TPTP Problem File: ITP095^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP095^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Liouville_Numbers problem prob_189__5867098_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Liouville_Numbers/prob_189__5867098_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 433 ( 186 unt; 79 typ; 0 def)
% Number of atoms : 956 ( 415 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2550 ( 66 ~; 22 |; 36 &;2130 @)
% ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 77 ( 77 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 69 usr; 26 con; 0-2 aty)
% Number of variables : 634 ( 11 ^; 612 !; 11 ?; 634 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:40:52.492
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
poly_poly_real: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
set_poly_real: $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__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (69)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
abs_abs_poly_real: poly_real > poly_real ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__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_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__Int__Oint_J,type,
zero_zero_poly_int: poly_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
zero_zero_poly_nat: poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
zero_z1423781445y_real: poly_poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
zero_zero_poly_real: poly_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
ring_1_Ints_int: set_int ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
ring_1690226883y_real: set_poly_real ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Int__Oint_J,type,
ring_12102921859ly_int: int > poly_int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
ring_11511526659y_real: int > poly_real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_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__Polynomial__Opoly_It__Real__Oreal_J,type,
ord_less_poly_real: poly_real > poly_real > $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__Polynomial__Opoly_It__Real__Oreal_J,type,
ord_le1180086932y_real: poly_real > poly_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Polynomial_Oalgebraic_001t__Real__Oreal,type,
algebraic_real: real > $o ).
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__Polynomial__Opoly_It__Real__Oreal_J,type,
degree_poly_real: poly_poly_real > nat ).
thf(sy_c_Polynomial_Odegree_001t__Real__Oreal,type,
degree_real: poly_real > nat ).
thf(sy_c_Polynomial_Opoly_001t__Int__Oint,type,
poly_int2: poly_int > int > int ).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
poly_nat2: poly_nat > nat > nat ).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_poly_real2: poly_poly_real > poly_real > poly_real ).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal,type,
poly_real2: poly_real > real > real ).
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__Polynomial__Opoly_It__Real__Oreal_J,type,
coeff_poly_real: poly_poly_real > nat > poly_real ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Real__Oreal,type,
coeff_real: poly_real > nat > real ).
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__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_2108872382y_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_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
field_1537545994s_real: set_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
divide1727078534y_real: poly_real > poly_real > poly_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
member_poly_real: poly_real > set_poly_real > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_A____,type,
a: real ).
thf(sy_v_M____,type,
m: real ).
thf(sy_v_a____,type,
a2: int ).
thf(sy_v_b____,type,
b: int ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p____,type,
p: poly_real ).
thf(sy_v_roots____,type,
roots: set_real ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_x,type,
x: real ).
% Relevant facts (353)
thf(fact_0_assms_I2_J,axiom,
algebraic_real @ x ).
% assms(2)
thf(fact_1_p_I2_J,axiom,
p != zero_zero_poly_real ).
% p(2)
thf(fact_2_p_I1_J,axiom,
! [I: nat] : ( member_real @ ( coeff_real @ p @ I ) @ ring_1_Ints_real ) ).
% p(1)
thf(fact_3_b,axiom,
ord_less_int @ zero_zero_int @ b ).
% b
thf(fact_4_p_I3_J,axiom,
( ( poly_real2 @ p @ x )
= zero_zero_real ) ).
% p(3)
thf(fact_5_no__root,axiom,
( ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) )
!= zero_zero_real ) ).
% no_root
thf(fact_6_n__def,axiom,
( n
= ( degree_real @ p ) ) ).
% n_def
thf(fact_7_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_8_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_9_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_10_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_11_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_12_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_13_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_14_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_15_poly__power,axiom,
! [P: poly_real,N: nat,X: real] :
( ( poly_real2 @ ( power_2108872382y_real @ P @ N ) @ X )
= ( power_power_real @ ( poly_real2 @ P @ X ) @ N ) ) ).
% poly_power
thf(fact_16_poly__power,axiom,
! [P: poly_nat,N: nat,X: nat] :
( ( poly_nat2 @ ( power_power_poly_nat @ P @ N ) @ X )
= ( power_power_nat @ ( poly_nat2 @ P @ X ) @ N ) ) ).
% poly_power
thf(fact_17_poly__power,axiom,
! [P: poly_int,N: nat,X: int] :
( ( poly_int2 @ ( power_power_poly_int @ P @ N ) @ X )
= ( power_power_int @ ( poly_int2 @ P @ X ) @ N ) ) ).
% poly_power
thf(fact_18_poly__1,axiom,
! [X: real] :
( ( poly_real2 @ one_one_poly_real @ X )
= one_one_real ) ).
% poly_1
thf(fact_19_poly__1,axiom,
! [X: int] :
( ( poly_int2 @ one_one_poly_int @ X )
= one_one_int ) ).
% poly_1
thf(fact_20_poly__1,axiom,
! [X: nat] :
( ( poly_nat2 @ one_one_poly_nat @ X )
= one_one_nat ) ).
% poly_1
thf(fact_21_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% of_int_abs
thf(fact_22_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
= ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% of_int_abs
thf(fact_23_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% of_int_power
thf(fact_24_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% of_int_power
thf(fact_25_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_26_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_27_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_28_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_29_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_30_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_31_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_32_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_33_irrationsl,axiom,
~ ( member_real @ x @ field_1537545994s_real ) ).
% irrationsl
thf(fact_34_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_35_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_int @ W )
= ( ring_1_of_int_int @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_36__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
~ ! [P2: poly_real] :
( ! [I2: nat] : ( member_real @ ( coeff_real @ P2 @ I2 ) @ ring_1_Ints_real )
=> ( ( P2 != zero_zero_poly_real )
=> ( ( poly_real2 @ P2 @ x )
!= zero_zero_real ) ) ) ).
% \<open>\<And>thesisa. (\<And>p. \<lbrakk>\<And>i. coeff p i \<in> \<int>; p \<noteq> 0; poly p x = 0\<rbrakk> \<Longrightarrow> thesisa) \<Longrightarrow> thesisa\<close>
thf(fact_37__092_060open_062x_A_092_060notin_062_Aroots_092_060close_062,axiom,
~ ( member_real @ x @ roots ) ).
% \<open>x \<notin> roots\<close>
thf(fact_38_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_11511526659y_real @ Z )
= zero_zero_poly_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_39_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_40_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_41_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_poly_real
= ( ring_11511526659y_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_42_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_43_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_44_of__int__0,axiom,
( ( ring_11511526659y_real @ zero_zero_int )
= zero_zero_poly_real ) ).
% of_int_0
thf(fact_45_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_46_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_47_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_48_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_49_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_50_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_51_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_real @ zero_z1423781445y_real @ N )
= zero_zero_poly_real ) ).
% coeff_0
thf(fact_52_coeff__0,axiom,
! [N: nat] :
( ( coeff_int @ zero_zero_poly_int @ N )
= zero_zero_int ) ).
% coeff_0
thf(fact_53_coeff__0,axiom,
! [N: nat] :
( ( coeff_nat @ zero_zero_poly_nat @ N )
= zero_zero_nat ) ).
% coeff_0
thf(fact_54_coeff__0,axiom,
! [N: nat] :
( ( coeff_real @ zero_zero_poly_real @ N )
= zero_zero_real ) ).
% coeff_0
thf(fact_55_poly__0,axiom,
! [X: poly_real] :
( ( poly_poly_real2 @ zero_z1423781445y_real @ X )
= zero_zero_poly_real ) ).
% poly_0
thf(fact_56_poly__0,axiom,
! [X: int] :
( ( poly_int2 @ zero_zero_poly_int @ X )
= zero_zero_int ) ).
% poly_0
thf(fact_57_poly__0,axiom,
! [X: nat] :
( ( poly_nat2 @ zero_zero_poly_nat @ X )
= zero_zero_nat ) ).
% poly_0
thf(fact_58_poly__0,axiom,
! [X: real] :
( ( poly_real2 @ zero_zero_poly_real @ X )
= zero_zero_real ) ).
% poly_0
thf(fact_59_lead__coeff__of__int,axiom,
! [K: int] :
( ( coeff_real @ ( ring_11511526659y_real @ K ) @ ( degree_real @ ( ring_11511526659y_real @ K ) ) )
= ( ring_1_of_int_real @ K ) ) ).
% lead_coeff_of_int
thf(fact_60_lead__coeff__of__int,axiom,
! [K: int] :
( ( coeff_int @ ( ring_12102921859ly_int @ K ) @ ( degree_int @ ( ring_12102921859ly_int @ K ) ) )
= ( ring_1_of_int_int @ K ) ) ).
% lead_coeff_of_int
thf(fact_61_leading__coeff__0__iff,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
= zero_zero_poly_real )
= ( P = zero_z1423781445y_real ) ) ).
% leading_coeff_0_iff
thf(fact_62_leading__coeff__0__iff,axiom,
! [P: poly_int] :
( ( ( coeff_int @ P @ ( degree_int @ P ) )
= zero_zero_int )
= ( P = zero_zero_poly_int ) ) ).
% leading_coeff_0_iff
thf(fact_63_leading__coeff__0__iff,axiom,
! [P: poly_real] :
( ( ( coeff_real @ P @ ( degree_real @ P ) )
= zero_zero_real )
= ( P = zero_zero_poly_real ) ) ).
% leading_coeff_0_iff
thf(fact_64_leading__coeff__0__iff,axiom,
! [P: poly_nat] :
( ( ( coeff_nat @ P @ ( degree_nat @ P ) )
= zero_zero_nat )
= ( P = zero_zero_poly_nat ) ) ).
% leading_coeff_0_iff
thf(fact_65_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_66_lead__coeff__1,axiom,
( ( coeff_int @ one_one_poly_int @ ( degree_int @ one_one_poly_int ) )
= one_one_int ) ).
% lead_coeff_1
thf(fact_67_lead__coeff__1,axiom,
( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
= one_one_nat ) ).
% lead_coeff_1
thf(fact_68_lead__coeff__1,axiom,
( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
= one_one_real ) ).
% lead_coeff_1
thf(fact_69_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_le1180086932y_real @ ( ring_11511526659y_real @ Z ) @ zero_zero_poly_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_70_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_71_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_72_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_le1180086932y_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_73_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_74_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_75_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_poly_real @ ( ring_11511526659y_real @ Z ) @ zero_zero_poly_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_76_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_77_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_78_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_poly_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_79_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_80_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_81_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_82_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_83_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_84_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_85_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_86_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_87_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_88_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_89_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_90_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_91_poly__eqI,axiom,
! [P: poly_real,Q: poly_real] :
( ! [N2: nat] :
( ( coeff_real @ P @ N2 )
= ( coeff_real @ Q @ N2 ) )
=> ( P = Q ) ) ).
% poly_eqI
thf(fact_92_mem__Collect__eq,axiom,
! [A: real,P3: real > $o] :
( ( member_real @ A @ ( collect_real @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_94_poly__eq__iff,axiom,
( ( ^ [Y: poly_real,Z2: poly_real] : Y = Z2 )
= ( ^ [P4: poly_real,Q2: poly_real] :
! [N3: nat] :
( ( coeff_real @ P4 @ N3 )
= ( coeff_real @ Q2 @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_95_Ints__0,axiom,
member_poly_real @ zero_zero_poly_real @ ring_1690226883y_real ).
% Ints_0
thf(fact_96_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_97_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_98_zero__poly_Orep__eq,axiom,
( ( coeff_poly_real @ zero_z1423781445y_real )
= ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% zero_poly.rep_eq
thf(fact_99_zero__poly_Orep__eq,axiom,
( ( coeff_int @ zero_zero_poly_int )
= ( ^ [Uu: nat] : zero_zero_int ) ) ).
% zero_poly.rep_eq
thf(fact_100_zero__poly_Orep__eq,axiom,
( ( coeff_nat @ zero_zero_poly_nat )
= ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% zero_poly.rep_eq
thf(fact_101_zero__poly_Orep__eq,axiom,
( ( coeff_real @ zero_zero_poly_real )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_poly.rep_eq
thf(fact_102_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_103_coeff__inject,axiom,
! [X: poly_real,Y2: poly_real] :
( ( ( coeff_real @ X )
= ( coeff_real @ Y2 ) )
= ( X = Y2 ) ) ).
% coeff_inject
thf(fact_104_le__degree,axiom,
! [P: poly_poly_real,N: nat] :
( ( ( coeff_poly_real @ P @ N )
!= zero_zero_poly_real )
=> ( ord_less_eq_nat @ N @ ( degree_poly_real @ P ) ) ) ).
% le_degree
thf(fact_105_le__degree,axiom,
! [P: poly_int,N: nat] :
( ( ( coeff_int @ P @ N )
!= zero_zero_int )
=> ( ord_less_eq_nat @ N @ ( degree_int @ P ) ) ) ).
% le_degree
thf(fact_106_le__degree,axiom,
! [P: poly_real,N: nat] :
( ( ( coeff_real @ P @ N )
!= zero_zero_real )
=> ( ord_less_eq_nat @ N @ ( degree_real @ P ) ) ) ).
% le_degree
thf(fact_107_le__degree,axiom,
! [P: poly_nat,N: nat] :
( ( ( coeff_nat @ P @ N )
!= zero_zero_nat )
=> ( ord_less_eq_nat @ N @ ( degree_nat @ P ) ) ) ).
% le_degree
thf(fact_108_poly__0__coeff__0,axiom,
! [P: poly_poly_real] :
( ( poly_poly_real2 @ P @ zero_zero_poly_real )
= ( coeff_poly_real @ P @ zero_zero_nat ) ) ).
% poly_0_coeff_0
thf(fact_109_poly__0__coeff__0,axiom,
! [P: poly_int] :
( ( poly_int2 @ P @ zero_zero_int )
= ( coeff_int @ P @ zero_zero_nat ) ) ).
% poly_0_coeff_0
thf(fact_110_poly__0__coeff__0,axiom,
! [P: poly_real] :
( ( poly_real2 @ P @ zero_zero_real )
= ( coeff_real @ P @ zero_zero_nat ) ) ).
% poly_0_coeff_0
thf(fact_111_poly__0__coeff__0,axiom,
! [P: poly_nat] :
( ( poly_nat2 @ P @ zero_zero_nat )
= ( coeff_nat @ P @ zero_zero_nat ) ) ).
% poly_0_coeff_0
thf(fact_112_algebraic__def,axiom,
( algebraic_real
= ( ^ [X2: real] :
? [P4: poly_real] :
( ! [I3: nat] : ( member_real @ ( coeff_real @ P4 @ I3 ) @ ring_1_Ints_real )
& ( P4 != zero_zero_poly_real )
& ( ( poly_real2 @ P4 @ X2 )
= zero_zero_real ) ) ) ) ).
% algebraic_def
thf(fact_113_algebraicI,axiom,
! [P: poly_real,X: real] :
( ! [I4: nat] : ( member_real @ ( coeff_real @ P @ I4 ) @ ring_1_Ints_real )
=> ( ( P != zero_zero_poly_real )
=> ( ( ( poly_real2 @ P @ X )
= zero_zero_real )
=> ( algebraic_real @ X ) ) ) ) ).
% algebraicI
thf(fact_114_algebraicE,axiom,
! [X: real] :
( ( algebraic_real @ X )
=> ~ ! [P2: poly_real] :
( ! [I2: nat] : ( member_real @ ( coeff_real @ P2 @ I2 ) @ ring_1_Ints_real )
=> ( ( P2 != zero_zero_poly_real )
=> ( ( poly_real2 @ P2 @ X )
!= zero_zero_real ) ) ) ) ).
% algebraicE
thf(fact_115_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_poly_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) ) ) ).
% of_int_pos
thf(fact_116_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_117_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_118_leading__coeff__neq__0,axiom,
! [P: poly_poly_real] :
( ( P != zero_z1423781445y_real )
=> ( ( coeff_poly_real @ P @ ( degree_poly_real @ P ) )
!= zero_zero_poly_real ) ) ).
% leading_coeff_neq_0
thf(fact_119_leading__coeff__neq__0,axiom,
! [P: poly_int] :
( ( P != zero_zero_poly_int )
=> ( ( coeff_int @ P @ ( degree_int @ P ) )
!= zero_zero_int ) ) ).
% leading_coeff_neq_0
thf(fact_120_leading__coeff__neq__0,axiom,
! [P: poly_nat] :
( ( P != zero_zero_poly_nat )
=> ( ( coeff_nat @ P @ ( degree_nat @ P ) )
!= zero_zero_nat ) ) ).
% leading_coeff_neq_0
thf(fact_121_leading__coeff__neq__0,axiom,
! [P: poly_real] :
( ( P != zero_zero_poly_real )
=> ( ( coeff_real @ P @ ( degree_real @ P ) )
!= zero_zero_real ) ) ).
% leading_coeff_neq_0
thf(fact_122_poly__all__0__iff__0,axiom,
! [P: poly_poly_real] :
( ( ! [X2: poly_real] :
( ( poly_poly_real2 @ P @ X2 )
= zero_zero_poly_real ) )
= ( P = zero_z1423781445y_real ) ) ).
% poly_all_0_iff_0
thf(fact_123_poly__all__0__iff__0,axiom,
! [P: poly_int] :
( ( ! [X2: int] :
( ( poly_int2 @ P @ X2 )
= zero_zero_int ) )
= ( P = zero_zero_poly_int ) ) ).
% poly_all_0_iff_0
thf(fact_124_poly__all__0__iff__0,axiom,
! [P: poly_real] :
( ( ! [X2: real] :
( ( poly_real2 @ P @ X2 )
= zero_zero_real ) )
= ( P = zero_zero_poly_real ) ) ).
% poly_all_0_iff_0
thf(fact_125_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_126_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_127_Ints__power,axiom,
! [A: real,N: nat] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( power_power_real @ A @ N ) @ ring_1_Ints_real ) ) ).
% Ints_power
thf(fact_128_Ints__power,axiom,
! [A: int,N: nat] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( power_power_int @ A @ N ) @ ring_1_Ints_int ) ) ).
% Ints_power
thf(fact_129_Ints__abs,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% Ints_abs
thf(fact_130_Ints__abs,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_abs
thf(fact_131_Ints__of__int,axiom,
! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% Ints_of_int
thf(fact_132_Ints__of__int,axiom,
! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% Ints_of_int
thf(fact_133_Ints__induct,axiom,
! [Q: real,P3: real > $o] :
( ( member_real @ Q @ ring_1_Ints_real )
=> ( ! [Z3: int] : ( P3 @ ( ring_1_of_int_real @ Z3 ) )
=> ( P3 @ Q ) ) ) ).
% Ints_induct
thf(fact_134_Ints__induct,axiom,
! [Q: int,P3: int > $o] :
( ( member_int @ Q @ ring_1_Ints_int )
=> ( ! [Z3: int] : ( P3 @ ( ring_1_of_int_int @ Z3 ) )
=> ( P3 @ Q ) ) ) ).
% Ints_induct
thf(fact_135_Ints__cases,axiom,
! [Q: real] :
( ( member_real @ Q @ ring_1_Ints_real )
=> ~ ! [Z3: int] :
( Q
!= ( ring_1_of_int_real @ Z3 ) ) ) ).
% Ints_cases
thf(fact_136_Ints__cases,axiom,
! [Q: int] :
( ( member_int @ Q @ ring_1_Ints_int )
=> ~ ! [Z3: int] :
( Q
!= ( ring_1_of_int_int @ Z3 ) ) ) ).
% Ints_cases
thf(fact_137_Ints__nonzero__abs__less1,axiom,
! [X: poly_real] :
( ( member_poly_real @ X @ ring_1690226883y_real )
=> ( ( ord_less_poly_real @ ( abs_abs_poly_real @ X ) @ one_one_poly_real )
=> ( X = zero_zero_poly_real ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_138_Ints__nonzero__abs__less1,axiom,
! [X: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
=> ( X = zero_zero_int ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_139_Ints__nonzero__abs__less1,axiom,
! [X: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( X = zero_zero_real ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_140_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_le1180086932y_real @ zero_zero_poly_real @ ( ring_11511526659y_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_141_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_nonneg
thf(fact_142_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_143_lead__coeff__power,axiom,
! [P: poly_real,N: nat] :
( ( coeff_real @ ( power_2108872382y_real @ P @ N ) @ ( degree_real @ ( power_2108872382y_real @ P @ N ) ) )
= ( power_power_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ N ) ) ).
% lead_coeff_power
thf(fact_144_lead__coeff__power,axiom,
! [P: poly_nat,N: nat] :
( ( coeff_nat @ ( power_power_poly_nat @ P @ N ) @ ( degree_nat @ ( power_power_poly_nat @ P @ N ) ) )
= ( power_power_nat @ ( coeff_nat @ P @ ( degree_nat @ P ) ) @ N ) ) ).
% lead_coeff_power
thf(fact_145_lead__coeff__power,axiom,
! [P: poly_int,N: nat] :
( ( coeff_int @ ( power_power_poly_int @ P @ N ) @ ( degree_int @ ( power_power_poly_int @ P @ N ) ) )
= ( power_power_int @ ( coeff_int @ P @ ( degree_int @ P ) ) @ N ) ) ).
% lead_coeff_power
thf(fact_146_Ints__nonzero__abs__ge1,axiom,
! [X: poly_real] :
( ( member_poly_real @ X @ ring_1690226883y_real )
=> ( ( X != zero_zero_poly_real )
=> ( ord_le1180086932y_real @ one_one_poly_real @ ( abs_abs_poly_real @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_147_Ints__nonzero__abs__ge1,axiom,
! [X: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( X != zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_148_Ints__nonzero__abs__ge1,axiom,
! [X: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( X != zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_149_poly__eq__poly__eq__iff,axiom,
! [P: poly_real,Q: poly_real] :
( ( ( poly_real2 @ P )
= ( poly_real2 @ Q ) )
= ( P = Q ) ) ).
% poly_eq_poly_eq_iff
thf(fact_150_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_151_of__int__lessD,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% of_int_lessD
thf(fact_152_of__int__leD,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% of_int_leD
thf(fact_153_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_154_int__poly__rat__no__root__ge,axiom,
! [P: poly_real,B: int,A: int] :
( ! [N2: nat] : ( member_real @ ( coeff_real @ P @ N2 ) @ ring_1_Ints_real )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ( poly_real2 @ P @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
!= zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ ( degree_real @ P ) ) ) @ ( abs_abs_real @ ( poly_real2 @ P @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) ) ) ) ) ) ) ).
% int_poly_rat_no_root_ge
thf(fact_155_power__decreasing__iff,axiom,
! [B: poly_real,M: nat,N: nat] :
( ( ord_less_poly_real @ zero_zero_poly_real @ B )
=> ( ( ord_less_poly_real @ B @ one_one_poly_real )
=> ( ( ord_le1180086932y_real @ ( power_2108872382y_real @ B @ M ) @ ( power_2108872382y_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_156_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_157_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_158_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_159_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_160_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_161_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_162_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_163__092_060open_062A_A_P_Areal__of__int_Ab_A_094_An_A_092_060le_062_AA_A_P_A1_092_060close_062,axiom,
ord_less_eq_real @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) @ ( divide_divide_real @ a @ one_one_real ) ).
% \<open>A / real_of_int b ^ n \<le> A / 1\<close>
thf(fact_164_zero__less__power__abs__iff,axiom,
! [A: poly_real,N: nat] :
( ( ord_less_poly_real @ zero_zero_poly_real @ ( power_2108872382y_real @ ( abs_abs_poly_real @ A ) @ N ) )
= ( ( A != zero_zero_poly_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_165_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
= ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_166_zero__less__power__abs__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= ( ( A != zero_zero_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_167_zero__le__divide__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_168_divide__le__0__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
= ( ( ord_less_eq_real @ A @ zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
thf(fact_169_A__pos,axiom,
ord_less_real @ zero_zero_real @ a ).
% A_pos
thf(fact_170_A__less_I1_J,axiom,
ord_less_real @ a @ one_one_real ).
% A_less(1)
thf(fact_171__092_060open_062A_A_092_060le_062_A1_092_060close_062,axiom,
ord_less_eq_real @ a @ one_one_real ).
% \<open>A \<le> 1\<close>
thf(fact_172_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_173_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_174_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_175_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_176_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_177_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_178_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_179_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_180_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_181_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_182_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_183_abs__divide,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_divide
thf(fact_184_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_185_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_186_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_187_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_188_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_189_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_190_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_191_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_192_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_193_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_194_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_195_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_196_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_197_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_nat @ X @ Y2 ) ) ) ).
% power_strict_increasing_iff
thf(fact_198_power__eq__0__iff,axiom,
! [A: poly_real,N: nat] :
( ( ( power_2108872382y_real @ A @ N )
= zero_zero_poly_real )
= ( ( A = zero_zero_poly_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_199_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_200_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_201_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_202_degree__0,axiom,
( ( degree_real @ zero_zero_poly_real )
= zero_zero_nat ) ).
% degree_0
thf(fact_203_degree__1,axiom,
( ( degree_real @ one_one_poly_real )
= zero_zero_nat ) ).
% degree_1
thf(fact_204_degree__of__int,axiom,
! [K: int] :
( ( degree_real @ ( ring_11511526659y_real @ K ) )
= zero_zero_nat ) ).
% degree_of_int
thf(fact_205_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_206_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_207_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_208_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_209_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_210_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_211_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_212_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_213_power__strict__decreasing__iff,axiom,
! [B: poly_real,M: nat,N: nat] :
( ( ord_less_poly_real @ zero_zero_poly_real @ B )
=> ( ( ord_less_poly_real @ B @ one_one_poly_real )
=> ( ( ord_less_poly_real @ ( power_2108872382y_real @ B @ M ) @ ( power_2108872382y_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_214_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_215_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_216_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_217_power__mono__iff,axiom,
! [A: poly_real,B: poly_real,N: nat] :
( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
=> ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_le1180086932y_real @ ( power_2108872382y_real @ A @ N ) @ ( power_2108872382y_real @ B @ N ) )
= ( ord_le1180086932y_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_218_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_219_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_220_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_221_power__increasing__iff,axiom,
! [B: real,X: nat,Y2: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_222_power__increasing__iff,axiom,
! [B: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_223_power__increasing__iff,axiom,
! [B: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_224__092_060open_062_092_060not_062_AA_A_P_Areal__of__int_Ab_A_094_An_A_060_A_092_060bar_062x_A_N_Areal__of__int_Aa_A_P_Areal__of__int_Ab_092_060bar_062_092_060close_062,axiom,
~ ( ord_less_real @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ).
% \<open>\<not> A / real_of_int b ^ n < \<bar>x - real_of_int a / real_of_int b\<bar>\<close>
thf(fact_225_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_226_div__poly__less,axiom,
! [X: poly_real,Y2: poly_real] :
( ( ord_less_nat @ ( degree_real @ X ) @ ( degree_real @ Y2 ) )
=> ( ( divide1727078534y_real @ X @ Y2 )
= zero_zero_poly_real ) ) ).
% div_poly_less
thf(fact_227_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_228_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_229_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_230_divide__poly__0,axiom,
! [F: poly_real] :
( ( divide1727078534y_real @ F @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% divide_poly_0
thf(fact_231_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
= zero_zero_poly_real ) ) ).
% zero_power
thf(fact_232_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_233_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_234_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_235_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_236_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_237_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_238_power__eq__iff__eq__base,axiom,
! [N: nat,A: poly_real,B: poly_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
=> ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
=> ( ( ( power_2108872382y_real @ A @ N )
= ( power_2108872382y_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_239_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_240_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_241_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_242_power__eq__imp__eq__base,axiom,
! [A: poly_real,N: nat,B: poly_real] :
( ( ( power_2108872382y_real @ A @ N )
= ( power_2108872382y_real @ B @ N ) )
=> ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
=> ( ( ord_le1180086932y_real @ zero_zero_poly_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_243_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_244_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_245_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_246_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_247_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_248_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_249_poly__IVT__neg,axiom,
! [A: real,B: real,P: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A ) )
=> ( ( ord_less_real @ ( poly_real2 @ P @ B ) @ zero_zero_real )
=> ? [X3: real] :
( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B )
& ( ( poly_real2 @ P @ X3 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_neg
thf(fact_250_poly__IVT__pos,axiom,
! [A: real,B: real,P: poly_real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( poly_real2 @ P @ A ) @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B ) )
=> ? [X3: real] :
( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B )
& ( ( poly_real2 @ P @ X3 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_pos
thf(fact_251_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_252_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).
% linordered_field_no_lb
thf(fact_253_coeff__eq__0,axiom,
! [P: poly_poly_real,N: nat] :
( ( ord_less_nat @ ( degree_poly_real @ P ) @ N )
=> ( ( coeff_poly_real @ P @ N )
= zero_zero_poly_real ) ) ).
% coeff_eq_0
thf(fact_254_coeff__eq__0,axiom,
! [P: poly_int,N: nat] :
( ( ord_less_nat @ ( degree_int @ P ) @ N )
=> ( ( coeff_int @ P @ N )
= zero_zero_int ) ) ).
% coeff_eq_0
thf(fact_255_coeff__eq__0,axiom,
! [P: poly_real,N: nat] :
( ( ord_less_nat @ ( degree_real @ P ) @ N )
=> ( ( coeff_real @ P @ N )
= zero_zero_real ) ) ).
% coeff_eq_0
thf(fact_256_coeff__eq__0,axiom,
! [P: poly_nat,N: nat] :
( ( ord_less_nat @ ( degree_nat @ P ) @ N )
=> ( ( coeff_nat @ P @ N )
= zero_zero_nat ) ) ).
% coeff_eq_0
thf(fact_257_less__degree__imp,axiom,
! [N: nat,P: poly_poly_real] :
( ( ord_less_nat @ N @ ( degree_poly_real @ P ) )
=> ? [I4: nat] :
( ( ord_less_nat @ N @ I4 )
& ( ( coeff_poly_real @ P @ I4 )
!= zero_zero_poly_real ) ) ) ).
% less_degree_imp
thf(fact_258_less__degree__imp,axiom,
! [N: nat,P: poly_int] :
( ( ord_less_nat @ N @ ( degree_int @ P ) )
=> ? [I4: nat] :
( ( ord_less_nat @ N @ I4 )
& ( ( coeff_int @ P @ I4 )
!= zero_zero_int ) ) ) ).
% less_degree_imp
thf(fact_259_less__degree__imp,axiom,
! [N: nat,P: poly_real] :
( ( ord_less_nat @ N @ ( degree_real @ P ) )
=> ? [I4: nat] :
( ( ord_less_nat @ N @ I4 )
& ( ( coeff_real @ P @ I4 )
!= zero_zero_real ) ) ) ).
% less_degree_imp
thf(fact_260_less__degree__imp,axiom,
! [N: nat,P: poly_nat] :
( ( ord_less_nat @ N @ ( degree_nat @ P ) )
=> ? [I4: nat] :
( ( ord_less_nat @ N @ I4 )
& ( ( coeff_nat @ P @ I4 )
!= zero_zero_nat ) ) ) ).
% less_degree_imp
thf(fact_261_degree__le,axiom,
! [N: nat,P: poly_poly_real] :
( ! [I4: nat] :
( ( ord_less_nat @ N @ I4 )
=> ( ( coeff_poly_real @ P @ I4 )
= zero_zero_poly_real ) )
=> ( ord_less_eq_nat @ ( degree_poly_real @ P ) @ N ) ) ).
% degree_le
thf(fact_262_degree__le,axiom,
! [N: nat,P: poly_int] :
( ! [I4: nat] :
( ( ord_less_nat @ N @ I4 )
=> ( ( coeff_int @ P @ I4 )
= zero_zero_int ) )
=> ( ord_less_eq_nat @ ( degree_int @ P ) @ N ) ) ).
% degree_le
thf(fact_263_degree__le,axiom,
! [N: nat,P: poly_real] :
( ! [I4: nat] :
( ( ord_less_nat @ N @ I4 )
=> ( ( coeff_real @ P @ I4 )
= zero_zero_real ) )
=> ( ord_less_eq_nat @ ( degree_real @ P ) @ N ) ) ).
% degree_le
thf(fact_264_degree__le,axiom,
! [N: nat,P: poly_nat] :
( ! [I4: nat] :
( ( ord_less_nat @ N @ I4 )
=> ( ( coeff_nat @ P @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( degree_nat @ P ) @ N ) ) ).
% degree_le
thf(fact_265_Liouville__Numbers__Misc_ORats__cases_H,axiom,
! [X: real] :
( ( member_real @ X @ field_1537545994s_real )
=> ~ ! [P2: int,Q3: int] :
( ( ord_less_int @ zero_zero_int @ Q3 )
=> ( X
!= ( divide_divide_real @ ( ring_1_of_int_real @ P2 ) @ ( ring_1_of_int_real @ Q3 ) ) ) ) ) ).
% Liouville_Numbers_Misc.Rats_cases'
thf(fact_266_coeff__0__power,axiom,
! [P: poly_real,N: nat] :
( ( coeff_real @ ( power_2108872382y_real @ P @ N ) @ zero_zero_nat )
= ( power_power_real @ ( coeff_real @ P @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_267_coeff__0__power,axiom,
! [P: poly_nat,N: nat] :
( ( coeff_nat @ ( power_power_poly_nat @ P @ N ) @ zero_zero_nat )
= ( power_power_nat @ ( coeff_nat @ P @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_268_coeff__0__power,axiom,
! [P: poly_int,N: nat] :
( ( coeff_int @ ( power_power_poly_int @ P @ N ) @ zero_zero_nat )
= ( power_power_int @ ( coeff_int @ P @ zero_zero_nat ) @ N ) ) ).
% coeff_0_power
thf(fact_269_eq__zero__or__degree__less,axiom,
! [P: poly_poly_real,N: nat] :
( ( ord_less_eq_nat @ ( degree_poly_real @ P ) @ N )
=> ( ( ( coeff_poly_real @ P @ N )
= zero_zero_poly_real )
=> ( ( P = zero_z1423781445y_real )
| ( ord_less_nat @ ( degree_poly_real @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_270_eq__zero__or__degree__less,axiom,
! [P: poly_int,N: nat] :
( ( ord_less_eq_nat @ ( degree_int @ P ) @ N )
=> ( ( ( coeff_int @ P @ N )
= zero_zero_int )
=> ( ( P = zero_zero_poly_int )
| ( ord_less_nat @ ( degree_int @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_271_eq__zero__or__degree__less,axiom,
! [P: poly_real,N: nat] :
( ( ord_less_eq_nat @ ( degree_real @ P ) @ N )
=> ( ( ( coeff_real @ P @ N )
= zero_zero_real )
=> ( ( P = zero_zero_poly_real )
| ( ord_less_nat @ ( degree_real @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_272_eq__zero__or__degree__less,axiom,
! [P: poly_nat,N: nat] :
( ( ord_less_eq_nat @ ( degree_nat @ P ) @ N )
=> ( ( ( coeff_nat @ P @ N )
= zero_zero_nat )
=> ( ( P = zero_zero_poly_nat )
| ( ord_less_nat @ ( degree_nat @ P ) @ N ) ) ) ) ).
% eq_zero_or_degree_less
thf(fact_273_power__not__zero,axiom,
! [A: poly_real,N: nat] :
( ( A != zero_zero_poly_real )
=> ( ( power_2108872382y_real @ A @ N )
!= zero_zero_poly_real ) ) ).
% power_not_zero
thf(fact_274_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_275_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_276_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_277_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_278_poly__pinfty__gt__lc,axiom,
! [P: poly_real] :
( ( ord_less_real @ zero_zero_real @ ( coeff_real @ P @ ( degree_real @ P ) ) )
=> ? [N2: real] :
! [X4: real] :
( ( ord_less_eq_real @ N2 @ X4 )
=> ( ord_less_eq_real @ ( coeff_real @ P @ ( degree_real @ P ) ) @ ( poly_real2 @ P @ X4 ) ) ) ) ).
% poly_pinfty_gt_lc
thf(fact_279_power__abs,axiom,
! [A: real,N: nat] :
( ( abs_abs_real @ ( power_power_real @ A @ N ) )
= ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% power_abs
thf(fact_280_power__abs,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ A @ N ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% power_abs
thf(fact_281_algebraic__altdef,axiom,
( algebraic_real
= ( ^ [X2: real] :
? [P4: poly_real] :
( ! [I3: nat] : ( member_real @ ( coeff_real @ P4 @ I3 ) @ field_1537545994s_real )
& ( P4 != zero_zero_poly_real )
& ( ( poly_real2 @ P4 @ X2 )
= zero_zero_real ) ) ) ) ).
% algebraic_altdef
thf(fact_282_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_283_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_284_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_285_divide__nonneg__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_286_divide__nonneg__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_287_divide__nonpos__nonneg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_288_divide__nonpos__nonpos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_289_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_290_power__mono,axiom,
! [A: poly_real,B: poly_real,N: nat] :
( ( ord_le1180086932y_real @ A @ B )
=> ( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
=> ( ord_le1180086932y_real @ ( power_2108872382y_real @ A @ N ) @ ( power_2108872382y_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_291_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_292_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_293_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_294_zero__le__power,axiom,
! [A: poly_real,N: nat] :
( ( ord_le1180086932y_real @ zero_zero_poly_real @ A )
=> ( ord_le1180086932y_real @ zero_zero_poly_real @ ( power_2108872382y_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_295_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_296_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_297_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_298_divide__neg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_299_divide__neg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_300_divide__pos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_301_divide__pos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_302_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_303_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_304_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_305_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_306_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_307_zero__less__power,axiom,
! [A: poly_real,N: nat] :
( ( ord_less_poly_real @ zero_zero_poly_real @ A )
=> ( ord_less_poly_real @ zero_zero_poly_real @ ( power_2108872382y_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_308_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_309_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_310_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_311_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_312_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
= one_one_poly_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_2108872382y_real @ zero_zero_poly_real @ N )
= zero_zero_poly_real ) ) ) ).
% power_0_left
thf(fact_313_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_314_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_315_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_316_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_317_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_318_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_319_power__increasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_320_power__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_321_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_322_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_323_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_324_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_325_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_326_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_327_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_328_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_329_nonzero__abs__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_330_frac__le,axiom,
! [Y2: real,X: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_331_frac__less,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_332_frac__less2,axiom,
! [X: real,Y2: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_333_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_334_divide__nonneg__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_335_divide__nonneg__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_336_divide__nonpos__neg,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y2 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_337_divide__nonpos__pos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_338_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_339_A__less_I2_J,axiom,
ord_less_real @ a @ ( divide_divide_real @ one_one_real @ m ) ).
% A_less(2)
thf(fact_340_ab,axiom,
ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) @ ( divide_divide_real @ a @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ n ) ) ).
% ab
thf(fact_341_ab_H,axiom,
ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) @ a ).
% ab'
thf(fact_342_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_343_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_344_M__pos,axiom,
ord_less_real @ zero_zero_real @ m ).
% M_pos
thf(fact_345_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_346_A__less_I3_J,axiom,
! [X5: real] :
( ( X5 != x )
=> ( ( ( poly_real2 @ p @ X5 )
= zero_zero_real )
=> ( ord_less_real @ a @ ( abs_abs_real @ ( minus_minus_real @ X5 @ x ) ) ) ) ) ).
% A_less(3)
thf(fact_347_that,axiom,
! [C: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ! [P2: int,Q3: int] :
( ( ord_less_int @ zero_zero_int @ Q3 )
=> ( ord_less_real @ ( divide_divide_real @ C @ ( power_power_real @ ( ring_1_of_int_real @ Q3 ) @ N ) ) @ ( abs_abs_real @ ( minus_minus_real @ x @ ( divide_divide_real @ ( ring_1_of_int_real @ P2 ) @ ( ring_1_of_int_real @ Q3 ) ) ) ) ) )
=> thesis ) ) ).
% that
thf(fact_348_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_349_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_350_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_351_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_352_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ).
%------------------------------------------------------------------------------