TPTP Problem File: ITP094^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP094^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Liouville_Numbers problem prob_128__5866194_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_128__5866194_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 450 ( 208 unt; 105 typ; 0 def)
% Number of atoms : 818 ( 457 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 1906 ( 173 ~; 17 |; 91 &;1377 @)
% ( 0 <=>; 248 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 223 ( 223 >; 0 *; 0 +; 0 <<)
% Number of symbols : 95 ( 94 usr; 19 con; 0-6 aty)
% Number of variables : 598 ( 66 ^; 485 !; 47 ?; 598 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:39:38.922
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
poly_poly_poly_real: $tType ).
thf(ty_n_t__Set__Oset_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
set_poly_poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
poly_poly_real: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J,type,
poly_poly_nat: $tType ).
thf(ty_n_t__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__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (94)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
finite1328464339y_real: set_poly_poly_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
finite1810960971y_real: set_poly_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
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__Nat__Onat_J,type,
one_one_poly_nat: poly_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
one_on501200385y_real: poly_poly_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
one_one_poly_real: poly_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
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__Nat__Onat_J_J,type,
zero_z1059985641ly_nat: poly_poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
zero_z935034829y_real: poly_poly_poly_real ).
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__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
ring_1897377867y_real: set_poly_poly_real ).
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_Nat_OSuc,type,
suc: nat > nat ).
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__Polynomial__Opoly_It__Real__Oreal_J_J,type,
ord_le38482960y_real: poly_poly_real > poly_poly_real > $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__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_Oalgebraic__int_001t__Real__Oreal,type,
algebraic_int_real: real > $o ).
thf(sy_c_Polynomial_Ocr__poly_001t__Real__Oreal,type,
cr_poly_real: ( nat > real ) > poly_real > $o ).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat,type,
degree_nat: poly_nat > nat ).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
degree_poly_nat: poly_poly_nat > nat ).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
degree360860553y_real: poly_poly_poly_real > 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_Odivide__poly__main_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
divide924636027y_real: poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > poly_poly_poly_real > nat > nat > poly_poly_poly_real ).
thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
divide1142363123y_real: poly_real > poly_poly_real > poly_poly_real > poly_poly_real > nat > nat > poly_poly_real ).
thf(sy_c_Polynomial_Odivide__poly__main_001t__Real__Oreal,type,
divide1561404011n_real: real > poly_real > poly_real > poly_real > nat > nat > poly_real ).
thf(sy_c_Polynomial_Ois__zero_001t__Nat__Onat,type,
is_zero_nat: poly_nat > $o ).
thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
is_zero_poly_real: poly_poly_real > $o ).
thf(sy_c_Polynomial_Ois__zero_001t__Real__Oreal,type,
is_zero_real: poly_real > $o ).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
order_poly_poly_real: poly_poly_real > poly_poly_poly_real > nat ).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
order_poly_real: poly_real > poly_poly_real > nat ).
thf(sy_c_Polynomial_Oorder_001t__Real__Oreal,type,
order_real: real > poly_real > nat ).
thf(sy_c_Polynomial_Opcr__poly_001t__Nat__Onat_001t__Nat__Onat,type,
pcr_poly_nat_nat: ( nat > nat > $o ) > ( nat > nat ) > poly_nat > $o ).
thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Nat__Onat_J_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
pcr_po273983709ly_nat: ( poly_nat > poly_nat > $o ) > ( nat > poly_nat ) > poly_poly_nat > $o ).
thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
pcr_po1200519205y_real: ( poly_poly_real > poly_poly_real > $o ) > ( nat > poly_poly_real ) > poly_poly_poly_real > $o ).
thf(sy_c_Polynomial_Opcr__poly_001t__Polynomial__Opoly_It__Real__Oreal_J_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
pcr_po1314690837y_real: ( poly_real > poly_real > $o ) > ( nat > poly_real ) > poly_poly_real > $o ).
thf(sy_c_Polynomial_Opcr__poly_001t__Real__Oreal_001t__Real__Oreal,type,
pcr_poly_real_real: ( real > real > $o ) > ( nat > real ) > poly_real > $o ).
thf(sy_c_Polynomial_Opderiv_001t__Nat__Onat,type,
pderiv_nat: poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opderiv_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
pderiv_poly_real: poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Opderiv_001t__Real__Oreal,type,
pderiv_real: poly_real > poly_real ).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat,type,
poly_nat2: poly_nat > nat > nat ).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_poly_nat2: poly_poly_nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
poly_poly_poly_real2: poly_poly_poly_real > poly_poly_real > poly_poly_real ).
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__Nat__Onat,type,
coeff_nat: poly_nat > nat > nat ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
coeff_poly_nat: poly_poly_nat > nat > poly_nat ).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
coeff_poly_poly_real: poly_poly_poly_real > nat > poly_poly_real ).
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_Polynomial_Opoly__cutoff_001t__Nat__Onat,type,
poly_cutoff_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_c1404107022y_real: nat > poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Real__Oreal,type,
poly_cutoff_real: nat > poly_real > poly_real ).
thf(sy_c_Polynomial_Opoly__shift_001t__Nat__Onat,type,
poly_shift_nat: nat > poly_nat > poly_nat ).
thf(sy_c_Polynomial_Opoly__shift_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_shift_poly_real: nat > poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Opoly__shift_001t__Real__Oreal,type,
poly_shift_real: nat > poly_real > poly_real ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat,type,
reflect_poly_nat: poly_nat > poly_nat ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
reflec781175074ly_nat: poly_poly_nat > poly_poly_nat ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
reflec144234502y_real: poly_poly_poly_real > poly_poly_poly_real ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
reflec1522834046y_real: poly_poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal,type,
reflect_poly_real: poly_real > poly_real ).
thf(sy_c_Polynomial_Orsquarefree_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
rsquar1555552848y_real: poly_poly_real > $o ).
thf(sy_c_Polynomial_Orsquarefree_001t__Real__Oreal,type,
rsquarefree_real: poly_real > $o ).
thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat,type,
synthetic_div_nat: poly_nat > nat > poly_nat ).
thf(sy_c_Polynomial_Osynthetic__div_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
synthe1498897281y_real: poly_poly_real > poly_real > poly_poly_real ).
thf(sy_c_Polynomial_Osynthetic__div_001t__Real__Oreal,type,
synthetic_div_real: poly_real > real > poly_real ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
field_1537545994s_real: set_real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
dvd_dvd_poly_nat: poly_nat > poly_nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
dvd_dv1946063458y_real: poly_poly_real > poly_poly_real > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
dvd_dvd_poly_real: poly_real > poly_real > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
collec927113489y_real: ( poly_poly_real > $o ) > set_poly_poly_real ).
thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
collect_poly_real: ( poly_real > $o ) > set_poly_real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
member1159720147y_real: poly_poly_real > set_poly_poly_real > $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_n____,type,
n: nat ).
thf(sy_v_p____,type,
p: poly_real ).
thf(sy_v_x,type,
x: real ).
% Relevant facts (344)
thf(fact_0_poly__roots__finite,axiom,
! [P: poly_poly_poly_real] :
( ( P != zero_z935034829y_real )
=> ( finite1328464339y_real
@ ( collec927113489y_real
@ ^ [X: poly_poly_real] :
( ( poly_poly_poly_real2 @ P @ X )
= zero_z1423781445y_real ) ) ) ) ).
% poly_roots_finite
thf(fact_1_poly__roots__finite,axiom,
! [P: poly_poly_real] :
( ( P != zero_z1423781445y_real )
=> ( finite1810960971y_real
@ ( collect_poly_real
@ ^ [X: poly_real] :
( ( poly_poly_real2 @ P @ X )
= zero_zero_poly_real ) ) ) ) ).
% poly_roots_finite
thf(fact_2_poly__roots__finite,axiom,
! [P: poly_real] :
( ( P != zero_zero_poly_real )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( poly_real2 @ P @ X )
= zero_zero_real ) ) ) ) ).
% poly_roots_finite
thf(fact_3_p_I2_J,axiom,
p != zero_zero_poly_real ).
% p(2)
thf(fact_4__092_060open_062pderiv_Ap_A_092_060noteq_062_A0_092_060close_062,axiom,
( ( pderiv_real @ p )
!= zero_zero_poly_real ) ).
% \<open>pderiv p \<noteq> 0\<close>
thf(fact_5_p_I3_J,axiom,
( ( poly_real2 @ p @ x )
= zero_zero_real ) ).
% p(3)
thf(fact_6_poly__0,axiom,
! [X2: poly_poly_real] :
( ( poly_poly_poly_real2 @ zero_z935034829y_real @ X2 )
= zero_z1423781445y_real ) ).
% poly_0
thf(fact_7_poly__0,axiom,
! [X2: poly_nat] :
( ( poly_poly_nat2 @ zero_z1059985641ly_nat @ X2 )
= zero_zero_poly_nat ) ).
% poly_0
thf(fact_8_poly__0,axiom,
! [X2: poly_real] :
( ( poly_poly_real2 @ zero_z1423781445y_real @ X2 )
= zero_zero_poly_real ) ).
% poly_0
thf(fact_9_poly__0,axiom,
! [X2: nat] :
( ( poly_nat2 @ zero_zero_poly_nat @ X2 )
= zero_zero_nat ) ).
% poly_0
thf(fact_10_poly__0,axiom,
! [X2: real] :
( ( poly_real2 @ zero_zero_poly_real @ X2 )
= zero_zero_real ) ).
% poly_0
thf(fact_11_finite__Collect__conjI,axiom,
! [P2: poly_real > $o,Q: poly_real > $o] :
( ( ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
| ( finite1810960971y_real @ ( collect_poly_real @ Q ) ) )
=> ( finite1810960971y_real
@ ( collect_poly_real
@ ^ [X: poly_real] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_12_finite__Collect__conjI,axiom,
! [P2: real > $o,Q: real > $o] :
( ( ( finite_finite_real @ ( collect_real @ P2 ) )
| ( finite_finite_real @ ( collect_real @ Q ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_13_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_14_finite__Collect__disjI,axiom,
! [P2: poly_real > $o,Q: poly_real > $o] :
( ( finite1810960971y_real
@ ( collect_poly_real
@ ^ [X: poly_real] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
& ( finite1810960971y_real @ ( collect_poly_real @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_15_finite__Collect__disjI,axiom,
! [P2: real > $o,Q: real > $o] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_real @ ( collect_real @ P2 ) )
& ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_16_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_17_poly__all__0__iff__0,axiom,
! [P: poly_poly_poly_real] :
( ( ! [X: poly_poly_real] :
( ( poly_poly_poly_real2 @ P @ X )
= zero_z1423781445y_real ) )
= ( P = zero_z935034829y_real ) ) ).
% poly_all_0_iff_0
thf(fact_18_poly__all__0__iff__0,axiom,
! [P: poly_real] :
( ( ! [X: real] :
( ( poly_real2 @ P @ X )
= zero_zero_real ) )
= ( P = zero_zero_poly_real ) ) ).
% poly_all_0_iff_0
thf(fact_19_poly__all__0__iff__0,axiom,
! [P: poly_poly_real] :
( ( ! [X: poly_real] :
( ( poly_poly_real2 @ P @ X )
= zero_zero_poly_real ) )
= ( P = zero_z1423781445y_real ) ) ).
% poly_all_0_iff_0
thf(fact_20_rsquarefree__roots,axiom,
( rsquarefree_real
= ( ^ [P3: poly_real] :
! [A: real] :
~ ( ( ( poly_real2 @ P3 @ A )
= zero_zero_real )
& ( ( poly_real2 @ ( pderiv_real @ P3 ) @ A )
= zero_zero_real ) ) ) ) ).
% rsquarefree_roots
thf(fact_21_not__finite__existsD,axiom,
! [P2: real > $o] :
( ~ ( finite_finite_real @ ( collect_real @ P2 ) )
=> ? [X_1: real] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_22_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_23_not__finite__existsD,axiom,
! [P2: poly_real > $o] :
( ~ ( finite1810960971y_real @ ( collect_poly_real @ P2 ) )
=> ? [X_1: poly_real] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_24_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B: set_real,R: real > real > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_real @ B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: real] :
( ( member_real @ X3 @ B )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A: real] :
( ( member_real @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_25_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A: real] :
( ( member_real @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_26_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B: set_poly_real,R: real > poly_real > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite1810960971y_real @ B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: poly_real] :
( ( member_poly_real @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: poly_real] :
( ( member_poly_real @ X3 @ B )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A: real] :
( ( member_real @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_27_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_real,R: nat > real > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_real @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: real] :
( ( member_real @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A: nat] :
( ( member_nat @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_28_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A: nat] :
( ( member_nat @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_29_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_poly_real,R: nat > poly_real > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite1810960971y_real @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: poly_real] :
( ( member_poly_real @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: poly_real] :
( ( member_poly_real @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A: nat] :
( ( member_nat @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_30_pigeonhole__infinite__rel,axiom,
! [A2: set_poly_real,B: set_real,R: poly_real > real > $o] :
( ~ ( finite1810960971y_real @ A2 )
=> ( ( finite_finite_real @ B )
=> ( ! [X3: poly_real] :
( ( member_poly_real @ X3 @ A2 )
=> ? [Xa: real] :
( ( member_real @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: real] :
( ( member_real @ X3 @ B )
& ~ ( finite1810960971y_real
@ ( collect_poly_real
@ ^ [A: poly_real] :
( ( member_poly_real @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_31_pigeonhole__infinite__rel,axiom,
! [A2: set_poly_real,B: set_nat,R: poly_real > nat > $o] :
( ~ ( finite1810960971y_real @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: poly_real] :
( ( member_poly_real @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite1810960971y_real
@ ( collect_poly_real
@ ^ [A: poly_real] :
( ( member_poly_real @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_32_pigeonhole__infinite__rel,axiom,
! [A2: set_poly_real,B: set_poly_real,R: poly_real > poly_real > $o] :
( ~ ( finite1810960971y_real @ A2 )
=> ( ( finite1810960971y_real @ B )
=> ( ! [X3: poly_real] :
( ( member_poly_real @ X3 @ A2 )
=> ? [Xa: poly_real] :
( ( member_poly_real @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: poly_real] :
( ( member_poly_real @ X3 @ B )
& ~ ( finite1810960971y_real
@ ( collect_poly_real
@ ^ [A: poly_real] :
( ( member_poly_real @ A @ A2 )
& ( R @ A @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_33_poly__eq__poly__eq__iff,axiom,
! [P: poly_real,Q2: poly_real] :
( ( ( poly_real2 @ P )
= ( poly_real2 @ Q2 ) )
= ( P = Q2 ) ) ).
% poly_eq_poly_eq_iff
thf(fact_34_poly__eq__poly__eq__iff,axiom,
! [P: poly_poly_real,Q2: poly_poly_real] :
( ( ( poly_poly_real2 @ P )
= ( poly_poly_real2 @ Q2 ) )
= ( P = Q2 ) ) ).
% poly_eq_poly_eq_iff
thf(fact_35_n__def,axiom,
( n
= ( degree_real @ p ) ) ).
% n_def
thf(fact_36_assms_I2_J,axiom,
algebraic_real @ x ).
% assms(2)
thf(fact_37_irrationsl,axiom,
~ ( member_real @ x @ field_1537545994s_real ) ).
% irrationsl
thf(fact_38_degree__0,axiom,
( ( degree_real @ zero_zero_poly_real )
= zero_zero_nat ) ).
% degree_0
thf(fact_39_degree__0,axiom,
( ( degree_poly_real @ zero_z1423781445y_real )
= zero_zero_nat ) ).
% degree_0
thf(fact_40_degree__0,axiom,
( ( degree_nat @ zero_zero_poly_nat )
= zero_zero_nat ) ).
% degree_0
thf(fact_41_pderiv__0,axiom,
( ( pderiv_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% pderiv_0
thf(fact_42_pderiv__0,axiom,
( ( pderiv_poly_real @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% pderiv_0
thf(fact_43_pderiv__0,axiom,
( ( pderiv_nat @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% pderiv_0
thf(fact_44_pderiv__eq__0__iff,axiom,
! [P: poly_real] :
( ( ( pderiv_real @ P )
= zero_zero_poly_real )
= ( ( degree_real @ P )
= zero_zero_nat ) ) ).
% pderiv_eq_0_iff
thf(fact_45_pderiv__eq__0__iff,axiom,
! [P: poly_poly_real] :
( ( ( pderiv_poly_real @ P )
= zero_z1423781445y_real )
= ( ( degree_poly_real @ P )
= zero_zero_nat ) ) ).
% pderiv_eq_0_iff
thf(fact_46_pderiv__eq__0__iff,axiom,
! [P: poly_nat] :
( ( ( pderiv_nat @ P )
= zero_zero_poly_nat )
= ( ( degree_nat @ P )
= zero_zero_nat ) ) ).
% pderiv_eq_0_iff
thf(fact_47_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_48_zero__reorient,axiom,
! [X2: poly_real] :
( ( zero_zero_poly_real = X2 )
= ( X2 = zero_zero_poly_real ) ) ).
% zero_reorient
thf(fact_49_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_50_zero__reorient,axiom,
! [X2: poly_poly_real] :
( ( zero_z1423781445y_real = X2 )
= ( X2 = zero_z1423781445y_real ) ) ).
% zero_reorient
thf(fact_51_zero__reorient,axiom,
! [X2: poly_nat] :
( ( zero_zero_poly_nat = X2 )
= ( X2 = zero_zero_poly_nat ) ) ).
% zero_reorient
thf(fact_52_is__zero__null,axiom,
( is_zero_real
= ( ^ [P3: poly_real] : P3 = zero_zero_poly_real ) ) ).
% is_zero_null
thf(fact_53_is__zero__null,axiom,
( is_zero_poly_real
= ( ^ [P3: poly_poly_real] : P3 = zero_z1423781445y_real ) ) ).
% is_zero_null
thf(fact_54_is__zero__null,axiom,
( is_zero_nat
= ( ^ [P3: poly_nat] : P3 = zero_zero_poly_nat ) ) ).
% is_zero_null
thf(fact_55_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_cutoff_0
thf(fact_56_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_c1404107022y_real @ N @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% poly_cutoff_0
thf(fact_57_poly__cutoff__0,axiom,
! [N: nat] :
( ( poly_cutoff_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_cutoff_0
thf(fact_58_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_real] :
( ( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
= zero_zero_real )
= ( P = zero_zero_poly_real ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_59_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_poly_real] :
( ( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
= zero_zero_poly_real )
= ( P = zero_z1423781445y_real ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_60_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_nat] :
( ( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
= zero_zero_nat )
= ( P = zero_zero_poly_nat ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_61_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_poly_poly_real] :
( ( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
= zero_z1423781445y_real )
= ( P = zero_z935034829y_real ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_62_reflect__poly__at__0__eq__0__iff,axiom,
! [P: poly_poly_nat] :
( ( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
= zero_zero_poly_nat )
= ( P = zero_z1059985641ly_nat ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_63__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,
~ ! [P4: poly_real] :
( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
=> ( ( P4 != zero_zero_poly_real )
=> ( ( poly_real2 @ P4 @ 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_64_synthetic__div__eq__0__iff,axiom,
! [P: poly_real,C: real] :
( ( ( synthetic_div_real @ P @ C )
= zero_zero_poly_real )
= ( ( degree_real @ P )
= zero_zero_nat ) ) ).
% synthetic_div_eq_0_iff
thf(fact_65_synthetic__div__eq__0__iff,axiom,
! [P: poly_poly_real,C: poly_real] :
( ( ( synthe1498897281y_real @ P @ C )
= zero_z1423781445y_real )
= ( ( degree_poly_real @ P )
= zero_zero_nat ) ) ).
% synthetic_div_eq_0_iff
thf(fact_66_synthetic__div__eq__0__iff,axiom,
! [P: poly_nat,C: nat] :
( ( ( synthetic_div_nat @ P @ C )
= zero_zero_poly_nat )
= ( ( degree_nat @ P )
= zero_zero_nat ) ) ).
% synthetic_div_eq_0_iff
thf(fact_67_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_real @ N @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% poly_shift_0
thf(fact_68_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_poly_real @ N @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% poly_shift_0
thf(fact_69_poly__shift__0,axiom,
! [N: nat] :
( ( poly_shift_nat @ N @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% poly_shift_0
thf(fact_70_order__root,axiom,
! [P: poly_real,A3: real] :
( ( ( poly_real2 @ P @ A3 )
= zero_zero_real )
= ( ( P = zero_zero_poly_real )
| ( ( order_real @ A3 @ P )
!= zero_zero_nat ) ) ) ).
% order_root
thf(fact_71_order__root,axiom,
! [P: poly_poly_real,A3: poly_real] :
( ( ( poly_poly_real2 @ P @ A3 )
= zero_zero_poly_real )
= ( ( P = zero_z1423781445y_real )
| ( ( order_poly_real @ A3 @ P )
!= zero_zero_nat ) ) ) ).
% order_root
thf(fact_72_order__root,axiom,
! [P: poly_poly_poly_real,A3: poly_poly_real] :
( ( ( poly_poly_poly_real2 @ P @ A3 )
= zero_z1423781445y_real )
= ( ( P = zero_z935034829y_real )
| ( ( order_poly_poly_real @ A3 @ P )
!= zero_zero_nat ) ) ) ).
% order_root
thf(fact_73_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_74_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_75_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_76_leading__coeff__0__iff,axiom,
! [P: poly_poly_poly_real] :
( ( ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) )
= zero_z1423781445y_real )
= ( P = zero_z935034829y_real ) ) ).
% leading_coeff_0_iff
thf(fact_77_leading__coeff__0__iff,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
= zero_zero_poly_nat )
= ( P = zero_z1059985641ly_nat ) ) ).
% leading_coeff_0_iff
thf(fact_78_divide__poly__main__0,axiom,
! [R2: poly_real,D: poly_real,Dr: nat,N: nat] :
( ( divide1561404011n_real @ zero_zero_real @ zero_zero_poly_real @ R2 @ D @ Dr @ N )
= zero_zero_poly_real ) ).
% divide_poly_main_0
thf(fact_79_divide__poly__main__0,axiom,
! [R2: poly_poly_real,D: poly_poly_real,Dr: nat,N: nat] :
( ( divide1142363123y_real @ zero_zero_poly_real @ zero_z1423781445y_real @ R2 @ D @ Dr @ N )
= zero_z1423781445y_real ) ).
% divide_poly_main_0
thf(fact_80_divide__poly__main__0,axiom,
! [R2: poly_poly_poly_real,D: poly_poly_poly_real,Dr: nat,N: nat] :
( ( divide924636027y_real @ zero_z1423781445y_real @ zero_z935034829y_real @ R2 @ D @ Dr @ N )
= zero_z935034829y_real ) ).
% divide_poly_main_0
thf(fact_81_zero__poly_Otransfer,axiom,
( pcr_poly_real_real
@ ^ [Y: real,Z: real] : Y = Z
@ ^ [Uu: nat] : zero_zero_real
@ zero_zero_poly_real ) ).
% zero_poly.transfer
thf(fact_82_zero__poly_Otransfer,axiom,
( pcr_po1314690837y_real
@ ^ [Y: poly_real,Z: poly_real] : Y = Z
@ ^ [Uu: nat] : zero_zero_poly_real
@ zero_z1423781445y_real ) ).
% zero_poly.transfer
thf(fact_83_zero__poly_Otransfer,axiom,
( pcr_poly_nat_nat
@ ^ [Y: nat,Z: nat] : Y = Z
@ ^ [Uu: nat] : zero_zero_nat
@ zero_zero_poly_nat ) ).
% zero_poly.transfer
thf(fact_84_zero__poly_Otransfer,axiom,
( pcr_po1200519205y_real
@ ^ [Y: poly_poly_real,Z: poly_poly_real] : Y = Z
@ ^ [Uu: nat] : zero_z1423781445y_real
@ zero_z935034829y_real ) ).
% zero_poly.transfer
thf(fact_85_zero__poly_Otransfer,axiom,
( pcr_po273983709ly_nat
@ ^ [Y: poly_nat,Z: poly_nat] : Y = Z
@ ^ [Uu: nat] : zero_zero_poly_nat
@ zero_z1059985641ly_nat ) ).
% zero_poly.transfer
thf(fact_86_p_I1_J,axiom,
! [I2: nat] : ( member_real @ ( coeff_real @ p @ I2 ) @ ring_1_Ints_real ) ).
% p(1)
thf(fact_87_reflect__poly__0,axiom,
( ( reflect_poly_real @ zero_zero_poly_real )
= zero_zero_poly_real ) ).
% reflect_poly_0
thf(fact_88_reflect__poly__0,axiom,
( ( reflec1522834046y_real @ zero_z1423781445y_real )
= zero_z1423781445y_real ) ).
% reflect_poly_0
thf(fact_89_reflect__poly__0,axiom,
( ( reflect_poly_nat @ zero_zero_poly_nat )
= zero_zero_poly_nat ) ).
% reflect_poly_0
thf(fact_90_synthetic__div__0,axiom,
! [C: real] :
( ( synthetic_div_real @ zero_zero_poly_real @ C )
= zero_zero_poly_real ) ).
% synthetic_div_0
thf(fact_91_synthetic__div__0,axiom,
! [C: poly_real] :
( ( synthe1498897281y_real @ zero_z1423781445y_real @ C )
= zero_z1423781445y_real ) ).
% synthetic_div_0
thf(fact_92_synthetic__div__0,axiom,
! [C: nat] :
( ( synthetic_div_nat @ zero_zero_poly_nat @ C )
= zero_zero_poly_nat ) ).
% synthetic_div_0
thf(fact_93_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_poly_real @ zero_z935034829y_real @ N )
= zero_z1423781445y_real ) ).
% coeff_0
thf(fact_94_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_nat @ zero_z1059985641ly_nat @ N )
= zero_zero_poly_nat ) ).
% coeff_0
thf(fact_95_coeff__0,axiom,
! [N: nat] :
( ( coeff_real @ zero_zero_poly_real @ N )
= zero_zero_real ) ).
% coeff_0
thf(fact_96_coeff__0,axiom,
! [N: nat] :
( ( coeff_poly_real @ zero_z1423781445y_real @ N )
= zero_zero_poly_real ) ).
% coeff_0
thf(fact_97_coeff__0,axiom,
! [N: nat] :
( ( coeff_nat @ zero_zero_poly_nat @ N )
= zero_zero_nat ) ).
% coeff_0
thf(fact_98_reflect__poly__reflect__poly,axiom,
! [P: poly_real] :
( ( ( coeff_real @ P @ zero_zero_nat )
!= zero_zero_real )
=> ( ( reflect_poly_real @ ( reflect_poly_real @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_99_reflect__poly__reflect__poly,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ P @ zero_zero_nat )
!= zero_zero_poly_real )
=> ( ( reflec1522834046y_real @ ( reflec1522834046y_real @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_100_reflect__poly__reflect__poly,axiom,
! [P: poly_nat] :
( ( ( coeff_nat @ P @ zero_zero_nat )
!= zero_zero_nat )
=> ( ( reflect_poly_nat @ ( reflect_poly_nat @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_101_reflect__poly__reflect__poly,axiom,
! [P: poly_poly_poly_real] :
( ( ( coeff_poly_poly_real @ P @ zero_zero_nat )
!= zero_z1423781445y_real )
=> ( ( reflec144234502y_real @ ( reflec144234502y_real @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_102_reflect__poly__reflect__poly,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ P @ zero_zero_nat )
!= zero_zero_poly_nat )
=> ( ( reflec781175074ly_nat @ ( reflec781175074ly_nat @ P ) )
= P ) ) ).
% reflect_poly_reflect_poly
thf(fact_103_coeff__0__reflect__poly,axiom,
! [P: poly_real] :
( ( coeff_real @ ( reflect_poly_real @ P ) @ zero_zero_nat )
= ( coeff_real @ P @ ( degree_real @ P ) ) ) ).
% coeff_0_reflect_poly
thf(fact_104_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_real] :
( ( ( coeff_real @ ( reflect_poly_real @ P ) @ zero_zero_nat )
= zero_zero_real )
= ( P = zero_zero_poly_real ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_105_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ ( reflec1522834046y_real @ P ) @ zero_zero_nat )
= zero_zero_poly_real )
= ( P = zero_z1423781445y_real ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_106_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_nat] :
( ( ( coeff_nat @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
= zero_zero_nat )
= ( P = zero_zero_poly_nat ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_107_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_poly_poly_real] :
( ( ( coeff_poly_poly_real @ ( reflec144234502y_real @ P ) @ zero_zero_nat )
= zero_z1423781445y_real )
= ( P = zero_z935034829y_real ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_108_coeff__0__reflect__poly__0__iff,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ ( reflec781175074ly_nat @ P ) @ zero_zero_nat )
= zero_zero_poly_nat )
= ( P = zero_z1059985641ly_nat ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_109_degree__reflect__poly__eq,axiom,
! [P: poly_real] :
( ( ( coeff_real @ P @ zero_zero_nat )
!= zero_zero_real )
=> ( ( degree_real @ ( reflect_poly_real @ P ) )
= ( degree_real @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_110_degree__reflect__poly__eq,axiom,
! [P: poly_poly_real] :
( ( ( coeff_poly_real @ P @ zero_zero_nat )
!= zero_zero_poly_real )
=> ( ( degree_poly_real @ ( reflec1522834046y_real @ P ) )
= ( degree_poly_real @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_111_degree__reflect__poly__eq,axiom,
! [P: poly_nat] :
( ( ( coeff_nat @ P @ zero_zero_nat )
!= zero_zero_nat )
=> ( ( degree_nat @ ( reflect_poly_nat @ P ) )
= ( degree_nat @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_112_degree__reflect__poly__eq,axiom,
! [P: poly_poly_poly_real] :
( ( ( coeff_poly_poly_real @ P @ zero_zero_nat )
!= zero_z1423781445y_real )
=> ( ( degree360860553y_real @ ( reflec144234502y_real @ P ) )
= ( degree360860553y_real @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_113_degree__reflect__poly__eq,axiom,
! [P: poly_poly_nat] :
( ( ( coeff_poly_nat @ P @ zero_zero_nat )
!= zero_zero_poly_nat )
=> ( ( degree_poly_nat @ ( reflec781175074ly_nat @ P ) )
= ( degree_poly_nat @ P ) ) ) ).
% degree_reflect_poly_eq
thf(fact_114_poly__reflect__poly__0,axiom,
! [P: poly_real] :
( ( poly_real2 @ ( reflect_poly_real @ P ) @ zero_zero_real )
= ( coeff_real @ P @ ( degree_real @ P ) ) ) ).
% poly_reflect_poly_0
thf(fact_115_poly__reflect__poly__0,axiom,
! [P: poly_poly_real] :
( ( poly_poly_real2 @ ( reflec1522834046y_real @ P ) @ zero_zero_poly_real )
= ( coeff_poly_real @ P @ ( degree_poly_real @ P ) ) ) ).
% poly_reflect_poly_0
thf(fact_116_poly__reflect__poly__0,axiom,
! [P: poly_nat] :
( ( poly_nat2 @ ( reflect_poly_nat @ P ) @ zero_zero_nat )
= ( coeff_nat @ P @ ( degree_nat @ P ) ) ) ).
% poly_reflect_poly_0
thf(fact_117_poly__reflect__poly__0,axiom,
! [P: poly_poly_poly_real] :
( ( poly_poly_poly_real2 @ ( reflec144234502y_real @ P ) @ zero_z1423781445y_real )
= ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) ) ) ).
% poly_reflect_poly_0
thf(fact_118_poly__reflect__poly__0,axiom,
! [P: poly_poly_nat] :
( ( poly_poly_nat2 @ ( reflec781175074ly_nat @ P ) @ zero_zero_poly_nat )
= ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) ) ) ).
% poly_reflect_poly_0
thf(fact_119_mem__Collect__eq,axiom,
! [A3: real,P2: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
! [A3: poly_real,P2: poly_real > $o] :
( ( member_poly_real @ A3 @ ( collect_poly_real @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A2: set_poly_real] :
( ( collect_poly_real
@ ^ [X: poly_real] : ( member_poly_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_125_Collect__cong,axiom,
! [P2: real > $o,Q: real > $o] :
( ! [X3: real] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_real @ P2 )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_126_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_127_Collect__cong,axiom,
! [P2: poly_real > $o,Q: poly_real > $o] :
( ! [X3: poly_real] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_poly_real @ P2 )
= ( collect_poly_real @ Q ) ) ) ).
% Collect_cong
thf(fact_128_poly__eqI,axiom,
! [P: poly_real,Q2: poly_real] :
( ! [N2: nat] :
( ( coeff_real @ P @ N2 )
= ( coeff_real @ Q2 @ N2 ) )
=> ( P = Q2 ) ) ).
% poly_eqI
thf(fact_129_poly__eq__iff,axiom,
( ( ^ [Y: poly_real,Z: poly_real] : Y = Z )
= ( ^ [P3: poly_real,Q3: poly_real] :
! [N3: nat] :
( ( coeff_real @ P3 @ N3 )
= ( coeff_real @ Q3 @ N3 ) ) ) ) ).
% poly_eq_iff
thf(fact_130_coeff__inject,axiom,
! [X2: poly_real,Y2: poly_real] :
( ( ( coeff_real @ X2 )
= ( coeff_real @ Y2 ) )
= ( X2 = Y2 ) ) ).
% coeff_inject
thf(fact_131_algebraicE,axiom,
! [X2: real] :
( ( algebraic_real @ X2 )
=> ~ ! [P4: poly_real] :
( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
=> ( ( P4 != zero_zero_poly_real )
=> ( ( poly_real2 @ P4 @ X2 )
!= zero_zero_real ) ) ) ) ).
% algebraicE
thf(fact_132_algebraicI,axiom,
! [P: poly_real,X2: real] :
( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
=> ( ( P != zero_zero_poly_real )
=> ( ( ( poly_real2 @ P @ X2 )
= zero_zero_real )
=> ( algebraic_real @ X2 ) ) ) ) ).
% algebraicI
thf(fact_133_algebraic__def,axiom,
( algebraic_real
= ( ^ [X: real] :
? [P3: poly_real] :
( ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ ring_1_Ints_real )
& ( P3 != zero_zero_poly_real )
& ( ( poly_real2 @ P3 @ X )
= zero_zero_real ) ) ) ) ).
% algebraic_def
thf(fact_134_algebraic__altdef,axiom,
( algebraic_real
= ( ^ [X: real] :
? [P3: poly_real] :
( ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ field_1537545994s_real )
& ( P3 != zero_zero_poly_real )
& ( ( poly_real2 @ P3 @ X )
= zero_zero_real ) ) ) ) ).
% algebraic_altdef
thf(fact_135_zero__poly_Orep__eq,axiom,
( ( coeff_poly_poly_real @ zero_z935034829y_real )
= ( ^ [Uu: nat] : zero_z1423781445y_real ) ) ).
% zero_poly.rep_eq
thf(fact_136_zero__poly_Orep__eq,axiom,
( ( coeff_poly_nat @ zero_z1059985641ly_nat )
= ( ^ [Uu: nat] : zero_zero_poly_nat ) ) ).
% zero_poly.rep_eq
thf(fact_137_zero__poly_Orep__eq,axiom,
( ( coeff_real @ zero_zero_poly_real )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_poly.rep_eq
thf(fact_138_zero__poly_Orep__eq,axiom,
( ( coeff_poly_real @ zero_z1423781445y_real )
= ( ^ [Uu: nat] : zero_zero_poly_real ) ) ).
% zero_poly.rep_eq
thf(fact_139_zero__poly_Orep__eq,axiom,
( ( coeff_nat @ zero_zero_poly_nat )
= ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% zero_poly.rep_eq
thf(fact_140_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_141_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_142_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_143_poly__0__coeff__0,axiom,
! [P: poly_poly_poly_real] :
( ( poly_poly_poly_real2 @ P @ zero_z1423781445y_real )
= ( coeff_poly_poly_real @ P @ zero_zero_nat ) ) ).
% poly_0_coeff_0
thf(fact_144_poly__0__coeff__0,axiom,
! [P: poly_poly_nat] :
( ( poly_poly_nat2 @ P @ zero_zero_poly_nat )
= ( coeff_poly_nat @ P @ zero_zero_nat ) ) ).
% poly_0_coeff_0
thf(fact_145_order__0I,axiom,
! [P: poly_real,A3: real] :
( ( ( poly_real2 @ P @ A3 )
!= zero_zero_real )
=> ( ( order_real @ A3 @ P )
= zero_zero_nat ) ) ).
% order_0I
thf(fact_146_order__0I,axiom,
! [P: poly_poly_real,A3: poly_real] :
( ( ( poly_poly_real2 @ P @ A3 )
!= zero_zero_poly_real )
=> ( ( order_poly_real @ A3 @ P )
= zero_zero_nat ) ) ).
% order_0I
thf(fact_147_order__0I,axiom,
! [P: poly_poly_poly_real,A3: poly_poly_real] :
( ( ( poly_poly_poly_real2 @ P @ A3 )
!= zero_z1423781445y_real )
=> ( ( order_poly_poly_real @ A3 @ P )
= zero_zero_nat ) ) ).
% order_0I
thf(fact_148_leading__coeff__neq__0,axiom,
! [P: poly_poly_poly_real] :
( ( P != zero_z935034829y_real )
=> ( ( coeff_poly_poly_real @ P @ ( degree360860553y_real @ P ) )
!= zero_z1423781445y_real ) ) ).
% leading_coeff_neq_0
thf(fact_149_leading__coeff__neq__0,axiom,
! [P: poly_poly_nat] :
( ( P != zero_z1059985641ly_nat )
=> ( ( coeff_poly_nat @ P @ ( degree_poly_nat @ P ) )
!= zero_zero_poly_nat ) ) ).
% leading_coeff_neq_0
thf(fact_150_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_151_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_152_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_153_Rats__infinite,axiom,
~ ( finite_finite_real @ field_1537545994s_real ) ).
% Rats_infinite
thf(fact_154_Rats__0,axiom,
member_real @ zero_zero_real @ field_1537545994s_real ).
% Rats_0
thf(fact_155_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_156_Ints__0,axiom,
member_poly_real @ zero_zero_poly_real @ ring_1690226883y_real ).
% Ints_0
thf(fact_157_Ints__0,axiom,
member1159720147y_real @ zero_z1423781445y_real @ ring_1897377867y_real ).
% Ints_0
thf(fact_158_rsquarefree__def,axiom,
( rsquarefree_real
= ( ^ [P3: poly_real] :
( ( P3 != zero_zero_poly_real )
& ! [A: real] :
( ( ( order_real @ A @ P3 )
= zero_zero_nat )
| ( ( order_real @ A @ P3 )
= one_one_nat ) ) ) ) ) ).
% rsquarefree_def
thf(fact_159_rsquarefree__def,axiom,
( rsquar1555552848y_real
= ( ^ [P3: poly_poly_real] :
( ( P3 != zero_z1423781445y_real )
& ! [A: poly_real] :
( ( ( order_poly_real @ A @ P3 )
= zero_zero_nat )
| ( ( order_poly_real @ A @ P3 )
= one_one_nat ) ) ) ) ) ).
% rsquarefree_def
thf(fact_160_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_cutoff_real @ N @ one_one_poly_real )
= zero_zero_poly_real ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_cutoff_real @ N @ one_one_poly_real )
= one_one_poly_real ) ) ) ).
% poly_cutoff_1
thf(fact_161_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_c1404107022y_real @ N @ one_on501200385y_real )
= zero_z1423781445y_real ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_c1404107022y_real @ N @ one_on501200385y_real )
= one_on501200385y_real ) ) ) ).
% poly_cutoff_1
thf(fact_162_poly__cutoff__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
= zero_zero_poly_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_cutoff_nat @ N @ one_one_poly_nat )
= one_one_poly_nat ) ) ) ).
% poly_cutoff_1
thf(fact_163_order__pderiv2,axiom,
! [P: poly_real,A3: real,N: nat] :
( ( ( pderiv_real @ P )
!= zero_zero_poly_real )
=> ( ( ( order_real @ A3 @ P )
!= zero_zero_nat )
=> ( ( ( order_real @ A3 @ ( pderiv_real @ P ) )
= N )
= ( ( order_real @ A3 @ P )
= ( suc @ N ) ) ) ) ) ).
% order_pderiv2
thf(fact_164_order__pderiv,axiom,
! [P: poly_real,A3: real] :
( ( ( pderiv_real @ P )
!= zero_zero_poly_real )
=> ( ( ( order_real @ A3 @ P )
!= zero_zero_nat )
=> ( ( order_real @ A3 @ P )
= ( suc @ ( order_real @ A3 @ ( pderiv_real @ P ) ) ) ) ) ) ).
% order_pderiv
thf(fact_165_cr__poly__def,axiom,
( cr_poly_real
= ( ^ [X: nat > real,Y3: poly_real] :
( X
= ( coeff_real @ Y3 ) ) ) ) ).
% cr_poly_def
thf(fact_166_dvd__pderiv__iff,axiom,
! [P: poly_real] :
( ( dvd_dvd_poly_real @ P @ ( pderiv_real @ P ) )
= ( ( degree_real @ P )
= zero_zero_nat ) ) ).
% dvd_pderiv_iff
thf(fact_167_degree__1,axiom,
( ( degree_real @ one_one_poly_real )
= zero_zero_nat ) ).
% degree_1
thf(fact_168_poly__1,axiom,
! [X2: real] :
( ( poly_real2 @ one_one_poly_real @ X2 )
= one_one_real ) ).
% poly_1
thf(fact_169_poly__1,axiom,
! [X2: poly_real] :
( ( poly_poly_real2 @ one_on501200385y_real @ X2 )
= one_one_poly_real ) ).
% poly_1
thf(fact_170_poly__1,axiom,
! [X2: nat] :
( ( poly_nat2 @ one_one_poly_nat @ X2 )
= one_one_nat ) ).
% poly_1
thf(fact_171_pderiv__1,axiom,
( ( pderiv_real @ one_one_poly_real )
= zero_zero_poly_real ) ).
% pderiv_1
thf(fact_172_pderiv__1,axiom,
( ( pderiv_poly_real @ one_on501200385y_real )
= zero_z1423781445y_real ) ).
% pderiv_1
thf(fact_173_pderiv__1,axiom,
( ( pderiv_nat @ one_one_poly_nat )
= zero_zero_poly_nat ) ).
% pderiv_1
thf(fact_174_lead__coeff__1,axiom,
( ( coeff_nat @ one_one_poly_nat @ ( degree_nat @ one_one_poly_nat ) )
= one_one_nat ) ).
% lead_coeff_1
thf(fact_175_lead__coeff__1,axiom,
( ( coeff_real @ one_one_poly_real @ ( degree_real @ one_one_poly_real ) )
= one_one_real ) ).
% lead_coeff_1
thf(fact_176_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_177_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_178_Rats__1,axiom,
member_real @ one_one_real @ field_1537545994s_real ).
% Rats_1
thf(fact_179_is__unit__iff__degree,axiom,
! [P: poly_real] :
( ( P != zero_zero_poly_real )
=> ( ( dvd_dvd_poly_real @ P @ one_one_poly_real )
= ( ( degree_real @ P )
= zero_zero_nat ) ) ) ).
% is_unit_iff_degree
thf(fact_180_not__dvd__pderiv,axiom,
! [P: poly_real] :
( ( ( degree_real @ P )
!= zero_zero_nat )
=> ~ ( dvd_dvd_poly_real @ P @ ( pderiv_real @ P ) ) ) ).
% not_dvd_pderiv
thf(fact_181_poly__shift__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_shift_real @ N @ one_one_poly_real )
= one_one_poly_real ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_shift_real @ N @ one_one_poly_real )
= zero_zero_poly_real ) ) ) ).
% poly_shift_1
thf(fact_182_poly__shift__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_shift_poly_real @ N @ one_on501200385y_real )
= one_on501200385y_real ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_shift_poly_real @ N @ one_on501200385y_real )
= zero_z1423781445y_real ) ) ) ).
% poly_shift_1
thf(fact_183_poly__shift__1,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( poly_shift_nat @ N @ one_one_poly_nat )
= one_one_poly_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( poly_shift_nat @ N @ one_one_poly_nat )
= zero_zero_poly_nat ) ) ) ).
% poly_shift_1
thf(fact_184_dvd__0__left__iff,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
= ( A3 = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_185_dvd__0__left__iff,axiom,
! [A3: poly_real] :
( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A3 )
= ( A3 = zero_zero_poly_real ) ) ).
% dvd_0_left_iff
thf(fact_186_dvd__0__left__iff,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_187_dvd__0__left__iff,axiom,
! [A3: poly_poly_real] :
( ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ A3 )
= ( A3 = zero_z1423781445y_real ) ) ).
% dvd_0_left_iff
thf(fact_188_dvd__0__left__iff,axiom,
! [A3: poly_nat] :
( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A3 )
= ( A3 = zero_zero_poly_nat ) ) ).
% dvd_0_left_iff
thf(fact_189_dvd__0__right,axiom,
! [A3: real] : ( dvd_dvd_real @ A3 @ zero_zero_real ) ).
% dvd_0_right
thf(fact_190_dvd__0__right,axiom,
! [A3: poly_real] : ( dvd_dvd_poly_real @ A3 @ zero_zero_poly_real ) ).
% dvd_0_right
thf(fact_191_dvd__0__right,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_192_dvd__0__right,axiom,
! [A3: poly_poly_real] : ( dvd_dv1946063458y_real @ A3 @ zero_z1423781445y_real ) ).
% dvd_0_right
thf(fact_193_dvd__0__right,axiom,
! [A3: poly_nat] : ( dvd_dvd_poly_nat @ A3 @ zero_zero_poly_nat ) ).
% dvd_0_right
thf(fact_194_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_195_not__is__unit__0,axiom,
~ ( dvd_dvd_poly_real @ zero_zero_poly_real @ one_one_poly_real ) ).
% not_is_unit_0
thf(fact_196_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_197_not__is__unit__0,axiom,
~ ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ one_on501200385y_real ) ).
% not_is_unit_0
thf(fact_198_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_199_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_200_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_201_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_202_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_203_dvd__trans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% dvd_trans
thf(fact_204_dvd__refl,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% dvd_refl
thf(fact_205_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_206_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_207_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_208_zero__neq__one,axiom,
zero_zero_poly_real != one_one_poly_real ).
% zero_neq_one
thf(fact_209_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_210_zero__neq__one,axiom,
zero_z1423781445y_real != one_on501200385y_real ).
% zero_neq_one
thf(fact_211_zero__neq__one,axiom,
zero_zero_poly_nat != one_one_poly_nat ).
% zero_neq_one
thf(fact_212_dvd__0__left,axiom,
! [A3: real] :
( ( dvd_dvd_real @ zero_zero_real @ A3 )
=> ( A3 = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_213_dvd__0__left,axiom,
! [A3: poly_real] :
( ( dvd_dvd_poly_real @ zero_zero_poly_real @ A3 )
=> ( A3 = zero_zero_poly_real ) ) ).
% dvd_0_left
thf(fact_214_dvd__0__left,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_215_dvd__0__left,axiom,
! [A3: poly_poly_real] :
( ( dvd_dv1946063458y_real @ zero_z1423781445y_real @ A3 )
=> ( A3 = zero_z1423781445y_real ) ) ).
% dvd_0_left
thf(fact_216_dvd__0__left,axiom,
! [A3: poly_nat] :
( ( dvd_dvd_poly_nat @ zero_zero_poly_nat @ A3 )
=> ( A3 = zero_zero_poly_nat ) ) ).
% dvd_0_left
thf(fact_217_dvd__unit__imp__unit,axiom,
! [A3: nat,B2: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_218_unit__imp__dvd,axiom,
! [B2: nat,A3: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ B2 @ A3 ) ) ).
% unit_imp_dvd
thf(fact_219_one__dvd,axiom,
! [A3: nat] : ( dvd_dvd_nat @ one_one_nat @ A3 ) ).
% one_dvd
thf(fact_220_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_221_old_Onat_Oinducts,axiom,
! [P2: nat > $o,Nat: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P2 @ Nat3 )
=> ( P2 @ ( suc @ Nat3 ) ) )
=> ( P2 @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_222_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_223_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_224_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_225_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_226_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_227_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P2 @ X3 @ Y4 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_228_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_229_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_230_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_231_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_232_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_233_algebraic__int_Oinducts,axiom,
! [X2: real,P2: real > $o] :
( ( algebraic_int_real @ X2 )
=> ( ! [P4: poly_real,X3: real] :
( ( ( coeff_real @ P4 @ ( degree_real @ P4 ) )
= one_one_real )
=> ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
=> ( ( ( poly_real2 @ P4 @ X3 )
= zero_zero_real )
=> ( P2 @ X3 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% algebraic_int.inducts
thf(fact_234_algebraic__int_Ointros,axiom,
! [P: poly_real,X2: real] :
( ( ( coeff_real @ P @ ( degree_real @ P ) )
= one_one_real )
=> ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
=> ( ( ( poly_real2 @ P @ X2 )
= zero_zero_real )
=> ( algebraic_int_real @ X2 ) ) ) ) ).
% algebraic_int.intros
thf(fact_235_algebraic__int_Osimps,axiom,
( algebraic_int_real
= ( ^ [A: real] :
? [P3: poly_real,X: real] :
( ( A = X )
& ( ( coeff_real @ P3 @ ( degree_real @ P3 ) )
= one_one_real )
& ! [I4: nat] : ( member_real @ ( coeff_real @ P3 @ I4 ) @ ring_1_Ints_real )
& ( ( poly_real2 @ P3 @ X )
= zero_zero_real ) ) ) ) ).
% algebraic_int.simps
thf(fact_236_algebraic__int_Ocases,axiom,
! [A3: real] :
( ( algebraic_int_real @ A3 )
=> ~ ! [P4: poly_real] :
( ( ( coeff_real @ P4 @ ( degree_real @ P4 ) )
= one_one_real )
=> ( ! [I: nat] : ( member_real @ ( coeff_real @ P4 @ I ) @ ring_1_Ints_real )
=> ( ( poly_real2 @ P4 @ A3 )
!= zero_zero_real ) ) ) ) ).
% algebraic_int.cases
thf(fact_237_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P2 @ X_12 )
=> ? [N2: nat] :
( ~ ( P2 @ N2 )
& ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_238_algebraic__int__0,axiom,
algebraic_int_real @ zero_zero_real ).
% algebraic_int_0
thf(fact_239_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_240_int__imp__algebraic__int,axiom,
! [X2: real] :
( ( member_real @ X2 @ ring_1_Ints_real )
=> ( algebraic_int_real @ X2 ) ) ).
% int_imp_algebraic_int
thf(fact_241_algebraic__int__imp__algebraic,axiom,
! [X2: real] :
( ( algebraic_int_real @ X2 )
=> ( algebraic_real @ X2 ) ) ).
% algebraic_int_imp_algebraic
thf(fact_242_rational__algebraic__int__is__int,axiom,
! [X2: real] :
( ( algebraic_int_real @ X2 )
=> ( ( member_real @ X2 @ field_1537545994s_real )
=> ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).
% rational_algebraic_int_is_int
thf(fact_243_algebraic__int__inverse,axiom,
! [P: poly_real,X2: real] :
( ( ( poly_real2 @ P @ X2 )
= zero_zero_real )
=> ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
=> ( ( ( coeff_real @ P @ zero_zero_nat )
= one_one_real )
=> ( algebraic_int_real @ ( inverse_inverse_real @ X2 ) ) ) ) ) ).
% algebraic_int_inverse
thf(fact_244_algebraic__int__root,axiom,
! [Y2: real,P: poly_real,X2: real] :
( ( algebraic_int_real @ Y2 )
=> ( ( ( poly_real2 @ P @ X2 )
= Y2 )
=> ( ! [I3: nat] : ( member_real @ ( coeff_real @ P @ I3 ) @ ring_1_Ints_real )
=> ( ( ( coeff_real @ P @ ( degree_real @ P ) )
= one_one_real )
=> ( ( ord_less_nat @ zero_zero_nat @ ( degree_real @ P ) )
=> ( algebraic_int_real @ X2 ) ) ) ) ) ) ).
% algebraic_int_root
thf(fact_245_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
=> ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_246_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ( dvd_dvd_nat @ A3 @ zero_zero_nat )
& ( A3 != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_247_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_248_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_249_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_250_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_251_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_252_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_253_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_254_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_255_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_256_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_257_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_258_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_259_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_260_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_261_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_262_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_263_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_264_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_265_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_266_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_267_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_268_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_269_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_270_strict__inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_271_less__Suc__induct,axiom,
! [I2: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P2 @ I3 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_272_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_273_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_274_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_275_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_276_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_277_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_278_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_279_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_280_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_281_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_282_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_283_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_284_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_285_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_286_Rats__inverse,axiom,
! [A3: real] :
( ( member_real @ A3 @ field_1537545994s_real )
=> ( member_real @ ( inverse_inverse_real @ A3 ) @ field_1537545994s_real ) ) ).
% Rats_inverse
thf(fact_287_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_288_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_289_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_290_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_291_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_292_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P2 @ M3 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_293_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_294_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_295_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_296_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_297_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_298_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_299_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_300_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_301_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_302_not__one__less__zero,axiom,
~ ( ord_less_poly_real @ one_one_poly_real @ zero_zero_poly_real ) ).
% not_one_less_zero
thf(fact_303_not__one__less__zero,axiom,
~ ( ord_le38482960y_real @ one_on501200385y_real @ zero_z1423781445y_real ) ).
% not_one_less_zero
thf(fact_304_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_305_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_306_zero__less__one,axiom,
ord_less_poly_real @ zero_zero_poly_real @ one_one_poly_real ).
% zero_less_one
thf(fact_307_zero__less__one,axiom,
ord_le38482960y_real @ zero_z1423781445y_real @ one_on501200385y_real ).
% zero_less_one
thf(fact_308_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_309_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_310_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_311_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_312_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_313_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_314_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_315_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_316_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_317_less__degree__imp,axiom,
! [N: nat,P: poly_poly_real] :
( ( ord_less_nat @ N @ ( degree_poly_real @ P ) )
=> ? [I3: nat] :
( ( ord_less_nat @ N @ I3 )
& ( ( coeff_poly_real @ P @ I3 )
!= zero_zero_poly_real ) ) ) ).
% less_degree_imp
thf(fact_318_less__degree__imp,axiom,
! [N: nat,P: poly_nat] :
( ( ord_less_nat @ N @ ( degree_nat @ P ) )
=> ? [I3: nat] :
( ( ord_less_nat @ N @ I3 )
& ( ( coeff_nat @ P @ I3 )
!= zero_zero_nat ) ) ) ).
% less_degree_imp
thf(fact_319_less__degree__imp,axiom,
! [N: nat,P: poly_poly_poly_real] :
( ( ord_less_nat @ N @ ( degree360860553y_real @ P ) )
=> ? [I3: nat] :
( ( ord_less_nat @ N @ I3 )
& ( ( coeff_poly_poly_real @ P @ I3 )
!= zero_z1423781445y_real ) ) ) ).
% less_degree_imp
thf(fact_320_less__degree__imp,axiom,
! [N: nat,P: poly_poly_nat] :
( ( ord_less_nat @ N @ ( degree_poly_nat @ P ) )
=> ? [I3: nat] :
( ( ord_less_nat @ N @ I3 )
& ( ( coeff_poly_nat @ P @ I3 )
!= zero_zero_poly_nat ) ) ) ).
% less_degree_imp
thf(fact_321_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A3: nat,B2: nat] :
( ( A3 != B2 )
=> ( ( dvd_dvd_nat @ A3 @ B2 )
=> ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_322_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( A3 != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_323_gcd__nat_Ostrict__implies__order,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( dvd_dvd_nat @ A3 @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_324_gcd__nat_Ostrict__iff__order,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
= ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_325_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A: nat,B3: nat] :
( ( ( dvd_dvd_nat @ A @ B3 )
& ( A != B3 ) )
| ( A = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_326_gcd__nat_Ostrict__trans2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( ( dvd_dvd_nat @ A3 @ C )
& ( A3 != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_327_gcd__nat_Ostrict__trans1,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A3 @ C )
& ( A3 != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_328_gcd__nat_Ostrict__trans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A3 @ C )
& ( A3 != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_329_gcd__nat_Oantisym,axiom,
! [A3: nat,B2: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_330_gcd__nat_Oirrefl,axiom,
! [A3: nat] :
~ ( ( dvd_dvd_nat @ A3 @ A3 )
& ( A3 != A3 ) ) ).
% gcd_nat.irrefl
thf(fact_331_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A: nat,B3: nat] :
( ( dvd_dvd_nat @ A @ B3 )
& ( dvd_dvd_nat @ B3 @ A ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_332_gcd__nat_Otrans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A3 @ C ) ) ) ).
% gcd_nat.trans
thf(fact_333_gcd__nat_Orefl,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).
% gcd_nat.refl
thf(fact_334_gcd__nat_Oasym,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ~ ( ( dvd_dvd_nat @ B2 @ A3 )
& ( B2 != A3 ) ) ) ).
% gcd_nat.asym
thf(fact_335_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_336_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
& ( zero_zero_nat != A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_337_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
= ( A3 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_338_poly__IVT__pos,axiom,
! [A3: real,B2: real,P: poly_real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_real @ ( poly_real2 @ P @ A3 ) @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ B2 ) )
=> ? [X3: real] :
( ( ord_less_real @ A3 @ X3 )
& ( ord_less_real @ X3 @ B2 )
& ( ( poly_real2 @ P @ X3 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_pos
thf(fact_339_poly__IVT__neg,axiom,
! [A3: real,B2: real,P: poly_real] :
( ( ord_less_real @ A3 @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ ( poly_real2 @ P @ A3 ) )
=> ( ( ord_less_real @ ( poly_real2 @ P @ B2 ) @ zero_zero_real )
=> ? [X3: real] :
( ( ord_less_real @ A3 @ X3 )
& ( ord_less_real @ X3 @ B2 )
& ( ( poly_real2 @ P @ X3 )
= zero_zero_real ) ) ) ) ) ).
% poly_IVT_neg
thf(fact_340_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P2 @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_341_Rats__dense__in__real,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [X3: real] :
( ( member_real @ X3 @ field_1537545994s_real )
& ( ord_less_real @ X2 @ X3 )
& ( ord_less_real @ X3 @ Y2 ) ) ) ).
% Rats_dense_in_real
thf(fact_342_Rats__no__bot__less,axiom,
! [X2: real] :
? [X3: real] :
( ( member_real @ X3 @ field_1537545994s_real )
& ( ord_less_real @ X3 @ X2 ) ) ).
% Rats_no_bot_less
thf(fact_343_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
% Conjectures (1)
thf(conj_0,conjecture,
( finite_finite_real
@ ( collect_real
@ ^ [X: real] :
( ( poly_real2 @ ( pderiv_real @ p ) @ X )
= zero_zero_real ) ) ) ).
%------------------------------------------------------------------------------