TPTP Problem File: SLH0451^1.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/0025_NTT_Scheme/prob_00132_003732__25778214_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1405 ( 647 unt; 129 typ; 0 def)
% Number of atoms : 3336 (1447 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 10227 ( 274 ~; 100 |; 173 &;8431 @)
% ( 0 <=>;1249 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 297 ( 297 >; 0 *; 0 +; 0 <<)
% Number of symbols : 120 ( 117 usr; 22 con; 0-4 aty)
% Number of variables : 2845 ( 114 ^;2595 !; 136 ?;2845 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:40:47.867
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
poly_F3299452240248304339ring_a: $tType ).
thf(ty_n_t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
list_F4626807571770296779ring_a: $tType ).
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $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__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (117)
thf(sy_c_Determinant_Odelete__index,type,
delete_index: nat > nat > nat ).
thf(sy_c_Determinant_Opermutation__delete,type,
permutation_delete: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Int__Oint,type,
permut3692553072317293667rt_int: int > nat > ( int > nat ) > int > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Nat__Onat,type,
permut3695043542826343943rt_nat: nat > nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Determinant_Opermutation__insert_001t__Real__Oreal,type,
permut4060954620988167523t_real: real > nat > ( real > nat ) > real > nat ).
thf(sy_c_Field__as__Ring_Ofield__class_Oeuclidean__size__field_001t__Real__Oreal,type,
field_5283244131969691238d_real: real > nat ).
thf(sy_c_Field__as__Ring_Ofield__class_Omod__field_001t__Real__Oreal,type,
field_341224784244110787d_real: real > real > real ).
thf(sy_c_Field__as__Ring_Ofield__class_Onormalize__field_001t__Finite____Field__Omod____ring_Itf__a_J,type,
field_3121160262079256089ring_a: finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Field__as__Ring_Ofield__class_Onormalize__field_001t__Real__Oreal,type,
field_8354674766439439704d_real: real > real ).
thf(sy_c_Fundamental__Theorem__Algebra_Opsize_001t__Finite____Field__Omod____ring_Itf__a_J,type,
fundam4324422711134394264ring_a: poly_F3299452240248304339ring_a > nat ).
thf(sy_c_Fundamental__Theorem__Algebra_Opsize_001t__Nat__Onat,type,
fundam7805642066694858301ze_nat: poly_nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
minus_3609261664126569004ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
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__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
minus_5354101470050066234ring_a: poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Int__Oint_J,type,
minus_minus_poly_int: poly_int > poly_int > poly_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
minus_minus_poly_nat: poly_nat > poly_nat > poly_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__a_J,type,
one_on2109788427901206336ring_a: finite_mod_ring_a ).
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__a_J_J,type,
one_on3394844594818161742ring_a: poly_F3299452240248304339ring_a ).
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__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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Finite____Field__Omod____ring_Itf__a_J,type,
times_5121417576591743744ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
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__a_J_J,type,
times_3242606764180207630ring_a: poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ).
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__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_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zero_z7902377541816115708ring_a: finite_mod_ring_a ).
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__a_J_J,type,
zero_z1830546546923837194ring_a: poly_F3299452240248304339ring_a ).
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__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__a_J,type,
if_Finite_mod_ring_a: $o > finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
map_na1928064127006292399ring_a: ( nat > finite_mod_ring_a ) > list_nat > list_F4626807571770296779ring_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Missing__Polynomial_Oleading__coeff_001t__Finite____Field__Omod____ring_Itf__a_J,type,
missin871596229213354819ring_a: poly_F3299452240248304339ring_a > finite_mod_ring_a ).
thf(sy_c_Missing__Polynomial_Oleading__coeff_001t__Nat__Onat,type,
missin2378734267208948434ff_nat: poly_nat > nat ).
thf(sy_c_NTT_Ontt__axioms_001tf__a,type,
ntt_axioms_a: nat > finite_mod_ring_a > finite_mod_ring_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
semiri9180929696517417892ring_a: nat > finite_mod_ring_a ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
semiri8000969770135892146ring_a: nat > poly_F3299452240248304339ring_a ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Int__Oint_J,type,
semiri6323754628967941525ly_int: nat > poly_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
semiri1278233611622362425ly_nat: nat > poly_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
semiri1039972187744592661y_real: nat > poly_real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
size_s7115545719440041015ring_a: list_F4626807571770296779ring_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Finite____Field__Omod____ring_Itf__a_J,type,
neg_nu5901776551076858996ring_a: finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Polynomial_OPoly_001t__Finite____Field__Omod____ring_Itf__a_J,type,
poly_F5739129160929385880ring_a: list_F4626807571770296779ring_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Polynomial_OPoly_001t__Nat__Onat,type,
poly_nat2: list_nat > poly_nat ).
thf(sy_c_Polynomial_Odegree_001t__Finite____Field__Omod____ring_Itf__a_J,type,
degree4881254707062955960ring_a: poly_F3299452240248304339ring_a > nat ).
thf(sy_c_Polynomial_Odegree_001t__Int__Oint,type,
degree_int: poly_int > nat ).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat,type,
degree_nat: poly_nat > nat ).
thf(sy_c_Polynomial_Odegree_001t__Real__Oreal,type,
degree_real: poly_real > nat ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Finite____Field__Omod____ring_Itf__a_J,type,
coeff_1607515655354303335ring_a: poly_F3299452240248304339ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Int__Oint,type,
coeff_int: poly_int > nat > int ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat,type,
coeff_nat: poly_nat > nat > nat ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
coeff_real: poly_real > nat > real ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Finite____Field__Omod____ring_Itf__a_J,type,
poly_c8149583573515411563ring_a: nat > poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ).
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__Real__Oreal,type,
poly_cutoff_real: nat > poly_real > poly_real ).
thf(sy_c_Polynomial_Opos__poly_001t__Int__Oint,type,
pos_poly_int: poly_int > $o ).
thf(sy_c_Polynomial_Opos__poly_001t__Nat__Onat,type,
pos_poly_nat: poly_nat > $o ).
thf(sy_c_Polynomial_Opos__poly_001t__Real__Oreal,type,
pos_poly_real: poly_real > $o ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Finite____Field__Omod____ring_Itf__a_J,type,
reflec4498816349307343611ring_a: poly_F3299452240248304339ring_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Int__Oint,type,
reflect_poly_int: poly_int > poly_int ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
reflect_poly_nat: poly_nat > poly_nat ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
reflect_poly_real: poly_real > poly_real ).
thf(sy_c_Polynomial_Osynthetic__div_001t__Finite____Field__Omod____ring_Itf__a_J,type,
synthe7653204327264257080ring_a: poly_F3299452240248304339ring_a > finite_mod_ring_a > poly_F3299452240248304339ring_a ).
thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat,type,
synthetic_div_nat: poly_nat > nat > poly_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Finite____Field__Omod____ring_Itf__a_J,type,
power_6826135765519566523ring_a: finite_mod_ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Finite____Field__Omod____ring_Itf__a_J_J,type,
power_6500929916544582089ring_a: poly_F3299452240248304339ring_a > nat > poly_F3299452240248304339ring_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Int__Oint_J,type,
power_power_poly_int: poly_int > nat > poly_int ).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
power_power_poly_nat: poly_nat > nat > poly_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
power_8994544051960338110y_real: poly_real > nat > poly_real ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v__092_060mu_062,type,
mu: finite_mod_ring_a ).
thf(sy_v__092_060omega_062,type,
omega: finite_mod_ring_a ).
thf(sy_v_f,type,
f: nat > finite_mod_ring_a ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_n_H,type,
n2: nat ).
% Relevant facts (1266)
thf(fact_0_n_H__gr__0,axiom,
ord_less_nat @ zero_zero_nat @ n2 ).
% n'_gr_0
thf(fact_1_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_2_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_3_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_4_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_5_Poly__map__coeff,axiom,
! [F: poly_nat,Num: nat] :
( ( ord_less_nat @ ( degree_nat @ F ) @ Num )
=> ( ( poly_nat2 @ ( map_nat_nat @ ( coeff_nat @ F ) @ ( upt @ zero_zero_nat @ Num ) ) )
= F ) ) ).
% Poly_map_coeff
thf(fact_6_Poly__map__coeff,axiom,
! [F: poly_F3299452240248304339ring_a,Num: nat] :
( ( ord_less_nat @ ( degree4881254707062955960ring_a @ F ) @ Num )
=> ( ( poly_F5739129160929385880ring_a @ ( map_na1928064127006292399ring_a @ ( coeff_1607515655354303335ring_a @ F ) @ ( upt @ zero_zero_nat @ Num ) ) )
= F ) ) ).
% Poly_map_coeff
thf(fact_7_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_8_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_9_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_10_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_11_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_12_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_13_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_14_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_15_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_16_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_17_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_18_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_19_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_20_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_21_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_22_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_23_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_24_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_25_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_26_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_27_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_28_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_29_leading__coeff__0__iff,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) )
= zero_z7902377541816115708ring_a )
= ( P2 = zero_z1830546546923837194ring_a ) ) ).
% leading_coeff_0_iff
thf(fact_30_leading__coeff__0__iff,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ P2 @ ( degree_nat @ P2 ) )
= zero_zero_nat )
= ( P2 = zero_zero_poly_nat ) ) ).
% leading_coeff_0_iff
thf(fact_31_leading__coeff__0__iff,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ P2 @ ( degree_int @ P2 ) )
= zero_zero_int )
= ( P2 = zero_zero_poly_int ) ) ).
% leading_coeff_0_iff
thf(fact_32_leading__coeff__0__iff,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ P2 @ ( degree_real @ P2 ) )
= zero_zero_real )
= ( P2 = zero_zero_poly_real ) ) ).
% leading_coeff_0_iff
thf(fact_33_degree__0,axiom,
( ( degree4881254707062955960ring_a @ zero_z1830546546923837194ring_a )
= zero_zero_nat ) ).
% degree_0
thf(fact_34_degree__0,axiom,
( ( degree_nat @ zero_zero_poly_nat )
= zero_zero_nat ) ).
% degree_0
thf(fact_35_coeff__0,axiom,
! [N: nat] :
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a @ N )
= zero_z7902377541816115708ring_a ) ).
% coeff_0
thf(fact_36_coeff__0,axiom,
! [N: nat] :
( ( coeff_nat @ zero_zero_poly_nat @ N )
= zero_zero_nat ) ).
% coeff_0
thf(fact_37_coeff__0,axiom,
! [N: nat] :
( ( coeff_int @ zero_zero_poly_int @ N )
= zero_zero_int ) ).
% coeff_0
thf(fact_38_coeff__0,axiom,
! [N: nat] :
( ( coeff_real @ zero_zero_poly_real @ N )
= zero_zero_real ) ).
% coeff_0
thf(fact_39_less__degree__imp,axiom,
! [N: nat,P2: poly_F3299452240248304339ring_a] :
( ( ord_less_nat @ N @ ( degree4881254707062955960ring_a @ P2 ) )
=> ? [I: nat] :
( ( ord_less_nat @ N @ I )
& ( ( coeff_1607515655354303335ring_a @ P2 @ I )
!= zero_z7902377541816115708ring_a ) ) ) ).
% less_degree_imp
thf(fact_40_less__degree__imp,axiom,
! [N: nat,P2: poly_nat] :
( ( ord_less_nat @ N @ ( degree_nat @ P2 ) )
=> ? [I: nat] :
( ( ord_less_nat @ N @ I )
& ( ( coeff_nat @ P2 @ I )
!= zero_zero_nat ) ) ) ).
% less_degree_imp
thf(fact_41_less__degree__imp,axiom,
! [N: nat,P2: poly_int] :
( ( ord_less_nat @ N @ ( degree_int @ P2 ) )
=> ? [I: nat] :
( ( ord_less_nat @ N @ I )
& ( ( coeff_int @ P2 @ I )
!= zero_zero_int ) ) ) ).
% less_degree_imp
thf(fact_42_less__degree__imp,axiom,
! [N: nat,P2: poly_real] :
( ( ord_less_nat @ N @ ( degree_real @ P2 ) )
=> ? [I: nat] :
( ( ord_less_nat @ N @ I )
& ( ( coeff_real @ P2 @ I )
!= zero_zero_real ) ) ) ).
% less_degree_imp
thf(fact_43_coeff__eq__0,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( ord_less_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N )
=> ( ( coeff_1607515655354303335ring_a @ P2 @ N )
= zero_z7902377541816115708ring_a ) ) ).
% coeff_eq_0
thf(fact_44_coeff__eq__0,axiom,
! [P2: poly_nat,N: nat] :
( ( ord_less_nat @ ( degree_nat @ P2 ) @ N )
=> ( ( coeff_nat @ P2 @ N )
= zero_zero_nat ) ) ).
% coeff_eq_0
thf(fact_45_coeff__eq__0,axiom,
! [P2: poly_int,N: nat] :
( ( ord_less_nat @ ( degree_int @ P2 ) @ N )
=> ( ( coeff_int @ P2 @ N )
= zero_zero_int ) ) ).
% coeff_eq_0
thf(fact_46_coeff__eq__0,axiom,
! [P2: poly_real,N: nat] :
( ( ord_less_nat @ ( degree_real @ P2 ) @ N )
=> ( ( coeff_real @ P2 @ N )
= zero_zero_real ) ) ).
% coeff_eq_0
thf(fact_47_leading__coeff__neq__0,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( P2 != zero_z1830546546923837194ring_a )
=> ( ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) )
!= zero_z7902377541816115708ring_a ) ) ).
% leading_coeff_neq_0
thf(fact_48_leading__coeff__neq__0,axiom,
! [P2: poly_nat] :
( ( P2 != zero_zero_poly_nat )
=> ( ( coeff_nat @ P2 @ ( degree_nat @ P2 ) )
!= zero_zero_nat ) ) ).
% leading_coeff_neq_0
thf(fact_49_leading__coeff__neq__0,axiom,
! [P2: poly_int] :
( ( P2 != zero_zero_poly_int )
=> ( ( coeff_int @ P2 @ ( degree_int @ P2 ) )
!= zero_zero_int ) ) ).
% leading_coeff_neq_0
thf(fact_50_leading__coeff__neq__0,axiom,
! [P2: poly_real] :
( ( P2 != zero_zero_poly_real )
=> ( ( coeff_real @ P2 @ ( degree_real @ P2 ) )
!= zero_zero_real ) ) ).
% leading_coeff_neq_0
thf(fact_51_zero__poly_Orep__eq,axiom,
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a )
= ( ^ [Uu: nat] : zero_z7902377541816115708ring_a ) ) ).
% zero_poly.rep_eq
thf(fact_52_zero__poly_Orep__eq,axiom,
( ( coeff_nat @ zero_zero_poly_nat )
= ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% zero_poly.rep_eq
thf(fact_53_zero__poly_Orep__eq,axiom,
( ( coeff_int @ zero_zero_poly_int )
= ( ^ [Uu: nat] : zero_zero_int ) ) ).
% zero_poly.rep_eq
thf(fact_54_zero__poly_Orep__eq,axiom,
( ( coeff_real @ zero_zero_poly_real )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_poly.rep_eq
thf(fact_55_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_56_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_57_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_58_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_59_n__gt__zero,axiom,
ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ n ) ).
% n_gt_zero
thf(fact_60_n__nonzero,axiom,
( ( semiri1314217659103216013at_int @ n )
!= zero_zero_int ) ).
% n_nonzero
thf(fact_61_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_62_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_63_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_64_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_65_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_66_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_degree__of__nat,axiom,
! [N: nat] :
( ( degree4881254707062955960ring_a @ ( semiri8000969770135892146ring_a @ N ) )
= zero_zero_nat ) ).
% degree_of_nat
thf(fact_69_degree__of__nat,axiom,
! [N: nat] :
( ( degree_nat @ ( semiri1278233611622362425ly_nat @ N ) )
= zero_zero_nat ) ).
% degree_of_nat
thf(fact_70_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_71_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_72_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_73_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_74_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_75_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_76_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_77_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_78_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_79_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_80_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_81_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_82_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_1607515655354303335ring_a @ ( semiri8000969770135892146ring_a @ N ) @ ( degree4881254707062955960ring_a @ ( semiri8000969770135892146ring_a @ N ) ) )
= ( semiri9180929696517417892ring_a @ N ) ) ).
% lead_coeff_of_nat
thf(fact_83_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_nat @ ( semiri1278233611622362425ly_nat @ N ) @ ( degree_nat @ ( semiri1278233611622362425ly_nat @ N ) ) )
= ( semiri1316708129612266289at_nat @ N ) ) ).
% lead_coeff_of_nat
thf(fact_84_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_int @ ( semiri6323754628967941525ly_int @ N ) @ ( degree_int @ ( semiri6323754628967941525ly_int @ N ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% lead_coeff_of_nat
thf(fact_85_lead__coeff__of__nat,axiom,
! [N: nat] :
( ( coeff_real @ ( semiri1039972187744592661y_real @ N ) @ ( degree_real @ ( semiri1039972187744592661y_real @ N ) ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% lead_coeff_of_nat
thf(fact_86_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_87_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_88_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_89_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_90_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_91_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_92_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_93_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_94_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_95_poly__eq__iff,axiom,
( ( ^ [Y2: poly_F3299452240248304339ring_a,Z: poly_F3299452240248304339ring_a] : ( Y2 = Z ) )
= ( ^ [P3: poly_F3299452240248304339ring_a,Q: poly_F3299452240248304339ring_a] :
! [N3: nat] :
( ( coeff_1607515655354303335ring_a @ P3 @ N3 )
= ( coeff_1607515655354303335ring_a @ Q @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_96_poly__eq__iff,axiom,
( ( ^ [Y2: poly_nat,Z: poly_nat] : ( Y2 = Z ) )
= ( ^ [P3: poly_nat,Q: poly_nat] :
! [N3: nat] :
( ( coeff_nat @ P3 @ N3 )
= ( coeff_nat @ Q @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_97_poly__eqI,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ! [N2: nat] :
( ( coeff_1607515655354303335ring_a @ P2 @ N2 )
= ( coeff_1607515655354303335ring_a @ Q2 @ N2 ) )
=> ( P2 = Q2 ) ) ).
% poly_eqI
thf(fact_98_poly__eqI,axiom,
! [P2: poly_nat,Q2: poly_nat] :
( ! [N2: nat] :
( ( coeff_nat @ P2 @ N2 )
= ( coeff_nat @ Q2 @ N2 ) )
=> ( P2 = Q2 ) ) ).
% poly_eqI
thf(fact_99_coeff__inject,axiom,
! [X: poly_F3299452240248304339ring_a,Y: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ X )
= ( coeff_1607515655354303335ring_a @ Y ) )
= ( X = Y ) ) ).
% coeff_inject
thf(fact_100_coeff__inject,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( ( coeff_nat @ X )
= ( coeff_nat @ Y ) )
= ( X = Y ) ) ).
% coeff_inject
thf(fact_101_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_102_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_103_n__gt__1,axiom,
ord_less_int @ one_one_int @ ( semiri1314217659103216013at_int @ n ) ).
% n_gt_1
thf(fact_104_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_105_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_106_nat__n,axiom,
( ( semiri1314217659103216013at_int @ ( nat2 @ ( semiri1314217659103216013at_int @ n ) ) )
= ( semiri1314217659103216013at_int @ n ) ) ).
% nat_n
thf(fact_107_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_108_coeff__poly__cutoff,axiom,
! [K: nat,N: nat,P2: poly_F3299452240248304339ring_a] :
( ( ( ord_less_nat @ K @ N )
=> ( ( coeff_1607515655354303335ring_a @ ( poly_c8149583573515411563ring_a @ N @ P2 ) @ K )
= ( coeff_1607515655354303335ring_a @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N )
=> ( ( coeff_1607515655354303335ring_a @ ( poly_c8149583573515411563ring_a @ N @ P2 ) @ K )
= zero_z7902377541816115708ring_a ) ) ) ).
% coeff_poly_cutoff
thf(fact_109_coeff__poly__cutoff,axiom,
! [K: nat,N: nat,P2: poly_nat] :
( ( ( ord_less_nat @ K @ N )
=> ( ( coeff_nat @ ( poly_cutoff_nat @ N @ P2 ) @ K )
= ( coeff_nat @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N )
=> ( ( coeff_nat @ ( poly_cutoff_nat @ N @ P2 ) @ K )
= zero_zero_nat ) ) ) ).
% coeff_poly_cutoff
thf(fact_110_coeff__poly__cutoff,axiom,
! [K: nat,N: nat,P2: poly_int] :
( ( ( ord_less_nat @ K @ N )
=> ( ( coeff_int @ ( poly_cutoff_int @ N @ P2 ) @ K )
= ( coeff_int @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N )
=> ( ( coeff_int @ ( poly_cutoff_int @ N @ P2 ) @ K )
= zero_zero_int ) ) ) ).
% coeff_poly_cutoff
thf(fact_111_coeff__poly__cutoff,axiom,
! [K: nat,N: nat,P2: poly_real] :
( ( ( ord_less_nat @ K @ N )
=> ( ( coeff_real @ ( poly_cutoff_real @ N @ P2 ) @ K )
= ( coeff_real @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N )
=> ( ( coeff_real @ ( poly_cutoff_real @ N @ P2 ) @ K )
= zero_zero_real ) ) ) ).
% coeff_poly_cutoff
thf(fact_112_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_113_pos__poly__def,axiom,
( pos_poly_nat
= ( ^ [P3: poly_nat] : ( ord_less_nat @ zero_zero_nat @ ( coeff_nat @ P3 @ ( degree_nat @ P3 ) ) ) ) ) ).
% pos_poly_def
thf(fact_114_pos__poly__def,axiom,
( pos_poly_int
= ( ^ [P3: poly_int] : ( ord_less_int @ zero_zero_int @ ( coeff_int @ P3 @ ( degree_int @ P3 ) ) ) ) ) ).
% pos_poly_def
thf(fact_115_pos__poly__def,axiom,
( pos_poly_real
= ( ^ [P3: poly_real] : ( ord_less_real @ zero_zero_real @ ( coeff_real @ P3 @ ( degree_real @ P3 ) ) ) ) ) ).
% pos_poly_def
thf(fact_116_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_117_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_118_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_119_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_120_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_121_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_122_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_123_of__nat__1,axiom,
( ( semiri9180929696517417892ring_a @ one_one_nat )
= one_on2109788427901206336ring_a ) ).
% of_nat_1
thf(fact_124_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_125_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_126_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_127_lead__coeff__1,axiom,
( ( coeff_int @ one_one_poly_int @ ( degree_int @ one_one_poly_int ) )
= one_one_int ) ).
% lead_coeff_1
thf(fact_128_lead__coeff__1,axiom,
( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
= one_one_real ) ).
% lead_coeff_1
thf(fact_129_lead__coeff__1,axiom,
( ( coeff_1607515655354303335ring_a @ one_on3394844594818161742ring_a @ ( degree4881254707062955960ring_a @ one_on3394844594818161742ring_a ) )
= one_on2109788427901206336ring_a ) ).
% lead_coeff_1
thf(fact_130_lead__coeff__1,axiom,
( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
= one_one_nat ) ).
% lead_coeff_1
thf(fact_131_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_132_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_133_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_134_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_135_one__reorient,axiom,
! [X: finite_mod_ring_a] :
( ( one_on2109788427901206336ring_a = X )
= ( X = one_on2109788427901206336ring_a ) ) ).
% one_reorient
thf(fact_136_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_137_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_138_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_139_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_140_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_141_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_142_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_143_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_144_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_145_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_146_split__nat,axiom,
! [P: nat > $o,I2: int] :
( ( P @ ( nat2 @ I2 ) )
= ( ! [N3: nat] :
( ( I2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I2 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_147_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_148_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_149_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_150_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_151_int__if,axiom,
! [P: $o,A: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_152_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_153_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= one_one_nat )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_154_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= one_one_int )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_155_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= one_one_real )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_156_monic__degree__0,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ P2 @ ( degree_int @ P2 ) )
= one_one_int )
=> ( ( ( degree_int @ P2 )
= zero_zero_nat )
= ( P2 = one_one_poly_int ) ) ) ).
% monic_degree_0
thf(fact_157_monic__degree__0,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ P2 @ ( degree_real @ P2 ) )
= one_one_real )
=> ( ( ( degree_real @ P2 )
= zero_zero_nat )
= ( P2 = one_one_poly_real ) ) ) ).
% monic_degree_0
thf(fact_158_monic__degree__0,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) )
= one_on2109788427901206336ring_a )
=> ( ( ( degree4881254707062955960ring_a @ P2 )
= zero_zero_nat )
= ( P2 = one_on3394844594818161742ring_a ) ) ) ).
% monic_degree_0
thf(fact_159_monic__degree__0,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ P2 @ ( degree_nat @ P2 ) )
= one_one_nat )
=> ( ( ( degree_nat @ P2 )
= zero_zero_nat )
= ( P2 = one_one_poly_nat ) ) ) ).
% monic_degree_0
thf(fact_160_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_161_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_162_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_163_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_164_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_165_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_166_zero__neq__one,axiom,
zero_z7902377541816115708ring_a != one_on2109788427901206336ring_a ).
% zero_neq_one
thf(fact_167_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_168_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_169_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_170_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_171_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5901776551076858996ring_a @ zero_z7902377541816115708ring_a )
= one_on2109788427901206336ring_a ) ).
% dbl_inc_simps(2)
thf(fact_172_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_173_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_174_degree__reflect__poly__eq,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ zero_zero_nat )
!= zero_z7902377541816115708ring_a )
=> ( ( degree4881254707062955960ring_a @ ( reflec4498816349307343611ring_a @ P2 ) )
= ( degree4881254707062955960ring_a @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_175_degree__reflect__poly__eq,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ P2 @ zero_zero_nat )
!= zero_zero_nat )
=> ( ( degree_nat @ ( reflect_poly_nat @ P2 ) )
= ( degree_nat @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_176_degree__reflect__poly__eq,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ P2 @ zero_zero_nat )
!= zero_zero_int )
=> ( ( degree_int @ ( reflect_poly_int @ P2 ) )
= ( degree_int @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_177_degree__reflect__poly__eq,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ P2 @ zero_zero_nat )
!= zero_zero_real )
=> ( ( degree_real @ ( reflect_poly_real @ P2 ) )
= ( degree_real @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_178_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ ( reflec4498816349307343611ring_a @ P2 ) @ zero_zero_nat )
= zero_z7902377541816115708ring_a )
= ( P2 = zero_z1830546546923837194ring_a ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_179_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ ( reflect_poly_nat @ P2 ) @ zero_zero_nat )
= zero_zero_nat )
= ( P2 = zero_zero_poly_nat ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_180_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ ( reflect_poly_int @ P2 ) @ zero_zero_nat )
= zero_zero_int )
= ( P2 = zero_zero_poly_int ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_181_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ ( reflect_poly_real @ P2 ) @ zero_zero_nat )
= zero_zero_real )
= ( P2 = zero_zero_poly_real ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_182_synthetic__div__eq__0__iff,axiom,
! [P2: poly_F3299452240248304339ring_a,C: finite_mod_ring_a] :
( ( ( synthe7653204327264257080ring_a @ P2 @ C )
= zero_z1830546546923837194ring_a )
= ( ( degree4881254707062955960ring_a @ P2 )
= zero_zero_nat ) ) ).
% synthetic_div_eq_0_iff
thf(fact_183_synthetic__div__eq__0__iff,axiom,
! [P2: poly_nat,C: nat] :
( ( ( synthetic_div_nat @ P2 @ C )
= zero_zero_poly_nat )
= ( ( degree_nat @ P2 )
= zero_zero_nat ) ) ).
% synthetic_div_eq_0_iff
thf(fact_184_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_185_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_186_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_187_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_188_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_189_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_190_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_191_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_192_degree__1,axiom,
( ( degree4881254707062955960ring_a @ one_on3394844594818161742ring_a )
= zero_zero_nat ) ).
% degree_1
thf(fact_193_degree__1,axiom,
( ( degree_nat @ one_one_poly_nat )
= zero_zero_nat ) ).
% degree_1
thf(fact_194_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_195_reflect__poly__reflect__poly,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ zero_zero_nat )
!= zero_z7902377541816115708ring_a )
=> ( ( reflec4498816349307343611ring_a @ ( reflec4498816349307343611ring_a @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_196_reflect__poly__reflect__poly,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ P2 @ zero_zero_nat )
!= zero_zero_nat )
=> ( ( reflect_poly_nat @ ( reflect_poly_nat @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_197_reflect__poly__reflect__poly,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ P2 @ zero_zero_nat )
!= zero_zero_int )
=> ( ( reflect_poly_int @ ( reflect_poly_int @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_198_reflect__poly__reflect__poly,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ P2 @ zero_zero_nat )
!= zero_zero_real )
=> ( ( reflect_poly_real @ ( reflect_poly_real @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_199_coeff__0__reflect__poly,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( coeff_1607515655354303335ring_a @ ( reflec4498816349307343611ring_a @ P2 ) @ zero_zero_nat )
= ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) ) ) ).
% coeff_0_reflect_poly
thf(fact_200_coeff__0__reflect__poly,axiom,
! [P2: poly_nat] :
( ( coeff_nat @ ( reflect_poly_nat @ P2 ) @ zero_zero_nat )
= ( coeff_nat @ P2 @ ( degree_nat @ P2 ) ) ) ).
% coeff_0_reflect_poly
thf(fact_201_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_202_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_203_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_204_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_205_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_206_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_207_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_208_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_209_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_210_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_211_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_212_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_213_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_214_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_215_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_216_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_217_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_218_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_219_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_220_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_221_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_222_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_223_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_224_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_225_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_226_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_227_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_228_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_229_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_230_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_231_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_232_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_233_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I: nat] :
( ( J2
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_234_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_235_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_236_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_237_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_238_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_239_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_240_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_241_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_242_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_243_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_244_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_245_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_246_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_247_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_248_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_249_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_250_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_251_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_252_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_253_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_254_leading__coeff__def,axiom,
( missin871596229213354819ring_a
= ( ^ [P3: poly_F3299452240248304339ring_a] : ( coeff_1607515655354303335ring_a @ P3 @ ( degree4881254707062955960ring_a @ P3 ) ) ) ) ).
% leading_coeff_def
thf(fact_255_leading__coeff__def,axiom,
( missin2378734267208948434ff_nat
= ( ^ [P3: poly_nat] : ( coeff_nat @ P3 @ ( degree_nat @ P3 ) ) ) ) ).
% leading_coeff_def
thf(fact_256_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_257_fib_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ ( suc @ N2 ) ) ) ) ) ).
% fib.cases
thf(fact_258_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_259_euclidean__size__field__def,axiom,
( field_5283244131969691238d_real
= ( ^ [X2: real] : ( if_nat @ ( X2 = zero_zero_real ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% euclidean_size_field_def
thf(fact_260_mu__properties_I2_J,axiom,
mu != one_on2109788427901206336ring_a ).
% mu_properties(2)
thf(fact_261_coeff__reflect__poly,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( ( ord_less_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N )
=> ( ( coeff_1607515655354303335ring_a @ ( reflec4498816349307343611ring_a @ P2 ) @ N )
= zero_z7902377541816115708ring_a ) )
& ( ~ ( ord_less_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N )
=> ( ( coeff_1607515655354303335ring_a @ ( reflec4498816349307343611ring_a @ P2 ) @ N )
= ( coeff_1607515655354303335ring_a @ P2 @ ( minus_minus_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N ) ) ) ) ) ).
% coeff_reflect_poly
thf(fact_262_coeff__reflect__poly,axiom,
! [P2: poly_nat,N: nat] :
( ( ( ord_less_nat @ ( degree_nat @ P2 ) @ N )
=> ( ( coeff_nat @ ( reflect_poly_nat @ P2 ) @ N )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( degree_nat @ P2 ) @ N )
=> ( ( coeff_nat @ ( reflect_poly_nat @ P2 ) @ N )
= ( coeff_nat @ P2 @ ( minus_minus_nat @ ( degree_nat @ P2 ) @ N ) ) ) ) ) ).
% coeff_reflect_poly
thf(fact_263_coeff__reflect__poly,axiom,
! [P2: poly_int,N: nat] :
( ( ( ord_less_nat @ ( degree_int @ P2 ) @ N )
=> ( ( coeff_int @ ( reflect_poly_int @ P2 ) @ N )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( degree_int @ P2 ) @ N )
=> ( ( coeff_int @ ( reflect_poly_int @ P2 ) @ N )
= ( coeff_int @ P2 @ ( minus_minus_nat @ ( degree_int @ P2 ) @ N ) ) ) ) ) ).
% coeff_reflect_poly
thf(fact_264_coeff__reflect__poly,axiom,
! [P2: poly_real,N: nat] :
( ( ( ord_less_nat @ ( degree_real @ P2 ) @ N )
=> ( ( coeff_real @ ( reflect_poly_real @ P2 ) @ N )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( degree_real @ P2 ) @ N )
=> ( ( coeff_real @ ( reflect_poly_real @ P2 ) @ N )
= ( coeff_real @ P2 @ ( minus_minus_nat @ ( degree_real @ P2 ) @ N ) ) ) ) ) ).
% coeff_reflect_poly
thf(fact_265_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_266_degree__less__if__less__eqI,axiom,
! [X: poly_F3299452240248304339ring_a,Y: poly_F3299452240248304339ring_a] :
( ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ X ) @ ( degree4881254707062955960ring_a @ Y ) )
=> ( ( ( coeff_1607515655354303335ring_a @ X @ ( degree4881254707062955960ring_a @ Y ) )
= zero_z7902377541816115708ring_a )
=> ( ( X != zero_z1830546546923837194ring_a )
=> ( ord_less_nat @ ( degree4881254707062955960ring_a @ X ) @ ( degree4881254707062955960ring_a @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_267_degree__less__if__less__eqI,axiom,
! [X: poly_F3299452240248304339ring_a,Y: poly_nat] :
( ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ X ) @ ( degree_nat @ Y ) )
=> ( ( ( coeff_1607515655354303335ring_a @ X @ ( degree_nat @ Y ) )
= zero_z7902377541816115708ring_a )
=> ( ( X != zero_z1830546546923837194ring_a )
=> ( ord_less_nat @ ( degree4881254707062955960ring_a @ X ) @ ( degree_nat @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_268_degree__less__if__less__eqI,axiom,
! [X: poly_nat,Y: poly_F3299452240248304339ring_a] :
( ( ord_less_eq_nat @ ( degree_nat @ X ) @ ( degree4881254707062955960ring_a @ Y ) )
=> ( ( ( coeff_nat @ X @ ( degree4881254707062955960ring_a @ Y ) )
= zero_zero_nat )
=> ( ( X != zero_zero_poly_nat )
=> ( ord_less_nat @ ( degree_nat @ X ) @ ( degree4881254707062955960ring_a @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_269_degree__less__if__less__eqI,axiom,
! [X: poly_nat,Y: poly_nat] :
( ( ord_less_eq_nat @ ( degree_nat @ X ) @ ( degree_nat @ Y ) )
=> ( ( ( coeff_nat @ X @ ( degree_nat @ Y ) )
= zero_zero_nat )
=> ( ( X != zero_zero_poly_nat )
=> ( ord_less_nat @ ( degree_nat @ X ) @ ( degree_nat @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_270_degree__less__if__less__eqI,axiom,
! [X: poly_int,Y: poly_F3299452240248304339ring_a] :
( ( ord_less_eq_nat @ ( degree_int @ X ) @ ( degree4881254707062955960ring_a @ Y ) )
=> ( ( ( coeff_int @ X @ ( degree4881254707062955960ring_a @ Y ) )
= zero_zero_int )
=> ( ( X != zero_zero_poly_int )
=> ( ord_less_nat @ ( degree_int @ X ) @ ( degree4881254707062955960ring_a @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_271_degree__less__if__less__eqI,axiom,
! [X: poly_int,Y: poly_nat] :
( ( ord_less_eq_nat @ ( degree_int @ X ) @ ( degree_nat @ Y ) )
=> ( ( ( coeff_int @ X @ ( degree_nat @ Y ) )
= zero_zero_int )
=> ( ( X != zero_zero_poly_int )
=> ( ord_less_nat @ ( degree_int @ X ) @ ( degree_nat @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_272_degree__less__if__less__eqI,axiom,
! [X: poly_real,Y: poly_F3299452240248304339ring_a] :
( ( ord_less_eq_nat @ ( degree_real @ X ) @ ( degree4881254707062955960ring_a @ Y ) )
=> ( ( ( coeff_real @ X @ ( degree4881254707062955960ring_a @ Y ) )
= zero_zero_real )
=> ( ( X != zero_zero_poly_real )
=> ( ord_less_nat @ ( degree_real @ X ) @ ( degree4881254707062955960ring_a @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_273_degree__less__if__less__eqI,axiom,
! [X: poly_real,Y: poly_nat] :
( ( ord_less_eq_nat @ ( degree_real @ X ) @ ( degree_nat @ Y ) )
=> ( ( ( coeff_real @ X @ ( degree_nat @ Y ) )
= zero_zero_real )
=> ( ( X != zero_zero_poly_real )
=> ( ord_less_nat @ ( degree_real @ X ) @ ( degree_nat @ Y ) ) ) ) ) ).
% degree_less_if_less_eqI
thf(fact_274_eq__zero__or__degree__less,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N )
=> ( ( ( coeff_1607515655354303335ring_a @ P2 @ N )
= zero_z7902377541816115708ring_a )
=> ( ( P2 = zero_z1830546546923837194ring_a )
| ( ord_less_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_275_eq__zero__or__degree__less,axiom,
! [P2: poly_nat,N: nat] :
( ( ord_less_eq_nat @ ( degree_nat @ P2 ) @ N )
=> ( ( ( coeff_nat @ P2 @ N )
= zero_zero_nat )
=> ( ( P2 = zero_zero_poly_nat )
| ( ord_less_nat @ ( degree_nat @ P2 ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_276_eq__zero__or__degree__less,axiom,
! [P2: poly_int,N: nat] :
( ( ord_less_eq_nat @ ( degree_int @ P2 ) @ N )
=> ( ( ( coeff_int @ P2 @ N )
= zero_zero_int )
=> ( ( P2 = zero_zero_poly_int )
| ( ord_less_nat @ ( degree_int @ P2 ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_277_eq__zero__or__degree__less,axiom,
! [P2: poly_real,N: nat] :
( ( ord_less_eq_nat @ ( degree_real @ P2 ) @ N )
=> ( ( ( coeff_real @ P2 @ N )
= zero_zero_real )
=> ( ( P2 = zero_zero_poly_real )
| ( ord_less_nat @ ( degree_real @ P2 ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_278_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_279_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_280_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_281_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_282_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_283_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_284_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_285_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_286_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_287_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_288_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_289_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_290_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_291_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_292_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_293_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_294_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_295_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_296_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_297_coeff__diff,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a,N: nat] :
( ( coeff_1607515655354303335ring_a @ ( minus_5354101470050066234ring_a @ P2 @ Q2 ) @ N )
= ( minus_3609261664126569004ring_a @ ( coeff_1607515655354303335ring_a @ P2 @ N ) @ ( coeff_1607515655354303335ring_a @ Q2 @ N ) ) ) ).
% coeff_diff
thf(fact_298_coeff__diff,axiom,
! [P2: poly_nat,Q2: poly_nat,N: nat] :
( ( coeff_nat @ ( minus_minus_poly_nat @ P2 @ Q2 ) @ N )
= ( minus_minus_nat @ ( coeff_nat @ P2 @ N ) @ ( coeff_nat @ Q2 @ N ) ) ) ).
% coeff_diff
thf(fact_299_coeff__diff,axiom,
! [P2: poly_int,Q2: poly_int,N: nat] :
( ( coeff_int @ ( minus_minus_poly_int @ P2 @ Q2 ) @ N )
= ( minus_minus_int @ ( coeff_int @ P2 @ N ) @ ( coeff_int @ Q2 @ N ) ) ) ).
% coeff_diff
thf(fact_300_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_301_diff__ge__0__iff__ge,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_eq_int @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_302_diff__ge__0__iff__ge,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
= ( ord_less_eq_real @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_303_diff__gt__0__iff__gt,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_304_diff__gt__0__iff__gt,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_305_diff__numeral__special_I9_J,axiom,
( ( minus_3609261664126569004ring_a @ one_on2109788427901206336ring_a @ one_on2109788427901206336ring_a )
= zero_z7902377541816115708ring_a ) ).
% diff_numeral_special(9)
thf(fact_306_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_307_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_308_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_309_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_310_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_311_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_312_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_313_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_314_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_315_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_316_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_317_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_318_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_319_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_320_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= zero_zero_nat )
= ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_321_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_322_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_323_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_324_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_325_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_326_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_327_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_328_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_329_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_330_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_331_verit__la__disequality,axiom,
! [A: int,B2: int] :
( ( A = B2 )
| ~ ( ord_less_eq_int @ A @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_332_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_333_verit__la__disequality,axiom,
! [A: real,B2: real] :
( ( A = B2 )
| ~ ( ord_less_eq_real @ A @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_334_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_335_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_336_diff__eq__diff__eq,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_337_diff__mono,axiom,
! [A: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_338_diff__mono,axiom,
! [A: real,B2: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_339_diff__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_340_diff__left__mono,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_341_diff__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_342_diff__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_343_diff__eq__diff__less__eq,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_344_diff__eq__diff__less__eq,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B2 )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_345_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_346_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_347_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_348_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_349_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_350_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_351_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_352_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_353_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_354_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_355_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_356_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_357_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_358_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_359_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_360_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_361_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_362_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_363_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_364_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_365_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_366_minus__poly_Orep__eq,axiom,
! [X: poly_F3299452240248304339ring_a,Xa: poly_F3299452240248304339ring_a] :
( ( coeff_1607515655354303335ring_a @ ( minus_5354101470050066234ring_a @ X @ Xa ) )
= ( ^ [N3: nat] : ( minus_3609261664126569004ring_a @ ( coeff_1607515655354303335ring_a @ X @ N3 ) @ ( coeff_1607515655354303335ring_a @ Xa @ N3 ) ) ) ) ).
% minus_poly.rep_eq
thf(fact_367_minus__poly_Orep__eq,axiom,
! [X: poly_nat,Xa: poly_nat] :
( ( coeff_nat @ ( minus_minus_poly_nat @ X @ Xa ) )
= ( ^ [N3: nat] : ( minus_minus_nat @ ( coeff_nat @ X @ N3 ) @ ( coeff_nat @ Xa @ N3 ) ) ) ) ).
% minus_poly.rep_eq
thf(fact_368_minus__poly_Orep__eq,axiom,
! [X: poly_int,Xa: poly_int] :
( ( coeff_int @ ( minus_minus_poly_int @ X @ Xa ) )
= ( ^ [N3: nat] : ( minus_minus_int @ ( coeff_int @ X @ N3 ) @ ( coeff_int @ Xa @ N3 ) ) ) ) ).
% minus_poly.rep_eq
thf(fact_369_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_370_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_371_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_372_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_373_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_374_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_375_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_376_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_377_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_378_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_379_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A3: real,B: real] :
( ( minus_minus_real @ A3 @ B )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_380_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B: int] :
( ( minus_minus_int @ A3 @ B )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_381_diff__strict__mono,axiom,
! [A: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_382_diff__strict__mono,axiom,
! [A: real,B2: real,D: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_383_diff__eq__diff__less,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_384_diff__eq__diff__less,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B2 )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_385_diff__strict__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_386_diff__strict__left__mono,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_387_diff__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_388_diff__strict__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_389_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_390_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_391_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_392_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_393_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_394_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_395_verit__comp__simplify1_I3_J,axiom,
! [B3: real,A4: real] :
( ( ~ ( ord_less_eq_real @ B3 @ A4 ) )
= ( ord_less_real @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_396_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_397_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_398_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_399_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_400_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_401_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_402_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_403_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_404_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_405_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_406_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_407_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_408_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_409_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_410_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_411_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_412_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_413_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_414_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_415_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_416_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_417_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_418_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_419_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_420_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_421_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_422_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_423_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_424_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_425_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( F @ K2 @ I4 ) )
=> ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_426_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_427_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_428_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_429_coeff__0__degree__minus__1,axiom,
! [Rrr: poly_F3299452240248304339ring_a,Dr: nat] :
( ( ( coeff_1607515655354303335ring_a @ Rrr @ Dr )
= zero_z7902377541816115708ring_a )
=> ( ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ Rrr ) @ Dr )
=> ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ Rrr ) @ ( minus_minus_nat @ Dr @ one_one_nat ) ) ) ) ).
% coeff_0_degree_minus_1
thf(fact_430_coeff__0__degree__minus__1,axiom,
! [Rrr: poly_nat,Dr: nat] :
( ( ( coeff_nat @ Rrr @ Dr )
= zero_zero_nat )
=> ( ( ord_less_eq_nat @ ( degree_nat @ Rrr ) @ Dr )
=> ( ord_less_eq_nat @ ( degree_nat @ Rrr ) @ ( minus_minus_nat @ Dr @ one_one_nat ) ) ) ) ).
% coeff_0_degree_minus_1
thf(fact_431_coeff__0__degree__minus__1,axiom,
! [Rrr: poly_int,Dr: nat] :
( ( ( coeff_int @ Rrr @ Dr )
= zero_zero_int )
=> ( ( ord_less_eq_nat @ ( degree_int @ Rrr ) @ Dr )
=> ( ord_less_eq_nat @ ( degree_int @ Rrr ) @ ( minus_minus_nat @ Dr @ one_one_nat ) ) ) ) ).
% coeff_0_degree_minus_1
thf(fact_432_coeff__0__degree__minus__1,axiom,
! [Rrr: poly_real,Dr: nat] :
( ( ( coeff_real @ Rrr @ Dr )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ ( degree_real @ Rrr ) @ Dr )
=> ( ord_less_eq_nat @ ( degree_real @ Rrr ) @ ( minus_minus_nat @ Dr @ one_one_nat ) ) ) ) ).
% coeff_0_degree_minus_1
thf(fact_433_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_434_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_435_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_436_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_437_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_438_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_439_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_440_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_441_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_442_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_443_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_444_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_445_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_446_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_447_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_448_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_449_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_450_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_451_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_452_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_453_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_454_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_455_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_456_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_457_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_458_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_459_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_460_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_461_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_462_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_463_eq__nat__nat__iff,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z4 ) )
= ( Z2 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_464_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X4: nat] : ( P4 @ X4 ) )
= ( ^ [P5: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_465_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X4: nat] : ( P4 @ X4 ) )
= ( ^ [P5: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_466_poly__eqI2,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( ( degree4881254707062955960ring_a @ P2 )
= ( degree4881254707062955960ring_a @ Q2 ) )
=> ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ ( degree4881254707062955960ring_a @ P2 ) )
=> ( ( coeff_1607515655354303335ring_a @ P2 @ I )
= ( coeff_1607515655354303335ring_a @ Q2 @ I ) ) )
=> ( P2 = Q2 ) ) ) ).
% poly_eqI2
thf(fact_467_poly__eqI2,axiom,
! [P2: poly_nat,Q2: poly_nat] :
( ( ( degree_nat @ P2 )
= ( degree_nat @ Q2 ) )
=> ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ ( degree_nat @ P2 ) )
=> ( ( coeff_nat @ P2 @ I )
= ( coeff_nat @ Q2 @ I ) ) )
=> ( P2 = Q2 ) ) ) ).
% poly_eqI2
thf(fact_468_degree__reflect__poly__le,axiom,
! [P2: poly_F3299452240248304339ring_a] : ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ ( reflec4498816349307343611ring_a @ P2 ) ) @ ( degree4881254707062955960ring_a @ P2 ) ) ).
% degree_reflect_poly_le
thf(fact_469_degree__reflect__poly__le,axiom,
! [P2: poly_nat] : ( ord_less_eq_nat @ ( degree_nat @ ( reflect_poly_nat @ P2 ) ) @ ( degree_nat @ P2 ) ) ).
% degree_reflect_poly_le
thf(fact_470_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_471_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_472_le__degree,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ N )
!= zero_z7902377541816115708ring_a )
=> ( ord_less_eq_nat @ N @ ( degree4881254707062955960ring_a @ P2 ) ) ) ).
% le_degree
thf(fact_473_le__degree,axiom,
! [P2: poly_nat,N: nat] :
( ( ( coeff_nat @ P2 @ N )
!= zero_zero_nat )
=> ( ord_less_eq_nat @ N @ ( degree_nat @ P2 ) ) ) ).
% le_degree
thf(fact_474_le__degree,axiom,
! [P2: poly_int,N: nat] :
( ( ( coeff_int @ P2 @ N )
!= zero_zero_int )
=> ( ord_less_eq_nat @ N @ ( degree_int @ P2 ) ) ) ).
% le_degree
thf(fact_475_le__degree,axiom,
! [P2: poly_real,N: nat] :
( ( ( coeff_real @ P2 @ N )
!= zero_zero_real )
=> ( ord_less_eq_nat @ N @ ( degree_real @ P2 ) ) ) ).
% le_degree
thf(fact_476_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_477_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_478_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_479_degree__synthetic__div,axiom,
! [P2: poly_F3299452240248304339ring_a,C: finite_mod_ring_a] :
( ( degree4881254707062955960ring_a @ ( synthe7653204327264257080ring_a @ P2 @ C ) )
= ( minus_minus_nat @ ( degree4881254707062955960ring_a @ P2 ) @ one_one_nat ) ) ).
% degree_synthetic_div
thf(fact_480_degree__synthetic__div,axiom,
! [P2: poly_nat,C: nat] :
( ( degree_nat @ ( synthetic_div_nat @ P2 @ C ) )
= ( minus_minus_nat @ ( degree_nat @ P2 ) @ one_one_nat ) ) ).
% degree_synthetic_div
thf(fact_481_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_482_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_483_degree__le,axiom,
! [N: nat,P2: poly_F3299452240248304339ring_a] :
( ! [I: nat] :
( ( ord_less_nat @ N @ I )
=> ( ( coeff_1607515655354303335ring_a @ P2 @ I )
= zero_z7902377541816115708ring_a ) )
=> ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N ) ) ).
% degree_le
thf(fact_484_degree__le,axiom,
! [N: nat,P2: poly_nat] :
( ! [I: nat] :
( ( ord_less_nat @ N @ I )
=> ( ( coeff_nat @ P2 @ I )
= zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( degree_nat @ P2 ) @ N ) ) ).
% degree_le
thf(fact_485_degree__le,axiom,
! [N: nat,P2: poly_int] :
( ! [I: nat] :
( ( ord_less_nat @ N @ I )
=> ( ( coeff_int @ P2 @ I )
= zero_zero_int ) )
=> ( ord_less_eq_nat @ ( degree_int @ P2 ) @ N ) ) ).
% degree_le
thf(fact_486_degree__le,axiom,
! [N: nat,P2: poly_real] :
( ! [I: nat] :
( ( ord_less_nat @ N @ I )
=> ( ( coeff_real @ P2 @ I )
= zero_zero_real ) )
=> ( ord_less_eq_nat @ ( degree_real @ P2 ) @ N ) ) ).
% degree_le
thf(fact_487_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_488_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_489_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_490_mu__prop,axiom,
! [M2: nat] :
( ( ( ( power_6826135765519566523ring_a @ mu @ M2 )
= one_on2109788427901206336ring_a )
& ( M2 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n @ M2 ) ) ).
% mu_prop
thf(fact_491_mu__prop_H,axiom,
! [M6: nat] :
( ( ( power_6826135765519566523ring_a @ mu @ M6 )
= one_on2109788427901206336ring_a )
=> ( ( M6 != zero_zero_nat )
=> ( ord_less_eq_nat @ n @ M6 ) ) ) ).
% mu_prop'
thf(fact_492_deg__Poly_H,axiom,
! [Xs: list_F4626807571770296779ring_a] :
( ( ( poly_F5739129160929385880ring_a @ Xs )
!= zero_z1830546546923837194ring_a )
=> ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ ( poly_F5739129160929385880ring_a @ Xs ) ) @ ( minus_minus_nat @ ( size_s7115545719440041015ring_a @ Xs ) @ one_one_nat ) ) ) ).
% deg_Poly'
thf(fact_493_deg__Poly_H,axiom,
! [Xs: list_nat] :
( ( ( poly_nat2 @ Xs )
!= zero_zero_poly_nat )
=> ( ord_less_eq_nat @ ( degree_nat @ ( poly_nat2 @ Xs ) ) @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).
% deg_Poly'
thf(fact_494_omega__properties_I2_J,axiom,
omega != one_on2109788427901206336ring_a ).
% omega_properties(2)
thf(fact_495_normalize__field__def,axiom,
( field_3121160262079256089ring_a
= ( ^ [X2: finite_mod_ring_a] : ( if_Finite_mod_ring_a @ ( X2 = zero_z7902377541816115708ring_a ) @ zero_z7902377541816115708ring_a @ one_on2109788427901206336ring_a ) ) ) ).
% normalize_field_def
thf(fact_496_normalize__field__def,axiom,
( field_8354674766439439704d_real
= ( ^ [X2: real] : ( if_real @ ( X2 = zero_zero_real ) @ zero_zero_real @ one_one_real ) ) ) ).
% normalize_field_def
thf(fact_497_permutation__insert__expand,axiom,
( permut3695043542826343943rt_nat
= ( ^ [I3: nat,J3: nat,P3: nat > nat,I5: nat] : ( if_nat @ ( ord_less_nat @ I5 @ I3 ) @ ( if_nat @ ( ord_less_nat @ ( P3 @ I5 ) @ J3 ) @ ( P3 @ I5 ) @ ( suc @ ( P3 @ I5 ) ) ) @ ( if_nat @ ( I5 = I3 ) @ J3 @ ( if_nat @ ( ord_less_nat @ ( P3 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) @ J3 ) @ ( P3 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) @ ( suc @ ( P3 @ ( minus_minus_nat @ I5 @ one_one_nat ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_498_permutation__insert__expand,axiom,
( permut3692553072317293667rt_int
= ( ^ [I3: int,J3: nat,P3: int > nat,I5: int] : ( if_nat @ ( ord_less_int @ I5 @ I3 ) @ ( if_nat @ ( ord_less_nat @ ( P3 @ I5 ) @ J3 ) @ ( P3 @ I5 ) @ ( suc @ ( P3 @ I5 ) ) ) @ ( if_nat @ ( I5 = I3 ) @ J3 @ ( if_nat @ ( ord_less_nat @ ( P3 @ ( minus_minus_int @ I5 @ one_one_int ) ) @ J3 ) @ ( P3 @ ( minus_minus_int @ I5 @ one_one_int ) ) @ ( suc @ ( P3 @ ( minus_minus_int @ I5 @ one_one_int ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_499_permutation__insert__expand,axiom,
( permut4060954620988167523t_real
= ( ^ [I3: real,J3: nat,P3: real > nat,I5: real] : ( if_nat @ ( ord_less_real @ I5 @ I3 ) @ ( if_nat @ ( ord_less_nat @ ( P3 @ I5 ) @ J3 ) @ ( P3 @ I5 ) @ ( suc @ ( P3 @ I5 ) ) ) @ ( if_nat @ ( I5 = I3 ) @ J3 @ ( if_nat @ ( ord_less_nat @ ( P3 @ ( minus_minus_real @ I5 @ one_one_real ) ) @ J3 ) @ ( P3 @ ( minus_minus_real @ I5 @ one_one_real ) ) @ ( suc @ ( P3 @ ( minus_minus_real @ I5 @ one_one_real ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_500_mod__field__def,axiom,
( field_341224784244110787d_real
= ( ^ [X2: real,Y5: real] : ( if_real @ ( Y5 = zero_zero_real ) @ X2 @ zero_zero_real ) ) ) ).
% mod_field_def
thf(fact_501_conj__le__cong,axiom,
! [X: int,X5: int,P: $o,P6: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_502_omega__properties_I1_J,axiom,
( ( power_6826135765519566523ring_a @ omega @ n )
= one_on2109788427901206336ring_a ) ).
% omega_properties(1)
thf(fact_503_length__map,axiom,
! [F: nat > finite_mod_ring_a,Xs: list_nat] :
( ( size_s7115545719440041015ring_a @ ( map_na1928064127006292399ring_a @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_504_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_505_omega__prop_H,axiom,
! [M6: nat] :
( ( ( power_6826135765519566523ring_a @ omega @ M6 )
= one_on2109788427901206336ring_a )
=> ( ( M6 != zero_zero_nat )
=> ( ord_less_eq_nat @ n @ M6 ) ) ) ).
% omega_prop'
thf(fact_506_omega__properties_I3_J,axiom,
! [M2: nat] :
( ( ( ( power_6826135765519566523ring_a @ omega @ M2 )
= one_on2109788427901206336ring_a )
& ( M2 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ n @ M2 ) ) ).
% omega_properties(3)
thf(fact_507_length__upt,axiom,
! [I2: nat,J2: nat] :
( ( size_size_list_nat @ ( upt @ I2 @ J2 ) )
= ( minus_minus_nat @ J2 @ I2 ) ) ).
% length_upt
thf(fact_508_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_509_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_510_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_511_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_512_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_513_coeff__0__power,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( coeff_1607515655354303335ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) @ zero_zero_nat )
= ( power_6826135765519566523ring_a @ ( coeff_1607515655354303335ring_a @ P2 @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_514_coeff__0__power,axiom,
! [P2: poly_nat,N: nat] :
( ( coeff_nat @ ( power_power_poly_nat @ P2 @ N ) @ zero_zero_nat )
= ( power_power_nat @ ( coeff_nat @ P2 @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_515_coeff__0__power,axiom,
! [P2: poly_int,N: nat] :
( ( coeff_int @ ( power_power_poly_int @ P2 @ N ) @ zero_zero_nat )
= ( power_power_int @ ( coeff_int @ P2 @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_516_coeff__0__power,axiom,
! [P2: poly_real,N: nat] :
( ( coeff_real @ ( power_8994544051960338110y_real @ P2 @ N ) @ zero_zero_nat )
= ( power_power_real @ ( coeff_real @ P2 @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_517_lead__coeff__power,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( coeff_1607515655354303335ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) @ ( degree4881254707062955960ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) ) )
= ( power_6826135765519566523ring_a @ ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) ) @ N ) ) ).
% lead_coeff_power
thf(fact_518_lead__coeff__power,axiom,
! [P2: poly_nat,N: nat] :
( ( coeff_nat @ ( power_power_poly_nat @ P2 @ N ) @ ( degree_nat @ ( power_power_poly_nat @ P2 @ N ) ) )
= ( power_power_nat @ ( coeff_nat @ P2 @ ( degree_nat @ P2 ) ) @ N ) ) ).
% lead_coeff_power
thf(fact_519_lead__coeff__power,axiom,
! [P2: poly_int,N: nat] :
( ( coeff_int @ ( power_power_poly_int @ P2 @ N ) @ ( degree_int @ ( power_power_poly_int @ P2 @ N ) ) )
= ( power_power_int @ ( coeff_int @ P2 @ ( degree_int @ P2 ) ) @ N ) ) ).
% lead_coeff_power
thf(fact_520_lead__coeff__power,axiom,
! [P2: poly_real,N: nat] :
( ( coeff_real @ ( power_8994544051960338110y_real @ P2 @ N ) @ ( degree_real @ ( power_8994544051960338110y_real @ P2 @ N ) ) )
= ( power_power_real @ ( coeff_real @ P2 @ ( degree_real @ P2 ) ) @ N ) ) ).
% lead_coeff_power
thf(fact_521_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_522_map__eq__imp__length__eq,axiom,
! [F: nat > finite_mod_ring_a,Xs: list_nat,G: nat > finite_mod_ring_a,Ys: list_nat] :
( ( ( map_na1928064127006292399ring_a @ F @ Xs )
= ( map_na1928064127006292399ring_a @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_523_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_524_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_525_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_526_degree__diff__less,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat,Q2: poly_F3299452240248304339ring_a] :
( ( ord_less_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N )
=> ( ( ord_less_nat @ ( degree4881254707062955960ring_a @ Q2 ) @ N )
=> ( ord_less_nat @ ( degree4881254707062955960ring_a @ ( minus_5354101470050066234ring_a @ P2 @ Q2 ) ) @ N ) ) ) ).
% degree_diff_less
thf(fact_527_degree__diff__le,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat,Q2: poly_F3299452240248304339ring_a] :
( ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N )
=> ( ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ Q2 ) @ N )
=> ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ ( minus_5354101470050066234ring_a @ P2 @ Q2 ) ) @ N ) ) ) ).
% degree_diff_le
thf(fact_528_int__le__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_529_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_530_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_531_pinf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_532_pinf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_533_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_534_pinf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_535_pinf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_536_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_537_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_538_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_539_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_540_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_541_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_542_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_543_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_544_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_545_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_546_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_547_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_548_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_549_minf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_550_minf_I1_J,axiom,
! [P: real > $o,P6: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_551_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_552_minf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q3: int > $o,Q4: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_553_minf_I2_J,axiom,
! [P: real > $o,P6: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q3 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q3 @ X6 ) )
= ( ( P6 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_554_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_555_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_556_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_557_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_558_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_559_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_560_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_561_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_562_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_563_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_564_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_565_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_566_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_567_degree__Poly,axiom,
! [Xs: list_F4626807571770296779ring_a] : ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ ( poly_F5739129160929385880ring_a @ Xs ) ) @ ( size_s7115545719440041015ring_a @ Xs ) ) ).
% degree_Poly
thf(fact_568_degree__Poly,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( degree_nat @ ( poly_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% degree_Poly
thf(fact_569_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_570_int__ops_I6_J,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_571_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_572_nat__diff__distrib,axiom,
! [Z4: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ Z4 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z4 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_573_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_574_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_575_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_576_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_577_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_578_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ X6 @ Z3 )
=> ( ord_less_eq_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_579_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_580_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_581_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ( ord_less_eq_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_582_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_583_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_584_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X6: real] :
( ( ord_less_real @ Z3 @ X6 )
=> ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_585_imp__le__cong,axiom,
! [X: int,X5: int,P: $o,P6: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_586_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_587_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_588_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_589_power__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_590_power__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_591_power__decreasing__iff,axiom,
! [B2: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_592_power__increasing__iff,axiom,
! [B2: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_593_power__increasing__iff,axiom,
! [B2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_594_power__increasing__iff,axiom,
! [B2: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_595_power__mono__iff,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_596_power__mono__iff,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_597_power__mono__iff,axiom,
! [A: real,B2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_598_power__strict__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_599_power__strict__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_600_power__strict__decreasing__iff,axiom,
! [B2: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_601_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_602_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_603_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_604_power__one,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ one_on2109788427901206336ring_a @ N )
= one_on2109788427901206336ring_a ) ).
% power_one
thf(fact_605_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_606_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_607_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_608_power__one__right,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_609_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_610_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_611_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_612_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_613_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_614_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_615_power__0__Suc,axiom,
! [N: nat] :
( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ ( suc @ N ) )
= zero_z7902377541816115708ring_a ) ).
% power_0_Suc
thf(fact_616_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_617_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_618_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_619_power__Suc0__right,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_620_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_621_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_622_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_623_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_624_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_625_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_626_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
= ( X
= ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_627_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
= ( X
= ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_628_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
= ( X
= ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_629_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B2 @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_630_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B2 @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_631_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B2 @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_632_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( power_power_nat @ M @ N ) )
= ( power_6826135765519566523ring_a @ ( semiri9180929696517417892ring_a @ M ) @ N ) ) ).
% of_nat_power
thf(fact_633_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_634_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_635_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_636_power__eq__0__iff,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( ( power_6826135765519566523ring_a @ A @ N )
= zero_z7902377541816115708ring_a )
= ( ( A = zero_z7902377541816115708ring_a )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_637_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_638_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_639_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_640_power__strict__increasing__iff,axiom,
! [B2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_641_power__strict__increasing__iff,axiom,
! [B2: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_642_power__strict__increasing__iff,axiom,
! [B2: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_643_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_644_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_645_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B2: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_646_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_647_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_648_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_649_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_650_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_651_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B2: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_652_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_653_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
= ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_654_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_655_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_656_monic__power,axiom,
! [P2: poly_int,N: nat] :
( ( ( coeff_int @ P2 @ ( degree_int @ P2 ) )
= one_one_int )
=> ( ( coeff_int @ ( power_power_poly_int @ P2 @ N ) @ ( degree_int @ ( power_power_poly_int @ P2 @ N ) ) )
= one_one_int ) ) ).
% monic_power
thf(fact_657_monic__power,axiom,
! [P2: poly_real,N: nat] :
( ( ( coeff_real @ P2 @ ( degree_real @ P2 ) )
= one_one_real )
=> ( ( coeff_real @ ( power_8994544051960338110y_real @ P2 @ N ) @ ( degree_real @ ( power_8994544051960338110y_real @ P2 @ N ) ) )
= one_one_real ) ) ).
% monic_power
thf(fact_658_monic__power,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) )
= one_on2109788427901206336ring_a )
=> ( ( coeff_1607515655354303335ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) @ ( degree4881254707062955960ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) ) )
= one_on2109788427901206336ring_a ) ) ).
% monic_power
thf(fact_659_power__not__zero,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( power_6826135765519566523ring_a @ A @ N )
!= zero_z7902377541816115708ring_a ) ) ).
% power_not_zero
thf(fact_660_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_661_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_662_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_663_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_664_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_665_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_666_power__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_667_power__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_668_power__mono,axiom,
! [A: real,B2: real,N: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_669_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_670_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_671_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_672_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_673_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_674_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_675_power__0,axiom,
! [A: finite_mod_ring_a] :
( ( power_6826135765519566523ring_a @ A @ zero_zero_nat )
= one_on2109788427901206336ring_a ) ).
% power_0
thf(fact_676_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_677_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_678_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_679_power__less__imp__less__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_680_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_681_power__less__imp__less__base,axiom,
! [A: real,N: nat,B2: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_682_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_683_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_684_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_685_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= one_on2109788427901206336ring_a ) )
& ( ( N != zero_zero_nat )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ) ).
% power_0_left
thf(fact_686_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_687_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_688_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_689_power__inject__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( A = B2 ) ) ) ) ).
% power_inject_base
thf(fact_690_power__inject__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( A = B2 ) ) ) ) ).
% power_inject_base
thf(fact_691_power__inject__base,axiom,
! [A: real,N: nat,B2: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( A = B2 ) ) ) ) ).
% power_inject_base
thf(fact_692_power__le__imp__le__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% power_le_imp_le_base
thf(fact_693_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% power_le_imp_le_base
thf(fact_694_power__le__imp__le__base,axiom,
! [A: real,N: nat,B2: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ A @ B2 ) ) ) ).
% power_le_imp_le_base
thf(fact_695_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_6826135765519566523ring_a @ zero_z7902377541816115708ring_a @ N )
= zero_z7902377541816115708ring_a ) ) ).
% zero_power
thf(fact_696_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_697_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_698_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_699_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_700_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_701_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_702_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_703_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_704_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_705_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_706_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_707_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_708_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_709_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_710_power__increasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_711_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_712_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_713_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_714_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_715_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_716_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_717_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_718_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_719_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_720_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_721_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_722_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_723_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_724_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_725_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B2: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B2 @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_726_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_727_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_728_power__decreasing,axiom,
! [N: nat,N5: nat,A: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_729_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_730_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_731_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_732_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_733_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_734_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_735_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_736_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_737_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_738_power__strict__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_739_power__strict__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_740_power__strict__mono,axiom,
! [A: real,B2: real,N: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_741_mu__properties_I1_J,axiom,
( ( times_5121417576591743744ring_a @ mu @ omega )
= one_on2109788427901206336ring_a ) ).
% mu_properties(1)
thf(fact_742_exp__tends__to__zero,axiom,
! [B2: real,C: real] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ? [X3: nat] : ( ord_less_eq_real @ ( power_power_real @ B2 @ X3 ) @ C ) ) ) ) ).
% exp_tends_to_zero
thf(fact_743_delete__index__def,axiom,
( delete_index
= ( ^ [I3: nat,I5: nat] : ( if_nat @ ( ord_less_nat @ I5 @ I3 ) @ I5 @ ( minus_minus_nat @ I5 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% delete_index_def
thf(fact_744_mult__cancel__right,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ C )
= ( times_5121417576591743744ring_a @ B2 @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_745_mult__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_746_mult__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_747_mult__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_748_mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A )
= ( times_5121417576591743744ring_a @ C @ B2 ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_749_mult__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_750_mult__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_751_mult__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_752_mult__eq__0__iff,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B2 )
= zero_z7902377541816115708ring_a )
= ( ( A = zero_z7902377541816115708ring_a )
| ( B2 = zero_z7902377541816115708ring_a ) ) ) ).
% mult_eq_0_iff
thf(fact_753_mult__eq__0__iff,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_754_mult__eq__0__iff,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_755_mult__eq__0__iff,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_756_mult__zero__right,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_right
thf(fact_757_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_758_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_759_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_760_mult__zero__left,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a @ A )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_left
thf(fact_761_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_762_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_763_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_764_mult__1,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A )
= A ) ).
% mult_1
thf(fact_765_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_766_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_767_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_768_mult_Oright__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ one_on2109788427901206336ring_a )
= A ) ).
% mult.right_neutral
thf(fact_769_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_770_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_771_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_772_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( times_times_nat @ M @ N ) )
= ( times_5121417576591743744ring_a @ ( semiri9180929696517417892ring_a @ M ) @ ( semiri9180929696517417892ring_a @ N ) ) ) ).
% of_nat_mult
thf(fact_773_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_774_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_775_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_776_mult__cancel__left1,axiom,
! [C: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ C @ B2 ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B2 = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left1
thf(fact_777_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_778_mult__cancel__left1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_779_mult__cancel__left2,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left2
thf(fact_780_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_781_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_782_mult__cancel__right1,axiom,
! [C: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ B2 @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B2 = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right1
thf(fact_783_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_784_mult__cancel__right1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_785_mult__cancel__right2,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ C )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right2
thf(fact_786_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_787_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_788_coeff__degree__mult,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( coeff_1607515655354303335ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) @ ( degree4881254707062955960ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) ) )
= ( times_5121417576591743744ring_a @ ( coeff_1607515655354303335ring_a @ Q2 @ ( degree4881254707062955960ring_a @ Q2 ) ) @ ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) ) ) ) ).
% coeff_degree_mult
thf(fact_789_coeff__degree__mult,axiom,
! [P2: poly_nat,Q2: poly_nat] :
( ( coeff_nat @ ( times_times_poly_nat @ P2 @ Q2 ) @ ( degree_nat @ ( times_times_poly_nat @ P2 @ Q2 ) ) )
= ( times_times_nat @ ( coeff_nat @ Q2 @ ( degree_nat @ Q2 ) ) @ ( coeff_nat @ P2 @ ( degree_nat @ P2 ) ) ) ) ).
% coeff_degree_mult
thf(fact_790_coeff__degree__mult,axiom,
! [P2: poly_int,Q2: poly_int] :
( ( coeff_int @ ( times_times_poly_int @ P2 @ Q2 ) @ ( degree_int @ ( times_times_poly_int @ P2 @ Q2 ) ) )
= ( times_times_int @ ( coeff_int @ Q2 @ ( degree_int @ Q2 ) ) @ ( coeff_int @ P2 @ ( degree_int @ P2 ) ) ) ) ).
% coeff_degree_mult
thf(fact_791_coeff__degree__mult,axiom,
! [P2: poly_real,Q2: poly_real] :
( ( coeff_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) @ ( degree_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) ) )
= ( times_times_real @ ( coeff_real @ Q2 @ ( degree_real @ Q2 ) ) @ ( coeff_real @ P2 @ ( degree_real @ P2 ) ) ) ) ).
% coeff_degree_mult
thf(fact_792_lead__coeff__mult,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( coeff_1607515655354303335ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) @ ( degree4881254707062955960ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) ) )
= ( times_5121417576591743744ring_a @ ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) ) @ ( coeff_1607515655354303335ring_a @ Q2 @ ( degree4881254707062955960ring_a @ Q2 ) ) ) ) ).
% lead_coeff_mult
thf(fact_793_lead__coeff__mult,axiom,
! [P2: poly_nat,Q2: poly_nat] :
( ( coeff_nat @ ( times_times_poly_nat @ P2 @ Q2 ) @ ( degree_nat @ ( times_times_poly_nat @ P2 @ Q2 ) ) )
= ( times_times_nat @ ( coeff_nat @ P2 @ ( degree_nat @ P2 ) ) @ ( coeff_nat @ Q2 @ ( degree_nat @ Q2 ) ) ) ) ).
% lead_coeff_mult
thf(fact_794_lead__coeff__mult,axiom,
! [P2: poly_int,Q2: poly_int] :
( ( coeff_int @ ( times_times_poly_int @ P2 @ Q2 ) @ ( degree_int @ ( times_times_poly_int @ P2 @ Q2 ) ) )
= ( times_times_int @ ( coeff_int @ P2 @ ( degree_int @ P2 ) ) @ ( coeff_int @ Q2 @ ( degree_int @ Q2 ) ) ) ) ).
% lead_coeff_mult
thf(fact_795_lead__coeff__mult,axiom,
! [P2: poly_real,Q2: poly_real] :
( ( coeff_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) @ ( degree_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) ) )
= ( times_times_real @ ( coeff_real @ P2 @ ( degree_real @ P2 ) ) @ ( coeff_real @ Q2 @ ( degree_real @ Q2 ) ) ) ) ).
% lead_coeff_mult
thf(fact_796_mult_Oleft__commute,axiom,
! [B2: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ B2 @ ( times_5121417576591743744ring_a @ A @ C ) )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_797_mult_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_798_mult_Oleft__commute,axiom,
! [B2: int,A: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_799_mult_Oleft__commute,axiom,
! [B2: real,A: real,C: real] :
( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_800_mult_Ocommute,axiom,
( times_5121417576591743744ring_a
= ( ^ [A3: finite_mod_ring_a,B: finite_mod_ring_a] : ( times_5121417576591743744ring_a @ B @ A3 ) ) ) ).
% mult.commute
thf(fact_801_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B: nat] : ( times_times_nat @ B @ A3 ) ) ) ).
% mult.commute
thf(fact_802_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B: int] : ( times_times_int @ B @ A3 ) ) ) ).
% mult.commute
thf(fact_803_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B: real] : ( times_times_real @ B @ A3 ) ) ) ).
% mult.commute
thf(fact_804_mult_Oassoc,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A @ B2 ) @ C )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_805_mult_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_806_mult_Oassoc,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_807_mult_Oassoc,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_808_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A @ B2 ) @ C )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_809_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_810_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_811_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_812_mult__right__cancel,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ A @ C )
= ( times_5121417576591743744ring_a @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_813_mult__right__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_814_mult__right__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_815_mult__right__cancel,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_816_mult__left__cancel,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ C @ A )
= ( times_5121417576591743744ring_a @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_817_mult__left__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_818_mult__left__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_819_mult__left__cancel,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_820_no__zero__divisors,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( B2 != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A @ B2 )
!= zero_z7902377541816115708ring_a ) ) ) ).
% no_zero_divisors
thf(fact_821_no__zero__divisors,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_822_no__zero__divisors,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_823_no__zero__divisors,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_824_divisors__zero,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B2 )
= zero_z7902377541816115708ring_a )
=> ( ( A = zero_z7902377541816115708ring_a )
| ( B2 = zero_z7902377541816115708ring_a ) ) ) ).
% divisors_zero
thf(fact_825_divisors__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_826_divisors__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_827_divisors__zero,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_828_mult__not__zero,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B2 )
!= zero_z7902377541816115708ring_a )
=> ( ( A != zero_z7902377541816115708ring_a )
& ( B2 != zero_z7902377541816115708ring_a ) ) ) ).
% mult_not_zero
thf(fact_829_mult__not__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_830_mult__not__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_831_mult__not__zero,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_832_mult_Ocomm__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ one_on2109788427901206336ring_a )
= A ) ).
% mult.comm_neutral
thf(fact_833_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_834_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_835_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_836_comm__monoid__mult__class_Omult__1,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_837_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_838_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_839_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_840_right__diff__distrib_H,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( minus_3609261664126569004ring_a @ B2 @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ B2 ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_841_right__diff__distrib_H,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_842_right__diff__distrib_H,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_843_right__diff__distrib_H,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_844_left__diff__distrib_H,axiom,
! [B2: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ B2 @ C ) @ A )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ B2 @ A ) @ ( times_5121417576591743744ring_a @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_845_left__diff__distrib_H,axiom,
! [B2: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_846_left__diff__distrib_H,axiom,
! [B2: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_847_left__diff__distrib_H,axiom,
! [B2: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_848_right__diff__distrib,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( minus_3609261664126569004ring_a @ B2 @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ B2 ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_849_right__diff__distrib,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_850_right__diff__distrib,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_851_left__diff__distrib,axiom,
! [A: finite_mod_ring_a,B2: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ A @ B2 ) @ C )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B2 @ C ) ) ) ).
% left_diff_distrib
thf(fact_852_left__diff__distrib,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% left_diff_distrib
thf(fact_853_left__diff__distrib,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% left_diff_distrib
thf(fact_854_mult__of__nat__commute,axiom,
! [X: nat,Y: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( semiri9180929696517417892ring_a @ X ) @ Y )
= ( times_5121417576591743744ring_a @ Y @ ( semiri9180929696517417892ring_a @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_855_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_856_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_857_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_858_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_859_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_860_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_861_zero__le__mult__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_862_zero__le__mult__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_863_mult__nonneg__nonpos2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_864_mult__nonneg__nonpos2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_865_mult__nonneg__nonpos2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_866_mult__nonpos__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_867_mult__nonpos__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_868_mult__nonpos__nonneg,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_869_mult__nonneg__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_870_mult__nonneg__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_871_mult__nonneg__nonpos,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_872_mult__nonneg__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_873_mult__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_874_mult__nonneg__nonneg,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_875_split__mult__neg__le,axiom,
! [A: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_876_split__mult__neg__le,axiom,
! [A: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_877_split__mult__neg__le,axiom,
! [A: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_878_mult__le__0__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_879_mult__le__0__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_880_mult__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_881_mult__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_882_mult__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_883_mult__right__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_884_mult__right__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_885_mult__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_886_mult__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_887_mult__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_888_mult__nonpos__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_889_mult__nonpos__nonpos,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_890_mult__left__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_891_mult__left__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_892_split__mult__pos__le,axiom,
! [A: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_893_split__mult__pos__le,axiom,
! [A: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_894_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_895_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_896_mult__mono_H,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_897_mult__mono_H,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_898_mult__mono_H,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_899_mult__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_900_mult__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_901_mult__mono,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_902_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_903_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_904_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_905_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_906_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B2 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_907_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_908_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_909_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_910_mult__strict__right__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_911_mult__strict__right__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_912_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_913_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B2 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_914_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_915_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_916_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_917_mult__strict__left__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_918_mult__strict__left__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_919_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_920_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_real @ A @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_921_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_922_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_923_zero__less__mult__pos2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_924_zero__less__mult__pos2,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_925_zero__less__mult__pos2,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_926_zero__less__mult__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_927_zero__less__mult__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_928_zero__less__mult__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_929_zero__less__mult__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_930_zero__less__mult__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_931_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_932_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_933_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_934_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_935_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_936_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_937_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_938_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_939_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_940_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_941_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_942_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_943_mult__less__0__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_944_mult__less__0__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_945_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_946_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_947_mult__neg__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_948_mult__neg__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_949_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_950_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_951_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_952_left__right__inverse__power,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a,N: nat] :
( ( ( times_5121417576591743744ring_a @ X @ Y )
= one_on2109788427901206336ring_a )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ X @ N ) @ ( power_6826135765519566523ring_a @ Y @ N ) )
= one_on2109788427901206336ring_a ) ) ).
% left_right_inverse_power
thf(fact_953_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_954_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_955_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_956_power__Suc2,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ N ) )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_957_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_958_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_959_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_960_power__Suc,axiom,
! [A: finite_mod_ring_a,N: nat] :
( ( power_6826135765519566523ring_a @ A @ ( suc @ N ) )
= ( times_5121417576591743744ring_a @ A @ ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_Suc
thf(fact_961_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_962_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_963_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_964_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_965_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_966_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_967_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_968_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_969_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_970_mult__right__le__imp__le,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_971_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_972_mult__right__le__imp__le,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_973_mult__left__le__imp__le,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_974_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_975_mult__left__le__imp__le,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_976_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_977_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_eq_real @ A @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_978_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ B2 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_979_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_eq_real @ B2 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_980_mult__less__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_981_mult__less__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_982_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_983_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_984_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_985_mult__right__less__imp__less,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_986_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_987_mult__right__less__imp__less,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_988_mult__less__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_989_mult__less__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_990_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_991_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_992_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_993_mult__left__less__imp__less,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_994_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_995_mult__left__less__imp__less,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_996_mult__le__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_997_mult__le__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_998_mult__le__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_999_mult__le__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1000_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1001_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1002_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1003_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1004_mult__le__one,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1005_mult__le__one,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1006_mult__le__one,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ B2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1007_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1008_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1009_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1010_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1011_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1012_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1013_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1014_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1015_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1016_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1017_mult__le__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1018_mult__le__cancel__left1,axiom,
! [C: real,B2: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1019_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1020_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1021_mult__le__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1022_mult__le__cancel__right1,axiom,
! [C: real,B2: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1023_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1024_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1025_mult__less__cancel__left1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1026_mult__less__cancel__left1,axiom,
! [C: real,B2: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1027_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1028_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1029_mult__less__cancel__right1,axiom,
! [C: int,B2: int] :
( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1030_mult__less__cancel__right1,axiom,
! [C: real,B2: real] :
( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1031_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1032_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1033_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1034_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1035_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1036_linear__exp__bound,axiom,
! [B2: real] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ? [P7: real] :
! [X6: nat] : ( ord_less_eq_real @ ( times_times_real @ ( power_power_real @ B2 @ X6 ) @ ( semiri5074537144036343181t_real @ X6 ) ) @ P7 ) ) ) ).
% linear_exp_bound
thf(fact_1037_poly__exp__bound,axiom,
! [B2: real,Deg: nat] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ? [P7: real] :
! [X6: nat] : ( ord_less_eq_real @ ( times_times_real @ ( power_power_real @ B2 @ X6 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ X6 ) @ Deg ) ) @ P7 ) ) ) ).
% poly_exp_bound
thf(fact_1038_power__eq__if,axiom,
( power_6826135765519566523ring_a
= ( ^ [P3: finite_mod_ring_a,M5: nat] : ( if_Finite_mod_ring_a @ ( M5 = zero_zero_nat ) @ one_on2109788427901206336ring_a @ ( times_5121417576591743744ring_a @ P3 @ ( power_6826135765519566523ring_a @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1039_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1040_power__eq__if,axiom,
( power_power_int
= ( ^ [P3: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1041_power__eq__if,axiom,
( power_power_real
= ( ^ [P3: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1042_power__minus__mult,axiom,
! [N: nat,A: finite_mod_ring_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_6826135765519566523ring_a @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1043_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1044_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1045_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1046_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_1047_mult__hom_Ohom__zero,axiom,
! [C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ C @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_hom.hom_zero
thf(fact_1048_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_1049_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_1050_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_1051_mult__le__cancel__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1052_mult__le__cancel__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1053_mult__le__cancel__iff2,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1054_mult__le__cancel__iff2,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1055_poly__cancel__eq__conv,axiom,
! [X: finite_mod_ring_a,A: finite_mod_ring_a,Y: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( X = zero_z7902377541816115708ring_a )
=> ( ( A != zero_z7902377541816115708ring_a )
=> ( ( Y = zero_z7902377541816115708ring_a )
= ( ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ Y ) @ ( times_5121417576591743744ring_a @ B2 @ X ) )
= zero_z7902377541816115708ring_a ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_1056_poly__cancel__eq__conv,axiom,
! [X: real,A: real,Y: real,B2: real] :
( ( X = zero_zero_real )
=> ( ( A != zero_zero_real )
=> ( ( Y = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A @ Y ) @ ( times_times_real @ B2 @ X ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_1057_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_1058_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1059_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1060_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1061_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_1062_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_1063_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_1064_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_1065_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1066_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_1067_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1068_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1069_int__ops_I7_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1070_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1071_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1072_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1073_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1074_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1075_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1076_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1077_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1078_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1079_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1080_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1081_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1082_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1083_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_1084_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_1085_nat__mult__distrib,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z4 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z4 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1086_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1087_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1088_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1089_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1090_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1091_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_1092_zmult__zless__mono2,axiom,
! [I2: int,J2: int,K: int] :
( ( ord_less_int @ I2 @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1093_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1094_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1095_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1096_linordered__field__no__lb,axiom,
! [X6: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X6 ) ).
% linordered_field_no_lb
thf(fact_1097_linordered__field__no__ub,axiom,
! [X6: real] :
? [X_12: real] : ( ord_less_real @ X6 @ X_12 ) ).
% linordered_field_no_ub
thf(fact_1098_degree__mult__eq__0,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( ( degree4881254707062955960ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) )
= zero_zero_nat )
= ( ( P2 = zero_z1830546546923837194ring_a )
| ( Q2 = zero_z1830546546923837194ring_a )
| ( ( P2 != zero_z1830546546923837194ring_a )
& ( Q2 != zero_z1830546546923837194ring_a )
& ( ( degree4881254707062955960ring_a @ P2 )
= zero_zero_nat )
& ( ( degree4881254707062955960ring_a @ Q2 )
= zero_zero_nat ) ) ) ) ).
% degree_mult_eq_0
thf(fact_1099_degree__mult__eq__0,axiom,
! [P2: poly_nat,Q2: poly_nat] :
( ( ( degree_nat @ ( times_times_poly_nat @ P2 @ Q2 ) )
= zero_zero_nat )
= ( ( P2 = zero_zero_poly_nat )
| ( Q2 = zero_zero_poly_nat )
| ( ( P2 != zero_zero_poly_nat )
& ( Q2 != zero_zero_poly_nat )
& ( ( degree_nat @ P2 )
= zero_zero_nat )
& ( ( degree_nat @ Q2 )
= zero_zero_nat ) ) ) ) ).
% degree_mult_eq_0
thf(fact_1100_monic__factor,axiom,
! [P2: poly_int,Q2: poly_int] :
( ( ( coeff_int @ ( times_times_poly_int @ P2 @ Q2 ) @ ( degree_int @ ( times_times_poly_int @ P2 @ Q2 ) ) )
= one_one_int )
=> ( ( ( coeff_int @ P2 @ ( degree_int @ P2 ) )
= one_one_int )
=> ( ( coeff_int @ Q2 @ ( degree_int @ Q2 ) )
= one_one_int ) ) ) ).
% monic_factor
thf(fact_1101_monic__factor,axiom,
! [P2: poly_real,Q2: poly_real] :
( ( ( coeff_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) @ ( degree_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) ) )
= one_one_real )
=> ( ( ( coeff_real @ P2 @ ( degree_real @ P2 ) )
= one_one_real )
=> ( ( coeff_real @ Q2 @ ( degree_real @ Q2 ) )
= one_one_real ) ) ) ).
% monic_factor
thf(fact_1102_monic__factor,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) @ ( degree4881254707062955960ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) ) )
= one_on2109788427901206336ring_a )
=> ( ( ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) )
= one_on2109788427901206336ring_a )
=> ( ( coeff_1607515655354303335ring_a @ Q2 @ ( degree4881254707062955960ring_a @ Q2 ) )
= one_on2109788427901206336ring_a ) ) ) ).
% monic_factor
thf(fact_1103_monic__mult,axiom,
! [P2: poly_int,Q2: poly_int] :
( ( ( coeff_int @ P2 @ ( degree_int @ P2 ) )
= one_one_int )
=> ( ( ( coeff_int @ Q2 @ ( degree_int @ Q2 ) )
= one_one_int )
=> ( ( coeff_int @ ( times_times_poly_int @ P2 @ Q2 ) @ ( degree_int @ ( times_times_poly_int @ P2 @ Q2 ) ) )
= one_one_int ) ) ) ).
% monic_mult
thf(fact_1104_monic__mult,axiom,
! [P2: poly_real,Q2: poly_real] :
( ( ( coeff_real @ P2 @ ( degree_real @ P2 ) )
= one_one_real )
=> ( ( ( coeff_real @ Q2 @ ( degree_real @ Q2 ) )
= one_one_real )
=> ( ( coeff_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) @ ( degree_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) ) )
= one_one_real ) ) ) ).
% monic_mult
thf(fact_1105_monic__mult,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) )
= one_on2109788427901206336ring_a )
=> ( ( ( coeff_1607515655354303335ring_a @ Q2 @ ( degree4881254707062955960ring_a @ Q2 ) )
= one_on2109788427901206336ring_a )
=> ( ( coeff_1607515655354303335ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) @ ( degree4881254707062955960ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) ) )
= one_on2109788427901206336ring_a ) ) ) ).
% monic_mult
thf(fact_1106_degree__mult__right__le,axiom,
! [Q2: poly_F3299452240248304339ring_a,P2: poly_F3299452240248304339ring_a] :
( ( Q2 != zero_z1830546546923837194ring_a )
=> ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ P2 ) @ ( degree4881254707062955960ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) ) ) ) ).
% degree_mult_right_le
thf(fact_1107_degree__mult__right__le,axiom,
! [Q2: poly_nat,P2: poly_nat] :
( ( Q2 != zero_zero_poly_nat )
=> ( ord_less_eq_nat @ ( degree_nat @ P2 ) @ ( degree_nat @ ( times_times_poly_nat @ P2 @ Q2 ) ) ) ) ).
% degree_mult_right_le
thf(fact_1108_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_1109_plusinfinity,axiom,
! [D: int,P6: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [X_1: int] : ( P6 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1110_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1111_degree__power__le,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] : ( ord_less_eq_nat @ ( degree4881254707062955960ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) ) @ ( times_times_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N ) ) ).
% degree_power_le
thf(fact_1112_degree__power__le,axiom,
! [P2: poly_nat,N: nat] : ( ord_less_eq_nat @ ( degree_nat @ ( power_power_poly_nat @ P2 @ N ) ) @ ( times_times_nat @ ( degree_nat @ P2 ) @ N ) ) ).
% degree_power_le
thf(fact_1113_Missing__Polynomial_Odegree__power__eq,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( P2 != zero_z1830546546923837194ring_a )
=> ( ( degree4881254707062955960ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) )
= ( times_times_nat @ ( degree4881254707062955960ring_a @ P2 ) @ N ) ) ) ).
% Missing_Polynomial.degree_power_eq
thf(fact_1114_Polynomial_Odegree__power__eq,axiom,
! [P2: poly_F3299452240248304339ring_a,N: nat] :
( ( P2 != zero_z1830546546923837194ring_a )
=> ( ( degree4881254707062955960ring_a @ ( power_6500929916544582089ring_a @ P2 @ N ) )
= ( times_times_nat @ N @ ( degree4881254707062955960ring_a @ P2 ) ) ) ) ).
% Polynomial.degree_power_eq
thf(fact_1115_zmult__zless__mono2__lemma,axiom,
! [I2: int,J2: int,K: nat] :
( ( ord_less_int @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1116_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1117_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_1118_mult__less__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_1119_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1120_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1121_poly__mod_Olead__coeff__monic__mult,axiom,
! [P2: poly_int,Q2: poly_int] :
( ( ( coeff_int @ P2 @ ( degree_int @ P2 ) )
= one_one_int )
=> ( ( coeff_int @ ( times_times_poly_int @ P2 @ Q2 ) @ ( degree_int @ ( times_times_poly_int @ P2 @ Q2 ) ) )
= ( coeff_int @ Q2 @ ( degree_int @ Q2 ) ) ) ) ).
% poly_mod.lead_coeff_monic_mult
thf(fact_1122_poly__mod_Olead__coeff__monic__mult,axiom,
! [P2: poly_real,Q2: poly_real] :
( ( ( coeff_real @ P2 @ ( degree_real @ P2 ) )
= one_one_real )
=> ( ( coeff_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) @ ( degree_real @ ( times_7914811829580426937y_real @ P2 @ Q2 ) ) )
= ( coeff_real @ Q2 @ ( degree_real @ Q2 ) ) ) ) ).
% poly_mod.lead_coeff_monic_mult
thf(fact_1123_poly__mod_Olead__coeff__monic__mult,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ ( degree4881254707062955960ring_a @ P2 ) )
= one_on2109788427901206336ring_a )
=> ( ( coeff_1607515655354303335ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) @ ( degree4881254707062955960ring_a @ ( times_3242606764180207630ring_a @ P2 @ Q2 ) ) )
= ( coeff_1607515655354303335ring_a @ Q2 @ ( degree4881254707062955960ring_a @ Q2 ) ) ) ) ).
% poly_mod.lead_coeff_monic_mult
thf(fact_1124_poly__mod_Olead__coeff__monic__mult,axiom,
! [P2: poly_nat,Q2: poly_nat] :
( ( ( coeff_nat @ P2 @ ( degree_nat @ P2 ) )
= one_one_nat )
=> ( ( coeff_nat @ ( times_times_poly_nat @ P2 @ Q2 ) @ ( degree_nat @ ( times_times_poly_nat @ P2 @ Q2 ) ) )
= ( coeff_nat @ Q2 @ ( degree_nat @ Q2 ) ) ) ) ).
% poly_mod.lead_coeff_monic_mult
thf(fact_1125_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1126_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_1127_complete__real,axiom,
! [S2: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S2 )
=> ( ? [Z5: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ? [Y3: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S2 )
=> ( ord_less_eq_real @ X6 @ Y3 ) )
& ! [Z5: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z5 ) )
=> ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_1128_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1129_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_1130_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1131_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1132_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1133_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1134_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_1135_less__eq__fract__respect,axiom,
! [B2: int,B3: int,D: int,D3: int,A: int,A4: int,C: int,C2: int] :
( ( B2 != zero_zero_int )
=> ( ( B3 != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( D3 != zero_zero_int )
=> ( ( ( times_times_int @ A @ B3 )
= ( times_times_int @ A4 @ B2 ) )
=> ( ( ( times_times_int @ C @ D3 )
= ( times_times_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A4 @ D3 ) @ ( times_times_int @ B3 @ D3 ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B3 ) @ ( times_times_int @ B3 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_1136_less__eq__fract__respect,axiom,
! [B2: real,B3: real,D: real,D3: real,A: real,A4: real,C: real,C2: real] :
( ( B2 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( D != zero_zero_real )
=> ( ( D3 != zero_zero_real )
=> ( ( ( times_times_real @ A @ B3 )
= ( times_times_real @ A4 @ B2 ) )
=> ( ( ( times_times_real @ C @ D3 )
= ( times_times_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ D ) ) @ ( times_times_real @ ( times_times_real @ C @ B2 ) @ ( times_times_real @ B2 @ D ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A4 @ D3 ) @ ( times_times_real @ B3 @ D3 ) ) @ ( times_times_real @ ( times_times_real @ C2 @ B3 ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_1137_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_1138_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1139_permutation__delete__expand,axiom,
( permutation_delete
= ( ^ [P3: nat > nat,I3: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ ( P3 @ ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) @ ( P3 @ I3 ) ) @ ( P3 @ ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) @ ( minus_minus_nat @ ( P3 @ ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% permutation_delete_expand
thf(fact_1140_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1141_psize__def,axiom,
( fundam4324422711134394264ring_a
= ( ^ [P3: poly_F3299452240248304339ring_a] : ( if_nat @ ( P3 = zero_z1830546546923837194ring_a ) @ zero_zero_nat @ ( suc @ ( degree4881254707062955960ring_a @ P3 ) ) ) ) ) ).
% psize_def
thf(fact_1142_psize__def,axiom,
( fundam7805642066694858301ze_nat
= ( ^ [P3: poly_nat] : ( if_nat @ ( P3 = zero_zero_poly_nat ) @ zero_zero_nat @ ( suc @ ( degree_nat @ P3 ) ) ) ) ) ).
% psize_def
thf(fact_1143_ntt__axioms_Ointro,axiom,
! [Omega: finite_mod_ring_a,N: nat,Mu: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega @ N )
= one_on2109788427901206336ring_a )
=> ( ( Omega != one_on2109788427901206336ring_a )
=> ( ! [M3: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega @ M3 )
= one_on2109788427901206336ring_a )
& ( M3 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ N @ M3 ) )
=> ( ( ( times_5121417576591743744ring_a @ Mu @ Omega )
= one_on2109788427901206336ring_a )
=> ( ntt_axioms_a @ N @ Omega @ Mu ) ) ) ) ) ).
% ntt_axioms.intro
thf(fact_1144_ntt__axioms__def,axiom,
( ntt_axioms_a
= ( ^ [N3: nat,Omega2: finite_mod_ring_a,Mu2: finite_mod_ring_a] :
( ( ( power_6826135765519566523ring_a @ Omega2 @ N3 )
= one_on2109788427901206336ring_a )
& ( Omega2 != one_on2109788427901206336ring_a )
& ! [M5: nat] :
( ( ( ( power_6826135765519566523ring_a @ Omega2 @ M5 )
= one_on2109788427901206336ring_a )
& ( M5 != zero_zero_nat ) )
=> ( ord_less_eq_nat @ N3 @ M5 ) )
& ( ( times_5121417576591743744ring_a @ Mu2 @ Omega2 )
= one_on2109788427901206336ring_a ) ) ) ) ).
% ntt_axioms_def
thf(fact_1145_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1146_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1147_abs__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_abs
thf(fact_1148_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_1149_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_1150_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_1151_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_1152_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_1153_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_1154_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_1155_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_1156_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_1157_abs__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
= ( times_times_int @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_1158_abs__mult__self__eq,axiom,
! [A: real] :
( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
= ( times_times_real @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_1159_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_1160_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_1161_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_1162_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_1163_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_1164_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_1165_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_1166_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_1167_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1168_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1169_nat__abs__mult__distrib,axiom,
! [W: int,Z2: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z2 ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_1170_nat__abs__int__diff,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ B2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ A @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1171_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_nat @ I @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1172_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1173_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1174_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1175_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1176_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1177_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1178_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1179_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1180_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1181_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1182_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1183_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1184_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1185_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1186_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1187_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1188_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1189_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1190_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1191_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1192_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1193_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1194_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1195_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1196_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1197_trans__le__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1198_trans__le__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1199_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M5 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1200_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1201_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1202_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1203_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1204_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1205_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1206_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1207_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1208_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1209_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_1210_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1211_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1212_trans__less__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_1213_trans__less__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1214_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1215_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1216_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1217_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1218_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1219_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1220_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1221_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
? [K5: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M5 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1222_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1223_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1224_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q5: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q5 ) ) ) ) ).
% less_natE
thf(fact_1225_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1226_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1227_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1228_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1229_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1230_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1231_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1232_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1233_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1234_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1235_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1236_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1237_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1238_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ( ( ord_less_nat @ A @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1239_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1240_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1241_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1242_nat__less__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1243_nat__less__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1244_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1245_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1246_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1247_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1248_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1249_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1250_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1251_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1252_int__ops_I5_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1253_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1254_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1255_int__ge__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1256_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1257_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1258_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1259_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1260_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1261_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1262_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1263_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1264_nat__int__add,axiom,
! [A: nat,B2: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
= ( plus_plus_nat @ A @ B2 ) ) ).
% nat_int_add
thf(fact_1265_int__induct,axiom,
! [P: int > $o,K: int,I2: int] :
( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_induct
% Helper facts (9)
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_3_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_nat @ ( degree4881254707062955960ring_a @ ( poly_F5739129160929385880ring_a @ ( map_na1928064127006292399ring_a @ f @ ( upt @ zero_zero_nat @ n ) ) ) ) @ n ).
%------------------------------------------------------------------------------