ring_n @ ( ring_18712857867054464081ring_n @ W ) @ ( ring_18712857867054464081ring_n @ Z ) ) ) ).
% of_int_mult
thf(fact_521_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_522_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_17892525584911698563ly_int @ ( times_times_int @ W @ Z ) )
= ( times_times_poly_int @ ( ring_17892525584911698563ly_int @ W ) @ ( ring_17892525584911698563ly_int @ Z ) ) ) ).
% of_int_mult
thf(fact_523_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_524_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_17892525584911698563ly_int @ Z )
= zero_zero_poly_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_525_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_12936506555246842115y_real @ Z )
= zero_zero_poly_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_526_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_14695796289142966411ly_int @ Z )
= zero_z799223564134138693ly_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_527_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_528_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_529_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_poly_int
= ( ring_17892525584911698563ly_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_530_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_poly_real
= ( ring_12936506555246842115y_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_531_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_z799223564134138693ly_int
= ( ring_14695796289142966411ly_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_532_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_533_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_534_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_17892525584911698563ly_int @ X )
= zero_zero_poly_int )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_535_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_12936506555246842115y_real @ X )
= zero_zero_poly_real )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_536_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_14695796289142966411ly_int @ X )
= zero_z799223564134138693ly_int )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_537_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= zero_zero_real )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_538_of__int__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= zero_zero_int )
= ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0_iff
thf(fact_539_of__int__hom_Ohom__zero,axiom,
( ( ring_18712857867054464081ring_n @ zero_zero_int )
= zero_z2753989067526334999ring_n ) ).
% of_int_hom.hom_zero
thf(fact_540_of__int__hom_Ohom__zero,axiom,
( ( ring_17892525584911698563ly_int @ zero_zero_int )
= zero_zero_poly_int ) ).
% of_int_hom.hom_zero
thf(fact_541_of__int__hom_Ohom__zero,axiom,
( ( ring_12936506555246842115y_real @ zero_zero_int )
= zero_zero_poly_real ) ).
% of_int_hom.hom_zero
thf(fact_542_of__int__hom_Ohom__zero,axiom,
( ( ring_14695796289142966411ly_int @ zero_zero_int )
= zero_z799223564134138693ly_int ) ).
% of_int_hom.hom_zero
thf(fact_543_of__int__hom_Ohom__zero,axiom,
( ( ring_14208964510912816607ring_n @ zero_zero_int )
= zero_z5482829069124612005ring_n ) ).
% of_int_hom.hom_zero
thf(fact_544_of__int__hom_Ohom__zero,axiom,
( ( ring_18169885536585341379ring_n @ zero_zero_int )
= zero_z7902377597758090121ring_n ) ).
% of_int_hom.hom_zero
thf(fact_545_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_hom.hom_zero
thf(fact_546_of__int__hom_Ohom__zero,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_hom.hom_zero
thf(fact_547_mult__1,axiom,
! [A: poly_real] :
( ( times_7914811829580426937y_real @ one_one_poly_real @ A )
= A ) ).
% mult_1
thf(fact_548_mult__1,axiom,
! [A: poly_poly_int] :
( ( times_4739760418287672641ly_int @ one_on1166514126663969025ly_int @ A )
= A ) ).
% mult_1
thf(fact_549_mult__1,axiom,
! [A: poly_nat] :
( ( times_times_poly_nat @ one_one_poly_nat @ A )
= A ) ).
% mult_1
thf(fact_550_mult__1,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ one_on5457780782968151273ring_n @ A )
= A ) ).
% mult_1
thf(fact_551_mult__1,axiom,
! [A: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ one_on2109788483843180749ring_n @ A )
= A ) ).
% mult_1
thf(fact_552_mult__1,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ one_on4318287115420659547ring_n @ A )
= A ) ).
% mult_1
thf(fact_553_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_554_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_555_mult__1,axiom,
! [A: poly_int] :
( ( times_times_poly_int @ one_one_poly_int @ A )
= A ) ).
% mult_1
thf(fact_556_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_557_mult_Oright__neutral,axiom,
! [A: poly_real] :
( ( times_7914811829580426937y_real @ A @ one_one_poly_real )
= A ) ).
% mult.right_neutral
thf(fact_558_mult_Oright__neutral,axiom,
! [A: poly_poly_int] :
( ( times_4739760418287672641ly_int @ A @ one_on1166514126663969025ly_int )
= A ) ).
% mult.right_neutral
thf(fact_559_mult_Oright__neutral,axiom,
! [A: poly_nat] :
( ( times_times_poly_nat @ A @ one_one_poly_nat )
= A ) ).
% mult.right_neutral
thf(fact_560_mult_Oright__neutral,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ A @ one_on5457780782968151273ring_n )
= A ) ).
% mult.right_neutral
thf(fact_561_mult_Oright__neutral,axiom,
! [A: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ A @ one_on2109788483843180749ring_n )
= A ) ).
% mult.right_neutral
thf(fact_562_mult_Oright__neutral,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ A @ one_on4318287115420659547ring_n )
= A ) ).
% mult.right_neutral
thf(fact_563_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_564_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_565_mult_Oright__neutral,axiom,
! [A: poly_int] :
( ( times_times_poly_int @ A @ one_one_poly_int )
= A ) ).
% mult.right_neutral
thf(fact_566_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_567_of__int__hom_Oeq__iff,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
= ( X = Y ) ) ).
% of_int_hom.eq_iff
thf(fact_568_of__int__hom_Oeq__iff,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_int @ X )
= ( ring_1_of_int_int @ Y ) )
= ( X = Y ) ) ).
% of_int_hom.eq_iff
thf(fact_569_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_570_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_int @ W )
= ( ring_1_of_int_int @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_571_mult__hom_Ohom__zero,axiom,
! [C2: poly_real] :
( ( times_7914811829580426937y_real @ C2 @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% mult_hom.hom_zero
thf(fact_572_mult__hom_Ohom__zero,axiom,
! [C2: poly_poly_int] :
( ( times_4739760418287672641ly_int @ C2 @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int ) ).
% mult_hom.hom_zero
thf(fact_573_mult__hom_Ohom__zero,axiom,
! [C2: poly_nat] :
( ( times_times_poly_nat @ C2 @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% mult_hom.hom_zero
thf(fact_574_mult__hom_Ohom__zero,axiom,
! [C2: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ C2 @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n ) ).
% mult_hom.hom_zero
thf(fact_575_mult__hom_Ohom__zero,axiom,
! [C2: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ C2 @ zero_z7902377597758090121ring_n )
= zero_z7902377597758090121ring_n ) ).
% mult_hom.hom_zero
thf(fact_576_mult__hom_Ohom__zero,axiom,
! [C2: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ C2 @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n ) ).
% mult_hom.hom_zero
thf(fact_577_mult__hom_Ohom__zero,axiom,
! [C2: int] :
( ( times_times_int @ C2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_578_mult__hom_Ohom__zero,axiom,
! [C2: nat] :
( ( times_times_nat @ C2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_579_mult__hom_Ohom__zero,axiom,
! [C2: poly_int] :
( ( times_times_poly_int @ C2 @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% mult_hom.hom_zero
thf(fact_580_mult__hom_Ohom__zero,axiom,
! [C2: real] :
( ( times_times_real @ C2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_581_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_582_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_583_zdvd__mult__cancel,axiom,
! [K2: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K2 @ M ) @ ( times_times_int @ K2 @ N ) )
=> ( ( K2 != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_584_zero__reorient,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( zero_z2753989067526334999ring_n = X )
= ( X = zero_z2753989067526334999ring_n ) ) ).
% zero_reorient
thf(fact_585_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_586_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_587_zero__reorient,axiom,
! [X: poly_int] :
( ( zero_zero_poly_int = X )
= ( X = zero_zero_poly_int ) ) ).
% zero_reorient
thf(fact_588_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_589_zero__reorient,axiom,
! [X: poly_real] :
( ( zero_zero_poly_real = X )
= ( X = zero_zero_poly_real ) ) ).
% zero_reorient
thf(fact_590_zero__reorient,axiom,
! [X: poly_poly_int] :
( ( zero_z799223564134138693ly_int = X )
= ( X = zero_z799223564134138693ly_int ) ) ).
% zero_reorient
thf(fact_591_zero__reorient,axiom,
! [X: poly_nat] :
( ( zero_zero_poly_nat = X )
= ( X = zero_zero_poly_nat ) ) ).
% zero_reorient
thf(fact_592_zero__reorient,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( zero_z5482829069124612005ring_n = X )
= ( X = zero_z5482829069124612005ring_n ) ) ).
% zero_reorient
thf(fact_593_zero__reorient,axiom,
! [X: finite_mod_ring_n] :
( ( zero_z7902377597758090121ring_n = X )
= ( X = zero_z7902377597758090121ring_n ) ) ).
% zero_reorient
thf(fact_594_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ C2 )
= ( times_4166049284782705435ring_n @ A @ ( times_4166049284782705435ring_n @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_595_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_596_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_597_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( times_times_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
= ( times_times_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_598_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_599_mult_Oassoc,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ ( times_4166049284782705435ring_n @ A @ B ) @ C2 )
= ( times_4166049284782705435ring_n @ A @ ( times_4166049284782705435ring_n @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_600_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_601_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_602_mult_Oassoc,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( times_times_poly_int @ ( times_times_poly_int @ A @ B ) @ C2 )
= ( times_times_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_603_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_604_mult_Ocommute,axiom,
( times_4166049284782705435ring_n
= ( ^ [A2: poly_F4222894760850802144ring_n,B2: poly_F4222894760850802144ring_n] : ( times_4166049284782705435ring_n @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_605_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_606_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_607_mult_Ocommute,axiom,
( times_times_poly_int
= ( ^ [A2: poly_int,B2: poly_int] : ( times_times_poly_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_608_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_609_mult_Oleft__commute,axiom,
! [B: poly_F4222894760850802144ring_n,A: poly_F4222894760850802144ring_n,C2: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ B @ ( times_4166049284782705435ring_n @ A @ C2 ) )
= ( times_4166049284782705435ring_n @ A @ ( times_4166049284782705435ring_n @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_610_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_611_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_612_mult_Oleft__commute,axiom,
! [B: poly_int,A: poly_int,C2: poly_int] :
( ( times_times_poly_int @ B @ ( times_times_poly_int @ A @ C2 ) )
= ( times_times_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_613_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_614_one__reorient,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( one_on4318287115420659547ring_n = X )
= ( X = one_on4318287115420659547ring_n ) ) ).
% one_reorient
thf(fact_615_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_616_one__reorient,axiom,
! [X: poly_int] :
( ( one_one_poly_int = X )
= ( X = one_one_poly_int ) ) ).
% one_reorient
thf(fact_617_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_618_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_619_one__reorient,axiom,
! [X: poly_real] :
( ( one_one_poly_real = X )
= ( X = one_one_poly_real ) ) ).
% one_reorient
thf(fact_620_one__reorient,axiom,
! [X: poly_poly_int] :
( ( one_on1166514126663969025ly_int = X )
= ( X = one_on1166514126663969025ly_int ) ) ).
% one_reorient
thf(fact_621_one__reorient,axiom,
! [X: poly_nat] :
( ( one_one_poly_nat = X )
= ( X = one_one_poly_nat ) ) ).
% one_reorient
thf(fact_622_one__reorient,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( one_on5457780782968151273ring_n = X )
= ( X = one_on5457780782968151273ring_n ) ) ).
% one_reorient
thf(fact_623_one__reorient,axiom,
! [X: finite_mod_ring_n] :
( ( one_on2109788483843180749ring_n = X )
= ( X = one_on2109788483843180749ring_n ) ) ).
% one_reorient
thf(fact_624_of__int__hom_Oinjectivity,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_real @ X )
= ( ring_1_of_int_real @ Y ) )
=> ( X = Y ) ) ).
% of_int_hom.injectivity
thf(fact_625_of__int__hom_Oinjectivity,axiom,
! [X: int,Y: int] :
( ( ( ring_1_of_int_int @ X )
= ( ring_1_of_int_int @ Y ) )
=> ( X = Y ) ) ).
% of_int_hom.injectivity
thf(fact_626_comm__monoid__mult__class_Omult__1,axiom,
! [A: poly_real] :
( ( times_7914811829580426937y_real @ one_one_poly_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_627_comm__monoid__mult__class_Omult__1,axiom,
! [A: poly_poly_int] :
( ( times_4739760418287672641ly_int @ one_on1166514126663969025ly_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_628_comm__monoid__mult__class_Omult__1,axiom,
! [A: poly_nat] :
( ( times_times_poly_nat @ one_one_poly_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_629_comm__monoid__mult__class_Omult__1,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ one_on5457780782968151273ring_n @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_630_comm__monoid__mult__class_Omult__1,axiom,
! [A: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ one_on2109788483843180749ring_n @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_631_comm__monoid__mult__class_Omult__1,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ one_on4318287115420659547ring_n @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_632_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_633_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_634_comm__monoid__mult__class_Omult__1,axiom,
! [A: poly_int] :
( ( times_times_poly_int @ one_one_poly_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_635_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_636_mult_Ocomm__neutral,axiom,
! [A: poly_real] :
( ( times_7914811829580426937y_real @ A @ one_one_poly_real )
= A ) ).
% mult.comm_neutral
thf(fact_637_mult_Ocomm__neutral,axiom,
! [A: poly_poly_int] :
( ( times_4739760418287672641ly_int @ A @ one_on1166514126663969025ly_int )
= A ) ).
% mult.comm_neutral
thf(fact_638_mult_Ocomm__neutral,axiom,
! [A: poly_nat] :
( ( times_times_poly_nat @ A @ one_one_poly_nat )
= A ) ).
% mult.comm_neutral
thf(fact_639_mult_Ocomm__neutral,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( times_2573333606529333417ring_n @ A @ one_on5457780782968151273ring_n )
= A ) ).
% mult.comm_neutral
thf(fact_640_mult_Ocomm__neutral,axiom,
! [A: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ A @ one_on2109788483843180749ring_n )
= A ) ).
% mult.comm_neutral
thf(fact_641_mult_Ocomm__neutral,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ A @ one_on4318287115420659547ring_n )
= A ) ).
% mult.comm_neutral
thf(fact_642_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_643_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_644_mult_Ocomm__neutral,axiom,
! [A: poly_int] :
( ( times_times_poly_int @ A @ one_one_poly_int )
= A ) ).
% mult.comm_neutral
thf(fact_645_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_646_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_17892525584911698563ly_int @ X )
= zero_zero_poly_int )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_647_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_12936506555246842115y_real @ X )
= zero_zero_poly_real )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_648_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_14695796289142966411ly_int @ X )
= zero_z799223564134138693ly_int )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_649_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= zero_zero_real )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_650_of__int__hom_Ohom__0,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= zero_zero_int )
=> ( X = zero_zero_int ) ) ).
% of_int_hom.hom_0
thf(fact_651_mult__of__int__commute,axiom,
! [X: int,Y: finite_mod_ring_n] :
( ( times_5121417632533718157ring_n @ ( ring_18169885536585341379ring_n @ X ) @ Y )
= ( times_5121417632533718157ring_n @ Y @ ( ring_18169885536585341379ring_n @ X ) ) ) ).
% mult_of_int_commute
thf(fact_652_mult__of__int__commute,axiom,
! [X: int,Y: poly_F4222894760850802144ring_n] :
( ( times_4166049284782705435ring_n @ ( ring_18712857867054464081ring_n @ X ) @ Y )
= ( times_4166049284782705435ring_n @ Y @ ( ring_18712857867054464081ring_n @ X ) ) ) ).
% mult_of_int_commute
thf(fact_653_mult__of__int__commute,axiom,
! [X: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_654_mult__of__int__commute,axiom,
! [X: int,Y: poly_int] :
( ( times_times_poly_int @ ( ring_17892525584911698563ly_int @ X ) @ Y )
= ( times_times_poly_int @ Y @ ( ring_17892525584911698563ly_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_655_mult__of__int__commute,axiom,
! [X: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_656_of__int__hom_Ohom__1,axiom,
! [X: int] :
( ( ( ring_17892525584911698563ly_int @ X )
= one_one_poly_int )
=> ( X = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_657_of__int__hom_Ohom__1,axiom,
! [X: int] :
( ( ( ring_12936506555246842115y_real @ X )
= one_one_poly_real )
=> ( X = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_658_of__int__hom_Ohom__1,axiom,
! [X: int] :
( ( ( ring_14695796289142966411ly_int @ X )
= one_on1166514126663969025ly_int )
=> ( X = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_659_of__int__hom_Ohom__1,axiom,
! [X: int] :
( ( ( ring_1_of_int_real @ X )
= one_one_real )
=> ( X = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_660_of__int__hom_Ohom__1,axiom,
! [X: int] :
( ( ( ring_1_of_int_int @ X )
= one_one_int )
=> ( X = one_one_int ) ) ).
% of_int_hom.hom_1
thf(fact_661_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_7914811829580426937y_real @ ( ring_12936506555246842115y_real @ X ) @ ( ring_12936506555246842115y_real @ Y ) )
= zero_zero_poly_real ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_662_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_4739760418287672641ly_int @ ( ring_14695796289142966411ly_int @ X ) @ ( ring_14695796289142966411ly_int @ Y ) )
= zero_z799223564134138693ly_int ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_663_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_2573333606529333417ring_n @ ( ring_14208964510912816607ring_n @ X ) @ ( ring_14208964510912816607ring_n @ Y ) )
= zero_z5482829069124612005ring_n ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_664_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_5121417632533718157ring_n @ ( ring_18169885536585341379ring_n @ X ) @ ( ring_18169885536585341379ring_n @ Y ) )
= zero_z7902377597758090121ring_n ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_665_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_4166049284782705435ring_n @ ( ring_18712857867054464081ring_n @ X ) @ ( ring_18712857867054464081ring_n @ Y ) )
= zero_z2753989067526334999ring_n ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_666_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) )
= zero_zero_int ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_667_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_poly_int @ ( ring_17892525584911698563ly_int @ X ) @ ( ring_17892525584911698563ly_int @ Y ) )
= zero_zero_poly_int ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_668_of__int__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) )
= zero_zero_real ) ) ).
% of_int_hom.hom_mult_eq_zero
thf(fact_669_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv8138414522854976442ring_n @ ( ring_18712857867054464081ring_n @ X ) @ one_on4318287115420659547ring_n ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_670_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_poly_int @ ( ring_17892525584911698563ly_int @ X ) @ one_one_poly_int ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_671_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_poly_real @ ( ring_12936506555246842115y_real @ X ) @ one_one_poly_real ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_672_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv6998304861263046114ly_int @ ( ring_14695796289142966411ly_int @ X ) @ one_on1166514126663969025ly_int ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_673_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv3135175980337127240ring_n @ ( ring_14208964510912816607ring_n @ X ) @ one_on5457780782968151273ring_n ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_674_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dv7258769396337835820ring_n @ ( ring_18169885536585341379ring_n @ X ) @ one_on2109788483843180749ring_n ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_675_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_real @ ( ring_1_of_int_real @ X ) @ one_one_real ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_676_of__int__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_int @ ( ring_1_of_int_int @ X ) @ one_one_int ) ) ).
% of_int_hom.hom_dvd_1
thf(fact_677_irred__inner__nz,axiom,
! [X: poly_real] :
( ( X != zero_zero_poly_real )
=> ( ( ! [B2: poly_real] :
( ( dvd_dvd_poly_real @ B2 @ X )
=> ( ~ ( dvd_dvd_poly_real @ X @ B2 )
=> ( dvd_dvd_poly_real @ B2 @ one_one_poly_real ) ) ) )
= ( ! [A2: poly_real,B2: poly_real] :
( ( X
= ( times_7914811829580426937y_real @ A2 @ B2 ) )
=> ( ( dvd_dvd_poly_real @ A2 @ one_one_poly_real )
| ( dvd_dvd_poly_real @ B2 @ one_one_poly_real ) ) ) ) ) ) ).
% irred_inner_nz
thf(fact_678_irred__inner__nz,axiom,
! [X: poly_poly_int] :
( ( X != zero_z799223564134138693ly_int )
=> ( ( ! [B2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ B2 @ X )
=> ( ~ ( dvd_dv6998304861263046114ly_int @ X @ B2 )
=> ( dvd_dv6998304861263046114ly_int @ B2 @ one_on1166514126663969025ly_int ) ) ) )
= ( ! [A2: poly_poly_int,B2: poly_poly_int] :
( ( X
= ( times_4739760418287672641ly_int @ A2 @ B2 ) )
=> ( ( dvd_dv6998304861263046114ly_int @ A2 @ one_on1166514126663969025ly_int )
| ( dvd_dv6998304861263046114ly_int @ B2 @ one_on1166514126663969025ly_int ) ) ) ) ) ) ).
% irred_inner_nz
thf(fact_679_irred__inner__nz,axiom,
! [X: int] :
( ( X != zero_zero_int )
=> ( ( ! [B2: int] :
( ( dvd_dvd_int @ B2 @ X )
=> ( ~ ( dvd_dvd_int @ X @ B2 )
=> ( dvd_dvd_int @ B2 @ one_one_int ) ) ) )
= ( ! [A2: int,B2: int] :
( ( X
= ( times_times_int @ A2 @ B2 ) )
=> ( ( dvd_dvd_int @ A2 @ one_one_int )
| ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ) ) ) ).
% irred_inner_nz
thf(fact_680_irred__inner__nz,axiom,
! [X: poly_int] :
( ( X != zero_zero_poly_int )
=> ( ( ! [B2: poly_int] :
( ( dvd_dvd_poly_int @ B2 @ X )
=> ( ~ ( dvd_dvd_poly_int @ X @ B2 )
=> ( dvd_dvd_poly_int @ B2 @ one_one_poly_int ) ) ) )
= ( ! [A2: poly_int,B2: poly_int] :
( ( X
= ( times_times_poly_int @ A2 @ B2 ) )
=> ( ( dvd_dvd_poly_int @ A2 @ one_one_poly_int )
| ( dvd_dvd_poly_int @ B2 @ one_one_poly_int ) ) ) ) ) ) ).
% irred_inner_nz
thf(fact_681_irred__inner__nz,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( ! [B2: real] :
( ( dvd_dvd_real @ B2 @ X )
=> ( ~ ( dvd_dvd_real @ X @ B2 )
=> ( dvd_dvd_real @ B2 @ one_one_real ) ) ) )
= ( ! [A2: real,B2: real] :
( ( X
= ( times_times_real @ A2 @ B2 ) )
=> ( ( dvd_dvd_real @ A2 @ one_one_real )
| ( dvd_dvd_real @ B2 @ one_one_real ) ) ) ) ) ) ).
% irred_inner_nz
thf(fact_682_zdvd__mono,axiom,
! [K2: int,M: int,T: int] :
( ( K2 != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K2 @ M ) @ ( times_times_int @ K2 @ T ) ) ) ) ).
% zdvd_mono
thf(fact_683_dvd__dvd__imp__unit__mult,axiom,
! [X: poly_real,Y: poly_real] :
( ( dvd_dvd_poly_real @ X @ Y )
=> ( ( dvd_dvd_poly_real @ Y @ X )
=> ? [Z2: poly_real] :
( ( dvd_dvd_poly_real @ Z2 @ one_one_poly_real )
& ( Y
= ( times_7914811829580426937y_real @ X @ Z2 ) ) ) ) ) ).
% dvd_dvd_imp_unit_mult
thf(fact_684_dvd__dvd__imp__unit__mult,axiom,
! [X: poly_poly_int,Y: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ X @ Y )
=> ( ( dvd_dv6998304861263046114ly_int @ Y @ X )
=> ? [Z2: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ Z2 @ one_on1166514126663969025ly_int )
& ( Y
= ( times_4739760418287672641ly_int @ X @ Z2 ) ) ) ) ) ).
% dvd_dvd_imp_unit_mult
thf(fact_685_dvd__dvd__imp__unit__mult,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ Y @ X )
=> ? [Z2: int] :
( ( dvd_dvd_int @ Z2 @ one_one_int )
& ( Y
= ( times_times_int @ X @ Z2 ) ) ) ) ) ).
% dvd_dvd_imp_unit_mult
thf(fact_686_dvd__dvd__imp__unit__mult,axiom,
! [X: poly_int,Y: poly_int] :
( ( dvd_dvd_poly_int @ X @ Y )
=> ( ( dvd_dvd_poly_int @ Y @ X )
=> ? [Z2: poly_int] :
( ( dvd_dvd_poly_int @ Z2 @ one_one_poly_int )
& ( Y
= ( times_times_poly_int @ X @ Z2 ) ) ) ) ) ).
% dvd_dvd_imp_unit_mult
thf(fact_687_dvd__dvd__imp__unit__mult,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ Y @ X )
=> ? [Z2: real] :
( ( dvd_dvd_real @ Z2 @ one_one_real )
& ( Y
= ( times_times_real @ X @ Z2 ) ) ) ) ) ).
% dvd_dvd_imp_unit_mult
thf(fact_688_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_c8149583629457385976ring_n @ N @ one_on4318287115420659547ring_n )
= zero_z2753989067526334999ring_n ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_c8149583629457385976ring_n @ N @ one_on4318287115420659547ring_n )
= one_on4318287115420659547ring_n ) ) ) ).
% poly_cutoff_1
thf(fact_689_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_cutoff_int @ N @ one_one_poly_int )
= zero_zero_poly_int ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_cutoff_int @ N @ one_one_poly_int )
= one_one_poly_int ) ) ) ).
% poly_cutoff_1
thf(fact_690_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_cutoff_real @ N @ one_one_poly_real )
= zero_zero_poly_real ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_cutoff_real @ N @ one_one_poly_real )
= one_one_poly_real ) ) ) ).
% poly_cutoff_1
thf(fact_691_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_cutoff_poly_int @ N @ one_on1166514126663969025ly_int )
= zero_z799223564134138693ly_int ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_cutoff_poly_int @ N @ one_on1166514126663969025ly_int )
= one_on1166514126663969025ly_int ) ) ) ).
% poly_cutoff_1
thf(fact_692_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
= zero_zero_poly_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
= one_one_poly_nat ) ) ) ).
% poly_cutoff_1
thf(fact_693_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_c5946732737480674950ring_n @ N @ one_on5457780782968151273ring_n )
= zero_z5482829069124612005ring_n ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_c5946732737480674950ring_n @ N @ one_on5457780782968151273ring_n )
= one_on5457780782968151273ring_n ) ) ) ).
% poly_cutoff_1
thf(fact_694_dvd__productE,axiom,
! [P: poly_real,A: poly_real,B: poly_real] :
( ( dvd_dvd_poly_real @ P @ ( times_7914811829580426937y_real @ A @ B ) )
=> ~ ! [X2: poly_real,Y2: poly_real] :
( ( P
= ( times_7914811829580426937y_real @ X2 @ Y2 ) )
=> ( ( dvd_dvd_poly_real @ X2 @ A )
=> ~ ( dvd_dvd_poly_real @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_695_dvd__productE,axiom,
! [P: int,A: int,B: int] :
( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B ) )
=> ~ ! [X2: int,Y2: int] :
( ( P
= ( times_times_int @ X2 @ Y2 ) )
=> ( ( dvd_dvd_int @ X2 @ A )
=> ~ ( dvd_dvd_int @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_696_dvd__productE,axiom,
! [P: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
=> ~ ! [X2: nat,Y2: nat] :
( ( P
= ( times_times_nat @ X2 @ Y2 ) )
=> ( ( dvd_dvd_nat @ X2 @ A )
=> ~ ( dvd_dvd_nat @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_697_dvd__productE,axiom,
! [P: poly_int,A: poly_int,B: poly_int] :
( ( dvd_dvd_poly_int @ P @ ( times_times_poly_int @ A @ B ) )
=> ~ ! [X2: poly_int,Y2: poly_int] :
( ( P
= ( times_times_poly_int @ X2 @ Y2 ) )
=> ( ( dvd_dvd_poly_int @ X2 @ A )
=> ~ ( dvd_dvd_poly_int @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_698_dvd__productE,axiom,
! [P: real,A: real,B: real] :
( ( dvd_dvd_real @ P @ ( times_times_real @ A @ B ) )
=> ~ ! [X2: real,Y2: real] :
( ( P
= ( times_times_real @ X2 @ Y2 ) )
=> ( ( dvd_dvd_real @ X2 @ A )
=> ~ ( dvd_dvd_real @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_699_division__decomp,axiom,
! [A: poly_real,B: poly_real,C2: poly_real] :
( ( dvd_dvd_poly_real @ A @ ( times_7914811829580426937y_real @ B @ C2 ) )
=> ? [B3: poly_real,C3: poly_real] :
( ( A
= ( times_7914811829580426937y_real @ B3 @ C3 ) )
& ( dvd_dvd_poly_real @ B3 @ B )
& ( dvd_dvd_poly_real @ C3 @ C2 ) ) ) ).
% division_decomp
thf(fact_700_division__decomp,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
=> ? [B3: int,C3: int] :
( ( A
= ( times_times_int @ B3 @ C3 ) )
& ( dvd_dvd_int @ B3 @ B )
& ( dvd_dvd_int @ C3 @ C2 ) ) ) ).
% division_decomp
thf(fact_701_division__decomp,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) )
=> ? [B3: nat,C3: nat] :
( ( A
= ( times_times_nat @ B3 @ C3 ) )
& ( dvd_dvd_nat @ B3 @ B )
& ( dvd_dvd_nat @ C3 @ C2 ) ) ) ).
% division_decomp
thf(fact_702_division__decomp,axiom,
! [A: poly_int,B: poly_int,C2: poly_int] :
( ( dvd_dvd_poly_int @ A @ ( times_times_poly_int @ B @ C2 ) )
=> ? [B3: poly_int,C3: poly_int] :
( ( A
= ( times_times_poly_int @ B3 @ C3 ) )
& ( dvd_dvd_poly_int @ B3 @ B )
& ( dvd_dvd_poly_int @ C3 @ C2 ) ) ) ).
% division_decomp
thf(fact_703_division__decomp,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) )
=> ? [B3: real,C3: real] :
( ( A
= ( times_times_real @ B3 @ C3 ) )
& ( dvd_dvd_real @ B3 @ B )
& ( dvd_dvd_real @ C3 @ C2 ) ) ) ).
% division_decomp
thf(fact_704_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A2: real,B2: real] :
( ( A2 = zero_zero_real )
=> ( B2 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_705_mult__delta__left,axiom,
! [B: $o,X: poly_real,Y: poly_real] :
( ( B
=> ( ( times_7914811829580426937y_real @ ( if_poly_real @ B @ X @ zero_zero_poly_real ) @ Y )
= ( times_7914811829580426937y_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_7914811829580426937y_real @ ( if_poly_real @ B @ X @ zero_zero_poly_real ) @ Y )
= zero_zero_poly_real ) ) ) ).
% mult_delta_left
thf(fact_706_mult__delta__left,axiom,
! [B: $o,X: poly_poly_int,Y: poly_poly_int] :
( ( B
=> ( ( times_4739760418287672641ly_int @ ( if_poly_poly_int @ B @ X @ zero_z799223564134138693ly_int ) @ Y )
= ( times_4739760418287672641ly_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_4739760418287672641ly_int @ ( if_poly_poly_int @ B @ X @ zero_z799223564134138693ly_int ) @ Y )
= zero_z799223564134138693ly_int ) ) ) ).
% mult_delta_left
thf(fact_707_mult__delta__left,axiom,
! [B: $o,X: poly_nat,Y: poly_nat] :
( ( B
=> ( ( times_times_poly_nat @ ( if_poly_nat @ B @ X @ zero_zero_poly_nat ) @ Y )
= ( times_times_poly_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_poly_nat @ ( if_poly_nat @ B @ X @ zero_zero_poly_nat ) @ Y )
= zero_zero_poly_nat ) ) ) ).
% mult_delta_left
thf(fact_708_mult__delta__left,axiom,
! [B: $o,X: poly_p6692042823160534382ring_n,Y: poly_p6692042823160534382ring_n] :
( ( B
=> ( ( times_2573333606529333417ring_n @ ( if_pol5337421645260315700ring_n @ B @ X @ zero_z5482829069124612005ring_n ) @ Y )
= ( times_2573333606529333417ring_n @ X @ Y ) ) )
& ( ~ B
=> ( ( times_2573333606529333417ring_n @ ( if_pol5337421645260315700ring_n @ B @ X @ zero_z5482829069124612005ring_n ) @ Y )
= zero_z5482829069124612005ring_n ) ) ) ).
% mult_delta_left
thf(fact_709_mult__delta__left,axiom,
! [B: $o,X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( B
=> ( ( times_5121417632533718157ring_n @ ( if_Finite_mod_ring_n @ B @ X @ zero_z7902377597758090121ring_n ) @ Y )
= ( times_5121417632533718157ring_n @ X @ Y ) ) )
& ( ~ B
=> ( ( times_5121417632533718157ring_n @ ( if_Finite_mod_ring_n @ B @ X @ zero_z7902377597758090121ring_n ) @ Y )
= zero_z7902377597758090121ring_n ) ) ) ).
% mult_delta_left
thf(fact_710_mult__delta__left,axiom,
! [B: $o,X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( B
=> ( ( times_4166049284782705435ring_n @ ( if_pol9129390727684501670ring_n @ B @ X @ zero_z2753989067526334999ring_n ) @ Y )
= ( times_4166049284782705435ring_n @ X @ Y ) ) )
& ( ~ B
=> ( ( times_4166049284782705435ring_n @ ( if_pol9129390727684501670ring_n @ B @ X @ zero_z2753989067526334999ring_n ) @ Y )
= zero_z2753989067526334999ring_n ) ) ) ).
% mult_delta_left
thf(fact_711_mult__delta__left,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_712_mult__delta__left,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_713_mult__delta__left,axiom,
! [B: $o,X: poly_int,Y: poly_int] :
( ( B
=> ( ( times_times_poly_int @ ( if_poly_int @ B @ X @ zero_zero_poly_int ) @ Y )
= ( times_times_poly_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_poly_int @ ( if_poly_int @ B @ X @ zero_zero_poly_int ) @ Y )
= zero_zero_poly_int ) ) ) ).
% mult_delta_left
thf(fact_714_mult__delta__left,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_715_mult__delta__right,axiom,
! [B: $o,X: poly_real,Y: poly_real] :
( ( B
=> ( ( times_7914811829580426937y_real @ X @ ( if_poly_real @ B @ Y @ zero_zero_poly_real ) )
= ( times_7914811829580426937y_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_7914811829580426937y_real @ X @ ( if_poly_real @ B @ Y @ zero_zero_poly_real ) )
= zero_zero_poly_real ) ) ) ).
% mult_delta_right
thf(fact_716_mult__delta__right,axiom,
! [B: $o,X: poly_poly_int,Y: poly_poly_int] :
( ( B
=> ( ( times_4739760418287672641ly_int @ X @ ( if_poly_poly_int @ B @ Y @ zero_z799223564134138693ly_int ) )
= ( times_4739760418287672641ly_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_4739760418287672641ly_int @ X @ ( if_poly_poly_int @ B @ Y @ zero_z799223564134138693ly_int ) )
= zero_z799223564134138693ly_int ) ) ) ).
% mult_delta_right
thf(fact_717_mult__delta__right,axiom,
! [B: $o,X: poly_nat,Y: poly_nat] :
( ( B
=> ( ( times_times_poly_nat @ X @ ( if_poly_nat @ B @ Y @ zero_zero_poly_nat ) )
= ( times_times_poly_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_poly_nat @ X @ ( if_poly_nat @ B @ Y @ zero_zero_poly_nat ) )
= zero_zero_poly_nat ) ) ) ).
% mult_delta_right
thf(fact_718_mult__delta__right,axiom,
! [B: $o,X: poly_p6692042823160534382ring_n,Y: poly_p6692042823160534382ring_n] :
( ( B
=> ( ( times_2573333606529333417ring_n @ X @ ( if_pol5337421645260315700ring_n @ B @ Y @ zero_z5482829069124612005ring_n ) )
= ( times_2573333606529333417ring_n @ X @ Y ) ) )
& ( ~ B
=> ( ( times_2573333606529333417ring_n @ X @ ( if_pol5337421645260315700ring_n @ B @ Y @ zero_z5482829069124612005ring_n ) )
= zero_z5482829069124612005ring_n ) ) ) ).
% mult_delta_right
thf(fact_719_mult__delta__right,axiom,
! [B: $o,X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( B
=> ( ( times_5121417632533718157ring_n @ X @ ( if_Finite_mod_ring_n @ B @ Y @ zero_z7902377597758090121ring_n ) )
= ( times_5121417632533718157ring_n @ X @ Y ) ) )
& ( ~ B
=> ( ( times_5121417632533718157ring_n @ X @ ( if_Finite_mod_ring_n @ B @ Y @ zero_z7902377597758090121ring_n ) )
= zero_z7902377597758090121ring_n ) ) ) ).
% mult_delta_right
thf(fact_720_mult__delta__right,axiom,
! [B: $o,X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( B
=> ( ( times_4166049284782705435ring_n @ X @ ( if_pol9129390727684501670ring_n @ B @ Y @ zero_z2753989067526334999ring_n ) )
= ( times_4166049284782705435ring_n @ X @ Y ) ) )
& ( ~ B
=> ( ( times_4166049284782705435ring_n @ X @ ( if_pol9129390727684501670ring_n @ B @ Y @ zero_z2753989067526334999ring_n ) )
= zero_z2753989067526334999ring_n ) ) ) ).
% mult_delta_right
thf(fact_721_mult__delta__right,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_722_mult__delta__right,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_723_mult__delta__right,axiom,
! [B: $o,X: poly_int,Y: poly_int] :
( ( B
=> ( ( times_times_poly_int @ X @ ( if_poly_int @ B @ Y @ zero_zero_poly_int ) )
= ( times_times_poly_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_poly_int @ X @ ( if_poly_int @ B @ Y @ zero_zero_poly_int ) )
= zero_zero_poly_int ) ) ) ).
% mult_delta_right
thf(fact_724_mult__delta__right,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_725_const__poly__dvd__1,axiom,
! [A: poly_real] :
( ( dvd_dv4532039564868358754y_real @ ( pCons_poly_real @ A @ zero_z5583686468110200389y_real ) @ one_on1191988272081909249y_real )
= ( dvd_dvd_poly_real @ A @ one_one_poly_real ) ) ).
% const_poly_dvd_1
thf(fact_726_const__poly__dvd__1,axiom,
! [A: poly_poly_int] :
( ( dvd_dv7705178354154678250ly_int @ ( pCons_poly_poly_int @ A @ zero_z240508265545053005ly_int ) @ one_on7423179019345326345ly_int )
= ( dvd_dv6998304861263046114ly_int @ A @ one_on1166514126663969025ly_int ) ) ).
% const_poly_dvd_1
thf(fact_727_const__poly__dvd__1,axiom,
! [A: poly_nat] :
( ( dvd_dv265015969339997062ly_nat @ ( pCons_poly_nat @ A @ zero_z3289306709065865449ly_nat ) @ one_on3656597271595695781ly_nat )
= ( dvd_dvd_poly_nat @ A @ one_one_poly_nat ) ) ).
% const_poly_dvd_1
thf(fact_728_const__poly__dvd__1,axiom,
! [A: poly_p6692042823160534382ring_n] :
( ( dvd_dv3919477662729673174ring_n @ ( pCons_2385395009258896524ring_n @ A @ zero_z3442457038203223091ring_n ) @ one_on281575345490252151ring_n )
= ( dvd_dv3135175980337127240ring_n @ A @ one_on5457780782968151273ring_n ) ) ).
% const_poly_dvd_1
thf(fact_729_const__poly__dvd__1,axiom,
! [A: finite_mod_ring_n] :
( ( dvd_dv8138414522854976442ring_n @ ( pCons_8126420873123957872ring_n @ A @ zero_z2753989067526334999ring_n ) @ one_on4318287115420659547ring_n )
= ( dvd_dv7258769396337835820ring_n @ A @ one_on2109788483843180749ring_n ) ) ).
% const_poly_dvd_1
thf(fact_730_const__poly__dvd__1,axiom,
! [A: int] :
( ( dvd_dvd_poly_int @ ( pCons_int @ A @ zero_zero_poly_int ) @ one_one_poly_int )
= ( dvd_dvd_int @ A @ one_one_int ) ) ).
% const_poly_dvd_1
thf(fact_731_const__poly__dvd__1,axiom,
! [A: real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ A @ zero_zero_poly_real ) @ one_one_poly_real )
= ( dvd_dvd_real @ A @ one_one_real ) ) ).
% const_poly_dvd_1
thf(fact_732_const__poly__dvd__1,axiom,
! [A: poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( pCons_poly_int @ A @ zero_z799223564134138693ly_int ) @ one_on1166514126663969025ly_int )
= ( dvd_dvd_poly_int @ A @ one_one_poly_int ) ) ).
% const_poly_dvd_1
thf(fact_733_const__poly__dvd__1,axiom,
! [A: nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ A @ zero_zero_poly_nat ) @ one_one_poly_nat )
= ( dvd_dvd_nat @ A @ one_one_nat ) ) ).
% const_poly_dvd_1
thf(fact_734_const__poly__dvd__1,axiom,
! [A: poly_F4222894760850802144ring_n] :
( ( dvd_dv3135175980337127240ring_n @ ( pCons_6246009715029582078ring_n @ A @ zero_z5482829069124612005ring_n ) @ one_on5457780782968151273ring_n )
= ( dvd_dv8138414522854976442ring_n @ A @ one_on4318287115420659547ring_n ) ) ).
% const_poly_dvd_1
thf(fact_735_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_p6692042823160534382ring_n,Y: poly_p6692042823160534382ring_n] :
( ( ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ X )
= ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_736_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( pCons_int @ zero_zero_int @ X )
= ( pCons_int @ zero_zero_int @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_737_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= ( pCons_nat @ zero_zero_nat @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_738_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_poly_int,Y: poly_poly_int] :
( ( ( pCons_poly_int @ zero_zero_poly_int @ X )
= ( pCons_poly_int @ zero_zero_poly_int @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_739_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_real,Y: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= ( pCons_real @ zero_zero_real @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_740_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_poly_real,Y: poly_poly_real] :
( ( ( pCons_poly_real @ zero_zero_poly_real @ X )
= ( pCons_poly_real @ zero_zero_poly_real @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_741_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_poly_poly_int,Y: poly_poly_poly_int] :
( ( ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ X )
= ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_742_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_poly_nat,Y: poly_poly_nat] :
( ( ( pCons_poly_nat @ zero_zero_poly_nat @ X )
= ( pCons_poly_nat @ zero_zero_poly_nat @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_743_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_p2743341848350813180ring_n,Y: poly_p2743341848350813180ring_n] :
( ( ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ X )
= ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_744_pCons__0__hom_Oeq__iff,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ X )
= ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ Y ) )
= ( X = Y ) ) ).
% pCons_0_hom.eq_iff
thf(fact_745_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c8149583629457385976ring_n @ N @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n ) ).
% poly_cutoff_0
thf(fact_746_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_int @ N @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% poly_cutoff_0
thf(fact_747_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_cutoff_0
thf(fact_748_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_poly_int @ N @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int ) ).
% poly_cutoff_0
thf(fact_749_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_cutoff_0
thf(fact_750_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c5946732737480674950ring_n @ N @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n ) ).
% poly_cutoff_0
thf(fact_751_pCons__eq__0__iff,axiom,
! [A: poly_real,P: poly_poly_real] :
( ( ( pCons_poly_real @ A @ P )
= zero_z5583686468110200389y_real )
= ( ( A = zero_zero_poly_real )
& ( P = zero_z5583686468110200389y_real ) ) ) ).
% pCons_eq_0_iff
thf(fact_752_pCons__eq__0__iff,axiom,
! [A: poly_poly_int,P: poly_poly_poly_int] :
( ( ( pCons_poly_poly_int @ A @ P )
= zero_z240508265545053005ly_int )
= ( ( A = zero_z799223564134138693ly_int )
& ( P = zero_z240508265545053005ly_int ) ) ) ).
% pCons_eq_0_iff
thf(fact_753_pCons__eq__0__iff,axiom,
! [A: poly_nat,P: poly_poly_nat] :
( ( ( pCons_poly_nat @ A @ P )
= zero_z3289306709065865449ly_nat )
= ( ( A = zero_zero_poly_nat )
& ( P = zero_z3289306709065865449ly_nat ) ) ) ).
% pCons_eq_0_iff
thf(fact_754_pCons__eq__0__iff,axiom,
! [A: poly_p6692042823160534382ring_n,P: poly_p2743341848350813180ring_n] :
( ( ( pCons_2385395009258896524ring_n @ A @ P )
= zero_z3442457038203223091ring_n )
= ( ( A = zero_z5482829069124612005ring_n )
& ( P = zero_z3442457038203223091ring_n ) ) ) ).
% pCons_eq_0_iff
thf(fact_755_pCons__eq__0__iff,axiom,
! [A: finite_mod_ring_n,P: poly_F4222894760850802144ring_n] :
( ( ( pCons_8126420873123957872ring_n @ A @ P )
= zero_z2753989067526334999ring_n )
= ( ( A = zero_z7902377597758090121ring_n )
& ( P = zero_z2753989067526334999ring_n ) ) ) ).
% pCons_eq_0_iff
thf(fact_756_pCons__eq__0__iff,axiom,
! [A: int,P: poly_int] :
( ( ( pCons_int @ A @ P )
= zero_zero_poly_int )
= ( ( A = zero_zero_int )
& ( P = zero_zero_poly_int ) ) ) ).
% pCons_eq_0_iff
thf(fact_757_pCons__eq__0__iff,axiom,
! [A: real,P: poly_real] :
( ( ( pCons_real @ A @ P )
= zero_zero_poly_real )
= ( ( A = zero_zero_real )
& ( P = zero_zero_poly_real ) ) ) ).
% pCons_eq_0_iff
thf(fact_758_pCons__eq__0__iff,axiom,
! [A: poly_int,P: poly_poly_int] :
( ( ( pCons_poly_int @ A @ P )
= zero_z799223564134138693ly_int )
= ( ( A = zero_zero_poly_int )
& ( P = zero_z799223564134138693ly_int ) ) ) ).
% pCons_eq_0_iff
thf(fact_759_pCons__eq__0__iff,axiom,
! [A: nat,P: poly_nat] :
( ( ( pCons_nat @ A @ P )
= zero_zero_poly_nat )
= ( ( A = zero_zero_nat )
& ( P = zero_zero_poly_nat ) ) ) ).
% pCons_eq_0_iff
thf(fact_760_pCons__eq__0__iff,axiom,
! [A: poly_F4222894760850802144ring_n,P: poly_p6692042823160534382ring_n] :
( ( ( pCons_6246009715029582078ring_n @ A @ P )
= zero_z5482829069124612005ring_n )
= ( ( A = zero_z2753989067526334999ring_n )
& ( P = zero_z5482829069124612005ring_n ) ) ) ).
% pCons_eq_0_iff
thf(fact_761_pCons__0__0,axiom,
( ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n ) ).
% pCons_0_0
thf(fact_762_pCons__0__0,axiom,
( ( pCons_int @ zero_zero_int @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% pCons_0_0
thf(fact_763_pCons__0__0,axiom,
( ( pCons_nat @ zero_zero_nat @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% pCons_0_0
thf(fact_764_pCons__0__0,axiom,
( ( pCons_poly_int @ zero_zero_poly_int @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int ) ).
% pCons_0_0
thf(fact_765_pCons__0__0,axiom,
( ( pCons_real @ zero_zero_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% pCons_0_0
thf(fact_766_pCons__0__0,axiom,
( ( pCons_poly_real @ zero_zero_poly_real @ zero_z5583686468110200389y_real )
= zero_z5583686468110200389y_real ) ).
% pCons_0_0
thf(fact_767_pCons__0__0,axiom,
( ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ zero_z240508265545053005ly_int )
= zero_z240508265545053005ly_int ) ).
% pCons_0_0
thf(fact_768_pCons__0__0,axiom,
( ( pCons_poly_nat @ zero_zero_poly_nat @ zero_z3289306709065865449ly_nat )
= zero_z3289306709065865449ly_nat ) ).
% pCons_0_0
thf(fact_769_pCons__0__0,axiom,
( ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ zero_z3442457038203223091ring_n )
= zero_z3442457038203223091ring_n ) ).
% pCons_0_0
thf(fact_770_pCons__0__0,axiom,
( ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n ) ).
% pCons_0_0
thf(fact_771_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ X )
= zero_z5482829069124612005ring_n )
= ( X = zero_z5482829069124612005ring_n ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_772_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_int] :
( ( ( pCons_int @ zero_zero_int @ X )
= zero_zero_poly_int )
= ( X = zero_zero_poly_int ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_773_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= zero_zero_poly_nat )
= ( X = zero_zero_poly_nat ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_774_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_poly_int] :
( ( ( pCons_poly_int @ zero_zero_poly_int @ X )
= zero_z799223564134138693ly_int )
= ( X = zero_z799223564134138693ly_int ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_775_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= zero_zero_poly_real )
= ( X = zero_zero_poly_real ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_776_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_poly_real] :
( ( ( pCons_poly_real @ zero_zero_poly_real @ X )
= zero_z5583686468110200389y_real )
= ( X = zero_z5583686468110200389y_real ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_777_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_poly_poly_int] :
( ( ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ X )
= zero_z240508265545053005ly_int )
= ( X = zero_z240508265545053005ly_int ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_778_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_poly_nat] :
( ( ( pCons_poly_nat @ zero_zero_poly_nat @ X )
= zero_z3289306709065865449ly_nat )
= ( X = zero_z3289306709065865449ly_nat ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_779_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_p2743341848350813180ring_n] :
( ( ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ X )
= zero_z3442457038203223091ring_n )
= ( X = zero_z3442457038203223091ring_n ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_780_pCons__0__hom_Ohom__0__iff,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ X )
= zero_z2753989067526334999ring_n )
= ( X = zero_z2753989067526334999ring_n ) ) ).
% pCons_0_hom.hom_0_iff
thf(fact_781_Missing__Polynomial_Omap__poly__pCons,axiom,
! [C2: int,P: poly_int,F: int > finite_mod_ring_n] :
( ( ( C2 != zero_zero_int )
| ( P != zero_zero_poly_int ) )
=> ( ( map_po1011533443592629756ring_n @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_8126420873123957872ring_n @ ( F @ C2 ) @ ( map_po1011533443592629756ring_n @ F @ P ) ) ) ) ).
% Missing_Polynomial.map_poly_pCons
thf(fact_782_Missing__Polynomial_Omap__poly__pCons,axiom,
! [C2: int,P: poly_int,F: int > poly_F4222894760850802144ring_n] :
( ( ( C2 != zero_zero_int )
| ( P != zero_zero_poly_int ) )
=> ( ( map_po7854272679927642762ring_n @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_6246009715029582078ring_n @ ( F @ C2 ) @ ( map_po7854272679927642762ring_n @ F @ P ) ) ) ) ).
% Missing_Polynomial.map_poly_pCons
thf(fact_783_Missing__Polynomial_Omap__poly__pCons,axiom,
! [C2: int,P: poly_int,F: int > poly_int] :
( ( ( C2 != zero_zero_int )
| ( P != zero_zero_poly_int ) )
=> ( ( map_po8616709625927008010ly_int @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_poly_int @ ( F @ C2 ) @ ( map_po8616709625927008010ly_int @ F @ P ) ) ) ) ).
% Missing_Polynomial.map_poly_pCons
thf(fact_784_Missing__Polynomial_Omap__poly__pCons,axiom,
! [C2: int,P: poly_int,F: int > real] :
( ( ( C2 != zero_zero_int )
| ( P != zero_zero_poly_int ) )
=> ( ( map_poly_int_real @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_real @ ( F @ C2 ) @ ( map_poly_int_real @ F @ P ) ) ) ) ).
% Missing_Polynomial.map_poly_pCons
thf(fact_785_Missing__Polynomial_Omap__poly__pCons,axiom,
! [C2: int,P: poly_int,F: int > nat] :
( ( ( C2 != zero_zero_int )
| ( P != zero_zero_poly_int ) )
=> ( ( map_poly_int_nat @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_nat @ ( F @ C2 ) @ ( map_poly_int_nat @ F @ P ) ) ) ) ).
% Missing_Polynomial.map_poly_pCons
thf(fact_786_Missing__Polynomial_Omap__poly__pCons,axiom,
! [C2: int,P: poly_int,F: int > int] :
( ( ( C2 != zero_zero_int )
| ( P != zero_zero_poly_int ) )
=> ( ( map_poly_int_int @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_int @ ( F @ C2 ) @ ( map_poly_int_int @ F @ P ) ) ) ) ).
% Missing_Polynomial.map_poly_pCons
thf(fact_787_pCons__one,axiom,
( ( pCons_poly_real @ one_one_poly_real @ zero_z5583686468110200389y_real )
= one_on1191988272081909249y_real ) ).
% pCons_one
thf(fact_788_pCons__one,axiom,
( ( pCons_poly_poly_int @ one_on1166514126663969025ly_int @ zero_z240508265545053005ly_int )
= one_on7423179019345326345ly_int ) ).
% pCons_one
thf(fact_789_pCons__one,axiom,
( ( pCons_poly_nat @ one_one_poly_nat @ zero_z3289306709065865449ly_nat )
= one_on3656597271595695781ly_nat ) ).
% pCons_one
thf(fact_790_pCons__one,axiom,
( ( pCons_2385395009258896524ring_n @ one_on5457780782968151273ring_n @ zero_z3442457038203223091ring_n )
= one_on281575345490252151ring_n ) ).
% pCons_one
thf(fact_791_pCons__one,axiom,
( ( pCons_8126420873123957872ring_n @ one_on2109788483843180749ring_n @ zero_z2753989067526334999ring_n )
= one_on4318287115420659547ring_n ) ).
% pCons_one
thf(fact_792_pCons__one,axiom,
( ( pCons_int @ one_one_int @ zero_zero_poly_int )
= one_one_poly_int ) ).
% pCons_one
thf(fact_793_pCons__one,axiom,
( ( pCons_real @ one_one_real @ zero_zero_poly_real )
= one_one_poly_real ) ).
% pCons_one
thf(fact_794_pCons__one,axiom,
( ( pCons_poly_int @ one_one_poly_int @ zero_z799223564134138693ly_int )
= one_on1166514126663969025ly_int ) ).
% pCons_one
thf(fact_795_pCons__one,axiom,
( ( pCons_nat @ one_one_nat @ zero_zero_poly_nat )
= one_one_poly_nat ) ).
% pCons_one
thf(fact_796_pCons__one,axiom,
( ( pCons_6246009715029582078ring_n @ one_on4318287115420659547ring_n @ zero_z5482829069124612005ring_n )
= one_on5457780782968151273ring_n ) ).
% pCons_one
thf(fact_797_one__poly__eq__simps_I1_J,axiom,
( one_on1191988272081909249y_real
= ( pCons_poly_real @ one_one_poly_real @ zero_z5583686468110200389y_real ) ) ).
% one_poly_eq_simps(1)
thf(fact_798_one__poly__eq__simps_I1_J,axiom,
( one_on7423179019345326345ly_int
= ( pCons_poly_poly_int @ one_on1166514126663969025ly_int @ zero_z240508265545053005ly_int ) ) ).
% one_poly_eq_simps(1)
thf(fact_799_one__poly__eq__simps_I1_J,axiom,
( one_on3656597271595695781ly_nat
= ( pCons_poly_nat @ one_one_poly_nat @ zero_z3289306709065865449ly_nat ) ) ).
% one_poly_eq_simps(1)
thf(fact_800_one__poly__eq__simps_I1_J,axiom,
( one_on281575345490252151ring_n
= ( pCons_2385395009258896524ring_n @ one_on5457780782968151273ring_n @ zero_z3442457038203223091ring_n ) ) ).
% one_poly_eq_simps(1)
thf(fact_801_one__poly__eq__simps_I1_J,axiom,
( one_on4318287115420659547ring_n
= ( pCons_8126420873123957872ring_n @ one_on2109788483843180749ring_n @ zero_z2753989067526334999ring_n ) ) ).
% one_poly_eq_simps(1)
thf(fact_802_one__poly__eq__simps_I1_J,axiom,
( one_one_poly_int
= ( pCons_int @ one_one_int @ zero_zero_poly_int ) ) ).
% one_poly_eq_simps(1)
thf(fact_803_one__poly__eq__simps_I1_J,axiom,
( one_one_poly_real
= ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) ).
% one_poly_eq_simps(1)
thf(fact_804_one__poly__eq__simps_I1_J,axiom,
( one_on1166514126663969025ly_int
= ( pCons_poly_int @ one_one_poly_int @ zero_z799223564134138693ly_int ) ) ).
% one_poly_eq_simps(1)
thf(fact_805_one__poly__eq__simps_I1_J,axiom,
( one_one_poly_nat
= ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) ).
% one_poly_eq_simps(1)
thf(fact_806_one__poly__eq__simps_I1_J,axiom,
( one_on5457780782968151273ring_n
= ( pCons_6246009715029582078ring_n @ one_on4318287115420659547ring_n @ zero_z5482829069124612005ring_n ) ) ).
% one_poly_eq_simps(1)
thf(fact_807_one__poly__eq__simps_I2_J,axiom,
( ( pCons_poly_real @ one_one_poly_real @ zero_z5583686468110200389y_real )
= one_on1191988272081909249y_real ) ).
% one_poly_eq_simps(2)
thf(fact_808_one__poly__eq__simps_I2_J,axiom,
( ( pCons_poly_poly_int @ one_on1166514126663969025ly_int @ zero_z240508265545053005ly_int )
= one_on7423179019345326345ly_int ) ).
% one_poly_eq_simps(2)
thf(fact_809_one__poly__eq__simps_I2_J,axiom,
( ( pCons_poly_nat @ one_one_poly_nat @ zero_z3289306709065865449ly_nat )
= one_on3656597271595695781ly_nat ) ).
% one_poly_eq_simps(2)
thf(fact_810_one__poly__eq__simps_I2_J,axiom,
( ( pCons_2385395009258896524ring_n @ one_on5457780782968151273ring_n @ zero_z3442457038203223091ring_n )
= one_on281575345490252151ring_n ) ).
% one_poly_eq_simps(2)
thf(fact_811_one__poly__eq__simps_I2_J,axiom,
( ( pCons_8126420873123957872ring_n @ one_on2109788483843180749ring_n @ zero_z2753989067526334999ring_n )
= one_on4318287115420659547ring_n ) ).
% one_poly_eq_simps(2)
thf(fact_812_one__poly__eq__simps_I2_J,axiom,
( ( pCons_int @ one_one_int @ zero_zero_poly_int )
= one_one_poly_int ) ).
% one_poly_eq_simps(2)
thf(fact_813_one__poly__eq__simps_I2_J,axiom,
( ( pCons_real @ one_one_real @ zero_zero_poly_real )
= one_one_poly_real ) ).
% one_poly_eq_simps(2)
thf(fact_814_one__poly__eq__simps_I2_J,axiom,
( ( pCons_poly_int @ one_one_poly_int @ zero_z799223564134138693ly_int )
= one_on1166514126663969025ly_int ) ).
% one_poly_eq_simps(2)
thf(fact_815_one__poly__eq__simps_I2_J,axiom,
( ( pCons_nat @ one_one_nat @ zero_zero_poly_nat )
= one_one_poly_nat ) ).
% one_poly_eq_simps(2)
thf(fact_816_one__poly__eq__simps_I2_J,axiom,
( ( pCons_6246009715029582078ring_n @ one_on4318287115420659547ring_n @ zero_z5482829069124612005ring_n )
= one_on5457780782968151273ring_n ) ).
% one_poly_eq_simps(2)
thf(fact_817_const__poly__dvd__const__poly__iff,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dv4532039564868358754y_real @ ( pCons_poly_real @ A @ zero_z5583686468110200389y_real ) @ ( pCons_poly_real @ B @ zero_z5583686468110200389y_real ) )
= ( dvd_dvd_poly_real @ A @ B ) ) ).
% const_poly_dvd_const_poly_iff
thf(fact_818_const__poly__dvd__const__poly__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_poly_int @ ( pCons_int @ A @ zero_zero_poly_int ) @ ( pCons_int @ B @ zero_zero_poly_int ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% const_poly_dvd_const_poly_iff
thf(fact_819_const__poly__dvd__const__poly__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ A @ zero_zero_poly_real ) @ ( pCons_real @ B @ zero_zero_poly_real ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% const_poly_dvd_const_poly_iff
thf(fact_820_const__poly__dvd__const__poly__iff,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( pCons_poly_int @ A @ zero_z799223564134138693ly_int ) @ ( pCons_poly_int @ B @ zero_z799223564134138693ly_int ) )
= ( dvd_dvd_poly_int @ A @ B ) ) ).
% const_poly_dvd_const_poly_iff
thf(fact_821_const__poly__dvd__const__poly__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ A @ zero_zero_poly_nat ) @ ( pCons_nat @ B @ zero_zero_poly_nat ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% const_poly_dvd_const_poly_iff
thf(fact_822_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_823_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_824_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_825_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_826_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_827_of__int__poly__hom_Obase_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ ( pCons_int @ A @ P ) )
= ( pCons_6246009715029582078ring_n @ ( ring_18712857867054464081ring_n @ A ) @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ P ) ) ) ).
% of_int_poly_hom.base.map_poly_pCons_hom
thf(fact_828_of__int__poly__hom_Obase_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ ( pCons_int @ A @ P ) )
= ( pCons_poly_int @ ( ring_17892525584911698563ly_int @ A ) @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ P ) ) ) ).
% of_int_poly_hom.base.map_poly_pCons_hom
thf(fact_829_of__int__poly__hom_Obase_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ ( pCons_int @ A @ P ) )
= ( pCons_8126420873123957872ring_n @ ( ring_18169885536585341379ring_n @ A ) @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ P ) ) ) ).
% of_int_poly_hom.base.map_poly_pCons_hom
thf(fact_830_of__int__poly__hom_Obase_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_poly_int_real @ ring_1_of_int_real @ ( pCons_int @ A @ P ) )
= ( pCons_real @ ( ring_1_of_int_real @ A ) @ ( map_poly_int_real @ ring_1_of_int_real @ P ) ) ) ).
% of_int_poly_hom.base.map_poly_pCons_hom
thf(fact_831_of__int__poly__hom_Obase_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_poly_int_int @ ring_1_of_int_int @ ( pCons_int @ A @ P ) )
= ( pCons_int @ ( ring_1_of_int_int @ A ) @ ( map_poly_int_int @ ring_1_of_int_int @ P ) ) ) ).
% of_int_poly_hom.base.map_poly_pCons_hom
thf(fact_832_of__int__hom_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ ( pCons_int @ A @ P ) )
= ( pCons_6246009715029582078ring_n @ ( ring_18712857867054464081ring_n @ A ) @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ P ) ) ) ).
% of_int_hom.map_poly_pCons_hom
thf(fact_833_of__int__hom_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ ( pCons_int @ A @ P ) )
= ( pCons_poly_int @ ( ring_17892525584911698563ly_int @ A ) @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ P ) ) ) ).
% of_int_hom.map_poly_pCons_hom
thf(fact_834_of__int__hom_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ ( pCons_int @ A @ P ) )
= ( pCons_8126420873123957872ring_n @ ( ring_18169885536585341379ring_n @ A ) @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ P ) ) ) ).
% of_int_hom.map_poly_pCons_hom
thf(fact_835_of__int__hom_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_poly_int_real @ ring_1_of_int_real @ ( pCons_int @ A @ P ) )
= ( pCons_real @ ( ring_1_of_int_real @ A ) @ ( map_poly_int_real @ ring_1_of_int_real @ P ) ) ) ).
% of_int_hom.map_poly_pCons_hom
thf(fact_836_of__int__hom_Omap__poly__pCons__hom,axiom,
! [A: int,P: poly_int] :
( ( map_poly_int_int @ ring_1_of_int_int @ ( pCons_int @ A @ P ) )
= ( pCons_int @ ( ring_1_of_int_int @ A ) @ ( map_poly_int_int @ ring_1_of_int_int @ P ) ) ) ).
% of_int_hom.map_poly_pCons_hom
thf(fact_837_of__int__poly__hom_Omap__poly__pCons__hom,axiom,
! [A: poly_int,P: poly_poly_int] :
( ( map_po2544297806126724112ring_n @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n ) @ ( pCons_poly_int @ A @ P ) )
= ( pCons_2385395009258896524ring_n @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n @ A ) @ ( map_po2544297806126724112ring_n @ ( map_po7854272679927642762ring_n @ ring_18712857867054464081ring_n ) @ P ) ) ) ).
% of_int_poly_hom.map_poly_pCons_hom
thf(fact_838_of__int__poly__hom_Omap__poly__pCons__hom,axiom,
! [A: poly_int,P: poly_poly_int] :
( ( map_po7381751157747918618ly_int @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int ) @ ( pCons_poly_int @ A @ P ) )
= ( pCons_poly_poly_int @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int @ A ) @ ( map_po7381751157747918618ly_int @ ( map_po8616709625927008010ly_int @ ring_17892525584911698563ly_int ) @ P ) ) ) ).
% of_int_poly_hom.map_poly_pCons_hom
thf(fact_839_of__int__poly__hom_Omap__poly__pCons__hom,axiom,
! [A: poly_int,P: poly_poly_int] :
( ( map_po1386818991929012354ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n ) @ ( pCons_poly_int @ A @ P ) )
= ( pCons_6246009715029582078ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ A ) @ ( map_po1386818991929012354ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n ) @ P ) ) ) ).
% of_int_poly_hom.map_poly_pCons_hom
thf(fact_840_of__int__poly__hom_Omap__poly__pCons__hom,axiom,
! [A: poly_int,P: poly_poly_int] :
( ( map_po9185419567230169618y_real @ ( map_poly_int_real @ ring_1_of_int_real ) @ ( pCons_poly_int @ A @ P ) )
= ( pCons_poly_real @ ( map_poly_int_real @ ring_1_of_int_real @ A ) @ ( map_po9185419567230169618y_real @ ( map_poly_int_real @ ring_1_of_int_real ) @ P ) ) ) ).
% of_int_poly_hom.map_poly_pCons_hom
thf(fact_841_of__int__poly__hom_Omap__poly__pCons__hom,axiom,
! [A: poly_int,P: poly_poly_int] :
( ( map_po136720853902459666ly_int @ ( map_poly_int_int @ ring_1_of_int_int ) @ ( pCons_poly_int @ A @ P ) )
= ( pCons_poly_int @ ( map_poly_int_int @ ring_1_of_int_int @ A ) @ ( map_po136720853902459666ly_int @ ( map_poly_int_int @ ring_1_of_int_int ) @ P ) ) ) ).
% of_int_poly_hom.map_poly_pCons_hom
thf(fact_842_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_p6692042823160534382ring_n,Y: poly_p6692042823160534382ring_n] :
( ( ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ X )
= ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_843_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( pCons_int @ zero_zero_int @ X )
= ( pCons_int @ zero_zero_int @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_844_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= ( pCons_nat @ zero_zero_nat @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_845_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_poly_int,Y: poly_poly_int] :
( ( ( pCons_poly_int @ zero_zero_poly_int @ X )
= ( pCons_poly_int @ zero_zero_poly_int @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_846_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_real,Y: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= ( pCons_real @ zero_zero_real @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_847_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_poly_real,Y: poly_poly_real] :
( ( ( pCons_poly_real @ zero_zero_poly_real @ X )
= ( pCons_poly_real @ zero_zero_poly_real @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_848_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_poly_poly_int,Y: poly_poly_poly_int] :
( ( ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ X )
= ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_849_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_poly_nat,Y: poly_poly_nat] :
( ( ( pCons_poly_nat @ zero_zero_poly_nat @ X )
= ( pCons_poly_nat @ zero_zero_poly_nat @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_850_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_p2743341848350813180ring_n,Y: poly_p2743341848350813180ring_n] :
( ( ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ X )
= ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_851_pCons__0__hom_Oinjectivity,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ X )
= ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ Y ) )
=> ( X = Y ) ) ).
% pCons_0_hom.injectivity
thf(fact_852_poly__induct2,axiom,
! [P2: poly_int > poly_int > $o,P: poly_int,Q: poly_int] :
( ( P2 @ zero_zero_poly_int @ zero_zero_poly_int )
=> ( ! [A3: int,P3: poly_int,B4: int,Q2: poly_int] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_int @ A3 @ P3 ) @ ( pCons_int @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_853_poly__induct2,axiom,
! [P2: poly_int > poly_real > $o,P: poly_int,Q: poly_real] :
( ( P2 @ zero_zero_poly_int @ zero_zero_poly_real )
=> ( ! [A3: int,P3: poly_int,B4: real,Q2: poly_real] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_int @ A3 @ P3 ) @ ( pCons_real @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_854_poly__induct2,axiom,
! [P2: poly_int > poly_nat > $o,P: poly_int,Q: poly_nat] :
( ( P2 @ zero_zero_poly_int @ zero_zero_poly_nat )
=> ( ! [A3: int,P3: poly_int,B4: nat,Q2: poly_nat] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_int @ A3 @ P3 ) @ ( pCons_nat @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_855_poly__induct2,axiom,
! [P2: poly_real > poly_int > $o,P: poly_real,Q: poly_int] :
( ( P2 @ zero_zero_poly_real @ zero_zero_poly_int )
=> ( ! [A3: real,P3: poly_real,B4: int,Q2: poly_int] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_real @ A3 @ P3 ) @ ( pCons_int @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_856_poly__induct2,axiom,
! [P2: poly_real > poly_real > $o,P: poly_real,Q: poly_real] :
( ( P2 @ zero_zero_poly_real @ zero_zero_poly_real )
=> ( ! [A3: real,P3: poly_real,B4: real,Q2: poly_real] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_real @ A3 @ P3 ) @ ( pCons_real @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_857_poly__induct2,axiom,
! [P2: poly_real > poly_nat > $o,P: poly_real,Q: poly_nat] :
( ( P2 @ zero_zero_poly_real @ zero_zero_poly_nat )
=> ( ! [A3: real,P3: poly_real,B4: nat,Q2: poly_nat] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_real @ A3 @ P3 ) @ ( pCons_nat @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_858_poly__induct2,axiom,
! [P2: poly_nat > poly_int > $o,P: poly_nat,Q: poly_int] :
( ( P2 @ zero_zero_poly_nat @ zero_zero_poly_int )
=> ( ! [A3: nat,P3: poly_nat,B4: int,Q2: poly_int] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_nat @ A3 @ P3 ) @ ( pCons_int @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_859_poly__induct2,axiom,
! [P2: poly_nat > poly_real > $o,P: poly_nat,Q: poly_real] :
( ( P2 @ zero_zero_poly_nat @ zero_zero_poly_real )
=> ( ! [A3: nat,P3: poly_nat,B4: real,Q2: poly_real] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_nat @ A3 @ P3 ) @ ( pCons_real @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_860_poly__induct2,axiom,
! [P2: poly_nat > poly_nat > $o,P: poly_nat,Q: poly_nat] :
( ( P2 @ zero_zero_poly_nat @ zero_zero_poly_nat )
=> ( ! [A3: nat,P3: poly_nat,B4: nat,Q2: poly_nat] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_nat @ A3 @ P3 ) @ ( pCons_nat @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_861_poly__induct2,axiom,
! [P2: poly_F4222894760850802144ring_n > poly_int > $o,P: poly_F4222894760850802144ring_n,Q: poly_int] :
( ( P2 @ zero_z2753989067526334999ring_n @ zero_zero_poly_int )
=> ( ! [A3: finite_mod_ring_n,P3: poly_F4222894760850802144ring_n,B4: int,Q2: poly_int] :
( ( P2 @ P3 @ Q2 )
=> ( P2 @ ( pCons_8126420873123957872ring_n @ A3 @ P3 ) @ ( pCons_int @ B4 @ Q2 ) ) )
=> ( P2 @ P @ Q ) ) ) ).
% poly_induct2
thf(fact_862_pCons__induct,axiom,
! [P2: poly_poly_real > $o,P: poly_poly_real] :
( ( P2 @ zero_z5583686468110200389y_real )
=> ( ! [A3: poly_real,P3: poly_poly_real] :
( ( ( A3 != zero_zero_poly_real )
| ( P3 != zero_z5583686468110200389y_real ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_poly_real @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_863_pCons__induct,axiom,
! [P2: poly_poly_poly_int > $o,P: poly_poly_poly_int] :
( ( P2 @ zero_z240508265545053005ly_int )
=> ( ! [A3: poly_poly_int,P3: poly_poly_poly_int] :
( ( ( A3 != zero_z799223564134138693ly_int )
| ( P3 != zero_z240508265545053005ly_int ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_poly_poly_int @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_864_pCons__induct,axiom,
! [P2: poly_poly_nat > $o,P: poly_poly_nat] :
( ( P2 @ zero_z3289306709065865449ly_nat )
=> ( ! [A3: poly_nat,P3: poly_poly_nat] :
( ( ( A3 != zero_zero_poly_nat )
| ( P3 != zero_z3289306709065865449ly_nat ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_poly_nat @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_865_pCons__induct,axiom,
! [P2: poly_p2743341848350813180ring_n > $o,P: poly_p2743341848350813180ring_n] :
( ( P2 @ zero_z3442457038203223091ring_n )
=> ( ! [A3: poly_p6692042823160534382ring_n,P3: poly_p2743341848350813180ring_n] :
( ( ( A3 != zero_z5482829069124612005ring_n )
| ( P3 != zero_z3442457038203223091ring_n ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_2385395009258896524ring_n @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_866_pCons__induct,axiom,
! [P2: poly_F4222894760850802144ring_n > $o,P: poly_F4222894760850802144ring_n] :
( ( P2 @ zero_z2753989067526334999ring_n )
=> ( ! [A3: finite_mod_ring_n,P3: poly_F4222894760850802144ring_n] :
( ( ( A3 != zero_z7902377597758090121ring_n )
| ( P3 != zero_z2753989067526334999ring_n ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_8126420873123957872ring_n @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_867_pCons__induct,axiom,
! [P2: poly_int > $o,P: poly_int] :
( ( P2 @ zero_zero_poly_int )
=> ( ! [A3: int,P3: poly_int] :
( ( ( A3 != zero_zero_int )
| ( P3 != zero_zero_poly_int ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_int @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_868_pCons__induct,axiom,
! [P2: poly_real > $o,P: poly_real] :
( ( P2 @ zero_zero_poly_real )
=> ( ! [A3: real,P3: poly_real] :
( ( ( A3 != zero_zero_real )
| ( P3 != zero_zero_poly_real ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_real @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_869_pCons__induct,axiom,
! [P2: poly_poly_int > $o,P: poly_poly_int] :
( ( P2 @ zero_z799223564134138693ly_int )
=> ( ! [A3: poly_int,P3: poly_poly_int] :
( ( ( A3 != zero_zero_poly_int )
| ( P3 != zero_z799223564134138693ly_int ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_poly_int @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_870_pCons__induct,axiom,
! [P2: poly_nat > $o,P: poly_nat] :
( ( P2 @ zero_zero_poly_nat )
=> ( ! [A3: nat,P3: poly_nat] :
( ( ( A3 != zero_zero_nat )
| ( P3 != zero_zero_poly_nat ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_nat @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_871_pCons__induct,axiom,
! [P2: poly_p6692042823160534382ring_n > $o,P: poly_p6692042823160534382ring_n] :
( ( P2 @ zero_z5482829069124612005ring_n )
=> ( ! [A3: poly_F4222894760850802144ring_n,P3: poly_p6692042823160534382ring_n] :
( ( ( A3 != zero_z2753989067526334999ring_n )
| ( P3 != zero_z5482829069124612005ring_n ) )
=> ( ( P2 @ P3 )
=> ( P2 @ ( pCons_6246009715029582078ring_n @ A3 @ P3 ) ) ) )
=> ( P2 @ P ) ) ) ).
% pCons_induct
thf(fact_872_pCons__0__hom_Ohom__0,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ X )
= zero_z5482829069124612005ring_n )
=> ( X = zero_z5482829069124612005ring_n ) ) ).
% pCons_0_hom.hom_0
thf(fact_873_pCons__0__hom_Ohom__0,axiom,
! [X: poly_int] :
( ( ( pCons_int @ zero_zero_int @ X )
= zero_zero_poly_int )
=> ( X = zero_zero_poly_int ) ) ).
% pCons_0_hom.hom_0
thf(fact_874_pCons__0__hom_Ohom__0,axiom,
! [X: poly_nat] :
( ( ( pCons_nat @ zero_zero_nat @ X )
= zero_zero_poly_nat )
=> ( X = zero_zero_poly_nat ) ) ).
% pCons_0_hom.hom_0
thf(fact_875_pCons__0__hom_Ohom__0,axiom,
! [X: poly_poly_int] :
( ( ( pCons_poly_int @ zero_zero_poly_int @ X )
= zero_z799223564134138693ly_int )
=> ( X = zero_z799223564134138693ly_int ) ) ).
% pCons_0_hom.hom_0
thf(fact_876_pCons__0__hom_Ohom__0,axiom,
! [X: poly_real] :
( ( ( pCons_real @ zero_zero_real @ X )
= zero_zero_poly_real )
=> ( X = zero_zero_poly_real ) ) ).
% pCons_0_hom.hom_0
thf(fact_877_pCons__0__hom_Ohom__0,axiom,
! [X: poly_poly_real] :
( ( ( pCons_poly_real @ zero_zero_poly_real @ X )
= zero_z5583686468110200389y_real )
=> ( X = zero_z5583686468110200389y_real ) ) ).
% pCons_0_hom.hom_0
thf(fact_878_pCons__0__hom_Ohom__0,axiom,
! [X: poly_poly_poly_int] :
( ( ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ X )
= zero_z240508265545053005ly_int )
=> ( X = zero_z240508265545053005ly_int ) ) ).
% pCons_0_hom.hom_0
thf(fact_879_pCons__0__hom_Ohom__0,axiom,
! [X: poly_poly_nat] :
( ( ( pCons_poly_nat @ zero_zero_poly_nat @ X )
= zero_z3289306709065865449ly_nat )
=> ( X = zero_z3289306709065865449ly_nat ) ) ).
% pCons_0_hom.hom_0
thf(fact_880_pCons__0__hom_Ohom__0,axiom,
! [X: poly_p2743341848350813180ring_n] :
( ( ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ X )
= zero_z3442457038203223091ring_n )
=> ( X = zero_z3442457038203223091ring_n ) ) ).
% pCons_0_hom.hom_0
thf(fact_881_pCons__0__hom_Ohom__0,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ X )
= zero_z2753989067526334999ring_n )
=> ( X = zero_z2753989067526334999ring_n ) ) ).
% pCons_0_hom.hom_0
thf(fact_882_Polynomial_Omap__poly__pCons,axiom,
! [F: int > int,C2: int,P: poly_int] :
( ( ( F @ zero_zero_int )
= zero_zero_int )
=> ( ( map_poly_int_int @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_int @ ( F @ C2 ) @ ( map_poly_int_int @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_883_Polynomial_Omap__poly__pCons,axiom,
! [F: int > nat,C2: int,P: poly_int] :
( ( ( F @ zero_zero_int )
= zero_zero_nat )
=> ( ( map_poly_int_nat @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_nat @ ( F @ C2 ) @ ( map_poly_int_nat @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_884_Polynomial_Omap__poly__pCons,axiom,
! [F: int > real,C2: int,P: poly_int] :
( ( ( F @ zero_zero_int )
= zero_zero_real )
=> ( ( map_poly_int_real @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_real @ ( F @ C2 ) @ ( map_poly_int_real @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_885_Polynomial_Omap__poly__pCons,axiom,
! [F: nat > int,C2: nat,P: poly_nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( map_poly_nat_int @ F @ ( pCons_nat @ C2 @ P ) )
= ( pCons_int @ ( F @ C2 ) @ ( map_poly_nat_int @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_886_Polynomial_Omap__poly__pCons,axiom,
! [F: nat > nat,C2: nat,P: poly_nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( map_poly_nat_nat @ F @ ( pCons_nat @ C2 @ P ) )
= ( pCons_nat @ ( F @ C2 ) @ ( map_poly_nat_nat @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_887_Polynomial_Omap__poly__pCons,axiom,
! [F: nat > real,C2: nat,P: poly_nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( map_poly_nat_real @ F @ ( pCons_nat @ C2 @ P ) )
= ( pCons_real @ ( F @ C2 ) @ ( map_poly_nat_real @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_888_Polynomial_Omap__poly__pCons,axiom,
! [F: real > int,C2: real,P: poly_real] :
( ( ( F @ zero_zero_real )
= zero_zero_int )
=> ( ( map_poly_real_int @ F @ ( pCons_real @ C2 @ P ) )
= ( pCons_int @ ( F @ C2 ) @ ( map_poly_real_int @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_889_Polynomial_Omap__poly__pCons,axiom,
! [F: real > nat,C2: real,P: poly_real] :
( ( ( F @ zero_zero_real )
= zero_zero_nat )
=> ( ( map_poly_real_nat @ F @ ( pCons_real @ C2 @ P ) )
= ( pCons_nat @ ( F @ C2 ) @ ( map_poly_real_nat @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_890_Polynomial_Omap__poly__pCons,axiom,
! [F: real > real,C2: real,P: poly_real] :
( ( ( F @ zero_zero_real )
= zero_zero_real )
=> ( ( map_poly_real_real @ F @ ( pCons_real @ C2 @ P ) )
= ( pCons_real @ ( F @ C2 ) @ ( map_poly_real_real @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_891_Polynomial_Omap__poly__pCons,axiom,
! [F: int > poly_int,C2: int,P: poly_int] :
( ( ( F @ zero_zero_int )
= zero_zero_poly_int )
=> ( ( map_po8616709625927008010ly_int @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_poly_int @ ( F @ C2 ) @ ( map_po8616709625927008010ly_int @ F @ P ) ) ) ) ).
% Polynomial.map_poly_pCons
thf(fact_892_of__int__poly,axiom,
( ring_14695796289142966411ly_int
= ( ^ [K: int] : ( pCons_poly_int @ ( ring_17892525584911698563ly_int @ K ) @ zero_z799223564134138693ly_int ) ) ) ).
% of_int_poly
thf(fact_893_of__int__poly,axiom,
( ring_14208964510912816607ring_n
= ( ^ [K: int] : ( pCons_6246009715029582078ring_n @ ( ring_18712857867054464081ring_n @ K ) @ zero_z5482829069124612005ring_n ) ) ) ).
% of_int_poly
thf(fact_894_of__int__poly,axiom,
( ring_18712857867054464081ring_n
= ( ^ [K: int] : ( pCons_8126420873123957872ring_n @ ( ring_18169885536585341379ring_n @ K ) @ zero_z2753989067526334999ring_n ) ) ) ).
% of_int_poly
thf(fact_895_of__int__poly,axiom,
( ring_12936506555246842115y_real
= ( ^ [K: int] : ( pCons_real @ ( ring_1_of_int_real @ K ) @ zero_zero_poly_real ) ) ) ).
% of_int_poly
thf(fact_896_of__int__poly,axiom,
( ring_17892525584911698563ly_int
= ( ^ [K: int] : ( pCons_int @ ( ring_1_of_int_int @ K ) @ zero_zero_poly_int ) ) ) ).
% of_int_poly
thf(fact_897_map__poly__simps,axiom,
! [C2: int,P: poly_int,F: int > int] :
( ( ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_poly_int_int @ F @ ( pCons_int @ C2 @ P ) )
= zero_zero_poly_int ) )
& ( ~ ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_poly_int_int @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_int @ ( F @ C2 ) @ ( map_poly_int_int @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_898_map__poly__simps,axiom,
! [C2: int,P: poly_int,F: int > real] :
( ( ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_poly_int_real @ F @ ( pCons_int @ C2 @ P ) )
= zero_zero_poly_real ) )
& ( ~ ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_poly_int_real @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_real @ ( F @ C2 ) @ ( map_poly_int_real @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_899_map__poly__simps,axiom,
! [C2: int,P: poly_int,F: int > nat] :
( ( ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_poly_int_nat @ F @ ( pCons_int @ C2 @ P ) )
= zero_zero_poly_nat ) )
& ( ~ ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_poly_int_nat @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_nat @ ( F @ C2 ) @ ( map_poly_int_nat @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_900_map__poly__simps,axiom,
! [C2: nat,P: poly_nat,F: nat > int] :
( ( ( ( C2 = zero_zero_nat )
& ( P = zero_zero_poly_nat ) )
=> ( ( map_poly_nat_int @ F @ ( pCons_nat @ C2 @ P ) )
= zero_zero_poly_int ) )
& ( ~ ( ( C2 = zero_zero_nat )
& ( P = zero_zero_poly_nat ) )
=> ( ( map_poly_nat_int @ F @ ( pCons_nat @ C2 @ P ) )
= ( pCons_int @ ( F @ C2 ) @ ( map_poly_nat_int @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_901_map__poly__simps,axiom,
! [C2: nat,P: poly_nat,F: nat > real] :
( ( ( ( C2 = zero_zero_nat )
& ( P = zero_zero_poly_nat ) )
=> ( ( map_poly_nat_real @ F @ ( pCons_nat @ C2 @ P ) )
= zero_zero_poly_real ) )
& ( ~ ( ( C2 = zero_zero_nat )
& ( P = zero_zero_poly_nat ) )
=> ( ( map_poly_nat_real @ F @ ( pCons_nat @ C2 @ P ) )
= ( pCons_real @ ( F @ C2 ) @ ( map_poly_nat_real @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_902_map__poly__simps,axiom,
! [C2: nat,P: poly_nat,F: nat > nat] :
( ( ( ( C2 = zero_zero_nat )
& ( P = zero_zero_poly_nat ) )
=> ( ( map_poly_nat_nat @ F @ ( pCons_nat @ C2 @ P ) )
= zero_zero_poly_nat ) )
& ( ~ ( ( C2 = zero_zero_nat )
& ( P = zero_zero_poly_nat ) )
=> ( ( map_poly_nat_nat @ F @ ( pCons_nat @ C2 @ P ) )
= ( pCons_nat @ ( F @ C2 ) @ ( map_poly_nat_nat @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_903_map__poly__simps,axiom,
! [C2: real,P: poly_real,F: real > int] :
( ( ( ( C2 = zero_zero_real )
& ( P = zero_zero_poly_real ) )
=> ( ( map_poly_real_int @ F @ ( pCons_real @ C2 @ P ) )
= zero_zero_poly_int ) )
& ( ~ ( ( C2 = zero_zero_real )
& ( P = zero_zero_poly_real ) )
=> ( ( map_poly_real_int @ F @ ( pCons_real @ C2 @ P ) )
= ( pCons_int @ ( F @ C2 ) @ ( map_poly_real_int @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_904_map__poly__simps,axiom,
! [C2: real,P: poly_real,F: real > real] :
( ( ( ( C2 = zero_zero_real )
& ( P = zero_zero_poly_real ) )
=> ( ( map_poly_real_real @ F @ ( pCons_real @ C2 @ P ) )
= zero_zero_poly_real ) )
& ( ~ ( ( C2 = zero_zero_real )
& ( P = zero_zero_poly_real ) )
=> ( ( map_poly_real_real @ F @ ( pCons_real @ C2 @ P ) )
= ( pCons_real @ ( F @ C2 ) @ ( map_poly_real_real @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_905_map__poly__simps,axiom,
! [C2: real,P: poly_real,F: real > nat] :
( ( ( ( C2 = zero_zero_real )
& ( P = zero_zero_poly_real ) )
=> ( ( map_poly_real_nat @ F @ ( pCons_real @ C2 @ P ) )
= zero_zero_poly_nat ) )
& ( ~ ( ( C2 = zero_zero_real )
& ( P = zero_zero_poly_real ) )
=> ( ( map_poly_real_nat @ F @ ( pCons_real @ C2 @ P ) )
= ( pCons_nat @ ( F @ C2 ) @ ( map_poly_real_nat @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_906_map__poly__simps,axiom,
! [C2: int,P: poly_int,F: int > finite_mod_ring_n] :
( ( ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_po1011533443592629756ring_n @ F @ ( pCons_int @ C2 @ P ) )
= zero_z2753989067526334999ring_n ) )
& ( ~ ( ( C2 = zero_zero_int )
& ( P = zero_zero_poly_int ) )
=> ( ( map_po1011533443592629756ring_n @ F @ ( pCons_int @ C2 @ P ) )
= ( pCons_8126420873123957872ring_n @ ( F @ C2 ) @ ( map_po1011533443592629756ring_n @ F @ P ) ) ) ) ) ).
% map_poly_simps
thf(fact_907_const__poly__dvd,axiom,
! [A: poly_real,B: poly_real] :
( ( dvd_dv4532039564868358754y_real @ ( pCons_poly_real @ A @ zero_z5583686468110200389y_real ) @ ( pCons_poly_real @ B @ zero_z5583686468110200389y_real ) )
= ( dvd_dvd_poly_real @ A @ B ) ) ).
% const_poly_dvd
thf(fact_908_const__poly__dvd,axiom,
! [A: finite_mod_ring_n,B: finite_mod_ring_n] :
( ( dvd_dv8138414522854976442ring_n @ ( pCons_8126420873123957872ring_n @ A @ zero_z2753989067526334999ring_n ) @ ( pCons_8126420873123957872ring_n @ B @ zero_z2753989067526334999ring_n ) )
= ( dvd_dv7258769396337835820ring_n @ A @ B ) ) ).
% const_poly_dvd
thf(fact_909_const__poly__dvd,axiom,
! [A: int,B: int] :
( ( dvd_dvd_poly_int @ ( pCons_int @ A @ zero_zero_poly_int ) @ ( pCons_int @ B @ zero_zero_poly_int ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% const_poly_dvd
thf(fact_910_const__poly__dvd,axiom,
! [A: real,B: real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ A @ zero_zero_poly_real ) @ ( pCons_real @ B @ zero_zero_poly_real ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% const_poly_dvd
thf(fact_911_const__poly__dvd,axiom,
! [A: poly_int,B: poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( pCons_poly_int @ A @ zero_z799223564134138693ly_int ) @ ( pCons_poly_int @ B @ zero_z799223564134138693ly_int ) )
= ( dvd_dvd_poly_int @ A @ B ) ) ).
% const_poly_dvd
thf(fact_912_const__poly__dvd,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ A @ zero_zero_poly_nat ) @ ( pCons_nat @ B @ zero_zero_poly_nat ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% const_poly_dvd
thf(fact_913_const__poly__dvd,axiom,
! [A: poly_F4222894760850802144ring_n,B: poly_F4222894760850802144ring_n] :
( ( dvd_dv3135175980337127240ring_n @ ( pCons_6246009715029582078ring_n @ A @ zero_z5482829069124612005ring_n ) @ ( pCons_6246009715029582078ring_n @ B @ zero_z5482829069124612005ring_n ) )
= ( dvd_dv8138414522854976442ring_n @ A @ B ) ) ).
% const_poly_dvd
thf(fact_914_ddvd__trans,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n,Z: poly_F4222894760850802144ring_n] :
( ( ( dvd_dv8138414522854976442ring_n @ X @ Y )
& ( dvd_dv8138414522854976442ring_n @ Y @ X ) )
=> ( ( ( dvd_dv8138414522854976442ring_n @ Y @ Z )
& ( dvd_dv8138414522854976442ring_n @ Z @ Y ) )
=> ( ( dvd_dv8138414522854976442ring_n @ X @ Z )
& ( dvd_dv8138414522854976442ring_n @ Z @ X ) ) ) ) ).
% ddvd_trans
thf(fact_915_ddvd__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ( dvd_dvd_int @ X @ Y )
& ( dvd_dvd_int @ Y @ X ) )
=> ( ( ( dvd_dvd_int @ Y @ Z )
& ( dvd_dvd_int @ Z @ Y ) )
=> ( ( dvd_dvd_int @ X @ Z )
& ( dvd_dvd_int @ Z @ X ) ) ) ) ).
% ddvd_trans
thf(fact_916_ddvd__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) )
=> ( ( ( dvd_dvd_nat @ Y @ Z )
& ( dvd_dvd_nat @ Z @ Y ) )
=> ( ( dvd_dvd_nat @ X @ Z )
& ( dvd_dvd_nat @ Z @ X ) ) ) ) ).
% ddvd_trans
thf(fact_917_ddvd__trans,axiom,
! [X: poly_int,Y: poly_int,Z: poly_int] :
( ( ( dvd_dvd_poly_int @ X @ Y )
& ( dvd_dvd_poly_int @ Y @ X ) )
=> ( ( ( dvd_dvd_poly_int @ Y @ Z )
& ( dvd_dvd_poly_int @ Z @ Y ) )
=> ( ( dvd_dvd_poly_int @ X @ Z )
& ( dvd_dvd_poly_int @ Z @ X ) ) ) ) ).
% ddvd_trans
thf(fact_918_ddvd__trans,axiom,
! [X: poly_real,Y: poly_real,Z: poly_real] :
( ( ( dvd_dvd_poly_real @ X @ Y )
& ( dvd_dvd_poly_real @ Y @ X ) )
=> ( ( ( dvd_dvd_poly_real @ Y @ Z )
& ( dvd_dvd_poly_real @ Z @ Y ) )
=> ( ( dvd_dvd_poly_real @ X @ Z )
& ( dvd_dvd_poly_real @ Z @ X ) ) ) ) ).
% ddvd_trans
thf(fact_919_ddvd__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ( dvd_dvd_real @ X @ Y )
& ( dvd_dvd_real @ Y @ X ) )
=> ( ( ( dvd_dvd_real @ Y @ Z )
& ( dvd_dvd_real @ Z @ Y ) )
=> ( ( dvd_dvd_real @ X @ Z )
& ( dvd_dvd_real @ Z @ X ) ) ) ) ).
% ddvd_trans
thf(fact_920_ddvd__sym,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( ( dvd_dv8138414522854976442ring_n @ X @ Y )
& ( dvd_dv8138414522854976442ring_n @ Y @ X ) )
=> ( ( dvd_dv8138414522854976442ring_n @ Y @ X )
& ( dvd_dv8138414522854976442ring_n @ X @ Y ) ) ) ).
% ddvd_sym
thf(fact_921_ddvd__sym,axiom,
! [X: int,Y: int] :
( ( ( dvd_dvd_int @ X @ Y )
& ( dvd_dvd_int @ Y @ X ) )
=> ( ( dvd_dvd_int @ Y @ X )
& ( dvd_dvd_int @ X @ Y ) ) ) ).
% ddvd_sym
thf(fact_922_ddvd__sym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) )
=> ( ( dvd_dvd_nat @ Y @ X )
& ( dvd_dvd_nat @ X @ Y ) ) ) ).
% ddvd_sym
thf(fact_923_ddvd__sym,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( dvd_dvd_poly_int @ X @ Y )
& ( dvd_dvd_poly_int @ Y @ X ) )
=> ( ( dvd_dvd_poly_int @ Y @ X )
& ( dvd_dvd_poly_int @ X @ Y ) ) ) ).
% ddvd_sym
thf(fact_924_ddvd__sym,axiom,
! [X: poly_real,Y: poly_real] :
( ( ( dvd_dvd_poly_real @ X @ Y )
& ( dvd_dvd_poly_real @ Y @ X ) )
=> ( ( dvd_dvd_poly_real @ Y @ X )
& ( dvd_dvd_poly_real @ X @ Y ) ) ) ).
% ddvd_sym
thf(fact_925_ddvd__sym,axiom,
! [X: real,Y: real] :
( ( ( dvd_dvd_real @ X @ Y )
& ( dvd_dvd_real @ Y @ X ) )
=> ( ( dvd_dvd_real @ Y @ X )
& ( dvd_dvd_real @ X @ Y ) ) ) ).
% ddvd_sym
thf(fact_926_pCons__0__as__mult,axiom,
! [P: poly_p6692042823160534382ring_n] :
( ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ P )
= ( times_2573333606529333417ring_n @ ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ ( pCons_6246009715029582078ring_n @ one_on4318287115420659547ring_n @ zero_z5482829069124612005ring_n ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_927_pCons__0__as__mult,axiom,
! [P: poly_nat] :
( ( pCons_nat @ zero_zero_nat @ P )
= ( times_times_poly_nat @ ( pCons_nat @ zero_zero_nat @ ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_928_pCons__0__as__mult,axiom,
! [P: poly_poly_int] :
( ( pCons_poly_int @ zero_zero_poly_int @ P )
= ( times_4739760418287672641ly_int @ ( pCons_poly_int @ zero_zero_poly_int @ ( pCons_poly_int @ one_one_poly_int @ zero_z799223564134138693ly_int ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_929_pCons__0__as__mult,axiom,
! [P: poly_real] :
( ( pCons_real @ zero_zero_real @ P )
= ( times_7914811829580426937y_real @ ( pCons_real @ zero_zero_real @ ( pCons_real @ one_one_real @ zero_zero_poly_real ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_930_pCons__0__as__mult,axiom,
! [P: poly_poly_real] :
( ( pCons_poly_real @ zero_zero_poly_real @ P )
= ( times_4423207553272384065y_real @ ( pCons_poly_real @ zero_zero_poly_real @ ( pCons_poly_real @ one_one_poly_real @ zero_z5583686468110200389y_real ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_931_pCons__0__as__mult,axiom,
! [P: poly_poly_poly_int] :
( ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ P )
= ( times_4100521150541653321ly_int @ ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ ( pCons_poly_poly_int @ one_on1166514126663969025ly_int @ zero_z240508265545053005ly_int ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_932_pCons__0__as__mult,axiom,
! [P: poly_poly_nat] :
( ( pCons_poly_nat @ zero_zero_poly_nat @ P )
= ( times_7229843563219399397ly_nat @ ( pCons_poly_nat @ zero_zero_poly_nat @ ( pCons_poly_nat @ one_one_poly_nat @ zero_z3289306709065865449ly_nat ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_933_pCons__0__as__mult,axiom,
! [P: poly_p2743341848350813180ring_n] :
( ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ P )
= ( times_4617534433836805431ring_n @ ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ ( pCons_2385395009258896524ring_n @ one_on5457780782968151273ring_n @ zero_z3442457038203223091ring_n ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_934_pCons__0__as__mult,axiom,
! [P: poly_F4222894760850802144ring_n] :
( ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ P )
= ( times_4166049284782705435ring_n @ ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ ( pCons_8126420873123957872ring_n @ one_on2109788483843180749ring_n @ zero_z2753989067526334999ring_n ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_935_pCons__0__as__mult,axiom,
! [P: poly_int] :
( ( pCons_int @ zero_zero_int @ P )
= ( times_times_poly_int @ ( pCons_int @ zero_zero_int @ ( pCons_int @ one_one_int @ zero_zero_poly_int ) ) @ P ) ) ).
% pCons_0_as_mult
thf(fact_936_map__poly__1,axiom,
! [F: int > int] :
( ( map_poly_int_int @ F @ one_one_poly_int )
= ( pCons_int @ ( F @ one_one_int ) @ zero_zero_poly_int ) ) ).
% map_poly_1
thf(fact_937_map__poly__1,axiom,
! [F: real > int] :
( ( map_poly_real_int @ F @ one_one_poly_real )
= ( pCons_int @ ( F @ one_one_real ) @ zero_zero_poly_int ) ) ).
% map_poly_1
thf(fact_938_map__poly__1,axiom,
! [F: nat > int] :
( ( map_poly_nat_int @ F @ one_one_poly_nat )
= ( pCons_int @ ( F @ one_one_nat ) @ zero_zero_poly_int ) ) ).
% map_poly_1
thf(fact_939_map__poly__1,axiom,
! [F: int > real] :
( ( map_poly_int_real @ F @ one_one_poly_int )
= ( pCons_real @ ( F @ one_one_int ) @ zero_zero_poly_real ) ) ).
% map_poly_1
thf(fact_940_map__poly__1,axiom,
! [F: real > real] :
( ( map_poly_real_real @ F @ one_one_poly_real )
= ( pCons_real @ ( F @ one_one_real ) @ zero_zero_poly_real ) ) ).
% map_poly_1
thf(fact_941_map__poly__1,axiom,
! [F: nat > real] :
( ( map_poly_nat_real @ F @ one_one_poly_nat )
= ( pCons_real @ ( F @ one_one_nat ) @ zero_zero_poly_real ) ) ).
% map_poly_1
thf(fact_942_map__poly__1,axiom,
! [F: int > nat] :
( ( map_poly_int_nat @ F @ one_one_poly_int )
= ( pCons_nat @ ( F @ one_one_int ) @ zero_zero_poly_nat ) ) ).
% map_poly_1
thf(fact_943_map__poly__1,axiom,
! [F: real > nat] :
( ( map_poly_real_nat @ F @ one_one_poly_real )
= ( pCons_nat @ ( F @ one_one_real ) @ zero_zero_poly_nat ) ) ).
% map_poly_1
thf(fact_944_map__poly__1,axiom,
! [F: nat > nat] :
( ( map_poly_nat_nat @ F @ one_one_poly_nat )
= ( pCons_nat @ ( F @ one_one_nat ) @ zero_zero_poly_nat ) ) ).
% map_poly_1
thf(fact_945_map__poly__1,axiom,
! [F: int > finite_mod_ring_n] :
( ( map_po1011533443592629756ring_n @ F @ one_one_poly_int )
= ( pCons_8126420873123957872ring_n @ ( F @ one_one_int ) @ zero_z2753989067526334999ring_n ) ) ).
% map_poly_1
thf(fact_946_is__unit__pCons__iff,axiom,
! [A: real,P: poly_real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ A @ P ) @ one_one_poly_real )
= ( ( P = zero_zero_poly_real )
& ( A != zero_zero_real ) ) ) ).
% is_unit_pCons_iff
thf(fact_947_is__unit__triv,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( dvd_dvd_poly_real @ ( pCons_real @ A @ zero_zero_poly_real ) @ one_one_poly_real ) ) ).
% is_unit_triv
thf(fact_948_is__unit__const__poly__iff,axiom,
! [C2: poly_real] :
( ( dvd_dv4532039564868358754y_real @ ( pCons_poly_real @ C2 @ zero_z5583686468110200389y_real ) @ one_on1191988272081909249y_real )
= ( dvd_dvd_poly_real @ C2 @ one_one_poly_real ) ) ).
% is_unit_const_poly_iff
thf(fact_949_is__unit__const__poly__iff,axiom,
! [C2: poly_poly_int] :
( ( dvd_dv7705178354154678250ly_int @ ( pCons_poly_poly_int @ C2 @ zero_z240508265545053005ly_int ) @ one_on7423179019345326345ly_int )
= ( dvd_dv6998304861263046114ly_int @ C2 @ one_on1166514126663969025ly_int ) ) ).
% is_unit_const_poly_iff
thf(fact_950_is__unit__const__poly__iff,axiom,
! [C2: poly_nat] :
( ( dvd_dv265015969339997062ly_nat @ ( pCons_poly_nat @ C2 @ zero_z3289306709065865449ly_nat ) @ one_on3656597271595695781ly_nat )
= ( dvd_dvd_poly_nat @ C2 @ one_one_poly_nat ) ) ).
% is_unit_const_poly_iff
thf(fact_951_is__unit__const__poly__iff,axiom,
! [C2: int] :
( ( dvd_dvd_poly_int @ ( pCons_int @ C2 @ zero_zero_poly_int ) @ one_one_poly_int )
= ( dvd_dvd_int @ C2 @ one_one_int ) ) ).
% is_unit_const_poly_iff
thf(fact_952_is__unit__const__poly__iff,axiom,
! [C2: real] :
( ( dvd_dvd_poly_real @ ( pCons_real @ C2 @ zero_zero_poly_real ) @ one_one_poly_real )
= ( dvd_dvd_real @ C2 @ one_one_real ) ) ).
% is_unit_const_poly_iff
thf(fact_953_is__unit__const__poly__iff,axiom,
! [C2: poly_int] :
( ( dvd_dv6998304861263046114ly_int @ ( pCons_poly_int @ C2 @ zero_z799223564134138693ly_int ) @ one_on1166514126663969025ly_int )
= ( dvd_dvd_poly_int @ C2 @ one_one_poly_int ) ) ).
% is_unit_const_poly_iff
thf(fact_954_is__unit__const__poly__iff,axiom,
! [C2: nat] :
( ( dvd_dvd_poly_nat @ ( pCons_nat @ C2 @ zero_zero_poly_nat ) @ one_one_poly_nat )
= ( dvd_dvd_nat @ C2 @ one_one_nat ) ) ).
% is_unit_const_poly_iff
thf(fact_955_is__unit__poly__iff,axiom,
! [P: poly_poly_real] :
( ( dvd_dv4532039564868358754y_real @ P @ one_on1191988272081909249y_real )
= ( ? [C4: poly_real] :
( ( P
= ( pCons_poly_real @ C4 @ zero_z5583686468110200389y_real ) )
& ( dvd_dvd_poly_real @ C4 @ one_one_poly_real ) ) ) ) ).
% is_unit_poly_iff
thf(fact_956_is__unit__poly__iff,axiom,
! [P: poly_poly_poly_int] :
( ( dvd_dv7705178354154678250ly_int @ P @ one_on7423179019345326345ly_int )
= ( ? [C4: poly_poly_int] :
( ( P
= ( pCons_poly_poly_int @ C4 @ zero_z240508265545053005ly_int ) )
& ( dvd_dv6998304861263046114ly_int @ C4 @ one_on1166514126663969025ly_int ) ) ) ) ).
% is_unit_poly_iff
thf(fact_957_is__unit__poly__iff,axiom,
! [P: poly_poly_nat] :
( ( dvd_dv265015969339997062ly_nat @ P @ one_on3656597271595695781ly_nat )
= ( ? [C4: poly_nat] :
( ( P
= ( pCons_poly_nat @ C4 @ zero_z3289306709065865449ly_nat ) )
& ( dvd_dvd_poly_nat @ C4 @ one_one_poly_nat ) ) ) ) ).
% is_unit_poly_iff
thf(fact_958_is__unit__poly__iff,axiom,
! [P: poly_int] :
( ( dvd_dvd_poly_int @ P @ one_one_poly_int )
= ( ? [C4: int] :
( ( P
= ( pCons_int @ C4 @ zero_zero_poly_int ) )
& ( dvd_dvd_int @ C4 @ one_one_int ) ) ) ) ).
% is_unit_poly_iff
thf(fact_959_is__unit__poly__iff,axiom,
! [P: poly_real] :
( ( dvd_dvd_poly_real @ P @ one_one_poly_real )
= ( ? [C4: real] :
( ( P
= ( pCons_real @ C4 @ zero_zero_poly_real ) )
& ( dvd_dvd_real @ C4 @ one_one_real ) ) ) ) ).
% is_unit_poly_iff
thf(fact_960_is__unit__poly__iff,axiom,
! [P: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ P @ one_on1166514126663969025ly_int )
= ( ? [C4: poly_int] :
( ( P
= ( pCons_poly_int @ C4 @ zero_z799223564134138693ly_int ) )
& ( dvd_dvd_poly_int @ C4 @ one_one_poly_int ) ) ) ) ).
% is_unit_poly_iff
thf(fact_961_is__unit__poly__iff,axiom,
! [P: poly_nat] :
( ( dvd_dvd_poly_nat @ P @ one_one_poly_nat )
= ( ? [C4: nat] :
( ( P
= ( pCons_nat @ C4 @ zero_zero_poly_nat ) )
& ( dvd_dvd_nat @ C4 @ one_one_nat ) ) ) ) ).
% is_unit_poly_iff
thf(fact_962_is__unit__polyE,axiom,
! [P: poly_poly_real] :
( ( dvd_dv4532039564868358754y_real @ P @ one_on1191988272081909249y_real )
=> ~ ! [C: poly_real] :
( ( P
= ( pCons_poly_real @ C @ zero_z5583686468110200389y_real ) )
=> ~ ( dvd_dvd_poly_real @ C @ one_one_poly_real ) ) ) ).
% is_unit_polyE
thf(fact_963_is__unit__polyE,axiom,
! [P: poly_poly_poly_int] :
( ( dvd_dv7705178354154678250ly_int @ P @ one_on7423179019345326345ly_int )
=> ~ ! [C: poly_poly_int] :
( ( P
= ( pCons_poly_poly_int @ C @ zero_z240508265545053005ly_int ) )
=> ~ ( dvd_dv6998304861263046114ly_int @ C @ one_on1166514126663969025ly_int ) ) ) ).
% is_unit_polyE
thf(fact_964_is__unit__polyE,axiom,
! [P: poly_poly_nat] :
( ( dvd_dv265015969339997062ly_nat @ P @ one_on3656597271595695781ly_nat )
=> ~ ! [C: poly_nat] :
( ( P
= ( pCons_poly_nat @ C @ zero_z3289306709065865449ly_nat ) )
=> ~ ( dvd_dvd_poly_nat @ C @ one_one_poly_nat ) ) ) ).
% is_unit_polyE
thf(fact_965_is__unit__polyE,axiom,
! [P: poly_int] :
( ( dvd_dvd_poly_int @ P @ one_one_poly_int )
=> ~ ! [C: int] :
( ( P
= ( pCons_int @ C @ zero_zero_poly_int ) )
=> ~ ( dvd_dvd_int @ C @ one_one_int ) ) ) ).
% is_unit_polyE
thf(fact_966_is__unit__polyE,axiom,
! [P: poly_real] :
( ( dvd_dvd_poly_real @ P @ one_one_poly_real )
=> ~ ! [C: real] :
( ( P
= ( pCons_real @ C @ zero_zero_poly_real ) )
=> ~ ( dvd_dvd_real @ C @ one_one_real ) ) ) ).
% is_unit_polyE
thf(fact_967_is__unit__polyE,axiom,
! [P: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ P @ one_on1166514126663969025ly_int )
=> ~ ! [C: poly_int] :
( ( P
= ( pCons_poly_int @ C @ zero_z799223564134138693ly_int ) )
=> ~ ( dvd_dvd_poly_int @ C @ one_one_poly_int ) ) ) ).
% is_unit_polyE
thf(fact_968_is__unit__polyE,axiom,
! [P: poly_nat] :
( ( dvd_dvd_poly_nat @ P @ one_one_poly_nat )
=> ~ ! [C: nat] :
( ( P
= ( pCons_nat @ C @ zero_zero_poly_nat ) )
=> ~ ( dvd_dvd_nat @ C @ one_one_nat ) ) ) ).
% is_unit_polyE
thf(fact_969_coeff__lift__hom_Ohom__dvd,axiom,
! [P: poly_real,Q: poly_real] :
( ( dvd_dvd_poly_real @ P @ Q )
=> ( dvd_dv4532039564868358754y_real @ ( pCons_poly_real @ P @ zero_z5583686468110200389y_real ) @ ( pCons_poly_real @ Q @ zero_z5583686468110200389y_real ) ) ) ).
% coeff_lift_hom.hom_dvd
thf(fact_970_coeff__lift__hom_Ohom__dvd,axiom,
! [P: finite_mod_ring_n,Q: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ P @ Q )
=> ( dvd_dv8138414522854976442ring_n @ ( pCons_8126420873123957872ring_n @ P @ zero_z2753989067526334999ring_n ) @ ( pCons_8126420873123957872ring_n @ Q @ zero_z2753989067526334999ring_n ) ) ) ).
% coeff_lift_hom.hom_dvd
thf(fact_971_coeff__lift__hom_Ohom__dvd,axiom,
! [P: int,Q: int] :
( ( dvd_dvd_int @ P @ Q )
=> ( dvd_dvd_poly_int @ ( pCons_int @ P @ zero_zero_poly_int ) @ ( pCons_int @ Q @ zero_zero_poly_int ) ) ) ).
% coeff_lift_hom.hom_dvd
thf(fact_972_coeff__lift__hom_Ohom__dvd,axiom,
! [P: real,Q: real] :
( ( dvd_dvd_real @ P @ Q )
=> ( dvd_dvd_poly_real @ ( pCons_real @ P @ zero_zero_poly_real ) @ ( pCons_real @ Q @ zero_zero_poly_real ) ) ) ).
% coeff_lift_hom.hom_dvd
thf(fact_973_coeff__lift__hom_Ohom__dvd,axiom,
! [P: poly_int,Q: poly_int] :
( ( dvd_dvd_poly_int @ P @ Q )
=> ( dvd_dv6998304861263046114ly_int @ ( pCons_poly_int @ P @ zero_z799223564134138693ly_int ) @ ( pCons_poly_int @ Q @ zero_z799223564134138693ly_int ) ) ) ).
% coeff_lift_hom.hom_dvd
thf(fact_974_coeff__lift__hom_Ohom__dvd,axiom,
! [P: nat,Q: nat] :
( ( dvd_dvd_nat @ P @ Q )
=> ( dvd_dvd_poly_nat @ ( pCons_nat @ P @ zero_zero_poly_nat ) @ ( pCons_nat @ Q @ zero_zero_poly_nat ) ) ) ).
% coeff_lift_hom.hom_dvd
thf(fact_975_coeff__lift__hom_Ohom__dvd,axiom,
! [P: poly_F4222894760850802144ring_n,Q: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ P @ Q )
=> ( dvd_dv3135175980337127240ring_n @ ( pCons_6246009715029582078ring_n @ P @ zero_z5482829069124612005ring_n ) @ ( pCons_6246009715029582078ring_n @ Q @ zero_z5482829069124612005ring_n ) ) ) ).
% coeff_lift_hom.hom_dvd
thf(fact_976_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: poly_real] :
( ( ( pCons_poly_real @ X @ zero_z5583686468110200389y_real )
= one_on1191988272081909249y_real )
= ( X = one_one_poly_real ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_977_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: poly_poly_int] :
( ( ( pCons_poly_poly_int @ X @ zero_z240508265545053005ly_int )
= one_on7423179019345326345ly_int )
= ( X = one_on1166514126663969025ly_int ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_978_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: poly_nat] :
( ( ( pCons_poly_nat @ X @ zero_z3289306709065865449ly_nat )
= one_on3656597271595695781ly_nat )
= ( X = one_one_poly_nat ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_979_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( ( pCons_2385395009258896524ring_n @ X @ zero_z3442457038203223091ring_n )
= one_on281575345490252151ring_n )
= ( X = one_on5457780782968151273ring_n ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_980_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: finite_mod_ring_n] :
( ( ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n )
= one_on4318287115420659547ring_n )
= ( X = one_on2109788483843180749ring_n ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_981_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: int] :
( ( ( pCons_int @ X @ zero_zero_poly_int )
= one_one_poly_int )
= ( X = one_one_int ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_982_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: real] :
( ( ( pCons_real @ X @ zero_zero_poly_real )
= one_one_poly_real )
= ( X = one_one_real ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_983_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: poly_int] :
( ( ( pCons_poly_int @ X @ zero_z799223564134138693ly_int )
= one_on1166514126663969025ly_int )
= ( X = one_one_poly_int ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_984_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: nat] :
( ( ( pCons_nat @ X @ zero_zero_poly_nat )
= one_one_poly_nat )
= ( X = one_one_nat ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_985_coeff__lift__hom_Ohom__1__iff,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n )
= one_on5457780782968151273ring_n )
= ( X = one_on4318287115420659547ring_n ) ) ).
% coeff_lift_hom.hom_1_iff
thf(fact_986_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: poly_real] :
( ( ( pCons_poly_real @ X @ zero_z5583686468110200389y_real )
= zero_z5583686468110200389y_real )
= ( X = zero_zero_poly_real ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_987_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: poly_poly_int] :
( ( ( pCons_poly_poly_int @ X @ zero_z240508265545053005ly_int )
= zero_z240508265545053005ly_int )
= ( X = zero_z799223564134138693ly_int ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_988_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: poly_nat] :
( ( ( pCons_poly_nat @ X @ zero_z3289306709065865449ly_nat )
= zero_z3289306709065865449ly_nat )
= ( X = zero_zero_poly_nat ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_989_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( ( pCons_2385395009258896524ring_n @ X @ zero_z3442457038203223091ring_n )
= zero_z3442457038203223091ring_n )
= ( X = zero_z5482829069124612005ring_n ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_990_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: finite_mod_ring_n] :
( ( ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n )
= ( X = zero_z7902377597758090121ring_n ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_991_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: int] :
( ( ( pCons_int @ X @ zero_zero_poly_int )
= zero_zero_poly_int )
= ( X = zero_zero_int ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_992_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: real] :
( ( ( pCons_real @ X @ zero_zero_poly_real )
= zero_zero_poly_real )
= ( X = zero_zero_real ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_993_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: poly_int] :
( ( ( pCons_poly_int @ X @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int )
= ( X = zero_zero_poly_int ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_994_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: nat] :
( ( ( pCons_nat @ X @ zero_zero_poly_nat )
= zero_zero_poly_nat )
= ( X = zero_zero_nat ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_995_coeff__lift__hom_Ohom__0__iff,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n )
= ( X = zero_z2753989067526334999ring_n ) ) ).
% coeff_lift_hom.hom_0_iff
thf(fact_996_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n ) ).
% coeff_lift_hom.hom_zero
thf(fact_997_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_int @ zero_zero_int @ zero_zero_poly_int )
= zero_zero_poly_int ) ).
% coeff_lift_hom.hom_zero
thf(fact_998_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_nat @ zero_zero_nat @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% coeff_lift_hom.hom_zero
thf(fact_999_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_poly_int @ zero_zero_poly_int @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int ) ).
% coeff_lift_hom.hom_zero
thf(fact_1000_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_real @ zero_zero_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% coeff_lift_hom.hom_zero
thf(fact_1001_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_poly_real @ zero_zero_poly_real @ zero_z5583686468110200389y_real )
= zero_z5583686468110200389y_real ) ).
% coeff_lift_hom.hom_zero
thf(fact_1002_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ zero_z240508265545053005ly_int )
= zero_z240508265545053005ly_int ) ).
% coeff_lift_hom.hom_zero
thf(fact_1003_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_poly_nat @ zero_zero_poly_nat @ zero_z3289306709065865449ly_nat )
= zero_z3289306709065865449ly_nat ) ).
% coeff_lift_hom.hom_zero
thf(fact_1004_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ zero_z3442457038203223091ring_n )
= zero_z3442457038203223091ring_n ) ).
% coeff_lift_hom.hom_zero
thf(fact_1005_coeff__lift__hom_Ohom__zero,axiom,
( ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n ) ).
% coeff_lift_hom.hom_zero
thf(fact_1006_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: poly_real] :
( ( dvd_dvd_poly_real @ X @ one_one_poly_real )
=> ( dvd_dv4532039564868358754y_real @ ( pCons_poly_real @ X @ zero_z5583686468110200389y_real ) @ one_on1191988272081909249y_real ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1007_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ X @ one_on1166514126663969025ly_int )
=> ( dvd_dv7705178354154678250ly_int @ ( pCons_poly_poly_int @ X @ zero_z240508265545053005ly_int ) @ one_on7423179019345326345ly_int ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1008_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: poly_nat] :
( ( dvd_dvd_poly_nat @ X @ one_one_poly_nat )
=> ( dvd_dv265015969339997062ly_nat @ ( pCons_poly_nat @ X @ zero_z3289306709065865449ly_nat ) @ one_on3656597271595695781ly_nat ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1009_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ X @ one_on5457780782968151273ring_n )
=> ( dvd_dv3919477662729673174ring_n @ ( pCons_2385395009258896524ring_n @ X @ zero_z3442457038203223091ring_n ) @ one_on281575345490252151ring_n ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1010_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: finite_mod_ring_n] :
( ( dvd_dv7258769396337835820ring_n @ X @ one_on2109788483843180749ring_n )
=> ( dvd_dv8138414522854976442ring_n @ ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n ) @ one_on4318287115420659547ring_n ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1011_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ( dvd_dvd_poly_int @ ( pCons_int @ X @ zero_zero_poly_int ) @ one_one_poly_int ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1012_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: real] :
( ( dvd_dvd_real @ X @ one_one_real )
=> ( dvd_dvd_poly_real @ ( pCons_real @ X @ zero_zero_poly_real ) @ one_one_poly_real ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1013_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: poly_int] :
( ( dvd_dvd_poly_int @ X @ one_one_poly_int )
=> ( dvd_dv6998304861263046114ly_int @ ( pCons_poly_int @ X @ zero_z799223564134138693ly_int ) @ one_on1166514126663969025ly_int ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1014_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: nat] :
( ( dvd_dvd_nat @ X @ one_one_nat )
=> ( dvd_dvd_poly_nat @ ( pCons_nat @ X @ zero_zero_poly_nat ) @ one_one_poly_nat ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1015_coeff__lift__hom_Ohom__dvd__1,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ X @ one_on4318287115420659547ring_n )
=> ( dvd_dv3135175980337127240ring_n @ ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n ) @ one_on5457780782968151273ring_n ) ) ).
% coeff_lift_hom.hom_dvd_1
thf(fact_1016_coeff__lift__hom_Oeq__iff,axiom,
! [X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n )
= ( pCons_8126420873123957872ring_n @ Y @ zero_z2753989067526334999ring_n ) )
= ( X = Y ) ) ).
% coeff_lift_hom.eq_iff
thf(fact_1017_coeff__lift__hom_Oeq__iff,axiom,
! [X: int,Y: int] :
( ( ( pCons_int @ X @ zero_zero_poly_int )
= ( pCons_int @ Y @ zero_zero_poly_int ) )
= ( X = Y ) ) ).
% coeff_lift_hom.eq_iff
thf(fact_1018_coeff__lift__hom_Oeq__iff,axiom,
! [X: real,Y: real] :
( ( ( pCons_real @ X @ zero_zero_poly_real )
= ( pCons_real @ Y @ zero_zero_poly_real ) )
= ( X = Y ) ) ).
% coeff_lift_hom.eq_iff
thf(fact_1019_coeff__lift__hom_Oeq__iff,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( pCons_poly_int @ X @ zero_z799223564134138693ly_int )
= ( pCons_poly_int @ Y @ zero_z799223564134138693ly_int ) )
= ( X = Y ) ) ).
% coeff_lift_hom.eq_iff
thf(fact_1020_coeff__lift__hom_Oeq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( pCons_nat @ X @ zero_zero_poly_nat )
= ( pCons_nat @ Y @ zero_zero_poly_nat ) )
= ( X = Y ) ) ).
% coeff_lift_hom.eq_iff
thf(fact_1021_coeff__lift__hom_Oeq__iff,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n )
= ( pCons_6246009715029582078ring_n @ Y @ zero_z5482829069124612005ring_n ) )
= ( X = Y ) ) ).
% coeff_lift_hom.eq_iff
thf(fact_1022_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_real,Y: poly_real] :
( ( ( times_7914811829580426937y_real @ X @ Y )
= zero_zero_poly_real )
=> ( ( times_4423207553272384065y_real @ ( pCons_poly_real @ X @ zero_z5583686468110200389y_real ) @ ( pCons_poly_real @ Y @ zero_z5583686468110200389y_real ) )
= zero_z5583686468110200389y_real ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1023_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_poly_int,Y: poly_poly_int] :
( ( ( times_4739760418287672641ly_int @ X @ Y )
= zero_z799223564134138693ly_int )
=> ( ( times_4100521150541653321ly_int @ ( pCons_poly_poly_int @ X @ zero_z240508265545053005ly_int ) @ ( pCons_poly_poly_int @ Y @ zero_z240508265545053005ly_int ) )
= zero_z240508265545053005ly_int ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1024_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( ( times_times_poly_nat @ X @ Y )
= zero_zero_poly_nat )
=> ( ( times_7229843563219399397ly_nat @ ( pCons_poly_nat @ X @ zero_z3289306709065865449ly_nat ) @ ( pCons_poly_nat @ Y @ zero_z3289306709065865449ly_nat ) )
= zero_z3289306709065865449ly_nat ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1025_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_p6692042823160534382ring_n,Y: poly_p6692042823160534382ring_n] :
( ( ( times_2573333606529333417ring_n @ X @ Y )
= zero_z5482829069124612005ring_n )
=> ( ( times_4617534433836805431ring_n @ ( pCons_2385395009258896524ring_n @ X @ zero_z3442457038203223091ring_n ) @ ( pCons_2385395009258896524ring_n @ Y @ zero_z3442457038203223091ring_n ) )
= zero_z3442457038203223091ring_n ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1026_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( ( times_5121417632533718157ring_n @ X @ Y )
= zero_z7902377597758090121ring_n )
=> ( ( times_4166049284782705435ring_n @ ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n ) @ ( pCons_8126420873123957872ring_n @ Y @ zero_z2753989067526334999ring_n ) )
= zero_z2753989067526334999ring_n ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1027_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( ( times_4166049284782705435ring_n @ X @ Y )
= zero_z2753989067526334999ring_n )
=> ( ( times_2573333606529333417ring_n @ ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n ) @ ( pCons_6246009715029582078ring_n @ Y @ zero_z5482829069124612005ring_n ) )
= zero_z5482829069124612005ring_n ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1028_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: int,Y: int] :
( ( ( times_times_int @ X @ Y )
= zero_zero_int )
=> ( ( times_times_poly_int @ ( pCons_int @ X @ zero_zero_poly_int ) @ ( pCons_int @ Y @ zero_zero_poly_int ) )
= zero_zero_poly_int ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1029_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: nat,Y: nat] :
( ( ( times_times_nat @ X @ Y )
= zero_zero_nat )
=> ( ( times_times_poly_nat @ ( pCons_nat @ X @ zero_zero_poly_nat ) @ ( pCons_nat @ Y @ zero_zero_poly_nat ) )
= zero_zero_poly_nat ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1030_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( times_times_poly_int @ X @ Y )
= zero_zero_poly_int )
=> ( ( times_4739760418287672641ly_int @ ( pCons_poly_int @ X @ zero_z799223564134138693ly_int ) @ ( pCons_poly_int @ Y @ zero_z799223564134138693ly_int ) )
= zero_z799223564134138693ly_int ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1031_coeff__lift__hom_Ohom__mult__eq__zero,axiom,
! [X: real,Y: real] :
( ( ( times_times_real @ X @ Y )
= zero_zero_real )
=> ( ( times_7914811829580426937y_real @ ( pCons_real @ X @ zero_zero_poly_real ) @ ( pCons_real @ Y @ zero_zero_poly_real ) )
= zero_zero_poly_real ) ) ).
% coeff_lift_hom.hom_mult_eq_zero
thf(fact_1032_coeff__lift__hom_Ohom__1,axiom,
! [X: poly_real] :
( ( ( pCons_poly_real @ X @ zero_z5583686468110200389y_real )
= one_on1191988272081909249y_real )
=> ( X = one_one_poly_real ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1033_coeff__lift__hom_Ohom__1,axiom,
! [X: poly_poly_int] :
( ( ( pCons_poly_poly_int @ X @ zero_z240508265545053005ly_int )
= one_on7423179019345326345ly_int )
=> ( X = one_on1166514126663969025ly_int ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1034_coeff__lift__hom_Ohom__1,axiom,
! [X: poly_nat] :
( ( ( pCons_poly_nat @ X @ zero_z3289306709065865449ly_nat )
= one_on3656597271595695781ly_nat )
=> ( X = one_one_poly_nat ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1035_coeff__lift__hom_Ohom__1,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( ( pCons_2385395009258896524ring_n @ X @ zero_z3442457038203223091ring_n )
= one_on281575345490252151ring_n )
=> ( X = one_on5457780782968151273ring_n ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1036_coeff__lift__hom_Ohom__1,axiom,
! [X: finite_mod_ring_n] :
( ( ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n )
= one_on4318287115420659547ring_n )
=> ( X = one_on2109788483843180749ring_n ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1037_coeff__lift__hom_Ohom__1,axiom,
! [X: int] :
( ( ( pCons_int @ X @ zero_zero_poly_int )
= one_one_poly_int )
=> ( X = one_one_int ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1038_coeff__lift__hom_Ohom__1,axiom,
! [X: real] :
( ( ( pCons_real @ X @ zero_zero_poly_real )
= one_one_poly_real )
=> ( X = one_one_real ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1039_coeff__lift__hom_Ohom__1,axiom,
! [X: poly_int] :
( ( ( pCons_poly_int @ X @ zero_z799223564134138693ly_int )
= one_on1166514126663969025ly_int )
=> ( X = one_one_poly_int ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1040_coeff__lift__hom_Ohom__1,axiom,
! [X: nat] :
( ( ( pCons_nat @ X @ zero_zero_poly_nat )
= one_one_poly_nat )
=> ( X = one_one_nat ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1041_coeff__lift__hom_Ohom__1,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n )
= one_on5457780782968151273ring_n )
=> ( X = one_on4318287115420659547ring_n ) ) ).
% coeff_lift_hom.hom_1
thf(fact_1042_coeff__lift__hom_Ohom__mult,axiom,
! [X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( pCons_8126420873123957872ring_n @ ( times_5121417632533718157ring_n @ X @ Y ) @ zero_z2753989067526334999ring_n )
= ( times_4166049284782705435ring_n @ ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n ) @ ( pCons_8126420873123957872ring_n @ Y @ zero_z2753989067526334999ring_n ) ) ) ).
% coeff_lift_hom.hom_mult
thf(fact_1043_coeff__lift__hom_Ohom__mult,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( pCons_6246009715029582078ring_n @ ( times_4166049284782705435ring_n @ X @ Y ) @ zero_z5482829069124612005ring_n )
= ( times_2573333606529333417ring_n @ ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n ) @ ( pCons_6246009715029582078ring_n @ Y @ zero_z5482829069124612005ring_n ) ) ) ).
% coeff_lift_hom.hom_mult
thf(fact_1044_coeff__lift__hom_Ohom__mult,axiom,
! [X: int,Y: int] :
( ( pCons_int @ ( times_times_int @ X @ Y ) @ zero_zero_poly_int )
= ( times_times_poly_int @ ( pCons_int @ X @ zero_zero_poly_int ) @ ( pCons_int @ Y @ zero_zero_poly_int ) ) ) ).
% coeff_lift_hom.hom_mult
thf(fact_1045_coeff__lift__hom_Ohom__mult,axiom,
! [X: nat,Y: nat] :
( ( pCons_nat @ ( times_times_nat @ X @ Y ) @ zero_zero_poly_nat )
= ( times_times_poly_nat @ ( pCons_nat @ X @ zero_zero_poly_nat ) @ ( pCons_nat @ Y @ zero_zero_poly_nat ) ) ) ).
% coeff_lift_hom.hom_mult
thf(fact_1046_coeff__lift__hom_Ohom__mult,axiom,
! [X: poly_int,Y: poly_int] :
( ( pCons_poly_int @ ( times_times_poly_int @ X @ Y ) @ zero_z799223564134138693ly_int )
= ( times_4739760418287672641ly_int @ ( pCons_poly_int @ X @ zero_z799223564134138693ly_int ) @ ( pCons_poly_int @ Y @ zero_z799223564134138693ly_int ) ) ) ).
% coeff_lift_hom.hom_mult
thf(fact_1047_coeff__lift__hom_Ohom__mult,axiom,
! [X: real,Y: real] :
( ( pCons_real @ ( times_times_real @ X @ Y ) @ zero_zero_poly_real )
= ( times_7914811829580426937y_real @ ( pCons_real @ X @ zero_zero_poly_real ) @ ( pCons_real @ Y @ zero_zero_poly_real ) ) ) ).
% coeff_lift_hom.hom_mult
thf(fact_1048_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_1049_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_1050_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C2 )
=> ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% gcd_nat.trans
thf(fact_1051_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( dvd_dvd_nat @ A2 @ B2 )
& ( dvd_dvd_nat @ B2 @ A2 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1052_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_1053_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_1054_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C2 )
& ( B != C2 ) )
=> ( ( dvd_dvd_nat @ A @ C2 )
& ( A != C2 ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1055_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C2 )
& ( B != C2 ) )
=> ( ( dvd_dvd_nat @ A @ C2 )
& ( A != C2 ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1056_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C2 )
=> ( ( dvd_dvd_nat @ A @ C2 )
& ( A != C2 ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1057_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1058_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A2: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A2 @ B2 )
& ( A2 != B2 ) )
| ( A2 = B2 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1059_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1060_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1061_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1062_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1063_coeff__lift__hom_Oinjectivity,axiom,
! [X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n )
= ( pCons_8126420873123957872ring_n @ Y @ zero_z2753989067526334999ring_n ) )
=> ( X = Y ) ) ).
% coeff_lift_hom.injectivity
thf(fact_1064_coeff__lift__hom_Oinjectivity,axiom,
! [X: int,Y: int] :
( ( ( pCons_int @ X @ zero_zero_poly_int )
= ( pCons_int @ Y @ zero_zero_poly_int ) )
=> ( X = Y ) ) ).
% coeff_lift_hom.injectivity
thf(fact_1065_coeff__lift__hom_Oinjectivity,axiom,
! [X: real,Y: real] :
( ( ( pCons_real @ X @ zero_zero_poly_real )
= ( pCons_real @ Y @ zero_zero_poly_real ) )
=> ( X = Y ) ) ).
% coeff_lift_hom.injectivity
thf(fact_1066_coeff__lift__hom_Oinjectivity,axiom,
! [X: poly_int,Y: poly_int] :
( ( ( pCons_poly_int @ X @ zero_z799223564134138693ly_int )
= ( pCons_poly_int @ Y @ zero_z799223564134138693ly_int ) )
=> ( X = Y ) ) ).
% coeff_lift_hom.injectivity
thf(fact_1067_coeff__lift__hom_Oinjectivity,axiom,
! [X: nat,Y: nat] :
( ( ( pCons_nat @ X @ zero_zero_poly_nat )
= ( pCons_nat @ Y @ zero_zero_poly_nat ) )
=> ( X = Y ) ) ).
% coeff_lift_hom.injectivity
thf(fact_1068_coeff__lift__hom_Oinjectivity,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n )
= ( pCons_6246009715029582078ring_n @ Y @ zero_z5482829069124612005ring_n ) )
=> ( X = Y ) ) ).
% coeff_lift_hom.injectivity
thf(fact_1069_coeff__lift__hom_Ohom__0,axiom,
! [X: poly_real] :
( ( ( pCons_poly_real @ X @ zero_z5583686468110200389y_real )
= zero_z5583686468110200389y_real )
=> ( X = zero_zero_poly_real ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1070_coeff__lift__hom_Ohom__0,axiom,
! [X: poly_poly_int] :
( ( ( pCons_poly_poly_int @ X @ zero_z240508265545053005ly_int )
= zero_z240508265545053005ly_int )
=> ( X = zero_z799223564134138693ly_int ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1071_coeff__lift__hom_Ohom__0,axiom,
! [X: poly_nat] :
( ( ( pCons_poly_nat @ X @ zero_z3289306709065865449ly_nat )
= zero_z3289306709065865449ly_nat )
=> ( X = zero_zero_poly_nat ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1072_coeff__lift__hom_Ohom__0,axiom,
! [X: poly_p6692042823160534382ring_n] :
( ( ( pCons_2385395009258896524ring_n @ X @ zero_z3442457038203223091ring_n )
= zero_z3442457038203223091ring_n )
=> ( X = zero_z5482829069124612005ring_n ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1073_coeff__lift__hom_Ohom__0,axiom,
! [X: finite_mod_ring_n] :
( ( ( pCons_8126420873123957872ring_n @ X @ zero_z2753989067526334999ring_n )
= zero_z2753989067526334999ring_n )
=> ( X = zero_z7902377597758090121ring_n ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1074_coeff__lift__hom_Ohom__0,axiom,
! [X: int] :
( ( ( pCons_int @ X @ zero_zero_poly_int )
= zero_zero_poly_int )
=> ( X = zero_zero_int ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1075_coeff__lift__hom_Ohom__0,axiom,
! [X: real] :
( ( ( pCons_real @ X @ zero_zero_poly_real )
= zero_zero_poly_real )
=> ( X = zero_zero_real ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1076_coeff__lift__hom_Ohom__0,axiom,
! [X: poly_int] :
( ( ( pCons_poly_int @ X @ zero_z799223564134138693ly_int )
= zero_z799223564134138693ly_int )
=> ( X = zero_zero_poly_int ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1077_coeff__lift__hom_Ohom__0,axiom,
! [X: nat] :
( ( ( pCons_nat @ X @ zero_zero_poly_nat )
= zero_zero_poly_nat )
=> ( X = zero_zero_nat ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1078_coeff__lift__hom_Ohom__0,axiom,
! [X: poly_F4222894760850802144ring_n] :
( ( ( pCons_6246009715029582078ring_n @ X @ zero_z5482829069124612005ring_n )
= zero_z5482829069124612005ring_n )
=> ( X = zero_z2753989067526334999ring_n ) ) ).
% coeff_lift_hom.hom_0
thf(fact_1079_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1080_poly__divides__pad__rule,axiom,
! [P: poly_p6692042823160534382ring_n,Q: poly_p6692042823160534382ring_n] :
( ( dvd_dv3135175980337127240ring_n @ P @ Q )
=> ( dvd_dv3135175980337127240ring_n @ P @ ( pCons_6246009715029582078ring_n @ zero_z2753989067526334999ring_n @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1081_poly__divides__pad__rule,axiom,
! [P: poly_int,Q: poly_int] :
( ( dvd_dvd_poly_int @ P @ Q )
=> ( dvd_dvd_poly_int @ P @ ( pCons_int @ zero_zero_int @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1082_poly__divides__pad__rule,axiom,
! [P: poly_poly_int,Q: poly_poly_int] :
( ( dvd_dv6998304861263046114ly_int @ P @ Q )
=> ( dvd_dv6998304861263046114ly_int @ P @ ( pCons_poly_int @ zero_zero_poly_int @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1083_poly__divides__pad__rule,axiom,
! [P: poly_real,Q: poly_real] :
( ( dvd_dvd_poly_real @ P @ Q )
=> ( dvd_dvd_poly_real @ P @ ( pCons_real @ zero_zero_real @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1084_poly__divides__pad__rule,axiom,
! [P: poly_poly_real,Q: poly_poly_real] :
( ( dvd_dv4532039564868358754y_real @ P @ Q )
=> ( dvd_dv4532039564868358754y_real @ P @ ( pCons_poly_real @ zero_zero_poly_real @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1085_poly__divides__pad__rule,axiom,
! [P: poly_poly_poly_int,Q: poly_poly_poly_int] :
( ( dvd_dv7705178354154678250ly_int @ P @ Q )
=> ( dvd_dv7705178354154678250ly_int @ P @ ( pCons_poly_poly_int @ zero_z799223564134138693ly_int @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1086_poly__divides__pad__rule,axiom,
! [P: poly_p2743341848350813180ring_n,Q: poly_p2743341848350813180ring_n] :
( ( dvd_dv3919477662729673174ring_n @ P @ Q )
=> ( dvd_dv3919477662729673174ring_n @ P @ ( pCons_2385395009258896524ring_n @ zero_z5482829069124612005ring_n @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1087_poly__divides__pad__rule,axiom,
! [P: poly_F4222894760850802144ring_n,Q: poly_F4222894760850802144ring_n] :
( ( dvd_dv8138414522854976442ring_n @ P @ Q )
=> ( dvd_dv8138414522854976442ring_n @ P @ ( pCons_8126420873123957872ring_n @ zero_z7902377597758090121ring_n @ Q ) ) ) ).
% poly_divides_pad_rule
thf(fact_1088_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1089_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1090_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1091_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1092_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1093_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_1094_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1095_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1096_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1097_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1098_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1099_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1100_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1101_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1102_euclidean__size__field__def,axiom,
( field_5283244131969691238d_real
= ( ^ [X3: real] : ( if_nat @ ( X3 = zero_zero_real ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% euclidean_size_field_def
thf(fact_1103_coeff__lift__hom_Ohom__signof,axiom,
! [P: nat > nat] :
( ( pCons_poly_int @ ( ring_17892525584911698563ly_int @ ( sign_nat @ P ) ) @ zero_z799223564134138693ly_int )
= ( ring_14695796289142966411ly_int @ ( sign_nat @ P ) ) ) ).
% coeff_lift_hom.hom_signof
thf(fact_1104_coeff__lift__hom_Ohom__signof,axiom,
! [P: nat > nat] :
( ( pCons_6246009715029582078ring_n @ ( ring_18712857867054464081ring_n @ ( sign_nat @ P ) ) @ zero_z5482829069124612005ring_n )
= ( ring_14208964510912816607ring_n @ ( sign_nat @ P ) ) ) ).
% coeff_lift_hom.hom_signof
thf(fact_1105_coeff__lift__hom_Ohom__signof,axiom,
! [P: nat > nat] :
( ( pCons_8126420873123957872ring_n @ ( ring_18169885536585341379ring_n @ ( sign_nat @ P ) ) @ zero_z2753989067526334999ring_n )
= ( ring_18712857867054464081ring_n @ ( sign_nat @ P ) ) ) ).
% coeff_lift_hom.hom_signof
thf(fact_1106_coeff__lift__hom_Ohom__signof,axiom,
! [P: nat > nat] :
( ( pCons_real @ ( ring_1_of_int_real @ ( sign_nat @ P ) ) @ zero_zero_poly_real )
= ( ring_12936506555246842115y_real @ ( sign_nat @ P ) ) ) ).
% coeff_lift_hom.hom_signof
thf(fact_1107_coeff__lift__hom_Ohom__signof,axiom,
! [P: nat > nat] :
( ( pCons_int @ ( ring_1_of_int_int @ ( sign_nat @ P ) ) @ zero_zero_poly_int )
= ( ring_17892525584911698563ly_int @ ( sign_nat @ P ) ) ) ).
% coeff_lift_hom.hom_signof
thf(fact_1108_psize__eq__0__iff,axiom,
! [P: poly_int] :
( ( ( fundam7803151596185808025ze_int @ P )
= zero_zero_nat )
= ( P = zero_zero_poly_int ) ) ).
% psize_eq_0_iff
thf(fact_1109_psize__eq__0__iff,axiom,
! [P: poly_real] :
( ( ( fundam22707326917796505e_real @ P )
= zero_zero_nat )
= ( P = zero_zero_poly_real ) ) ).
% psize_eq_0_iff
thf(fact_1110_psize__eq__0__iff,axiom,
! [P: poly_poly_int] :
( ( ( fundam3750135516382849185ly_int @ P )
= zero_zero_nat )
= ( P = zero_z799223564134138693ly_int ) ) ).
% psize_eq_0_iff
thf(fact_1111_psize__eq__0__iff,axiom,
! [P: poly_nat] :
( ( ( fundam7805642066694858301ze_nat @ P )
= zero_zero_nat )
= ( P = zero_zero_poly_nat ) ) ).
% psize_eq_0_iff
thf(fact_1112_psize__eq__0__iff,axiom,
! [P: poly_p6692042823160534382ring_n] :
( ( ( fundam881691483918006195ring_n @ P )
= zero_zero_nat )
= ( P = zero_z5482829069124612005ring_n ) ) ).
% psize_eq_0_iff
thf(fact_1113__C_K_C,axiom,
kyber_5366887534115960522ucible @ ( finite_card_n @ top_top_set_n ) @ p ).
% "*"
thf(fact_1114_card__nat,axiom,
( ( finite_card_nat @ top_top_set_nat )
= zero_zero_nat ) ).
% card_nat
thf(fact_1115_card__literal,axiom,
( ( finite_card_literal @ top_top_set_literal )
= zero_zero_nat ) ).
% card_literal
thf(fact_1116_mod__poly__irreducible__def,axiom,
( kyber_5366887534115960522ucible
= ( ^ [M2: nat,Q3: poly_int] :
( ~ ( kyber_mod_poly_rel @ M2 @ Q3 @ zero_zero_poly_int )
& ~ ( kyber_6337687584560828108s_unit @ M2 @ Q3 )
& ! [A2: poly_int,B2: poly_int] :
( ( kyber_mod_poly_rel @ M2 @ Q3 @ ( times_times_poly_int @ A2 @ B2 ) )
=> ( ( kyber_6337687584560828108s_unit @ M2 @ A2 )
| ( kyber_6337687584560828108s_unit @ M2 @ B2 ) ) ) ) ) ) ).
% mod_poly_irreducible_def
thf(fact_1117_mod__poly__is__unit__def,axiom,
( kyber_6337687584560828108s_unit
= ( ^ [M2: nat,P4: poly_int] :
? [R: poly_int] : ( kyber_mod_poly_rel @ M2 @ ( times_times_poly_int @ P4 @ R ) @ one_one_poly_int ) ) ) ).
% mod_poly_is_unit_def
thf(fact_1118_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1119_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1120_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1121_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_1122_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1123_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1124_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1125_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1126_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1127_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1128_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1129_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1130_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1131_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1132_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1133_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1134_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1135_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1136_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1137_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1138_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P2 @ M3 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_1139_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_1140_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1141_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1142_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1143_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1144_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1145_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1146_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1147_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1148_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1149_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1150_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1151_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1152_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1153_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1154_dvd__diff__nat,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ M )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1155_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1156_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1157_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1158_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1159_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D2: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X2 ) @ ( times_times_nat @ B @ Y2 ) )
= D2 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X2 ) @ ( times_times_nat @ A @ Y2 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_1160_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1161_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1162_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1163_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1164_dvd__mult__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1165_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1166_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1167_int__less__induct,axiom,
! [I: int,K2: int,P2: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P2 @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_1168_minusinfinity,axiom,
! [D: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P1 @ X2 )
= ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P2 @ X2 )
= ( P1 @ X2 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1169_plusinfinity,axiom,
! [D: int,P5: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int,K3: int] :
( ( P5 @ X2 )
= ( P5 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P2 @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [X_1: int] : ( P5 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1170_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_1171_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1172_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1173_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1174_zdvd__zdiffD,axiom,
! [K2: int,M: int,N: int] :
( ( dvd_dvd_int @ K2 @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K2 @ N )
=> ( dvd_dvd_int @ K2 @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1175_zmult__zless__mono2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1176_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1177_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1178_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_1179_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1180_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1181_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1182_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K2: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1183_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_1184_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1185_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1186_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1187_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_1188_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ( power_power_real @ X2 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1189_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_1190_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1191_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1192_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1193_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1194_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1195_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1196_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1197_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1198_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1199_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1200_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1201_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1202_real__sup__exists,axiom,
! [P2: real > $o] :
( ? [X_1: real] : ( P2 @ X_1 )
=> ( ? [Z4: real] :
! [X2: real] :
( ( P2 @ X2 )
=> ( ord_less_real @ X2 @ Z4 ) )
=> ? [S2: real] :
! [Y4: real] :
( ( ? [X3: real] :
( ( P2 @ X3 )
& ( ord_less_real @ Y4 @ X3 ) ) )
= ( ord_less_real @ Y4 @ S2 ) ) ) ) ).
% real_sup_exists
thf(fact_1203_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1204_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
= ( ? [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 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_1205_forall__pos__mono,axiom,
! [P2: real > $o,E: real] :
( ! [D2: real,E2: real] :
( ( ord_less_real @ D2 @ E2 )
=> ( ( P2 @ D2 )
=> ( P2 @ E2 ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P2 @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_1206_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1207_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1208_real__divide__square__eq,axiom,
! [R3: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R3 @ A ) @ ( times_times_real @ R3 @ R3 ) )
= ( divide_divide_real @ A @ R3 ) ) ).
% real_divide_square_eq
thf(fact_1209_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1210_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1211_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1212_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1213_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1214_real__of__int__div,axiom,
! [D: int,N: int] :
( ( dvd_dvd_int @ D @ N )
=> ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
= ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% real_of_int_div
thf(fact_1215_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( ( dvd_dvd_nat @ D @ N )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_1216_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1217_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_1218_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_1219_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_1220_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_1221_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X3: real,Y5: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y5 ) ) ) ) ).
% divide_real_def
thf(fact_1222_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1223_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1224_nat__mult__div__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1225_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1226_int__div__less__self,axiom,
! [X: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K2 )
=> ( ord_less_int @ ( divide_divide_int @ X @ K2 ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1227_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_1228_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K3 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1229_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1230_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1231_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1232_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1233_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1234_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1235_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1236_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1237_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1238_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1239_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1240_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1241_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1242_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).
% dvd_1_left
thf(fact_1243_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1244_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1245_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1246_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1247_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1248_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1249_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1250_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1251_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1252_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1253_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_1254_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1255_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1256_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1257_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1258_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1259_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1260_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1261_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I: nat] :
( ( P2 @ K2 )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1262_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1263_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1264_int__of__nat__induct,axiom,
! [P2: int > $o,Z: int] :
( ! [N2: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P2 @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1265_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1266_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1267_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int,T2: int] : ( dvd_dvd_int @ D3 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_1268_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D3 ) ) ) ) ).
% uminus_dvd_conv(1)
% Helper facts (21)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Int__Oint_J_T,axiom,
! [X: poly_int,Y: poly_int] :
( ( if_poly_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Int__Oint_J_T,axiom,
! [X: poly_int,Y: poly_int] :
( ( if_poly_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Nat__Onat_J_T,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( if_poly_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Nat__Onat_J_T,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( if_poly_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Real__Oreal_J_T,axiom,
! [X: poly_real,Y: poly_real] :
( ( if_poly_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Real__Oreal_J_T,axiom,
! [X: poly_real,Y: poly_real] :
( ( if_poly_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Finite____Field__Omod____ring_Itf__n_J_T,axiom,
! [X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( if_Finite_mod_ring_n @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Field__Omod____ring_Itf__n_J_T,axiom,
! [X: finite_mod_ring_n,Y: finite_mod_ring_n] :
( ( if_Finite_mod_ring_n @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J_T,axiom,
! [X: poly_poly_int,Y: poly_poly_int] :
( ( if_poly_poly_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Int__Oint_J_J_T,axiom,
! [X: poly_poly_int,Y: poly_poly_int] :
( ( if_poly_poly_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_T,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( if_pol9129390727684501670ring_n @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_T,axiom,
! [X: poly_F4222894760850802144ring_n,Y: poly_F4222894760850802144ring_n] :
( ( if_pol9129390727684501670ring_n @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_T,axiom,
! [X: poly_p6692042823160534382ring_n,Y: poly_p6692042823160534382ring_n] :
( ( if_pol5337421645260315700ring_n @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__n_J_J_J_T,axiom,
! [X: poly_p6692042823160534382ring_n,Y: poly_p6692042823160534382ring_n] :
( ( if_pol5337421645260315700ring_n @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ p )
!= zero_z2753989067526334999ring_n )
& ~ ( dvd_dv8138414522854976442ring_n @ ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ p ) @ one_on4318287115420659547ring_n )
& ! [A3: poly_F4222894760850802144ring_n,B4: poly_F4222894760850802144ring_n] :
( ( ( map_po1011533443592629756ring_n @ ring_18169885536585341379ring_n @ p )
!= ( times_4166049284782705435ring_n @ A3 @ B4 ) )
| ( dvd_dv8138414522854976442ring_n @ A3 @ one_on4318287115420659547ring_n )
| ( dvd_dv8138414522854976442ring_n @ B4 @ one_on4318287115420659547ring_n ) ) ) ).
%------------------------------------------------------------------------------